JP4947131B2 - Data transfer network, data transfer network configuration method, and data transfer network evaluation method - Google Patents

Description

  The present invention relates to a data transfer network, a data transfer network configuration method, and a data transfer network evaluation method.

  A data transfer network in which a plurality of nodes are connected by a link is, for example, a general Internet network, a public communication network, a corporate communication network, a LAN, a computer network, a distributed computer network, a distributed router network, an exchange network, a router, and various devices. Are widely used today, such as various switch ropes used in the world, data communication networks connecting CPUs and memories, and data communication networks inside LSIs such as CPUs.

  The conventional data transfer network configuration method that combines multiple nodes and distributes node processing includes (a) bus network, (b) ring network, (c) hub network (star network), and (d) complete. There were various regular network configuration methods such as mesh networks, (e) bus and ring hyper-extended (hypercube) networks.

  Some networks that are actually used as information communication networks have a network structure called a complex network such as the Internet network structure.

  In addition, although not used practically, random networks and generalized random networks have been actively studied as network structures from an academic point of view.

  A network that adopts these various shapes, structures, and construction methods (hereinafter, the shapes, structures, and construction methods are simply referred to as “structure”) is assumed to have the same number of nodes with the same performance and the same number of nodes. Even if a link is used, a difference occurs in the total amount of traffic that can be processed in the entire network due to a difference in the structure, and a difference also occurs in the average number of relays (average number of hops) (Section 2 of Non-Patent Document 1) (See simulation results for

  In the future, as will be seen in computer IT networks and the like, a large number of data transfer networks will appear that have a large number of terminals and whose terminal speed is much higher than the current level. For such a large and high-speed network, it is important to increase the traffic capacity of the entire network and reduce the number of relays in the network. Therefore, optimization and efficiency of the network structure are important.

  In general, if only the number of links is increased with the same number of nodes, the total amount of traffic that can be processed in the entire network increases and the average number of hops also decreases, but the cost for the links increases. Therefore, there are the optimal number of nodes and the optimal number of links in terms of cost, and the optimal traffic total value and the optimal average are included in the data transfer network to which the optimal number of nodes and the optimal number of links are applied. There must be something that achieves the number of hops.

  In addition, when the number of nodes and the number of links are given, there is a network structure showing the maximum traffic amount maxXeff that satisfies this condition. The average number of hops (hereinafter referred to as the maximum number of NW average hops) maxH of the network indicating the maximum traffic amount maxXeff is relatively large, and just because the maximum traffic amount maxXeff is taken, it is not always excellent in practical use. Absent. Practically, in a network with a given number of nodes and links, clarify the network structure that shows the maximum traffic volume under the condition that the average hop count is less than the maximum NW average hop count maxH. Is desired.

  When the average number of hops is equal to or less than H and indicates the maximum traffic total amount, the maximum traffic total amount is written as X (H), and the relationship between H and X (H) is referred to as H-Xeff limit performance. In this case, X (maxH) = maxXeff. In Non-Patent Document 2, in a network where the number of nodes is N and the number of links is L, the minimum average hop number minH may be minH = 2-2 * L / N / (N−1). It is shown. Now, X (minH) = minXeff.

  H-Xeff limit performance X (H) is an average of each network structure in a two-dimensional coordinate system in which the horizontal axis represents the number of hops (average relay number performance) H and the vertical axis represents the traffic volume (network bandwidth performance) Xeff. When plotted according to the number of hops and the traffic volume, a function (step function) that monotonously increases (stepwise) from (minH, minXeff) to (maxH, maxXeff).

  FIG. 18 shows the distribution of the network bandwidth performance Xeff and the average number of relays performance H by taking a network with N = 50 and L = 75 as an example. FIG. 18 plots the network bandwidth performance Xeff and the average number of relays performance H obtained for a network with N = 50 and L = 75, and plots such a plot for a number of different networks with N = 50 and L = 75. Is what I did.

  In FIG. 18, a point PM represents a point (maxH, maxXeff), and a point PL represents a point (minH, minXeff). A step in which the boundary at the left end of the distribution from the point PL of (minH, minXeff) to the point of the maximum point (maxH, maxXeff) represents the H-Xcff limit performance X (H).

Among a plurality of networks having the same average number of relays performance H, a network having the best network bandwidth performance Xeff is preferable. The network bandwidth performance Xeff ₁ in the smaller average number of relays performance H ₁ is larger in the network bandwidth performance Xeff ₂ in the larger average number of relays performance H ₂ in the two average relay number performances H ₁ and H ₂ that follow _{each other.} If worse, the (max ₁ , Xeff ₁ ) network is preferred. In this way, there is no network suitable for a network whose average number of relays performance is greater than maxH. In addition, a suitable network exists on the H-Xeff limit performance X (H) that monotonously increases (in a stepped manner) from (minH, minXeff) to (maxH, maxXeff).

Sponsored by the 5th Network Ecology Symposium, Network Ecology Research Group of Information Processing Society of Japan, Masato Kawahara, Noriyoshi Imanaka, Masami Date, Yoshihiro Nakahira, Toshihisa Nakai, presentation document “The Internet as a Complex Network and Its Network Bandwidth Performance "March 2009 The Institute of Electronics, Information and Communication Engineers "Presentation of Complex Network Time Study Group, Presenter Masato Kawahara, Presentation Document" Network Structure and Network Performance-Regular Large Scale Structure, Random Network, Scale Free Network-", April 2007

  In fact, the problem of how N nodes can be connected to build a more economical, energy-saving, large-capacity, high-speed network is an important issue.

  In order to improve the economical efficiency and energy saving of the network, the average number of hops H must be reduced with the smallest number of links L for a given number of nodes N. Lowering the average hop count H also increases the network speed. At the same time, in order to increase the capacity, the network structure may be a network structure having a limit performance X (H) determined by the average hop count H.

  However, it is not impossible to derive the network structure on the H-Xeff limit performance X (H) using various optimization methods, but it takes a considerable amount of time. In practice, it is difficult to determine all the structures under various existing constraints.

  Therefore, even if the network structure is not strictly on the H-Xeff limit performance X (H), the network structure on the H-Xeff limit performance X (H) has performance close to the network bandwidth performance and the average number of relays performance. If the network structure can be easily derived, it will be very effective in practice.

  The present invention has been made in view of the above points, and easily exhibits the network bandwidth performance and the average number of relays performance of the network structure on the H-Xeff limit performance X (H). An attempt was made to provide a confirmed data transfer network.

  In addition, the present invention has sought to provide a method for configuring a data transfer network that can guarantee that the network structure on the H-Xeff limit performance X (H) has performance close to the network bandwidth performance and the average number of relays performance. Is.

  Furthermore, the present invention provides data that can be evaluated whether or not the data transfer network to be evaluated exhibits performance close to the network bandwidth performance and average number of relays performance of the network structure on the H-Xeff limit performance X (H). An attempt was made to provide a transport network evaluation method.

  It is well known that network structures having structures such as cubic regular graphs, quadratic regular graphs, quintic regular graphs, bipartite graphs, etc. are efficient network structures, without needing to be studied. In the present invention, it is excluded.

In the first aspect of the present invention, when the number of nodes is N and the number of links is L, a third order regular graph, a fourth order regular graph, a fifth order regular graph satisfying a relationship of N> 10 and N <L <3N, in concatenated data transfer network having a structure other than the structure of the bipartite graph, and characterized by being configured so as to satisfy all of the following conditions C1, C2 and C3.

  The second aspect of the present invention provides a third order regular graph, a fourth order regular graph, a fifth order regular graph satisfying the relationship of N> 10 and N <L <3N, where N is the number of nodes and L is the number of links. In the method for configuring a connected data transfer network having a structure other than the structure of the bipartite graph, the data transfer network is configured to satisfy three or more conditions including the conditions C1 to C3 among the conditions C1 to C7. It is characterized by.

  According to a third aspect of the present invention, there is provided a data transfer network for evaluating a connected data transfer network to be evaluated that satisfies a relationship of N> 10 and N <L <3N, where N is the number of nodes and L is the number of links. The evaluation method is characterized in that an evaluation result is determined based on satisfaction of three or more conditions including the conditions C1 to C3 among the conditions C1 to C7.

  (Condition C1) In the network, a node having degree 1 is always connected to a core node having degree 3 or higher, and is not connected to a node having degree 2 or lower.

  (Condition C2) In a network in which the network is linearly reduced, the number of core nodes Ncore is an integer satisfying 2 ≦ Ncore ≦ N, and the average relative deviation dkcore around the average value Kcore of Ncore core nodes in the primary reduced network Is dkcore <0.6.

  (Condition C3) The network limit in a network having a virtual cage graph structure in which the average number of relays performance value H and the network bandwidth performance value Xeff of the network are p, the average node order is k, and k is the node order. When the network bandwidth performance value is cageXeff, the relationship of the formula (A) is satisfied.

Xeff> (cageXeff / N-1) (H + p-2)
/ {2 (1 / cageXeff + p-3)} + 1 (A)
(Condition C4) The number of core nodes Ncore is a divisor of 2 (L−N).

(Condition C5) The core node number Ncore is the number of closed shell cores N _cc (h _shell ) with respect to the outermost shell layer number h _shell in a virtual cage structure having an average node order.

  (Condition C6) S in the range of 0 ≦ S ≦ max (0, N-Ncore-Dmax) is the number of nodes whose degree is 1 in the network, and D obtained from D = N-Ncore-S Is the number of nodes with degree 2 in the network.

(However, Dmax is a maximum integer not exceeding Ncore {2 (L−N) / Ncore + 2−X} / 2)
(Condition C7) Assuming that S degree 1 nodes in the network are distributed to Ncore core nodes in the primary contracted network, the order of the core nodes after distributing the S order 1 nodes is the distribution method. Assuming that an ideal distribution method in which the variation in distribution is the smallest is applied, the way in which the nodes of degree 1 and the core nodes are connected in the network matches. Alternatively, it is assumed that S degree 1 nodes in the network are distributed to Ncore core nodes in the primary contracted network, and the distribution method is as follows. When it is assumed that a distribution method is applied in which part or all of the nodes of degree 1 connected to the lower-order core nodes are moved to higher-order core nodes after being distributed once so that the variation becomes the smallest As a result, the way of connection of the node of degree 1 and the core node in the network matches.

  According to the first aspect of the present invention, it is possible to provide a data transfer network that exhibits performance close to the network bandwidth performance and the average number of relays performance of the network structure on the H-Xeff limit performance X (H).

  Further, according to the second aspect of the present invention, the data transfer network configured by the configuration method exhibits performance close to the network bandwidth performance and the average number of relays performance of the network structure on the H-Xeff limit performance X (H). Can be guaranteed.

  Furthermore, according to the third aspect of the present invention, whether or not the data transfer network to be evaluated exhibits performance close to the network bandwidth performance and the average number of relays performance of the network structure on the H-Xeff limit performance X (H). Can be evaluated.

It is explanatory drawing which shows the network structure of a tree structure with a node degree of 3. It is explanatory drawing which shows distribution of the average number-of-relays performance H-network bandwidth performance Xeff characteristic of each type of network, dividing various networks in the number of nodes N = 50 and the number of links L = 75 into three types. It is explanatory drawing which shows the distribution of the average relay number performance H-network band performance Xeff characteristic which satisfy | fills the condition C1, and the distribution of the average relay number performance H-network band performance Xeff characteristic which does not satisfy | fill the condition C1. In a network with the number of nodes N = 50 and the number of links L = 75, the network bandwidth performance Xeff is inferior to the H-Xeff limit performance value. The average value of the core node order in the primary reduced network of the network near the limit performance value less than 1. It is explanatory drawing which shows the appearance frequency of the average relative deviation dkcore around Kcore. Description showing average relay number performance H-network bandwidth performance Xeff characteristics of a network with the number of nodes N = 50 and the number of links L = 75 included in each region when the average relative deviation dkcore is divided into four regions FIG. In a network with the number of nodes N = 50 and the number of links L = 75, the core node number Ncore satisfies the condition C4 or C5, and thus shows an outstanding characteristic in the average relay number performance H-network bandwidth performance Xeff characteristic. It is explanatory drawing shown. The figure shows how (maxH, maxXeff) changes when the number of links is changed with the number of nodes N = 50 and the average node order k from 3 to 9. It is explanatory drawing of a reason why D edge node is engaged only in traffic relay. It is explanatory drawing which shows distribution of the average value of the distribution number of the S edge node to a core node, average number-of-relays performance H, and network bandwidth performance Xeff. It is explanatory drawing which shows the network structure in one Example of the data transfer network of the number of nodes N = 50 and the number of links L = 75 which satisfies all the conditions C1-C7. It is explanatory drawing which shows typically the example of a network structure of the number of nodes N = 30000 and the number of links L = 61500. It is explanatory drawing which shows distribution for every number of core nodes of the average relay number performance H-network band performance Xeff characteristic which the network provided with conditions C1-C7 shows, respectively. It is a flowchart which shows the flow of a process of the simple trial method which is 1st Embodiment of the structure method of a data transfer network. It is a flowchart which shows the flow of a process of the trial selection method which is 2nd Embodiment of the structure method of a data transfer network. It is a flowchart which shows the flow of a process of the random link replacement method which is 3rd Embodiment of the structure method of a data transfer network. It is a flowchart which shows the flow of a process of GA optimization method which is 4th Embodiment of the structure method of a data transfer network. It is a flowchart which shows the flow of the process in the evaluation method of a data transfer network. It is explanatory drawing which shows distribution of network bandwidth performance Xeff and average number-of-relays performance H about a network with the number of nodes N = 50 and the number of links L = 75.

(A) Embodiment of Data Transfer Network Hereinafter, an embodiment of a data transfer network according to the present invention will be described in detail with reference to the drawings.

The data transfer network of this embodiment is a concatenated network, and when the number of nodes is N and the number of links is L, the relationship of N> 10, N <L <3N, 4L ² <N ³ is satisfied. It has a network structure other than the network structure of the third order regular graph, the fourth order regular graph, the fifth order regular graph, and the bipartite graph, and satisfies the following conditions (C1) to (C7).

  (C1) In the network NW, a node having degree 1 is always connected to a node having degree 3 or higher, and is not connected to a node having degree 2 or lower.

  (C2) In the network NW * obtained by linearly reducing the network NW, the number of core nodes Ncore is an integer satisfying 2 ≦ Ncore ≦ N, and the average around the average value Kcore in the primary reduced network NW * of Ncore core nodes The relative deviation dkcore is dkcore <0.6.

  (C3) The network of limit networks in a network having a virtual cage graph structure in which the average relay number performance value H and the network bandwidth performance value Xeff of the network NW are p, the average node order is k, and k is the node order. When the bandwidth performance value is cageXeff, the relationship of the expression (1) is satisfied.

Xeff> (cageXeff / N-1) (H + p-2)
/ {2 (1 / cageXeff + p-3)} + 1 (1)
(C4) The number of core nodes Ncore is a divisor of 2 (L−N).

(C5) The number of core nodes Ncore is the number N _cc (h _shell ) of closed _shells relative to the number h _shell of the outermost shell layer.

  (C6) S in the range of 0 ≦ S ≦ max (0, N−Ncore−Dmax) is the number of nodes whose node order k is 1 in the network NW, and is obtained from D = N−Ncore−S. D is the number of nodes whose node order k is 2 in the network NW.

  (C7) Assuming that S S edge nodes in the network NW are distributed to Ncore core nodes in the primary contracted network NW *, the distribution method of the core nodes after distributing the S S edge nodes as the distribution method As a result of assuming that an ideal distribution method in which the variation of the network is the smallest is applied, the connection method between the S edge node and the core node in the network NW matches. Alternatively, it is assumed that S S edge nodes in the network NW are distributed to Ncore core nodes in the primary contracted network NW *, and as a distribution method, the order distribution of the core nodes after the S S edge nodes are distributed When it is assumed that a distribution method is applied in which a part or all of S edge nodes connected to a lower-order core node are moved to a higher-order core node after being distributed so that the variation is minimized. As a result, the connection method between the S edge node and the core node in the network NW is consistent.

  The data transfer network according to the embodiment satisfies all of the conditions C1 to C7. However, as a modification, only one of the conditions C1 to C3 and the conditions C4 to C7 is satisfied. That satisfy the above conditions (excluding all cases of conditions C1 to C7).

The reason for satisfying N> 10 is that an attempt was made to target a large-scale network structure having a certain number of nodes N. Also, the premise of satisfying the relationship of N <L <3N, 4L ² <N ³ is to define an efficient data transfer network with a network structure having a smaller number of links L than the number of nodes N. This is because. Note that a network structure having a smaller number of links L than the number of nodes N may be defined only by the relationship N <L <3N.

(A-1) Conditions C1 to C3 (A-1-1) Definition of terms and parameters First, definitions of terms and parameters will be described.

  Consider a network NW in which the number of nodes is N and the number of links is L. The number of links connected to a node is referred to as the “node order” or simply “order” of the node.

  A node whose node order is 1 is called an S edge node, the number of S edge nodes is expressed as S, a node whose node order is 2 is called a D edge node, and the number of D edge nodes is expressed as D. A node having a node degree of 3 or more is called a core node, and the number of core nodes is expressed as Ncore. At this time, the relationship of N = S + D + Ncore is established.

  The reason why a node having a node degree of 3 or more is important as a core node is as follows. The route to a node with a node degree of 1 is uniquely determined if the route to the node to which the node is directly connected is determined, and there is no room for route selection with the directly connected node. This is because the route selectivity is less important than a node having a node degree of 3 or more. For the same reason, a primary contracted network NW * described later is introduced. A node with node order 2 has two traffic input / output ports. Traffic input from one input / output port is output to the other input / output port and input from the other input / output port. Since the output traffic is output to one of the input / output ports, the route cannot be selected by the node having the node order of 2, and the route selectivity is less important than the node having the node order of 3 or more. It is.

  Here, a network obtained by removing all S links having a degree of 1 and S links connected to the nodes having a degree of 1 from the network is referred to as a network obtained by linearly reducing the original network. When the original network is represented as NW, the first reduced network is represented as NW *. The primary contracted network (hereinafter referred to as a primary contracted network) NW * includes D + Ncore nodes having a node degree of 2 or more in the original network NW.

  The node order in the primary contracted network NW * is expressed using K (upper case), and the node order in the original network NW is expressed separately using k (lower case). Note that k itself represents the average node order in the network NW, as will be described later.

  Of the Ncore core nodes, Ki is the node order in the primary contracted network NW * of the i-th (i is an integer from 1 to Ncore) core node. When the average value of the node degree K in the primary reduced network NM * of the Ncore core nodes is Kcore, the average relative deviation dkcore around the average value Kcore is determined as in the following equation (2). However, the sum Σ in the equation (2) is for i from 1 to Ncore, and √ {} represents the square root.

dkcore
= √ {Σ (Ki−Kcore) ² / (Ncore−1)} / Kcore (2)
For the network NW, the average number of relays performance value H, the network bandwidth performance value Xeff, the average link rate p, and the cage limit network bandwidth performance value cageXeff are determined as follows.

  The average number-of-relays performance value H is an average of the number of shortest path hops between all node pairs. When the number of hops of the shortest route from the node i to the node j is expressed as Hi, j (1 ≦ i <j ≦ N), the average number of relays performance value H can be expressed by equation (3). . However, the sum Σ in the equation (3) is a sum for a node pair of N (N−1) / 2. The hop count Hi, j of the shortest path can be obtained by the Dijkstra method or the like when the network structure is determined.

H = ΣHi, j / {N (N−1) / 2} (3)
Further, the number of the shortest paths from the node i to the node j is represented as Si, j. The number of shortest paths between nodes Si, j can also be obtained using the Dijkstra method. The node load Tk of each node k is defined by equation (4) using the shortest number of paths between nodes Si, j, and the maximum value of node load Tk is defined as Y as in equation (5). The sum Σ in equation (4) is the sum of combinations where 1 ≦ i, j, k ≦ N and Hi, j = Hi, k + Hk, j. Further, max in the equation (5) is a maximum value when 1 ≦ k ≦ N.

Tk = ΣSi, k Sk, j / Si, j (4)
Y = max (Tk) (5)
The node load Tk of the node k represented by the equation (4) passes through the node k, and two nodes i of all combinations of the route that becomes the shortest route of the nodes i and k and the route that becomes the shortest route of the nodes k and j. , J with respect to all the paths that are the shortest paths, for all pairs of two nodes i, j.

  Using the maximum value Y of the node load Tk, the network bandwidth performance Xeff is defined by equation (6).

Xeff = {N (N−1) / 2} / Y (6)
In the expression (6) and the expression (7) described later, N (N−1) / 2 is the number of combinations of two nodes when the number of nodes is N. When all the nodes have the same capacity, the node having the maximum value Y of the node load Tk defines the network processing capacity. The capability of the node (upper limit of processing amount) depends on the product of the communicable bandwidth and the load per communication. Expression (6) defines the network bandwidth performance Xeff with a constant value of the node capability (upper limit of processing amount) being “1”, and the network structure can be evaluated in this way as well.

  Further, the average link rate p is defined by equation (7). The average node order k in the network NW is defined by equation (8).

p = 2L / N / (N-1) (7)
k = 2L / N (8)
Two nodes are connected to one link. Therefore, as shown in equation (7), a value obtained by dividing the number of links L by the number of combinations of two nodes N (N−1) / 2 can be used as the average link rate (note that equation (7) Represents the division by N (N-1) / 2 by multiplication with the numerator denominator swapped). As described above, since two nodes are connected to one link, the total number of link ends connected to the node is twice the number of links (2L) and is divided by the number of nodes. Thus, it is possible to obtain the average node order which is the number of link ends per node.

Using the average node order k in the network NW, the sequence {f _h (k)} shown in equation (9) in which the h-th term becomes f _h (k), and the h-th sum from the first term is F _h (k) (10) consider the series _{F h (k)} in the expression.

{F _h (k)} = {1, k, k (k−1), k (k−1) ² ,... K (k−1) ^{h− 2} ,.
... (9)
{F _h (k)} = {1, 1 + k, 1 + k + k (k−1), 1 + k + k (k−1) + k (k−1) ² ,...} (10)
FIG. 1 is an explanatory diagram of the meanings represented by the sequence {f _h (k)} and the series {F _h (k)}, and shows an example in which the average node order k is 3. FIG. 1 shows a tree-structured network (so-called “cage graph-structured network”) in which the node order of other nodes is k (= 3) except for the outermost shell node, and FIG. FIG. 1B shows a network that can be reached from the top node NT of the tree structure in a maximum of 4 hops, and FIG. 1B shows a network that can be reached from the top node NT of the tree structure in a maximum of 5 hops. Note that the network illustrated in FIG. 1A is a primary contracted network of the network illustrated in FIG.

Focusing on the top node NT of the tree structure, the number of nodes that can be reached in 0 hops is “1” related to the node, and the number of nodes that can be reached in 1 hop is equal to the number of links connected to the node “ k ”, and the number of nodes that can be reached in two hops is“ k ”obtained by multiplying the number k of nodes that can be reached in one hop by the number (k−1) branching outward from each node that can be reached in one hop. (K-1) ", and the number of nodes that can be reached in 3 hops is the number of nodes that can be reached in 2 hops k (k-1). “k (k−1) ² ” multiplied by k−1), and similarly represents the number of nodes that can be reached with an arbitrary number of hops. For each hop number, the number sequence that lists the number of reachable nodes matches the number sequence {f _h (k)} shown in equation (9).

Since the sequence {f _h (k)} shown in equation (9) has the above-mentioned meaning, the first term of the series {F _h (k)} shown in equation (10) is 0 hops from the top node NT. The number of reachable nodes represents “1”, the second term represents the number of nodes reachable within 1 hop from the top node NT, “1 + k”, and the third term represents within 2 hops from the top node NT. The number of nodes that can be reached is “1 + k + k (k−1)”, and the fourth term is the number of nodes that can be reached within 3 hops from the top node NT “1 + k + k (k−1) + k (k−1) ² ”. The same applies hereinafter.

When h is the maximum integer satisfying F _h (k) ≦ N, h−1 is referred to as the cage limit outermost layer number cageh _shell in the number of nodes N and the number of links L. Since the h-th term represents the number of nodes that can be reached from the top node NT within (h-1) hops, the cage limit outermost layer number cageh _shell is the number of nodes that can be reached with the number of hops less than this number. Represents the number of nodes (total number of nodes) N or less, but the number of nodes that can be reached by the number of hops larger by 1 than this number exceeds the number of nodes (total number of nodes) N.

In the case of the example in FIG. 1, {F _h (k)} is {1, 4, 10, 22, 46, 94,...}, So if the number of nodes (total number of nodes) N is 50, The maximum integer h that satisfies F _h (k) ≦ N is 5, and the cage limit outermost shell layer number cageh _shell is “4”. FIG. 1A shows a structure similar to that described for the nodes up to the number of the cage limit outermost shell layers, the cageh _shell , and 46 nodes are described. If the number of nodes (total number of nodes) N is 50, four nodes can be provided outside the 46 nodes.

Now, consider the general term of (h-1) k (k -1) represented by ^h-2 (11) sequence shown in the expression {(h-1) k ( k-1) h-2}.

{(H-1) k (k-1) ^h-2 } = {(1-1) 1, (2-1) k, (3-1) k (k-1), (4-1) k (K-1) ² , ..., (h-1) k (k-1) ^h-2 , ...} ... (11)
In the tree structure as shown in FIG. 1, a case is considered in which traffic occurs from the top node NT to the N nodes the same number of times (here, once). (H-1) k (k -1) k in ^{h-2 (k-1)} h-2 , in ^the tree structure shown in FIG. 1, the node that reaches the top node NT in (h-1) hop (H-1) k (k-1) (h-1) in ^h-2 represents the number of hops. Therefore, (h-1) k (k-1) ^h-2 is a value obtained by multiplying the number of nodes reached by (h-1) hops from the top node NT by the number of hops, and (h-1) hops. Represents the total number of hops to the node that can be reached by.

cage limit the number of nodes that can reach one greater in number of hops than the outermost layer number _{Cageh shell} from the node number N, the number of nodes reachable cage limit outermost layer number _{Cageh shell} less than the number of hops _F h (k) N-Fh (k) minus. cage number of hops to the node which can be reached one greater in number of hops than the limit outermost layer number _{Cageh shell} is, of _course, is _{cageh shell} +1.

The cage limit average hop count cageh is expressed as the sum of the terms from the first term of the sequence {(h-1) k (k-1) ^h-2 } to the cageh _shell + 1th as shown in the equation (12): A value obtained by adding (N−F _h (k)) (cageh _shell +1) is defined by dividing by N−1. This is because 1 in N-1 does not consider a hop to the top node NT. In addition, the cage limit outermost shell layer number cageh _shell can be represented by h-1 from its definition.

cageh = {(1-1) 1+ (2-1) k + (3-1) k (k-1) + (4-1) k (k-1) ² + ... + (h-1) k (k _{^{-1) h-2 + (N}} -F h (k)) (cageh shell +1)} / (N-1) ... (12)
The “cage limit network bandwidth performance value“ cageXeH ”is defined as shown in equation (13) using the“ cage limit average hop count “cageh”.

cageXeff = N / (cageh + 1) (13)
The various parameters determined as described above, that is, the cage limit network bandwidth performance value cageXeff, the average link rate p, the average node order k, the cage limit outermost shell layer number cageh _shell , and the cage limit average hop count cageh are all nodes. If the number N and the link number L are determined, the values are determined, and the values can be actually obtained.

  The above-described conditions C1 to C3 are expressed by the terms and parameters described above.

(A-1-2) Outline of data transfer network satisfying conditions C1 to C3 The data transfer network satisfying conditions C1 to C3 has a number of links where the number of nodes N is 11 or more and the number of links L is N <L <3N. In a rare network, the network has a simple configuration and excellent network bandwidth performance and average relay number performance. Since N <L <3N, the number of links is rare with respect to the number of nodes, so it is expected that an economical and energy-saving network can be built if the network bandwidth performance and the average number of relays are improved.

  The present inventor has found that any network structure in the vicinity of the H-Xeff limit performance X (H) takes a structure of a core / edge separation structure. The core edge separation structure is characterized in that the S edge node is always connected to the core node, and the primary contracted network NW * does not have a link with the number of connections of one.

  When the core / edge separation structure is used, the average order of the core nodes in the primary contracted network NW * is 2 (L−N) / Ncore + 2, and conversely, the average order of the core nodes in the primary contracted network NW *. Is 2 (L−N) / Ncore + 2, the network has a core / edge separation structure.

  In this way, when looking at the network structure, the nodes are distinguished into the S edge node, the D edge node, and the core node, the network NW is linearly reduced, and the order distribution in the primary reduced network NW * is observed. It is possible to easily grasp a data transfer network having a configuration excellent in network bandwidth performance and average number of relays performance.

(A-1-3) Reasons for Adopting Conditions C1 to C3 Next, the reasons for adopting the conditions C1 to C3, in other words, the effects obtained by the data transfer network that satisfies the conditions C1 to C3 will be described.

  As shown in FIG. 3 in Section 4 of Non-Patent Document 1, in a network with relatively few links where N <L <3N, a random network is more difficult than an actual network such as the Internet. There is much better network bandwidth performance.

  FIG. 2 shows, as an example, the average number of relays performance H of various networks at N = 50 and L = 75—the network bandwidth performance Xeff characteristic. The horizontal axis represents the average number of relays performance H, and the vertical axis represents the network. Band performance Xeff is shown. 2B shows the H-Xeff characteristics of the random network, FIG. 2C shows the H-Xeff characteristics of the complex network, and FIG. 2A shows the H-Xeff characteristics of the other networks. Is shown.

  The structure of a complex network generally used as a structure of the Internet or the like has a small average relay number performance H but a large network bandwidth performance Xeff, and a random network can take a larger network bandwidth performance Xeff but an average relay number performance. It can be seen that H also tends to increase. Each point (plot) shown in FIG. 2 (A) is a point (network) whose H-Xeff characteristics have been examined so far, and the upper left boundary line indicates the above-described H-Xeff limit performance. If an appropriate network on the H-Xeff limit performance is selected, an excellent network such as a network having better average relay number performance and network bandwidth performance than a complex network or a random network can be selected.

  FIG. 3 shows the network on the H-Xeff characteristic examined in FIG. 2 that satisfies the condition C1 and does not satisfy the condition C1. FIG. 3A shows what satisfies the condition C1, and FIG. 3B shows that which does not satisfy the condition C1.

  FIG. 3 shows that a network that satisfies the condition C1 exists in the vicinity of the H-Xeff limit performance, and a network that does not satisfy the condition C1 does not exist in the vicinity of the H-Xeff limit performance. Thus, it can be understood that the condition C1 is almost necessary in terms of probability statistics in order for the network to exhibit a characteristic value in the vicinity of the H-Xeff limit performance.

  Here, an example in which N = 50 and L = 75 is shown, but any combination of the number of nodes N and the number of links L is the same.

  The links can be classified into six types as follows, and the number of each link is distinguished and described. The number of links stretched between core nodes is Lcc, the number of links spanned between core nodes and D edge nodes is Lcd, the number of links stretched between core nodes and S edge nodes is Lcs, and D edge nodes The number of links stretched between the S edge nodes is denoted by Ldd, the number of links spanned between the D edge node and the S edge node is denoted by Lds, and the number of links stretched between the S edge nodes is denoted by Lss. At this time, there is a relationship shown in the equations (14) to (16) between the various node numbers and the various link numbers.

S = 2Lss + Lds + Lcs (14)
2D = Lds + 2Ldd + Lcd (15)
L = Lcc + Lcd + Lcs + Ldd + Lds + Lss (16)
Since Lss = 0 is generally established because the network is connected between all nodes, equation (16) can be transformed into equation (17).

L = Lcc + Lcd + Lcs + Ldd + Lds (17)
When the condition C1 is expressed by an equation, Lds = 0, and thus the equations (14), (15), and (17) are transformed into equations (18), (19), and (20), respectively. be able to.

S = Lcs (18)
2D = 2Ldd + Lcd (19)
L = Lcs + Ldd + Lcd + Lcc (20)
From equation (19), Ldd = D−Lcd / 2 is obtained. By applying this and equation (18), equation (20) can be transformed into equation (21).

L = S + D + (Lcd + 2Lcc) / 2 (21)
Considering the total number of all the core nodes of the number of links connected to the core nodes of the primary contracted network NW * using the average value Kcore of the node orders in the primary contracted network NW * of the Ncore core nodes, (22) The formula holds. Further, with respect to the number of nodes, the equation (23) is established. By applying the formulas (22) and (23) to the formula (21), the formula (24) is obtained.

Ncore · Kcore = Lcd + 2Lcc (22)
N = Ncore + S + D (23)
L = N-Ncore + Ncore · Kcore / 2 (24)
From (24), it can be seen that the node order average value Kcore in the primary contracted network NW * is expressed by equation (25). From the condition C1, the condition of the expression (25) for the average value Kcore of the node orders in the primary contracted network NW * is obtained. Conversely, since the condition C1 can be derived from the condition of the equation (25), the average value Kcore of the node order in the condition C1 and the primary contracted network NW * is Kcore = 2 (L−N) / Ncore + 2. It turns out that they are completely equivalent.

Kcore = 2 (L-N) / Ncore + 2 (25)
As can be seen from FIG. 2, the network in the vicinity of the H-Xeff limit performance value is much higher in performance than the generally used network such as a complex network or a random network, and FIG. It can be seen that among the networks with high performance, the network satisfying the condition C1 tends to have particularly excellent performance. The structural condition necessary to obtain such an excellent network is the condition C1, and the relational expression Kcore = 2 (LN) / Ncore + 2 between the number of core nodes and the average value of the core nodes when the network is linearly reduced. Is to satisfy.

  FIG. 4 shows, for example, N = 50 and L = 75, where the network bandwidth performance Xeff is inferior to the H-Xeff limit performance value. The frequency of appearance of the average relative deviation dkcore expressed by the above-described equation (2) around Kcore is shown. In FIG. 4, the horizontal axis represents the average relative deviation dkcore, the curve freq is the appearance frequency of the network taking the average relative deviation dkcore, and the curve accum is the cumulative frequency of the appearance frequency freq.

  FIG. 5 shows average relative deviations dkcore as follows: (R1) dkcore = 0.0, (R2) 0.0 <dkcore ≦ 0.2, (R3) 0.2 <dkcore ≦ 0.4, (R4) 0. The figure shows the average relay number performance H-network bandwidth performance Xeff characteristics of the networks included in each of the four regions R1 to R4 where 5 <dkcore ≦ 0.6. 5A to 5D respectively show the average number of relays performance H-network bandwidth performance Xeff characteristic of the regions R1 to R4.

  FIG. 5 shows that when the average relative deviation dkcore is a network having values in the four regions R1 to R4 described above, the network has characteristics in the H-Xeff limit performance value and its vicinity.

  On the other hand, among the networks having dkcore not included in the above four regions R1 to R4 (0.6 <dkcore), there is no network that takes the H-Xeff limit performance value and its vicinity.

  As seen in the example of N = 50 and L = 75, when the average relative deviation dkcore around the average value Kcore is dkcore <0.6, the average number of relays performance H-network that is stochastically superior It is expected to exhibit bandwidth performance Xeff characteristics. This is the basis for the above-described condition C2.

  As can be seen from FIG. 3, just because the condition C1 is satisfied, it is not always in the vicinity of the H-Xeff limit performance value. The same applies to condition C2.

  The condition C3 that defines that the above-described expression (1) is satisfied is a condition that is always near the H-Xeff limit performance value.

  The region represented by the condition C3 is a region above the straight line connecting (minH, minXeff) and (cageh, cageXeff / 2). (Cageh, cageXeff / 2) is a network bandwidth performance characteristic that is half the value of the cage limit (cageh, cageXeff), and is sufficiently excellent as a network bandwidth performance characteristic. H-Xeff limit performance It is close to the value. In any combination of the number of nodes N and the number of links L targeted by the first embodiment, the limit performance value is in the region represented by the condition C3, and the region represented by the condition C3 is the H-Xeff limit performance. This is an area that defines the vicinity of the value.

  For the reasons described above, the conditions C1 to C3 are adopted, and the conditions C1 to C3 are satisfied, so that the data transfer network has the network bandwidth of the network structure on the H-Xeff limit performance X (H). The performance and the performance close to the average number of relays can be guaranteed.

(A-2) Conditions C4 and C5 (A-2-1) Contents of Condition C4 Among the networks satisfying the conditions C1 to C3, the number of core nodes for exhibiting the outstanding average number of relays performance H-network bandwidth performance Xeff characteristics Conditions for Ncore are a condition C4 and a condition C5 described later.

  When the network having the number N of nodes and the number L of links satisfies the condition C1, the average value Kcore of the node degree K in the primary reduced network NW * of the Ncore core nodes is expressed as Kcore = 2 (L-N) / Ncore + 2. When the distribution around the average value Kcore of the core node order K in the primary reduced network NW * is dkcore <0.6 that satisfies the condition C2, any integer Ncore where 2 ≦ Ncore ≦ N is satisfied. The network bandwidth performance and the average number of relays performance are excellent. Especially, when the condition C4 that “the number of core nodes Ncore is a divisor of 2 (L−N)” is satisfied, the network bandwidth performance is further improved. And the average number of relays is expected statistically.

(A-2-2) Contents of Condition C5 Similarly, in the network satisfying the conditions C1 to C3, in particular, “the number of core nodes Ncore is the number of closed shell cores N _cc (h _shell ) with respect to the outermost shell number h _shell” . When the condition C5 is satisfied, a further excellent network bandwidth performance and average number of relays performance are expected statistically.

Here, the outermost shell layer number h _shell and the closed shell core number N _cc (h _shell ) are defined as follows.

For the real number x, the sequence {f _h (x)} shown in the equation (26) in which the h-th term becomes f _h (x), and the h-th sum F _h (x) from the first term. Consider the series {F _h (x)} shown in equation (27). Expressions (26) and (27) are similar to the expressions (9) and (10) described above.

{F _h (x)} = {1, x, x (x−1), x (x−1) ² ,... X (x−1) ^{h− 2} ,.
... (26)
{F _h (x)} = {1, 1 + x, 1 + x + x (x-1), 1 + x + x (x-1) + x (x-1) ² , ...} (27)
Here, for each core node number Ncore, the maximum integer h satisfying F _h {2 (L−N) / Ncore + 2} ≦ N is referred to as the outermost shell layer number h _shell corresponding to the core node number Ncore.

When the real number x is an integer, the sequence {f _h (x)} represents the maximum number of nodes that can be reached from one node by h−1 hops in a network in which the node orders are all x. The series {F _h (x)} represents the maximum value of the sum of the number of nodes that can be reached from one node to h−1 hops.

When the real number x is not an integer, the sequence {f _h (x)} represents an expected value of the maximum number of nodes that can be reached from one node by h−1 hops in the network in which the average node order is x. It is thought that. Further, the series {F _h (x)} is considered to represent an expected value of the maximum value of the sum of the number of nodes that can be reached from one node to h−1 hops.

Based on f _h (x), F _h (x), and the outermost shell layer number h _shell , the closed core number N _cc (h _shell ) is considered as follows.

In any integer Ncore where 2 ≦ Ncore ≦ N, the outermost shell layer number h _shell determined as described above corresponds to each Ncore. The outermost shell layer number h _shell monotonically increases with respect to the core node number Ncore. The maximum value of the outermost shell layer number h _shell is when Ncore = N, and is obtained by obtaining the maximum integer h that satisfies F _h (2L / N) ≦ N. This maximum outermost shell layer number h _shell is defined as maxh _shell . In the largest outermost layer number _{maxh shell} following each outermost layer number _{h shell,} the largest core node number nCore, number closed shell core against the outermost shell layer number _{h shell} the outermost shell layer number _{h shell} corresponding Called N _cc (h _shell ).

The condition C5 described above is expressed by using the outermost shell layer number h _shell and the closed shell core number N _cc (h _shell ) defined as described above.

(A-2-3) Reasons for Adopting Conditions C4 and C5 Next, the reasons for adopting the conditions C4 and C5, in other words, the effects obtained by the data transfer network satisfying the condition C4 or C5 will be described.

  As described above, among the networks satisfying the conditions C1 to C3, the condition for the core node number Ncore for showing the outstanding average relay number performance H-network bandwidth performance Xeff characteristic is the condition C4 or C5. These conditions C4 and C5 are both independent, and either one of them may be satisfied or both may be satisfied at the same time. The time when both hold simultaneously is the time when it can be further expected to show excellent characteristics as the average number of relays performance H-network bandwidth performance Xeff characteristics.

  FIG. 6 shows an example of a network with the number of nodes N = 50 and the number of links L = 75. When the core node number Ncore satisfies the condition C4 or C5, the average number of relays performance H-network bandwidth performance Xeff characteristics Is shown. The condition C4 or C5 corresponding to the core node number Ncore corresponding to the protruding portion of FIG. 6 is shown.

  In addition, the network which protrudes to the maximum of the number of core nodes Ncore = 50 in FIG. 6 has a network structure of a regular graph of the third order, and is a structure outside the examination in this specification. Although this structure is not considered, this maximally protruding network satisfies the conditions C4 and C5, although the conditions C1 to C3 are not satisfied.

  As described above, in the vicinity of the H-Xeff limit performance X (H) value, the network structure is a core / edge separation structure. Satisfying the condition C1 is synonymous with taking a core / edge separation structure. By satisfying the condition C1, the core node order average value Kcore in the primary contraction network is Kcore = 2 (L−N) / Ncore + 2. The condition C2 indicates that there are many networks having excellent average relay number performance H-network bandwidth performance Xeff characteristics, although the distribution of the core node order is distributed near the average value Kcore. As shown in FIG. 5, when the network bandwidth performance Xeff is large and the core node number Ncore is also large, the core node orders are concentrated near the average value. When the number of core nodes Ncore is small, the structure number density near the H-Xeff limit performance is small, and even if the core node order is somewhat deviated from the average value, it is possible to show the characteristics near the H-Xeff limit performance. . The condition C2 indicates an allowable range that is allowed as a deviation from the average value of the core node orders when the number of core nodes Ncore is small.

  In order to convince the reason why the condition C4 or C5 is satisfied due to the average number-of-relays performance H-network bandwidth performance Xeff characteristic, it is necessary to understand the following.

  In order to increase the network bandwidth performance Xeff, it is necessary to reduce the traffic processing at each node as much as possible. In particular, when the processing amount of the node performing the maximum processing in the network is reduced, the network bandwidth performance Xeff of the entire network is increased.

  Incidentally, there is a close relationship between the average number of relays performance H and the amount of node processing. The relationship is that the total processing amount of nodes in the entire network is proportional to the average relay number performance H (strictly, the total processing amount of nodes is proportional to the average relay number performance H plus one). It is.

  The network bandwidth performance Xeff is determined by the maximum node processing amount, and the average relay number performance H is determined by the total processing amount of all nodes. For this reason, if the average number of relays performance value H is the same, the maximum network bandwidth performance can be obtained when variations in node throughput are eliminated. Since the average processing amount is strictly proportional to (H + 1) / N, the network bandwidth performance Xeff is N / (H + 1). This value is the best value of the network bandwidth performance when the average number of relays H can eliminate the node processing variation, and is lower than this value in a general network.

Incidentally, in a network with a large number of links such that L ≧ N ^1.5 , it is possible to easily take this best value. However, in the region of N <L <3N, which is the subject of this specification, links are rare. Therefore, the variation in node processing is not lost, and the value is less than this best value.

  FIG. 6 shows a curve indicating the best value N / (H + 1) of the network bandwidth performance Xeff in a network with the number of nodes N = 50 and the number of links L = 75. The point of (maxH, maxXeff) in a network with the number of nodes N = 50 and the number of links L = 75 is located in the immediate vicinity of this best value curve, but the node processing is not even a little and there are variations. Indicates that the best value cannot be reached.

  Here, consider a method of reducing variations in node processing in a network of (maxH, maxXeff). It is necessary to arrange the processing amount as much as possible in all the nodes with N nodes and to suppress the average relay number performance H as much as possible. For this reason, it is considered that there is no difference in the processing amount when the orders of the respective nodes are aligned as much as possible.

  When 2L / N is an integer, all node orders can be made uniform. For example, when N = 50 and L = 75, the average node order is exactly 3, and all node orders can be made uniform. When 2L / N is an integer, the network is a “regular graph” in “graph theory”. By the way, regular graphs have been studied in a wide range of research fields including “graph theory” research, and many structures are already known, so they are out of the scope of this specification. Even when N = 50 and L = 75, all node orders are not 3 and are not regular graphs, and in this specification, those not regular graphs are targeted.

  FIG. 7 shows changes in (maxH, maxXeff) when the number of links is changed from the number of nodes N = 50 to the average node degree k ranging from 3 to 9. As the number of links L increases, the average node order increases, maxH decreases, and maxXeff increases. However, maxXeff does not increase uniformly, but takes a local maximum value when the average node order k is an integer value and fluctuates up and down. While increasing. Thus, it can be seen that when the average node order has an integer value, the orders of all the nodes are aligned and take a large maxXeff.

  In particular, if there is a structure in which the average node order is an integer and the entire network looks the same from any node (automorphic structure), the processing amount of all the nodes is strictly equal. Such a structure has not been found so far in a network with the number of nodes N = 50 and the number of links L = 75. Whether such an automorphic structure exists or not depends on the combination of the number of nodes N and the number of links L, and the number of links L is rare compared to the number of nodes N. The automorphic structure has been found in the study of the “cage graph” and “Moore graph” in “graph theory”. As an example, when the number of links L is larger than the number of nodes N, N = 50 and L = 175, the degree 7 Hoffman Singleton graph is known, which is a regular graph of

  In the Hoffman Singleton graph, there are 7 nearest nodes from any node, 42 secondary neighbors with 2 hops, and 2 or fewer hops between all nodes. . As a result, the average relay number performance H and network bandwidth performance Xeff of Hoffman Singleton graph are H = (1 × 7 + 2 × 42) / (N−1) = 13/7, Xeff = 50 / (H + 1) = 17 It is calculated as .5.

In a regular network called a general cage graph or Moore graph, when the degree is k, the number of hops from the 0th hop, the 1st hop, the 2nd hop, ... is increased from any node. In this case, the maximum number of nodes that can be reached for the first time is the first, second, third in the sequence {f _h (k)} shown in equation (9) (or equation (26)), respectively. , ... h + 1 corresponds to the first term.

In a regular network with node order k, if the number of nodes that can be reached for the first time from any node with hop count h is the maximum number k (k−1) ^h−1 , this network structure is Here we say “h closed shell”. The Hoffman Singleton graph is two closed shells. In a network with the number of nodes N = 50 and the number of links L = 75, a four-shell structure of {1, 3, 6, 12, 24, 4} is conceivable. If such a structure exists, the maximum number of hops is five. In a network with the number of nodes N = 50 and the number of links L = 75, such a four-closed shell structure has not yet been found, but a three-closed shell structure has been found. The structure having the largest number of closed shells is called the maximum closed shell structure. In a network with a given number of nodes N and a link number L, the maximum number of hops between nodes is the smallest, and the minimum value is the maximum value in a network with a given number of nodes N and links L. It is called the outer shell layer number h _shell . The outermost layer number h _shell in the network with the number of nodes N = 50 and the number of links L = 75 is 5. A closed _shell with the outermost shell layer number h _shell is called a completely closed shell. The Hoffman Singleton graph is completely closed. Even if four closed shells exist in a network with the number of nodes N = 50 and the number of links L = 75, they are not completely closed. Fully closed shell and Moore graph are synonymous.

  In a regular network with node order k, it is shown that the average number of relays H is small when the number of closed shells is large. Moreover, since the symmetry of a node increases when the number of closed shells is large, the variation in node processing is also considered to be small. Therefore, in a network in which 2L / N is an integer, the network having (maxH, maxXeff) has a maximum closed shell structure without exception among all the network structures examined so far.

  When 2L / N is not an integer, all node orders cannot be made uniform, and the average core node order Kcore when Ncore = N is not an integer. Therefore, strictly speaking, the argument made up of “regular graphs” does not hold as it is. However, in fact, it has been experienced that such a network structure in which the average core node order is not an integer shows the same tendency as the “regular graph” statistically.

Even if 2L / N is not an integer and the average core node order k when Ncore = N is a non-integer, the sequence {f _h (k)} shown in the above equation (9) (or equation (26)) The first, second, third,..., H + 1th term arrives only when the number of hops from the 0th hop, the first hop, the second hop,. The average of the maximum number of nodes that can be considered.

  When 2L / N is an integer, 2L / N is a non-integer so that excellent average relay number performance H-network bandwidth performance Xeff characteristics can be expected if the maximum closed shell structure or the completely closed shell structure is taken. However, when the number sequence shown in the above equation (9) (or equation (26)) increases the number of hops from the 0th hop, 1st hop, 2nd hop, ... h hop from all nodes Similarly, an excellent average relay number performance H-network bandwidth performance Xeff characteristic can be expected when the distribution has an average value of the number of nodes that can reach for the first time.

  The above is a tendency in the vicinity of (max, maxXeff). In the primary reduced network NW * of the network NW in the vicinity of the H-Xeff limit performance, the number of nodes D + Ncore of the primary reduced network NW * and the primary reduced network NW There is a similar tendency in the average node order k * in *. Actually, as shown in FIG. 6, the average relay number performance H-network bandwidth performance Xeff characteristic is prominent with a certain limited number of cores Ncore.

  In other words, in correspondence with the explanation given above with reference to FIG. 7, when the node order k * in the primary contracted network NW * is an integer, excellent average relay number performance H-network bandwidth performance Xeff characteristic is expected. Can do. Condition C4 corresponds to such contents.

  Also, in the primary contracted network NW *, it can be reached only when the number of hops is increased from the respective nodes of the number of nodes D + Ncore to the 0th hop, the 1st hop, the 2nd hop,. When the average value of the number of nodes is distributed as shown in the equation (28), excellent average relay number performance H-network bandwidth performance Xeff characteristics can be expected as in the maximum closed shell structure. In particular, a completely closed shell structure is expected when this node order is extended to a non-integer. Condition C5 corresponds to such contents.

1, k *, k * (k * -1), ..., k * (k * -1) ^h-1 (28)
(A-3) Condition C6 (A-3-1) Contents of Condition C6 Condition C1 to C3, Condition C1 to C4, Condition C1 to C3, C5, or the network satisfying Condition C1 to C5, Condition C6 network satisfying the better network bandwidth performance and the average number of relays performance is probabilistic statistically expected.

  The above-described conditions C1 to C5 are determined by the number of nodes N, the number of links L, the number of core nodes Ncore, and the like, and do not depend on the values actually taken by the S edge node number S and the D edge node number D. Among the networks satisfying the conditions C1 to C3, the conditions C1 to C4, the conditions C1 to C3, C5, or the conditions C1 to C5, S + D = N-Ncore, 0 ≦ S ≦ N-Ncore, 0 ≦ D ≦ N-Ncore In any combination of S edge node number S and D edge node number D, excellent network bandwidth performance and average number of relays performance can be expected stochastically. By satisfying the condition C6 for the number of edge nodes S and the number D of edge nodes at the same time, further excellent network bandwidth performance and average number of relays performance are probabilistically expected.

Using the above-described concepts of the sequence {f _h (x)}, the series {F _h (x)}, and the outermost shell layer number h _shell , the following can be considered.

Using the series {F _h (x)} expressed by the equation (27), for each core node number Ncore, a maximum integer h satisfying F _h {2 (L−N) / Ncore + 2} ≦ N is determined as a core node. The outermost shell layer number h _shell corresponding to the number Ncore was used. At this time, there is X where Fhmax (X) = N with X ≦ 2 (L−N) / Ncore + 2. Using such X, Dmax is determined as the maximum integer not exceeding Ncore {2 (L−N) / Ncore + 2−X} / 2. Thus, Dmax can be obtained from the number of nodes N, the number of links L, and the number of core nodes Ncore.

  The condition C6 is described by using the Dmax thus obtained, and the range of the S edge node number S and the D edge node number D capable of realizing excellent network bandwidth performance and average relay number performance is defined. Yes.

(A-3-2) Reason for Adoption of Condition C6 Next, the reason for adopting the condition C6, in other words, the effect obtained by the data transfer network that satisfies the condition C6 will be described.

Table 1 shows various parameters related to the network showing the limit performance in the network having the number of nodes N = 50 and the number of links L = 75 with respect to the typical number of core nodes Ncore. The first row of Table 1 describes the number of core nodes Ncore, the core node degree distribution, the number D of edge nodes D, the maximum value (upper limit value) Dmax that the number of edge D nodes can take, the number S of edge nodes, and the conditions that are satisfied. Has been. For example, “18K3 + 16K4” in the column of the core node degree distribution represents “the total node order of 18 core nodes having a node degree of 3 and 16 core nodes having a node degree of 4.”

  The average value Kcore in the primary contraction network NW * of the core node is determined according to the size of the core node number Ncore, but the values that the D edge node number D and S edge node number S can take are arbitrary. . From the maximum value Dmax that can be obtained as described above, it can be seen that the actual value of the D edge node number D is in the range of 0 ≦ D ≦ Dmax. The effect of the network satisfying the condition C6 is shown in Table 1.

  As can be seen from Table 1, the number D of edge nodes tends to take either 0 or the maximum possible value Dmax.

  The reason why the possible range of the number D of edge nodes D is expressed as 0 ≦ D ≦ Dmax by the maximum value Dmax that can be obtained as described above will be described below.

  To understand why, you need to keep the following in mind:

  Most of the traffic flowing through the network NW passes through the primary reduced network NW *. The exception is only when the node that sent the traffic or the destination node is an S edge node, and passes through a link that is not on the primary contracted network NW * only once at the beginning or at the end.

  The reason why the S edge node is necessary for the entire network is to realize a larger core node average order Kcore by reducing its own node order together with the D edge node. For this reason, both the S edge node and the D edge node are less involved in traffic relaying in the network.

  Unlike the S edge node, the D edge node is a node in the primary contracted network NW * and may be involved in traffic relaying. The reason why only a small amount is involved will be described with reference to FIG.

FIG. 8 (A) shows a state in which one of D edge node D ₁ is connected to the other nodes in the primary contraction network NW *. In this case, the primary contracted network NW * is considered to exhibit performance near the limit performance. The node D the edge node _{D 1} are connected is set to _{n a} and _{n b.} If n _a and n _b are not directly linked, D edge node D ₁ is engaged in relaying traffic between at least n _a n _b . Relay traffic D edge node _{D 1} is engaged, is indicated by _{f 1} in FIG. 8 (A). The traffic D edge node _{D 1} is involved, in addition to _{f 1,} as shown in FIG. 8 (C), traffic _f a, is _{f b} exists that is terminated at D the edge node _{D 1.}

When there is more relay traffic f ₁ than terminal traffic such as f _a and f _b , the arrangement of FIG. 8B is more advantageous than FIG. 8A. Conversely, when the relay traffic f ₁ is less than the terminal traffic f _a and f _b , it can be seen that the arrangement of FIG. 8A is advantageous.

  As described above, although the D edge node is also a node in the primary contracted network NW *, it is generally not involved in traffic relay in general, and it is generally considered that the network relay is performed exclusively by a network formed by links between core nodes. (The reason I refused to be general here is that there are very few exceptions, but they exist).

  Since network relay is performed exclusively by a network formed by links between core nodes, the structure of the network formed by links between core nodes has an average number of relays performance H-network bandwidth performance Xeff characteristic of the entire network. It is the key to decide.

  The average node order k * in the primary reduced network NW * composed of Ncore core nodes and D D edge nodes is (29) because the average node order of the core nodes is kcore and the node order of the D edge nodes is 2. It is expressed by an expression.

k * = (Ncore · kcore + 2D) / (Ncore + D) (29)
Since the number D of edge nodes D is arbitrary in the conditions such as S + D = N−Ncore, 0 ≦ S ≦ N−Ncore, 0 ≦ D ≦ N−Ncore, the expression (28) is completely (extended). D can be adjusted by adjusting so as to satisfy the condition of the closed shell structure. The value of D obtained by this adjustment is Dmax obtained by the calculation method described above.

  From a numerical condition, a (closed) fully closed shell structure may be obtained when D is exactly Dmax, but such a structure does not necessarily exist. Therefore, in order to obtain the maximum number of closed shells, it suffices that D is equal to or less than Dmax, so that the value of D is in the range of 0 ≦ D ≦ Dmax.

  In the general case, according to the above discussion, it is considered that the number of D edge nodes is more advantageous than the number of S edge nodes. There is a small difference, and depending on the conditions of the actual structure, it is advantageous to have more S edge nodes or more D edge nodes. Therefore, the actual number D of D edge nodes is 0 ≦ D ≦ Dmax. It is understood that the number in the range is often 0 or Dmax.

(A-4) Condition C7 (A-4-1) Contents of Condition C7 Among the conditions C1 to C6, among the networks satisfying three or more conditions including the conditions C1 to C3, the network satisfying the condition C7 is Even better network bandwidth performance and average number of relays are expected statistically .

  Of the conditions C1 to C6, if there is a combination of the S edge node number S, the D edge node number D, and the core node number Ncore that satisfy three or more conditions including the conditions C1 to C3, the S S in the network NW An edge node is arbitrarily connected to any one of the Ncore core nodes in the primary contracted network NW *. Regardless of the connection method, the conditions C1 to C3 are included in the conditions C1 to C6. If the number of S edge nodes, D edge nodes, and core nodes satisfying three or more conditions is satisfied, it can be expected that the network bandwidth performance and the average number of relays are excellent in terms of probability statistics.

  However, if it is assumed that the S edge nodes in the network NW are distributed to the Ncore core nodes in the primary contracted network NW * according to the ideal distribution method shown below, the actual connection of the core nodes of the S edge node If it is the same as that distributed by such an ideal distribution method, better network bandwidth performance and average number-of-relays performance are expected statistically.

"Ideal distribution method of S edge node"
Suppose that one S edge node in the network NW is sequentially connected to each of the Ncore core nodes in the primary contracted network NW *. Let s be the number of S edge nodes connected to each core node.

  (A) s is zero until the S edge node is started for each core node.

  (A) When the node degree of each core node in the primary contracted network NW * is K, any one node is arbitrarily selected from the core nodes having the smallest K + s, and one S edge node is connected thereto. By this operation, s of the core nodes connected to the S edge node is increased by 1, and the number of remaining S edge nodes is decreased by 1.

  (C) The operations (a) are repeated until there are no S edge nodes connected to the core node.

  By performing the operations (a) to (c), it is possible to finally distribute S S edge nodes in the network NW to Ncore core nodes in the primary contracted network NW *.

  An ideal distribution method for the S edge nodes may be any method that minimizes the variation in the degree distribution of the core nodes after distributing all the S edge nodes, and is not limited to the method described above.

  Condition C7 described above is based on the assumption that the S edge nodes in the network NW are distributed to the Ncore core nodes in the primary contracted network NW * by applying an ideal distribution method. This is a condition that the connection method between the node and the core node is the same. Note that the condition C7 is such that the S S edge nodes in the network NW are once distributed by applying an ideal distribution method to the Ncore core nodes in the primary contracted network NW * and then connected to the lower order core nodes. Assuming that a distribution method of moving some or all of the S edge nodes to higher-order core nodes is applied, the connection between the S edge nodes and the core nodes in the network NW is identical. It is supposed to include.

(A-4-2) Reason for Adoption of Condition C7 Next, the reason for adopting the condition C7, in other words, the effect obtained by the data transfer network satisfying the condition C7 will be described. In the following, a case where the number of nodes N = 50, the number of links L = 75, and the number of core nodes Ncore = 20 will be described as an example.

When the number of nodes N = 50, the number of links L = 75, and the number of core nodes Ncore = 20, the node order average value Kcore of the core nodes in the primary contracted network NW * is Kcore = 2 (L−N) /Ncore+2=4.5. Become. The distribution of the average order in which dkcore has the smallest average relative deviation around the average value Kcore is a distribution of 10 nodes of node order 4 and 10 nodes of node order 5. This distribution is expressed as 10K4 + 10K5. The maximum value Dmax that can be taken as the number of D edge nodes is 1. A network having this distribution satisfies the conditions C1 to C3 and C5, and satisfies the condition C4 if the number D of edge nodes D is 0 or 1. In the following, D = 0 is taken as an example. At this time, the number S of S edge nodes becomes 30, and it becomes a problem how to distribute the S edge nodes to the core nodes. In this case, since there are two types of core nodes in terms of node order, there are 31 ways shown in Table 2 for distributing the 30 S edge nodes to the two types of core nodes. In Table 2, S _K4 represents the number of S edge nodes connected to the core node having the node degree of 4, and S _K5 represents the number of S edge nodes connected to the core node of the node order of 5.

  There are various networks in which there are 10 nodes of node order 4 and 10 nodes of node order 5, and the average values of average relay number performance H and network bandwidth performance Xeff are described in Table 2.

FIG. 9 shows the distribution of the average value of the distribution number S _K4 of the S edge node to the core node of the node degree 4 and the average number of relays performance H and the network bandwidth performance Xeff on the H-Xeff plane.

  As can be seen from FIG. 9, the method of allocating 20 S edge nodes to the 4th-order core node and 10 S edge nodes to the 5th-order core node, respectively, is the average number of relays performance H-network bandwidth performance Xeff characteristics of the entire network. It turns out that it is excellent in view. More specifically, this distribution method is a method of distributing two S edge nodes to ten fourth-order core nodes and one S edge node to ten fifth-order core nodes. Yes, the degree of the distributed core nodes is 6 for all the core nodes. This distribution method coincides with the above-described “ideal distribution method of S edge nodes”.

In addition, the distribution method in which the S edge node distribution number (S _K4 ) to the fourth-order core node is less than 20 is inferior to the network bandwidth performance Xeff at 20, but is superior in the average relay number performance H. . That is, if the weight of the average number of relays performance H is large compared with the network bandwidth performance Xeff, the number of distribution to the lower-order core nodes is reduced compared to the “ideal distribution method of S edge nodes”, What is necessary is just to increase the number of distribution to a high-order core node. The latter half of the condition C7 defines such a case.

  In the above description, the case where the number of nodes N = 50, the number of links L = 75, and the number of core nodes Ncore = 20 has been described as an example, but the same applies to other cases.

  The contents of the explanation of the effect of the network satisfying the conditions C4 to C6 can be summarized as an excellent network in terms of the average number of relays performance H-network bandwidth performance Xeff characteristics in the H-Xeff limit performance and nearby networks. The network structural characteristics obtained can be summarized as follows (i) and (ii).

  (I) The network takes a core / edge separation structure, which is a network structure in which the S edge node is not connected to the D edge node. The average value Kcore of the node orders of the core nodes in the primary contracted network NW * is Kcore = 2 (L−N) / Ncore + 2. When the number of core nodes Ncore is large, the average value Kcore is small, and when the number of core nodes Ncore is small, the average value Kcore is large.

(Ii) Almost all traffic flowing in the network is relayed via the core node. In other words, the S edge node only terminates the traffic, does not relay the traffic, and the D edge node does not almost relay, and the termination is the main function. Therefore, the maximum processing node becomes a core node, and the effective maximum node number Xeff, which is the network bandwidth performance, is about Xeff to Ncore / 3 at most. If the core node number Ncore is large, the network bandwidth performance Xeff is large, and if the core node number Ncore is small, the network bandwidth performance Xeff is small. The traffic is averaged by the 0th hop, the 1st hop, the 2nd hop,... The hth hop, via the number of nodes described in each term of the above equation (28) on the average, and the primary reduced network NW. * Travel through. Therefore, the average number of relays H is equal to or more than the second item of equation (28), and is the average number of hops up to the maximum outermost shell layer number h _shell . This is almost determined by the number h _shell of the outermost shell layer. When the node order average value Kcore of the core nodes in the primary contracted network NW * is small, the outermost shell layer number h _shell is large, and when the average value Kcore is large, the outermost shell layer number h _shell is small. For this reason, if the network has a large core node number Ncore, the average relay number H is large, and if the core node number Ncore is small, the average relay number H is small.

  Traffic is probabilistically concentrated on nodes with a high degree. If the traffic processing is biased to some nodes, the network bandwidth performance Xeff decreases. On the other hand, when the traffic is relayed between nodes having a large degree, the average number of traffic relays is reduced.

  The reason why the condition C7 is superior from the viewpoint of the average number of relays performance H-network bandwidth performance Xeff characteristic of the entire network as shown in FIG. 9 can be considered as follows.

  When there are core nodes having different node orders in the primary contracted network NW *, the larger the node order average value Kcore, the greater the amount of relay traffic. In such a case, it is possible to distribute the S edge nodes well and to equalize the processing amount among the core nodes without variation. By performing such equalization, the maximum network bandwidth performance Xeff can be realized. When the condition of the first half of the condition C7 is satisfied, it can be seen that the S edge nodes are well allocated, the node processing amount is equalized, and the maximum network bandwidth performance Xeff is obtained.

  The network bandwidth performance Xeff is lowered by shifting the S edge node to the core node having a higher node order than the arrangement that gives the maximum network bandwidth performance Xeff. However, a network structure with an excellent average relay number performance H is obtained. It is done.

(A-5) Effect of Data Transfer Network of Embodiment According to the data transfer network of the embodiment, the network bandwidth performance and the average number of relays are satisfied in a network with a small number of links because at least the conditions C1 to C3 are satisfied. The performance can be excellent. When one or more of the conditions C4 to C7 are satisfied, the network bandwidth performance and the average number-of-relays performance are further improved.

(A-6) First Example of Data Transfer Network of Embodiment FIG. 10 shows an example of a data transfer network that satisfies all of the conditions C1 to C7 and has the number of nodes N = 50 and the number of links L = 75. It shows an irregular network structure.

  FIG. 10A shows the position of the network of this embodiment on the H-Xeff plane.

  FIG. 10C describes the connection relationship of each core node in a layout diagram. In FIG. 10C, illustration of the S edge node and the D edge node is omitted, and all 10 core nodes and direct links (solid lines) between the core nodes are shown. The broken line in FIG. 10C indicates that there is an indirect link via the D edge node between the core nodes at both ends of the broken line.

  FIG. 10B describes the connection relationship of each core node in a table. Each row includes the number of core nodes, the number of connected S edge nodes, the number of other core nodes connected via a direct link, and the numbers and indirect links of other core nodes connected via an indirect link. It consists of a set of numbers. The first row is the row of the core node with the number “1”, is connected to one S edge node, is directly linked to the core nodes with the numbers 3, 7, 9, 10 and the core node with the number 2 Are connected by four indirect links, connected to the core node of number 4 by two indirect links, and one core node of number 5 (notation is omitted in the case of one). It shows that it is connected with an indirect link.

(A-7) About a network with the number of nodes N = 50 and the number of links L = 75 Hereinafter, with respect to the network with the number of nodes N = 50 and the number of links L = 75 used in the above description, the relationship with the above-described conditions Organize and explain.

Table 3 shows the core node order K that minimizes the average value Kcore and the average relative deviation dkcore of the core node order in the primary contracted network NW * of the core edge separation structure in the network having the number of nodes N = 50 and the number of links L = 75. , The outermost shell layer number h _shell , the maximum value Dmax that can be taken as the number of D edge nodes, the maximum D value, and the S value at that time.

The average value Kcore is given by Kcore = 2 (L−N) / Ncore + 2 in the range of 2 ≦ Ncore ≦ N.

If the average relative deviation dkcore is within 0.6 around the average value Kcore, the conditions C1 to C3 are satisfied. The distribution of the core node degree K that minimizes the average relative deviation dkcore in Table 3 satisfies all of the conditions C1 to C3 except that the core node number Ncore is 50 (first row in Table 3).

If the core node number Ncore is not equal to the node number N and the average value Kcore is an integer, the condition C4 is satisfied. The number of core nodes Ncore is 25, 10, 5, 2 when the condition C4 is satisfied.

As the number of core nodes Ncore increases, the number of outermost shell layers h _shell also increases. When the number of nodes N = 50 and the number of links L = 75, the outermost shell layer number h _shell takes an integer value from 2 to 5. When each of the outermost shell layer values h _shell from 2 to 5 has the maximum core node number Ncore, the condition C5 is satisfied. Condition C5 is satisfied when the number of core nodes Ncore is 48, 34, 20, or 8.

The condition C6 is satisfied when the D value is equal to or less than the Dmax value.

  When the core node degree K distribution has the minimum Ncore value and the average relative deviation dkcore in Table 3 and the D value is equal to or less than the Dmax value, the network structure satisfies the conditions C1 to C3 and C6.

  As an example of a specific network structure, two network structures with a core node number Ncore = 34, a core node degree K distribution of 18K3, 16K4, D = 0, and S = 16 are shown. The former corresponds to the second example of the data transfer network of the embodiment, and the latter corresponds to the third example of the data transfer network of the embodiment.

[Ncore = 34, 18K3, 16K4, D = 0, S = 16 example (1)]
Table 4 shows the number of each core node, the number of S edge nodes connected to the core node, and the numbers of other core nodes to which each core node is connected. The number after # in each row is the core node number. The number of S edge nodes connected to the core node is represented by the number between the core node numbers; The expression method as shown in Table 4 is called a link table.

  The link table in Table 4 defines the actual network structure.

  The network structure of this example is Ncore = 34, 18K3, 16K4, D = 0, S = 16.

  The order distribution of the entire network is [0, 16, 0, 2, 32]. Here, the method of expressing the degree distribution such as [0, 16, 0, 2, 32] used is expressed by a sequence of the number of nodes when the order is increased one by one, starting from the number of the zeroth order nodes. Yes. The arrangement of the S edge nodes of this network is an arrangement that satisfies the first half of the condition C7.

  The average number of relays performance H-network bandwidth performance Xeff characteristic of this network is H = 3.2620408, Xeff = 8.968883, and shows the maximum network bandwidth performance Xeff among the Ncore = 34 networks examined so far. Structure.

[Ncore = 34, 18K3, 16K4, D = 0, S = 16 example (2)]
Table 5 shows another structure with Ncore = 34, 18K3, 16K4, D = 0, S = 16 in the link table.

  The degree distribution of the entire network is also [0, 16, 0, 2, 32] as in the example (1). Also, as in the example (1), the arrangement of the S edge nodes of this network is the condition C7. Satisfy the first half of.

  The average number of relays H of this network-the network bandwidth performance Xeff characteristics are H = 3.2359184 and Xeff = 8.812950, and the minimum average number of relays performance H among the Ncore = 34 networks investigated so far. In the average relay number performance H, the maximum network bandwidth performance Xeff is shown.

  As described above, two embodiments (second embodiment and third embodiment) in which the number of nodes N = 50, the number of links L = 75, and Ncore = 34, 18K3, 16K4, D = 0, and S = 16 are shown. As can be seen from the two examples, in order to find an actual network structure having excellent average number of relays performance H-network bandwidth performance Xeff characteristics, some of conditions C1 to C7 are combined (however, (Contain at least conditions C1 to C3), Ncore number of the link tables in Table 4 and Table 5, and the number of link destinations of each core node, and determine how to connect the actual core nodes within the framework You can see that it is good. Such a determination method will be described later.

(A-8) Regarding a network with the number of nodes N = 30000 and the number of links L = 61500 Next, the relationship between the above-described conditions will be described for a network having the same number of nodes N and the same number of links L as the Internet. To do.

  The scale of the Internet is N = 30000 and L = 61500 at the AS (application server) level. As shown in Section 4 of Non-Patent Document 1 described above, the Internet structure is a complex network, and the average number of relays performance H-network bandwidth performance Xeff characteristics are H≈3.8 and Xeff≈6. Estimated.

Tables corresponding to Table 3 above when the number of nodes N = 30000 and the number of links L = 61500 are divided into Tables 6 and 7. Tables 6 and 7 show the number of core nodes Ncore that satisfies the condition C4 or C5.

  Here, as an example of a specific network structure, a network structure in which Ncore = 252, the distribution of the core node degree K is 252K252, D = 126, and S = 29622 is shown.

[Example of Ncore = 252, 252K252, D = 126, S = 29622]
The 252 core nodes can form a full mesh using 251 links. The remaining one link is applied as a link for interposing a D edge node between 126 arbitrary core node pairs. Thus, the structure of the primary contracted network NW * is obtained. If the S edge node is distributed to one of the core nodes with respect to the structure of the primary contracted network NW *, an actual network structure can be obtained.

  Depending on how the S edge nodes are distributed, characteristics are born in the average number of relays performance H-network bandwidth performance Xeff characteristics.

  In order to obtain a structure with good network bandwidth performance Xeff, the structure should be made as uniform as possible. Since the number of S edge nodes 29622 is decomposed as 29622 = 117 × 114 + 118 × (252−114), 117 S edge nodes are connected to 114 core nodes, and 118 more than the other 138 core nodes are added. The network structure can be determined by connecting S edge nodes.

  FIG. 11 schematically shows the structure of this network.

  FIG. 11A shows 252 core nodes out of 30000 nodes represented by point PCs arranged in a circle, and FIG. 11B shows a state in which the core nodes are linked so as to form a full mesh. Is represented in a simplified manner, representing one core node. In FIG. 11C, the D edge node is represented by a point PD, and every other core node PC is connected. FIG. 11C shows a state in which a plurality of S edge nodes are connected with one core node PC as a representative. Here, the S edge node is represented by a point PS. In this structure, there are actually 114 core nodes to which 117 S edge nodes are connected, and 118 S edge nodes are connected to the remaining 138 core nodes.

  The average number of relays performance H-network bandwidth performance Xeff characteristics of this network are H = 2.979316 and Xeff = 125.773515.

  The average number of relays performance H is almost 3 because the core nodes are fully meshed so that traffic can pass between any core nodes in one hop. Since most nodes are S edge nodes, much of the traffic requires one hop from the starting S edge node to the core node, one hop between core nodes, and one hop from the core node to the egress S edge node. In short, you can get up to 3 hops. Since there is some traffic that reaches from the departure node to the egress node with the hop count less than that, the average hop count is interpreted as a value of H = 2.979316, which is slightly lower than 3.

  The network bandwidth performance Xeff is a value close to 126, which is almost half of the number of core nodes Ncore 252. The reason why almost half of the Ncore number is the network bandwidth performance Xeff is that most of traffic is traffic between S edge nodes, most of which is due to using the core node as a relay node twice.

  In the current Internet, H≈3.8 and Xeff≈6, so that the network structure shown in FIG. 11 has much better performance than the current Internet. That is, the network structure shown in FIG. 11 has the same number of nodes and links as the current Internet, but the average number of relays performance H is small, and the network bandwidth performance Xeff is about 20 times the traffic. It is.

  The average number of hops (average number of relays) affects transmission delay and signal quality in terms of network performance. The smaller the average number of hops, the lower the transmission delay and the higher the signal quality. In addition to these signal characteristics, the average number of hops is an index that directly indicates network operation costs and network power consumption.

  H / Xeff, which is obtained by dividing the average number of relays performance H by the network bandwidth performance Xeff, is an index simply indicating the network operation cost and network power consumption required per unit traffic. The example of the network structure shown in FIG. The index is about 26 times higher than the current Internet.

(B) Embodiments of Method for Configuring Data Transfer Network Next, four embodiments of a method for configuring a data transfer network according to the present invention will be described in detail with reference to the drawings.

  The configuration method of each embodiment is intended to configure a data transfer network that satisfies at least three conditions including at least the conditions C1 to C3 among the above-described conditions C1 to C7. In the configuration method of each embodiment, a network structure in an initial state is determined after determining a framework such as the number of nodes N, the number of links, and the number of core nodes Ncore, and the network bandwidth performance value or the average number of relays performance value of the entire network is determined. By optimizing either or both of them as an index, a data transfer network with good performance is finally constructed.

  As a method of deciding how to connect between core nodes in the framework, for example, a method of deciding at random first is conceivable. By randomly deciding, a network distribution as shown in FIG. When the connection method obtained by the random method is changed little by little, when the average relay number performance H and the network bandwidth performance Xeff are improved, the optimization of repeating the connection is repeated. By applying such an optimization method, it is possible to obtain a network on or near the H-Xeff limit performance.

  The effect of imposing the conditions C1 to C7 can lead to a network structure having an excellent average number of relays performance H-network bandwidth performance Xeff characteristic as shown in FIG. In addition, when the optimization method is applied, the amount of work required for the optimization work is greatly increased by using a network structure that satisfies some of the conditions C1 to C7 at the starting point of the optimization. It can be shortened. These effects are considered to increase as the number of nodes N and the number of links L increase.

  The merits of applying the optimization method will be specifically described by taking the case where the number of nodes N = 50 and the number of links L = 75 as an example.

  FIG. 12 shows the distribution of the average number of relays performance H-network bandwidth performance Xeff characteristic shown by the networks having the conditions C1 to C7. FIG. 12 shows all Ncores where 2 ≦ Ncore ≦ 50. A typical number of core nodes Ncore is indicated by a numerical value, and the distribution of the number of core nodes extends vertically below the numerical value. The distribution in FIG. 12 is obtained by randomly connecting the nodes under the conditions C1 to C7. By optimizing the network structure at a certain position on such distribution with the network bandwidth performance value Xeff and the average number-of-relays performance value H of the entire network as indices, the H-Xeff limit performance or the H-Xeff limit performance A network structure having excellent average number of relays performance H-network bandwidth performance Xeff characteristics in the vicinity can be derived.

(B-1) First Embodiment of Data Transfer Network Configuration Method Next, a simple trial method will be described as a first embodiment of a data transfer network configuration method. FIG. 13 is a flowchart showing a flow of processing of the simple trial method.

  For example, a program for executing a simple trial method (simple trial method program) is installed in a computer, and the simple trial method is executed by interaction between the operator and the simple trial method program. The same applies to the trial selection method, random link replacement method, and GA optimization method described later.

First, the operator determines and inputs the number of nodes N and the number of links L (step 100). Here, whether traffic meets the necessary environmental conditions such as total required traffic (lower limit of traffic bandwidth performance Xeff), required hop characteristics (upper limit of average number of relays performance H), network construction cost, network operation cost, network power consumption, etc. While examining the bandwidth performance and the average number of relays performance, it is determined so as to be consistent with network member requirements such as used node capacity and used transmission amount (capacity of link transmission). In this input step, the computer may check the relationship of N <L <3N, 4L ² <N ³ .

  Next, candidates for the number of core nodes Ncore of the network are calculated according to the items related to the necessary environmental conditions (step 101). For example, a calculation formula for calculating the core node number Ncore is determined in advance from the values of the node number N, the link number L, and the necessary environment items, and the core node number Ncore is calculated. The operator may also input the number of core nodes Ncore.

  When a candidate for the number of core nodes Ncore is obtained, it is determined whether or not there is a value that satisfies the conditions C4 and C5 in the vicinity of the candidate. If so, the order distribution of the core nodes is determined, and the process proceeds to step 103 described later. If there is no value that satisfies the conditions C4 and C5, a reexamination is performed, such as whether the range to be regarded as a neighborhood is appropriate, or whether the conditions C4 and C5 do not have to be satisfied, and the appropriate number of core nodes cannot be determined even by the reexamination. In this case, the process returns to step 100 described above, and when an appropriate number of core nodes is determined by reexamination, the order distribution of core nodes is determined, and the process proceeds to step 103 described later (step 102). For example, the computer automatically determines whether there is a value that satisfies the conditions C4 and C5 in the vicinity of the candidate for the core node number Ncore. The determination of the core node order distribution may also be automatically executed by the computer. For example, after calculating an average node order K determined by the number of nodes N and the number of links L, a predetermined multiple thereof is set as an average node order related to the core node, and the average node order related to the core node is achieved with as few node orders as possible. The order distribution of the core node is determined as follows.

  As described above, when the number of applied nodes N, the number of links L, and the number of core nodes Ncore are aligned and the degree distribution of the core nodes is determined, the condition C6 is satisfied from the number of nodes N, the number of links L, and the number of core nodes Ncore. The upper limit Dmax of the number of D edge nodes is determined, D within a range of 0 ≦ D ≦ Dmax is determined at random, then the number of S edge nodes is determined as S = N−D−Ncore, and the total degree distribution is determined. (Step 103). The processing after step 103 is executed by, for example, the computer. In addition, when determining the degree distribution of the core node or when determining the total degree distribution, it is confirmed that the condition C2 is satisfied.

  Next, for the nodes (core node and D edge node) excluding the S edge node, the link destination of the node is randomly determined so as not to be a multiple link in order from the smallest node order, and the primary contraction is repeated. A network is formed (step 104). If it is impossible to avoid the occurrence of multiple links during the formation of the primary reduced network, or if the primary reduced network is not connected, the process returns to step 104 to return to the primary reduced network. Is started again from the beginning (step 105).

  When the primary contracted network is formed, the S S edge nodes are randomly linked to the Ncore core nodes to complete the network structure (step 106). The condition C1 is satisfied by the process of step 106. It is confirmed that the condition C3 is satisfied for the completed network structure. If the satisfaction of the condition C7 is also a requirement, it is confirmed that the condition C7 is satisfied for the completed network structure. If the conditions C3 and C7 are not satisfied, for example, the process returns to step 104, and the formation of the primary contraction network is performed again from the beginning.

(B-2) Second Embodiment of Data Transfer Network Configuration Method Next, a trial selection method will be described as a second embodiment of the data transfer network configuration method. FIG. 14 is a flowchart showing a flow of processing of the trial selection method, and the same reference numerals are assigned to the same and corresponding steps as in FIG. 13 described above.

  In the trial selection method, the processing (steps 100 to 102) until the core node number Ncore is determined from the inputted node number N and link number L and thereafter the order distribution of the core nodes is determined is the same as in the simple trial method.

  In this trial selection method, the network structure determination process in steps 103 to 106 is executed a plurality of times as a loop process (step 107). From the obtained network structures, the best network structure is selected based on an evaluation formula that reflects at least one of traffic bandwidth performance and average hop performance (for example, weighted addition value of both performance values) (Step 108).

  Here, the number of repetitions of the network structure determination processing in steps 103 to 106 may be a fixed number, or the number of repetitions is repeated until a network structure equal to or higher than the best standard is found (however, the number of times of censoring) May be set). In the latter case, when a new network structure is determined, it is determined whether or not the evaluation value is greater than or equal to the best standard, and if it is greater than or equal to the best standard, the iteration is terminated. It is determined which is the best network structure, and the best network structure is determined at that time.

(B-3) Third Embodiment of Data Transfer Network Configuration Method Next, a random link replacement method will be described as a third embodiment of the data transfer network configuration method. FIG. 15 is a flowchart showing a flow of processing of the random link replacement method, and the same reference numerals are assigned to the same and corresponding steps as in FIG. 13 described above.

  In the random link replacement method, the processing (steps 100 to 102) until the core node number Ncore is determined from the inputted node number N and link number L and then the order distribution of the core nodes is determined is the same as in the simple trial method.

  As described above, when the number of applied nodes N, the number of links L, and the number of core nodes Ncore are aligned and the degree distribution of the core nodes is determined, the condition C6 is satisfied from the number of nodes N, the number of links L, and the number of core nodes Ncore. After determining the upper limit Dmax of the number of D edge nodes and randomly determining D within the range of 0 ≦ D ≦ Dmax, the number of S edge nodes is determined by S = N−D−Ncore, and the total degree distribution is determined. Furthermore, S S edge nodes are randomly linked to Ncore core nodes (step 150). Note that the processing in step 150 may be divided into step 103 and step 106 as in the simple trial method.

  Next, for the nodes (core node and D edge node) excluding the S edge node, the link destination of the node is randomly determined so as not to be a multiple link in order from the smallest node order, and the primary contraction is repeated. A part of the network is formed (step 104). If it is impossible to avoid the occurrence of multiple links during the formation of the primary contracted network portion, or if the primary contracted network is not connected, the process returns to step 104 to perform the primary contraction. The formation of the network portion is started again from the beginning (step 105).

  When the portion of the primary contracted network is formed, the network structure is completed since the S S edge nodes have already been randomly linked to the Ncore core nodes.

  The condition C1 is satisfied by the processing subsequent to step 150. It is confirmed that the condition C3 is satisfied for the completed network structure. If the satisfaction of the condition C7 is also a requirement, it is confirmed that the condition C7 is satisfied for the completed network structure. If the conditions C3 and C7 are not satisfied, for example, the process returns to step 104 and the formation of the primary contracted network portion is started again.

  When the network structure is completed, the loop process consisting of steps 151 to 153 is repeatedly executed (step 154).

  That is, one link is randomly selected from the links other than the links connected to the S edge node, and the one determined by the random number is executed out of the movement or replacement (step 151). If the state after the link movement or after the link change becomes a multiple link or the network structure is not connected, the link movement or link change is canceled and the process returns to step 151 (step 152). If the network structure after link movement or after link relocation does not have multiple links and is connected, link movement is based on an evaluation formula that reflects at least one of traffic bandwidth performance and average hop performance. It is determined whether or not the network structure after or after the link replacement is improved from the network structure before the movement or replacement, and if not improved, the link movement or link replacement is canceled and the process returns to step 151. If it has been improved, the improved network structure is determined as a decision candidate at that time (step 152). When the result of the improvement in step 153 occurs continuously for a predetermined number of times, the loop processing in steps 151 to 153 is terminated, and the network structure is finally determined as the network structure determination candidate at that time. To do.

(B-4) Fourth Embodiment of Data Transfer Network Configuration Method Next, a GA (genetic algorithm) optimization method will be described as a fourth embodiment of the data transfer network configuration method. FIG. 16 is a flowchart showing the processing flow of the GA optimization method, in which the same reference numerals are assigned to the same and corresponding steps as in FIG.

  In this GA optimization method, the processing up to step 150 is the same as the random link replacement method.

  In the GA optimization method, the processes in steps 104 and 105 described above are repeatedly executed a plurality of times to form a plurality of network structures (step 200). Each gene sample formed in this way is used (step 201). The link tables shown in Table 4 and Table 5 described above uniquely represent the network structure, and the connection relationship of each core node number can be regarded as a gene sequence associated with each core node number. . That is, the network structure can be used as a gene sample.

  Next, a plurality of gene samples are randomly selected as initial gene samples (step 202). Next, using a random number or the like, a new gene sample is formed by genetic manipulation such as mating, crossover, and mutation (step 203). Such genetic manipulation corresponds to the movement or replacement of links. If a new gene sample (network structure) as a result of genetic manipulation becomes a multiple link or the network structure is not connected, the process returns to step 203 to re-form a new gene sample. (Step 204).

  Next, the most excellent gene sample is determined based on an evaluation formula reflecting at least one of traffic bandwidth performance and average hop performance from a plurality of gene samples including a new gene sample. It is confirmed whether or not the sample is superior (improved) to the gene sample that is a candidate at that time. If the sample is superior, the determined gene sample is set as a candidate gene sample (step 205). . Whether the determined gene sample is superior or not superior to the candidate gene sample at that time, a good one is selected from the plurality of gene samples that have been compared and judged, and A selection is also made from among the gene samples in 201, and the process returns to step 203.

  When the result that the gene sample determined to be the most excellent gene sample is not superior (improved) than the candidate gene sample at that time has occurred continuously for a predetermined number of times, the genetic algorithm is The network structure is finally determined for the gene sample (network structure) that is a candidate at that time.

(C) Embodiment of Method for Evaluating Data Transfer Network Next, an embodiment of a method for evaluating a data transfer network according to the present invention will be described in detail with reference to the drawings.

  In this evaluation method, an evaluation target network having a predetermined network structure is evaluated. For example, an evaluation program for a data transfer network is installed in a computer, and the computer executes the evaluation program by inputting information on the network to be evaluated in which the network structure is described like the above-described link table. Evaluate the target network.

  FIG. 17 is a flowchart illustrating a processing flow in the data transfer network evaluation method according to the embodiment.

First, information on the network to be evaluated is fetched (step 250). And it is discriminate | determined whether the network of evaluation object can be evaluated (step 251). For example, if the relationship between the number of nodes N and the number of links L does not satisfy the relationship of N> 10, N <L <3N, 4L ² <N ³ , it cannot be evaluated. In this case, the fact is notified (step 252).

  If the network to be evaluated can be evaluated, it is detected whether or not each of the above-described conditions C1 to C7 is satisfied (step 253). Then, the evaluation result determined by the combination of satisfied conditions is extracted from the table in which the combination of satisfied conditions and the evaluation result are associated (step 254), and the evaluation result is output (step 255). Terminate the process.

  For example, if any one of the conditions C1 to C3 is not satisfied, the evaluation result indicates that the network structure is not preferable, and those satisfying the conditions C1 to C3 are preferable as the network structure, The more satisfactory the other conditions C4,..., C7, the greater the preferable degree.

  Note that the evaluation result may be output in a score or in stages. Further, the average relay number performance H and the network bandwidth performance Xeff may be output together.

  By applying the evaluation method of the embodiment, the data transfer network can be evaluated from the aspect of the average relay number performance H and the network bandwidth performance Xeff.

(D) Other Embodiments The use of the data transfer network of the present invention is not limited. The link may be one that transmits an optical signal, may be one that allows an electrical signal to pass, and may be wired or wireless. In the case of an optical network, a node may be one that relays an optical signal as it is, or may be one that returns an optical signal to an electrical signal and then relays it by capturing a relay destination or the like.

Claims (10)

When the number of nodes is N and the number of links is L, other than the structure of the cubic regular graph, quadratic regular graph, fifth regular graph, and bipartite graph satisfying the relationship of N> 10 and N <L <3N In a linked data transfer network having a structure,
A data transfer network configured to satisfy all of the following conditions C1, C2, and C3.
(Condition C1) In the network, a node having degree 1 is always connected to a core node having degree 3 or higher, and is not connected to a node having degree 2 or lower.
(Condition C2) In a network in which the network is linearly reduced, the number of core nodes Ncore is an integer satisfying 2 ≦ Ncore ≦ N, and the average relative deviation dkcore around the average value Kcore of Ncore core nodes in the primary reduced network Is dkcore <0.6.
(Condition C3) The network limit in a network having a virtual cage graph structure in which the average number of relays performance value H and the network bandwidth performance value Xeff of the network are p, the average node order is k, and k is the node order. When the network bandwidth performance value is cageXeff, the relationship of the formula (A) is satisfied.
Xeff> (cageXeff / N-1) (H + p-2)
/ {2 (1 / cageXeff + p-3)} + 1 (A) The data transfer network according to claim 1, wherein
Further, the data transfer network, characterized by being configured to also satisfy the following condition C4.
(Condition C4) The number of core nodes Ncore is a divisor of 2 (L−N). In the data transfer network according to claim 1 or 2,
Further, the data transfer network, characterized by being configured to also satisfy the following condition C5.
(Condition C5) The core node number Ncore is the number of closed shell cores N _cc (h _shell ) with respect to the outermost shell layer number h _shell in a virtual cage structure having an average node order. In the data transfer network according to any one of claims 1 to 3,
Furthermore, the data transfer network characterized by satisfy | filling also the following conditions C6.
(Condition C6) S in the range of 0 ≦ S ≦ max (0, N-Ncore-Dmax) is the number of nodes whose degree is 1 in the network, and D obtained from D = N-Ncore-S Is the number of nodes with degree 2 in the network.
(However, Dmax is a maximum integer not exceeding Ncore {2 (L−N) / Ncore + 2−X} / 2) In the data transfer network according to any one of claims 1 to 4,
Further, the data transfer network, characterized by being configured to also satisfy the following condition C7.
(Condition C7) Assuming that S degree 1 nodes in the network are distributed to Ncore core nodes in the primary contracted network, the order of the core nodes after distributing the S order 1 nodes is the distribution method. Assuming that an ideal distribution method in which the variation in distribution is the smallest is applied, the way in which the nodes of degree 1 and the core nodes are connected in the network matches. Alternatively, it is assumed that S degree 1 nodes in the network are distributed to Ncore core nodes in the primary contracted network, and the distribution method is as follows. When it is assumed that a distribution method is applied in which part or all of the nodes of degree 1 connected to the lower-order core nodes are moved to higher-order core nodes after being distributed once so that the variation becomes the smallest As a result, the way of connection of the node of degree 1 and the core node in the network matches. When the number of nodes is N and the number of links is L, other than the structure of the cubic regular graph, quadratic regular graph, fifth regular graph, and bipartite graph satisfying the relationship of N> 10 and N <L <3N In a method for configuring a concatenated data transfer network having a structure,
A method for configuring a data transfer network, wherein the data transfer network is configured to satisfy three or more conditions including the conditions C1 to C3 among the conditions C1 to C7.
(Condition C1) In the network, a node having degree 1 is always connected to a core node having degree 3 or higher, and is not connected to a node having degree 2 or lower.
(Condition C2) In a network in which the network is linearly reduced, the number of core nodes Ncore is an integer satisfying 2 ≦ Ncore ≦ N, and the average relative deviation dkcore around the average value Kcore of Ncore core nodes in the primary reduced network Is dkcore <0.6.
(Condition C3) The network limit in a network having a virtual cage graph structure in which the average number of relays performance value H and the network bandwidth performance value Xeff of the network are p, the average node order is k, and k is the node order. When the network bandwidth performance value is cageXeff, the relationship of the formula (A) is satisfied.
Xeff> (cageXeff / N-1) (H + p-2)
/ {2 (1 / cageXeff + p-3)} + 1 (A)
(Condition C4) The number of core nodes Ncore is a divisor of 2 (L−N).
(Condition C5) The core node number Ncore is the number of closed shell cores N _cc (h _shell ) with respect to the outermost shell layer number h _shell in a virtual cage structure having an average node order.
(Condition C6) S in the range of 0 ≦ S ≦ max (0, N-Ncore-Dmax) is the number of nodes whose degree is 1 in the network, and D obtained from D = N-Ncore-S Is the number of nodes with degree 2 in the network.
(However, Dmax is a maximum integer not exceeding Ncore {2 (L−N) / Ncore + 2−X} / 2)
(Condition C7) Assuming that S degree 1 nodes in the network are distributed to Ncore core nodes in the primary contracted network, the order of the core nodes after distributing the S order 1 nodes is the distribution method. Assuming that an ideal distribution method in which the variation in distribution is the smallest is applied, the way in which the nodes of degree 1 and the core nodes are connected in the network matches. Alternatively, it is assumed that S degree 1 nodes in the network are distributed to Ncore core nodes in the primary contracted network, and the distribution method is as follows. When it is assumed that a distribution method is applied in which part or all of the nodes of degree 1 connected to the lower-order core nodes are moved to higher-order core nodes after being distributed once so that the variation becomes the smallest As a result, the way of connection of the node of degree 1 and the core node in the network matches. The method for configuring a data transfer network according to claim 6,
Among the conditions C1 to C7, a data transfer network that satisfies three or more conditions including the conditions C1 to C3 is configured as a plurality of candidates, and at least the network bandwidth performance and the average number of relays performance are selected from the plurality of configured candidates. A method for configuring a data transfer network, wherein one having a high evaluation value based on one is selected. The method for configuring a data transfer network according to claim 6,
A data transfer network that satisfies three or more conditions including the conditions C1 to C3 among the conditions C1 to C7 is configured as a temporary candidate.
The random candidate is applied to the temporary candidate to move or relink the link, and the network structure after the movement or replacement is evaluated based on at least one of the network bandwidth performance and the average number of relays to optimize the temporary candidate. And determining a final network structure. A method for configuring a data transfer network. The method for configuring a data transfer network according to claim 6,
A plurality of data transfer networks satisfying three or more conditions including the conditions C1 to C3 among the conditions C1 to C7 are configured as gene samples, and at least the network bandwidth performance and the average number of relays performance for the configured gene samples A method for configuring a data transfer network, wherein a final network structure is determined by applying a genetic algorithm using an evaluation value based on one side. A data transfer network evaluation method for evaluating a connected data transfer network to be evaluated that satisfies a relationship of N> 10 and N <L <3N, where N is the number of nodes and L is the number of links,
An evaluation method for a data transfer network, characterized in that an evaluation result is determined based on satisfaction of three or more conditions including conditions C1 to C3 among conditions C1 to C7.
(Condition C1) In the network, a node having degree 1 is always connected to a core node having degree 3 or higher, and is not connected to a node having degree 2 or lower.
(Condition C2) In a network in which the network is linearly reduced, the number of core nodes Ncore is an integer satisfying 2 ≦ Ncore ≦ N, and the average relative deviation dkcore around the average value Kcore of Ncore core nodes in the primary reduced network Is dkcore <0.6.
(Condition C3) The network limit in a network having a virtual cage graph structure in which the average number of relays performance value H and the network bandwidth performance value Xeff of the network are p, the average node order is k, and k is the node order. When the network bandwidth performance value is cageXeff, the relationship of the formula (A) is satisfied.
Xeff> (cageXeff / N-1) (H + p-2)
/ {2 (1 / cageXeff + p-3)} + 1 (A)
(Condition C4) The number of core nodes Ncore is a divisor of 2 (L−N).
(Condition C5) The core node number Ncore is the number of closed shell cores N _cc (h _shell ) with respect to the outermost shell layer number h _shell in a virtual cage structure having an average node order.
(Condition C6) S in the range of 0 ≦ S ≦ max (0, N-Ncore-Dmax) is the number of nodes whose degree is 1 in the network, and D obtained from D = N-Ncore-S Is the number of nodes with degree 2 in the network.
(However, Dmax is a maximum integer not exceeding Ncore {2 (L−N) / Ncore + 2−X} / 2)
(Condition C7) Assuming that S degree 1 nodes in the network are distributed to Ncore core nodes in the primary contracted network, the order of the core nodes after distributing the S order 1 nodes is the distribution method. Assuming that an ideal distribution method in which the variation in distribution is the smallest is applied, the way in which the nodes of degree 1 and the core nodes are connected in the network matches. Alternatively, it is assumed that S degree 1 nodes in the network are distributed to Ncore core nodes in the primary contracted network, and the distribution method is as follows. When it is assumed that a distribution method is applied in which part or all of the nodes of degree 1 connected to the lower-order core nodes are moved to higher-order core nodes after being distributed once so that the variation becomes the smallest As a result, the way of connection of the node of degree 1 and the core node in the network matches.

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