JP2013070300A - Network configuration apparatus, processing method, and program - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a network configuration apparatus which composes a graph suitable for a communication network, a processing method, and a program.SOLUTION: A network configuration apparatus composes a graph G in which a communication network is configured with V nodes and E links (V, E=numbers). The graph G thus composed is such that regarding mutually different primes p and q which satisfy a quadratic residue (p|q)=1, a Ramanujan graph where the number of nodes is q+1 and node degree is p+1 is composed, and that when one or more nodes are removed from the Ramanujan graph, the Ramanujan graph does not become non-linked, and before and after a node goes down, the node degree remains unchanged at an even number equal to or greater than 4 and also a second minimum Laplacian eigenvalue of the graph G assumes a largest possible value within the range in which a prescribed relational expression is satisfied.

Description

  The present invention relates to a network configuration device, a processing method, and a program, and more particularly, to a network configuration device, a processing method, and a program that configure a communication network and system using graph theory.

  A communication terminal of a node constituting a graph structure (hereinafter, also simply referred to as “graph”) of a system including a network and a network (hereinafter, referred to as “NW”) may fail. In this case, the NW graph structure configured after the failure of the node maintains the “three characteristics suitable for the communication NW” that the original NW graph structure had, thereby enhancing the robustness of the communication NW. It is important to keep.

  The three characteristics suitable for the communication NW mean 1) short communication time between any two nodes, 2) multiple paths between any two nodes, and 3) high scalability. To do. These three conditions are not sufficient conditions for “characteristics suitable for communication NW”, but are particularly important conditions among necessary conditions.

  Regarding the above three conditions, there are conditions on the graph corresponding to the above three conditions (hereinafter referred to as “graph conditions”) (see, for example, Non-Patent Documents 1 to 3).

  The graph condition of 1) is that the first passage time in the random walk on the graph is short as shown in Non-Patent Document 1, and the graph diameter is as shown in Non-Patent Document 2. Is small. It is known that the first passage time becomes shorter as the second minimum Laplacian eigenvalue of the graph is larger (see Non-Patent Documents 1 and 3).

  The graph condition of 2) is that the graph is connected and the expansion constant is large as shown in Non-Patent Document 2.

  The graph condition of 3) is that the number of links constituting the graph structure increases linearly as the number of nodes (NW size) increases, or increases below the number that increases linearly. The complete graph satisfies the graph conditions 1) and 2), but the number of links increases in proportion to the square of the number of nodes as the number of nodes increases, so the condition of 3) is satisfied. Not.

  Non-Patent Document 2 also describes that a graph satisfying “three characteristics suitable for communication NW” is a Ramanujan graph. However, Non-Patent Document 2 does not describe the structure and conditions of a Ramanujan graph that can recursively construct a Ramanujan graph when a node fails.

  In Non-Patent Document 1 and Non-Patent Documents 4 to 6, Entangled NW is described. This is a graph that minimizes the ratio (λN / λ2) between the second minimum Laplacian eigenvalue λ2 and the maximum Laplacian eigenvalue λN. This graph satisfies only the graph condition 1).

  In Non-Patent Documents 7 to 9, Bimodal NW is described. In this graph, even when nodes or links are removed randomly or selectively, the connectivity of the original graph may be maintained. It is shown quantitatively.

Luca Donetti, Franco Neri and Miguel A Munoz. “Optical network topologies: expanders, cages, Ramanu graphs, entangled networks and all that,” Journal of Stochastic, 6 and 200, VEC. Co-authored by Toyoichi Hiramatsu and Koji Chinen. “Introduction to finite mathematics, finite upper half plane and Ramanujan graph” Makino Publishing (first edition in August 2003), ISBN 4-434-03407-3 C3041. P Zumstein. "Comparison of Spectral Methods Through the Adjacency Matrix and the Laplacian of a Graph," submitted 28. February 2005 at ETH Zurich, 2005. L. Donetti, P.M. I. Hurtado, and M.M. A. Munoz. “Entangled Networks, Synchronization, and Optimal Network Topology,” Phys. Rev. Lett. 95, 188701, 2005. L. Donetti, P.M. I. Hurtado, and M.M. A. Munoz. “Entangled networks, super-homogeneity and optimal network topology,” Arxiv preprint cond-mat / 0502230, 2005. L. Donetti, P.M. I. Hurtado, and M.M. A. Munoz. “Network synchronization: optimal and pessimistic scale-free topologies,” Journal of Physics A: Mathematical and Theoretical, Vol. 41, no. 22, 2008. G. Paul, T.W. Tanizawa, S .; Havlin, and H.H. E. Stanley. “Optimization of Robustness of Complex Networks,” [Proc. 2003 International Conference on Growing Networks and Graphs], Euro. Phys. J. et al. B 38, pp-187-191, 2004. T Tanizawa, G Paul and R Cohen et al. “Optimization of network robustness to waves of targeted and random attacks,” Phys. Rev. E. Vol. 71, no. 4, 2005. T.A. Tanizawa, G .; Paul, S.M. Havlin, and H.M. E. Stanley. “Optimization of the robustness of multimodal networks,” Phys. Rev. E, Vol. 74, no. 1,2006.

  In the graph of the communication NW configured after the failure of the node, maintaining the “three characteristics suitable for the communication NW” that the original graph had is important for maintaining the robustness of the communication NW.

  However, in Non-Patent Documents 1 to 9, there is no specific description for realizing the above-described graph configuration, and a graph configuration method that can maintain three characteristics suitable for the communication NW even after a node failure is clarified. Was not.

  An object of the present invention is to provide a network configuration apparatus, a processing method, and a program for solving the above-described problems.

  The network configuration apparatus of the present invention is a network configuration apparatus that creates a graph G that configures a communication network with V nodes and E links. In the graph G, the square remainder (p | q) = 1. For different primes p and primes q satisfying the above, the Ramanujan graph is not connected when one or more nodes are removed from a connected Ramanujan graph with the number of nodes being q + 1 and the node degree being p + 1. Before and after the failure of the node, the node order remains an even number equal to or greater than 4, and the second minimum Laplacian eigenvalue of the graph G is as large as possible within a range satisfying the following relational expression. .

Here, k is the node order, λN (L (G)) is the maximum Laplacian eigenvalue of the graph G that satisfies the above, and λ2 (L (G)) is the graph G of the graph G that satisfies the above. The second minimum Laplacian eigenvalue, λ2 (L (G (V−1, E−k / 2))) is connected to the one node and the node from the graph G that satisfies the above The second minimum Laplacian eigenvalue of the graph from which (k / 2) links of the links are removed.

  The processing method of the present invention is a processing method performed by the network configuration apparatus, and in the graph G in which the communication network is composed of V nodes and E links, the square remainder (p | q) = 1. For a prime p and a prime q different from each other, a first composition step for constructing a connected Ramanujan graph having a node number of q + 1 and a node degree of p + 1, and removing the node or nodes from the Ramanujan graph In this case, the Ramanujan graph is not disconnected, the node order remains an even number of 4 or more before and after the failure of the node, and the second minimum Laplacian eigenvalue of the graph G is And a second configuration step of configuring the graph G as large as possible within a range satisfying the relational expression.

Here, k is the node order, λN (L (G)) is the maximum Laplacian eigenvalue of the graph G that satisfies the above, and λ2 (L (G)) is the graph G of the graph G that satisfies the above. The second minimum Laplacian eigenvalue, λ2 (L (G (V−1, E−k / 2))) is connected to the one node and the node from the graph G that satisfies the above The second minimum Laplacian eigenvalue of the graph from which (k / 2) links of the links are removed.

  In addition, the program of the present invention allows a computer to have different prime numbers p and prime numbers q satisfying a square remainder (p | q) = 1 in a graph G in which a communication network is composed of V nodes and E links. A first configuration procedure for constructing a connected Ramanujan graph having a node number of q + 1 and a node degree of p + 1, and the Ramanujan graph is disconnected when one or more of the nodes are removed from the Ramanujan graph Before and after the failure of the node, the node order remains an even number of 4 or more, and the second minimum Laplacian eigenvalue of the graph G can be within a range satisfying the following relational expression. And a second configuration procedure that configures the largest graph G.

Here, k is the node order, λN (L (G)) is the maximum Laplacian eigenvalue of the graph G that satisfies the above, and λ2 (L (G)) is the graph G of the graph G that satisfies the above. The second minimum Laplacian eigenvalue, λ2 (L (G (V−1, E−k / 2))) is connected to the one node and the node from the graph G that satisfies the above The second minimum Laplacian eigenvalue of the graph from which (k / 2) links of the links are removed.

  According to the present invention, it is possible to construct a graph suitable for a communication network.

1 is a block diagram illustrating a network configuration device according to a first embodiment of the present invention. It is a figure which shows the network which the network structure apparatus comprised. It is a flowchart which shows the processing method of a network structure apparatus. It is a figure which shows the network which the network configuration apparatus of 2nd Embodiment comprised. It is a flowchart which shows the processing method of a network structure apparatus. It is a figure which shows a network when a node in a network fails. It is a figure which shows an example of the reconstruction of the communication path performed when a node fails. It is a flowchart which shows the rebuilding method of a network. It is a figure for demonstrating operation | movement of the node which communicates using the protocol in which high speed communication is possible. It is a figure which shows the communication network of 3rd Embodiment. It is a flowchart for demonstrating operation | movement of the server for eigenvalue calculation. It is a flowchart which shows the process of the step A shown to FIG. 11a. It is a flowchart which shows the process of the step B shown in FIG. 11b. It is a block diagram which shows the structural example of the server for eigenvalue calculation.

  Embodiments of the present invention will be described below with reference to the drawings.

  FIG. 1 is a block diagram showing a network configuration apparatus according to the first embodiment of the present invention.

  The network configuration apparatus 10 configures a communication system including a communication network or a communication network using a graph G constructed by V nodes and E links. The network configuration device 10 configures a graph G that satisfies graph conditions regarding three characteristics suitable for the communication network.

  Here, the graph G configured by the network configuration device 10 will be described.

  There is no clear lower limit for the minimum Laplacian eigenvalue of a general graph. On the other hand, the second minimum Laplacian eigenvalue of the Ramanujan graph, which is one of k-regular graphs, has a lower limit (see Non-Patent Document 1). Therefore, in the k-regular graph in which the degree of the node is k, there is an eigenvalue having a clear lower limit, and therefore it is guaranteed that an eigenvalue equal to or greater than the lower limit value exists.

  In addition, in the k-regular graph, it is possible to configure a graph that minimizes the first passage time (see Non-Patent Document 1) of a random walk following a Markov chain between any two nodes in the graph.

  The expression indicating the first passage time described in Non-Patent Document 1 is expressed by the following equation using Theorem 2.20 indicating the relationship between the normalized Laplacian eigenvalue and the Laplacian eigenvalue described in Non-Patent Document 3. Rewritten as follows.

Here, M is the total number of links in the network, 0 <λv (L) ≦ 2 is the vth eigenvalue (v is a natural number of 2 or more) from the smaller Laplacian matrix of the network, and u_ ( l, s) is a value of the t-th row from the top of the eigenvector having a magnitude of 1 for the l-th eigenvalue (l is a natural number of 2 or more) from the smaller normalized Laplacian matrix of the network.

  The Ramanujan graph, which is one of k-regular graphs, maximizes the graph diameter by maximizing the absolute value of the non-obvious eigenvalue of the graph in order to maximize the expansion constant.

  In addition, in the k-regular graph, a graph excluding a complete graph generally increases the number of links linearly as the number of nodes increases.

  Therefore, in this embodiment, the “Ramanujan graph” is used as the graph G that satisfies the graph condition of “three characteristics suitable for the communication NW”.

  Therefore, the network configuration device 10 configures a connected Ramanujan graph G1 in which the number of nodes is q + 1 and the node degree is p + 1 for different prime numbers p and prime numbers q satisfying the square remainder (p | q) = 1. . The node order of the Ramanujan graph G1 is an even number of 4 or more and is unchanged before and after the removal of the node.

  The network configuration device 10 forms a graph G that is as large as possible within a range in which the second minimum Laplacian eigenvalue of the Ramanujan graph G1 satisfies the inequality and the equation of Equation (2).

Here, k is the node order, λN (L (G)) is the maximum Laplacian eigenvalue of the graph G that satisfies the above, and λ2 (L (G)) is the second minimum of the graph G that satisfies the above. The Laplacian eigenvalue, λ2 (L (G (V-1, Ek / 2))) is one of the nodes from the graph G satisfying the above and the link connected to the node. The (k / 2) number of links and the second minimum Laplacian eigenvalue of the graph from which are removed. The reason why k / 2 links of links connected to the node are removed together with one node is to prevent the node order of the graph G from changing even after the node is removed.

  FIG. 2 is a diagram illustrating an example of a network configured by the network configuration apparatus 10.

  In the network NW1, a system including a communication network or a structure of a graph G in which the communication network includes a plurality of nodes N and a plurality of links L is shown. The network NW1 includes a network NW1-a, a network NW1-b, a network NW1-c, and a network NW1-d.

  In the present embodiment, the network configuration device 10 is used in the design stage of the network NW1. The network configuration device 10 includes a communication unit 11, a graph configuration unit 12, and a graph determination unit 13.

  The communication unit 11 receives the maximum node number N1 of allowable nodes in the network NW1 and the maximum link number E1 of allowable links. For example, the communication unit 11 receives the maximum node number N1 and the maximum link number E1 input by the network administrator as design information from an external communication device. The communication unit 11 supplies the maximum number of nodes N1 and the maximum number of links E1 to the graph configuration unit 12.

  When the graph configuration unit 12 receives the maximum number of nodes N1 and the maximum number of links E1, the graph configuration unit 12 satisfies the maximum number of nodes N1 and the maximum number of links E1 and can realize a graph condition regarding “three characteristics suitable for a communication network”. The temporary graph G1 of the network NW1-a is created (constructed). For example, the graph configuration unit 12 creates a plurality of temporary graphs G1 of the network NW1-a.

The graph configuration unit 12 satisfies the number of nodes | V1 | satisfying the equation (3) and the equation (4) using the prime number p1 and the prime number q1 so that the graph of the network NW1-a becomes the Ramanujan graph G1. Set node order k ₁ . The prime number p1 and the prime number q1 are different prime numbers that satisfy the square remainder (p1 | q1) = 1.

| V1 | = q1 + 1 ≦ N1 (3)
_{k 1 = p1 + 1 ≦ E1} ··· formula (4)
That is, when the graph configuration unit 12 receives the maximum number of nodes N1 and the maximum number of links E1, the graph configuration unit 12 is configured by the number of nodes | V1 | that satisfies Equation (3) and the node order k ₁ that satisfies Equation (4). The Ramanujan graph G1 (V1, E1) is configured as a physical network or a logical network. Here, V1 is a node set representing all nodes in the Ramanujan graph G1, and E1 is a link set representing all links in the Ramanujan graph G1.

Graph determination unit 13, a Ramanujan graph G1 composed of a graph structure 12, from the Ramanujan graph G1, 1 nodes, and, (k _1/2) present among the links that were connected to the node The Ramanujan graph G1-v1 from which the link is removed and the Ramanujan graph G in which the respective second minimum Laplacian eigenvalues are unchanged are taken. This Ramanujan graph G remains unchanged even before and after the removal of the node, with the node order k ₁ being an even number of 4 or more, and Equation (2) holds.

  The graph determination unit 13 includes a matrix calculation unit 131, an eigenvalue calculation unit 132, a determination unit 133, and an eigenvalue selection unit 134.

  The matrix calculation unit 131 calculates the Laplacian matrix of the Ramanujan graph G1 configured by the graph configuration unit 12. The matrix calculation unit 131 supplies the Laplacian matrix of the Ramanujan graph G <b> 1 to the eigenvalue calculation unit 132.

  When the eigenvalue calculation unit 132 receives the Laplacian matrix from the matrix calculation unit 131, the eigenvalue calculation unit 132 calculates a second minimum Laplacian eigenvalue using the Laplacian matrix. The eigenvalue calculation unit 14 supplies the second minimum Laplacian eigenvalue to the determination unit 133.

When the determination unit 133 receives the second minimum Laplacian eigenvalue from the eigenvalue calculation unit 132, the determination unit 133 controls the graph configuration unit 12 to select one node from the Ramanujan graph G1 and the links connected to the node. to the a (k _1/2) this link removal. Determining section 133 specifically, the Ramanujan graph G1 created by the graph component 12 is removed and one of the nodes, the (k _1/2) This links among the links connected to the node A removal instruction signal instructing the effect is supplied to the graph construction unit 12. When the node and the link are removed by the graph construction unit 12, the matrix calculation unit 131 calculates the Laplacian matrix of the Ramanujan graph G1-v1 after the removal. Then, the eigenvalue calculation unit 132 calculates the minimum Laplacian eigenvalue of the Laplacian matrix of the Ramanujan graph G1-v1, and supplies the calculated value to the determination unit 133.

  The determination unit 133 determines whether the Ramanujan graph G1-v1 after removal and the original Ramanujan graph G1 satisfy Expression (2). That is, the determination unit 133 determines whether or not the second minimum Laplacian eigenvalue of the Ramanujan graph G1-v1 after removal is the same as the second minimum Laplacian eigenvalue of the original Ramanujan graph G1.

  When the removed Ramanujan graph G1-v1 and the original Ramanujan graph G1 satisfy Equation (2), that is, the determination unit 133 has the same second minimum Laplacian eigenvalue before and after node removal. In this case, the second minimum Laplacian eigenvalue is supplied to the eigenvalue selection unit 134.

  For example, the determination unit 133 removes one node from the Ramanujan graph G1 from one to a predetermined number one by one. Then, each time the node in the Ramanujan graph G1 is removed one by one, the determination unit 133 determines whether or not the Ramanujan graph G1-v1 after the removal and the original Ramanujan graph G1 satisfy Expression (2). Determine. The determination unit 133 removes one node from the Ramanujan graph G1 one by one to a predetermined number and then satisfies the expression (2) for all determinations, and then determines the second minimum Laplacian eigenvalue as an eigenvalue selection unit. 134.

  When the eigenvalue selection unit 134 receives the second minimum Laplacian eigenvalue within the range satisfying Expression (2) from the determination unit 133, the eigenvalue selection unit 134 selects the maximum second minimum Laplacian eigenvalue from the second minimum Laplacian eigenvalue. Alternatively, the eigenvalue selection unit 16 selects the second minimum Laplacian eigenvalue having the largest possible value from the second minimum Laplacian eigenvalues.

  In addition, the eigenvalue selection unit 134 receives, for example, the second minimum Laplacian eigenvalue from the determination unit 133, and when the second minimum Laplacian eigenvalue is smaller than a predetermined eigenvalue threshold, the eigenvalue selection unit 134 adds a new graph to the graph configuration unit 12. You may instruct construction.

  The eigenvalue selection unit 134 uses the selected second minimum Laplacian eigenvalue to generate graph configuration information including the number of nodes in the graph G of the network NW1-a, the number of links, and the connection relationship between the nodes and links. To the communication unit 11. Upon receiving the graph configuration information from the eigenvalue selection unit 134, the communication unit 11 transmits the graph configuration information to an external device.

  FIG. 3 is a flowchart illustrating a processing method of the network configuration apparatus 10.

  First, the communication unit 11 receives, for example, the maximum node number N1 and the maximum link number E1 as design information input by a network administrator (step S1-1). The communication unit 11 supplies the maximum number of nodes N1 and the maximum number of links E1 to the graph configuration unit 12.

  When the graph configuration unit 12 receives the maximum number of nodes N1 and the maximum number of links E1, the graph configuration unit 12 satisfies the maximum number of nodes N1 and the maximum number of links E1, and can realize graph conditions regarding three characteristics suitable for the communication network. A graph of the network NW1-a is constructed (step S1-2).

The graph configuration unit 12 configures a Ramanujan graph G1 (V1, E1) in which the graph of the network NW1-a has the number of nodes | V1 | that satisfies Expression 3 and the node order k ₁ that satisfies Expression 4. Step S1-3).

Graph determination unit 13, a Ramanujan graph G1 composed of a graph structure 12, from the Ramanujan graph G1, 1 nodes, and, (k _1/2) present among the links that were connected to the node Are removed (step S1-4).

The graph determination unit 13 uses the Ramanujan graph G1-v1 from which the _one node and (k 1/2) links are removed and the original Ramanujan graph G1, and the node order k ₁ remains an even number of 4 or more. And it is confirmed that the formula (2) holds (step S1-5). If the node order k ₁ is not an even number equal to or greater than 4, or if the equation (2) does not hold, the graph determination unit 13 returns to step S1-2.

On the other hand, the graph determining unit 13 is invariant with the node order k ₁ being an even number of 4 or more, and the number of nodes removed from the Ramanujan graph G1 is less than a predetermined number when Expression (2) holds. (Step S1-6). If the number of removed nodes is less than the predetermined number, the graph determination unit 13 returns to step S1-4 and repeats steps S1-4 and 1- 1 until the number of removed nodes reaches the predetermined number. Process 5 is performed.

  The graph determination unit 13 selects the second smallest Laplacia eigenvalue that is as large as possible within the range satisfying the expression (2) of the Ramanujan graph G1 (step S1-7), and ends the processing method of the network configuration device 10 To do.

  Note that if the network NW1-a alone does not satisfy the conditions of the maximum number of nodes N1 and the maximum number of links E1, the network constituent device 10 uses the processing method shown in FIG. 3 to connect the networks NW1-b to NW1-c. You may comprise and connect these.

  According to the present embodiment, the network configuration device 10 configures a graph G in which a communication network is constructed with V nodes and E links. The network configuration device 10 forms a connected Ramanujan graph G1 in which the number of nodes is q + 1 and the node order is p + 1 for different prime numbers p and prime numbers q satisfying the square remainder (p | q) = 1, When one or more nodes are removed from the Ramanujan graph G1, the Ramanujan graph G1-v1 does not become unconnected, and the node degree remains unchanged evenly before and after the node fails, and A graph G having the second smallest Laplacian eigenvalue of the graph G as large as possible within a range satisfying the relational expression of the equation (2) is formed.

  Therefore, the network configuration apparatus 10 can create a graph G that satisfies the graph conditions 1) to 3) regarding the three characteristics suitable for the communication network. The graph G may be included in a connected Ramanujan graph, or may be included in a Ramanujan graph that is not a connected bipartite graph.

  Next, a second embodiment of the present invention will be described. In the second embodiment, in a situation where an existing network exists, the network configuration device configures the graph G on the network. The network configuration apparatus according to the second embodiment is the same as the configuration shown in FIG.

  FIG. 4 is a diagram illustrating a network configured by the network configuration apparatus 10 according to the second embodiment.

  In the existing network NW2, as in FIG. 2, a system including a communication network or a structure of a graph G in which the communication network is configured by a plurality of nodes N and a plurality of links L is shown. The network NW2 includes a network NW2-a, a network NW2-b, a network NW2-c, and a network NW2-d.

  FIG. 5 is a flowchart illustrating a processing method according to the second embodiment.

  First, the communication unit 11 receives, as design information, designation information for designating the network NW2, the maximum number of nodes N2 in the NW2-a, and the maximum number of links E2 (step S2-1). The communication unit 11 supplies the maximum number of nodes N1 and the maximum number of links E1 to the graph configuration unit 12.

  When the graph configuration unit 12 receives the maximum number of nodes N1 and the maximum number of links E1, the graph configuration unit 12 satisfies the maximum number of nodes N1 and the maximum number of links E1, and can realize graph conditions regarding three characteristics suitable for the communication network. A graph of the network NW2-a is constructed (step S2-2).

Graph component 12 is a graph of the network NW2-a is the number of nodes satisfying the equation (5) | V2 | a is the formula (6) is a node degree k ₂ satisfying connecting a Ramanujan graph G1 (V2, E2) (Step S2-3).

| V2 | = q2 + 1 ≦ N2 (5)
k ₂ = p2 + 1 ≦ E2 (6)
Note that q2 and p2 are different prime numbers satisfying the square remainder (p2 | q2) = 1.

Graph determination unit 13, a Ramanujan graph G2 composed of a graph structure 12, from the Ramanujan graph G1, 1 nodes, and, (k _2/2) present among the links that were connected to the node Are removed (step S2-4).

Graph determining unit 13, one of the nodes and (k _2/2) in this link and have been Ramanujan graph G2-v2 and the original Ramanujan graph G2 removed, node degree k ₂ is still four or more even number unchanged And it is confirmed that Expression (2) holds (step S2-5). If the node order k ₂ is not an even number equal to or greater than 4, or if the equation (2) does not hold, the graph determination unit 13 returns to step S2-2.

On the other hand, the graph determination unit 13 does not change even if the node order k ₂ is an even number of 4 or more, and the number of nodes removed from the Ramanujan graph G2 is less than a predetermined number when the equation (2) holds. (Step S2-6). If the number of removed nodes is less than the predetermined number, the graph determination unit 13 returns to step S2-4 and repeats steps S2-4 and S2- until the number of removed nodes reaches the predetermined number. Process 5 is performed.

  The graph determination unit 13 selects the second smallest Laplacia eigenvalue that is as large as possible within the range satisfying the expression (2) of the Ramanujan graph G2 (step S2-7), and ends the processing method of the network configuration device 10 To do.

  If the network NW2-a alone does not satisfy the conditions of the maximum number of nodes N2 and the maximum number of links E2, the network configuration device 10 newly configures the networks NW2-b to NW2-d and connects them. May be.

  Next, in the state where the network NW1 or NW2 is configured, when a node in the network NW1 or NW2 fails, the shortest communication path among the communication paths that pass through the nodes other than the failed node is reconstructed. A method for this will be described.

  FIG. 6 is a diagram illustrating the network NW3 configured by the network configuration device 10. As illustrated in FIG.

  The network NW3 includes a network NW3-a, a network NW3-b, a network NW3-c, and a network NW3-d. Here, it is assumed that the node a has failed in the network NW3-a.

FIG. 7 is a diagram illustrating an example of a communication path reconfiguration performed when the node a fails in the network NW3-a. 7 includes a network NW3-a (V3, E3) , the network NW3-a-a (V3- a, E3-k 3/2) to node a node a has been removed by faults and is shown ing.

The network NW3-a includes a partial graph S that is not directly connected to the node a and a partial graph U that is directly connected to the node a. Subgraph U in the network NW3-a-a 'is the destination of k _3/2 pieces of link among the k ₃ pieces of link is connected to the node a is switched from the node a to another node It is a partial graph after.

  FIG. 8 is a flowchart showing a method for reconfiguring the network NW3-a shown in FIG.

  First, when a failure occurs in the node a in the network NW-3a (step S3-0), all the nodes adjacent to the node a transmit the failure information of the node a to nodes other than the node a. The failure information is transmitted to all the nodes in the NW3-a (step S3-1). Since the network NW-3a has three characteristics suitable for a communication network, the communication time between any two nodes is short, and there are a plurality of paths between any two nodes. For this reason, in the network NW3-a according to the present embodiment, the failure information reaches all the nodes in a short time.

  The failure information includes the identification information of the node a, the time when the failure occurred in the node a, and the identification information of the node a. The failure information is distributed to each route in the NW3-a using a communication protocol P2 that enables higher-speed communication than a general route control protocol. As the communication protocol, for example, a protocol that performs path control based on a flooding algorithm is used.

When the failure information is transmitted to all the nodes in the NW3-a, the node a is removed and the link is replaced. In this case, the second minimum Laplacian eigenvalue of NW3-a is to reconstruct the same value as the original NW3-a (V3, E3) NW3-a-a (V3-a, E3-k 3/2) Is possible. When changing a communication path in a short time, it is only necessary to connect two nodes connected to the node a among the nodes on the communication path.

  FIG. 9 is a flowchart for explaining the operation of each node that performs communication using the communication protocol P2.

  First, in the network NWa-3 in which link replacement is performed using a communication protocol, a shortest spanning tree having all nodes as roots is configured (step S3-3).

  When each node in the network NW3-a detects a failure in the node a, it transmits the failure information to an adjacent node on the shortest spanning tree rooted in its own node according to the flooding algorithm (step S3-4). ).

  When each node in the network NW3-a receives the failure information (step S3-4-2), if there is only one failure information, all adjacent nodes on the shortest spanning tree having its own node as the root (Step S3-4-3). In the network NW3-a, failure information is transmitted until it is transmitted to all nodes.

  When each node in the network NW3-a receives a plurality of pieces of failure information (step S3-5), identification information or occurrence times included in the plurality of pieces of failure information are different from each other (step S3-6). Each of the plurality of pieces of failure information is transmitted to an adjacent node on the shortest spanning tree having its own node as a root (step S3-6-1).

  On the other hand, when each node in the network NW3-a receives a plurality of pieces of failure information (step S3-5) and the plurality of pieces of failure information are the same (step S3-6), And the failure information is transmitted to an adjacent node on the shortest spanning tree rooted in its own node (step S3-6-2).

  Next, as a third embodiment of the present invention, a method of configuring the network NW1 or NW2 suitable for the communication NW by each node in the existing communication network autonomously switching the communication path will be described.

FIG. 10 is a diagram illustrating a network NW4 according to the third embodiment. Network NW4 is
Each node in the existing communication network has an autonomous function for autonomously controlling the communication path.

  The network NW4 includes a communication node CN and a server CS for calculating eigenvalues in addition to a large number of nodes N and a large number of links L. The network NW4 includes a network NW4-a, a network NW4-b, a network NW4-c, and a network NW4-d.

  FIG. 11a is a flowchart for explaining the operation of the server CS for eigenvalue calculation.

  The network NW4-a has a function of transmitting failure information to each node in the network NW4-a using the communication protocol P2 described in FIG. 8 when a failure occurs in the node. The server CS connected to the node in the network NW4 determines the node connection destination using an autonomous distributed technique. As an autonomously distributed method, for example, a Designated Router selection method described in RFC 2328-OSPF Version 2 can be cited.

  The server CS accepts, as setting information input by the network administrator, a maximum period Tmax from when a node has failed until it starts control of the network NW4. The maximum period Tmax is a maximum value of time for propagating the failure information.

  Upon receiving the maximum period Tmax, the server CS divides the maximum period Tmax by the required time TP2 required for communication between the two nodes using the communication protocol P2, and the maximum period Tmax is divided by the required time TP2. The maximum natural number Nmax that does not exceed the value is calculated (step S4-2-3). Nmax indicates the maximum number of nodes that can transmit the failure information through the physical link between the nodes using the communication protocol P2.

  The server CS selects an appropriate communication node N0 from the nodes in the network NW4, and collects all nodes from the communication node N0 to the Nmaxth node that can be reached by following the physical link (step S4- 2-4). Here, the number of all nodes from the communication node N0 to the Nmaxth node is n.

  The server CS determines the partial network composed of the identified n nodes as a temporary network range (hereinafter referred to as “NW4-a”) through which failure information is propagated (step S4-2-5). ).

  The server CS generates an adjacency matrix A (NW4-a) of NW4-a and a Laplacian matrix L (NW4-a) (step S4-3). The adjacency matrix A and Laplacian matrix L are n rows and n columns since the total number of nodes of NW4-a is n.

  The server CS obtains the maximum eigenvalue λN of the adjacency matrix A (NW4-a) and the second minimum Laplacian eigenvalue λ2 of the Laplacian matrix L (NW4-a) (step S4-4-1). The server CS compares the first arrival time π (NW4-a), which is the calculation result of the equation (1) when v = 2, and Nmax, and if the equation (7) is satisfied, Proceed to step A. If the expression (7) is not satisfied, the number of nodes is increased (step S4-4-4).

π (NW4-a) <Nmax (7)
FIG. 11b is a flowchart showing the process of step A shown in FIG. 11a.

In the process of step A, server CS, the number of nodes satisfying the equation (8) | V4 | a is, the connection of Ramanujan graph G4 such that node degree k ₄ satisfying the equation (9), a physical NW or logical As a target NW (step S4-5-1). Here, V4 is a node set of the Ramanujan graph G4, and E4 is a link set of the Ramanujan graph G4.

| V4 | = q4 + 1 ≦ N4 Expression (8)
k ₄ = p4 + 1 ≦ E4 (9)
Note that the prime number p4 and the prime number q4 are different prime numbers that satisfy the square remainder (p4 | q4) = 1.

Server CS, a Ramanujan graph G4, from its Ramanujan graph G4, 1 nodes, and to remove the (k _4/2) This link of the link is connected to the node (step S4-5 -2).

Server CS, in its one node and (k _4/2) Ramanujan This link is removed graph G4-v4 and original Ramanujan graph G4, node degree k ₄ are remained unchanged even number of 4 or more Yes, and it is confirmed that Expression (2) holds (step S4-5-3). If the node order k ₂ is not an even number equal to or greater than 4, or if the equation (2) does not hold, the graph determination unit 13 returns to step S4-5-1.

If the node order k ₂ remains an even number of 4 or more and is unchanged, and the expression (2) holds, the server CS determines whether or not the number of nodes removed from the Ramanujan graph G2 is less than a predetermined number. Is confirmed (step S4-6). If the number of removed nodes is less than the predetermined number, the server CS returns to step S4-5-2 and repeats steps S4-5-2 and until the number of removed nodes reaches the predetermined number. The process of S4-5-3 is performed.

  The server CS selects the second smallest Laplacia eigenvalue that is as large as possible within the range satisfying the expression (2) of the Ramanujan graph G4 (step S4-7), and proceeds to the process of step B.

  FIG. 11c is a flowchart showing the process of step B shown in FIG. 11b.

  In the process of Step B, the server CS uses steps of S4-5-1, S4-5-2, 4-5-3, S4-6, and S4-7 using an optimization algorithm called simulated annealing. The NW4-a graph G is constructed by repeatedly executing the series of processes (step S4-8).

  Then, the server CS transmits the configuration information of the NW4-a to all nodes belonging to the NW4-a determined in step S4-2-5 using the communication protocol P2 via the communication node CN (step S4-9).

  Also for the network excluding NW4-a from the network NW4, a series of processing procedures from step S4-2-3 to S4-9 are executed to configure NW4-b to 4-d, and the processing of the server CS The method ends. For this reason, the network NW4 is divided by the NWs 4-a to 4-d.

  FIG. 12 is a block diagram illustrating an exemplary configuration of the communication node CN and the server CS.

  The communication node CN receives the configuration information related to the nodes and links in the NW4, and transmits the configuration information related to NW4- * (for example, NW4-a) among the configuration information of the NW4 to the server CS.

  The server CS includes a communication unit 21, a Laplacian matrix calculation unit 22, an eigenvalue λ2 calculation unit 23, an adjacency matrix calculation unit 24, an eigenvalue λN calculation unit 25, an initial arrival time π calculation unit 26, and a comparison unit 27. , NW configuration unit 28. Server CS can be generally referred to as a network configuration device.

  The communication unit 21 receives, for example, the configuration information of NW4-a from the communication node CN. The communication unit 21 supplies the configuration information of the NW4-a to the Laplacian matrix calculation unit 22 and the adjacency matrix calculation unit 24.

  The Laplacian matrix calculation unit 22 calculates a Laplacian matrix L using the configuration information of NW4-a, and supplies the Laplacian matrix L to the eigenvalue λ2 calculation unit 23.

  When receiving the Laplacian matrix L of NW4-a, the eigenvalue λ2 calculation unit 23 calculates the second minimum Laplacian eigenvalue λ2 of the Laplacian matrix.

  When receiving the configuration information of NW4-a from the communication unit 21, the adjacency matrix calculation unit 24 calculates an adjacency matrix of the NW4-a and supplies the adjacency matrix to the eigenvalue λN calculation unit 25.

  When receiving the adjacency matrix of NW4-a, the eigenvalue λN calculation unit 25 calculates the maximum Laplacian eigenvalue λN of the adjacency matrix and supplies the maximum Laplacian eigenvalue λN to the first arrival time π calculation unit 26.

  Upon receiving the maximum Laplacian eigenvalue λN of NW4-a, the initial arrival time π calculating unit 26 calculates the initial arrival time π when v = 2 in the equation (1), and the initial arrival time π is compared with the comparison unit 27. To supply.

  When the comparison unit 27 receives the first arrival time π, the comparison unit 27 confirms that the first arrival time π satisfies Expression (7). The comparison unit 27 supplies the NW4-a configuration information to the NW configuration unit 28 when the first arrival time π satisfies the equation (7).

The NW configuration unit 28 configures the graph specified by the configuration information of the NW4-a to be a Ramanujan graph G4. Specifically, the NW constructing unit 28 converts the Ramanujan graph G4 (V4, E4) having the node order k ₄ satisfying the equation (4) into the physical network or the logical number | V4 | Configure as a network.

NW configuration unit 28, a Ramanujan graph G4, from the Ramanujan graph G4, 1 nodes, and, among the links that are connected to the node (k _1/2) Ramanujan graph G4 of this link was removed Take the Ramanujan graph G where the second minimum Laplacian eigenvalue is unchanged at −v4. This Ramanujan graph G is invariant with the node order k ₄ being an even number of 4 or more before and after the failure of the node, and Equation (2) holds. The NW configuration unit 28 generates NW configuration information indicating the Ramanujan graph G of the NW4-a, and supplies the NW configuration information of the NW4-a to the communication unit 21.

  Upon receiving the NW configuration information of NW4-a from the NW configuration unit 28, the communication unit 21 transmits the NW configuration information of NW4-a to the node of NW4-a using the communication protocol P2 via the communication node CN. introduce. For this reason, the node of NW4-a can switch the communication path in NW4-a autonomously.

  According to the third embodiment, the Ramanujan graph G4 of the NW4-a is composed of one or more nodes that autonomously control the communication path of the communication network. For this reason, the communication node CN and the server CS can configure the communication network NW4-a that satisfies the graph conditions 1) to 3).

  In each embodiment described above, the illustrated configuration is merely an example, and the present invention is not limited to the configuration.

DESCRIPTION OF SYMBOLS 10 Network component apparatus 11 Communication part 12 Graph structure part 13 Graph determination part 131 Matrix calculation part 132 Eigenvalue calculation part 133 Judgment part 134 Eigenvalue selection part

Claims (6)

A network configuration device that creates a graph G that configures a communication network with V nodes and E links,
In the graph G, with respect to different prime numbers p and prime numbers q satisfying the quadratic remainder (p | q) = 1, the node number is q + 1 and the node number is one or more from the connected Ramanujan graph having the node degree p + 1. When the node is removed, the Ramanujan graph is not disconnected, the node order remains an even number of 4 or more before and after the failure of the node, and the second minimum Laplacian eigenvalue of the graph G is A network component that is as large as possible within a range that satisfies the following relational expression.

Here, k is the node order, λN (L (G)) is the maximum Laplacian eigenvalue of the graph G that satisfies the above, and λ2 (L (G)) is the graph G of the graph G that satisfies the above. The second minimum Laplacian eigenvalue, λ2 (L (G (V−1, E−k / 2))) is connected to the one node and the node from the graph G that satisfies the above The second minimum Laplacian eigenvalue of the graph from which (k / 2) links of the links are removed. The network configuration device according to claim 1,
The graph G is a network configuration device included in a connected Ramanujan graph. The network configuration device according to claim 1 or 2,
The graph G is a network configuration device included in a Ramanujan graph that is not a connected bipartite graph. The network configuration device according to any one of claims 1 to 3,
The graph G is a network configuration device configured by one or a plurality of nodes that autonomously control a communication path of the communication network. A processing method performed by a network configuration device,
In a graph G in which the communication network is composed of V nodes and E links, for different prime numbers p and prime numbers q satisfying the square remainder (p | q) = 1, the number of nodes is q + 1, and the node order A first composing step of constructing a connected Ramanujan graph with p + 1
When one or more of the nodes are removed from the Ramanujan graph, the Ramanujan graph does not become unconnected, and the node degree remains unchanged evenly before and after the node fails, and And a second configuration step of configuring the graph G as large as possible within a range in which the second minimum Laplacian eigenvalue of the graph G satisfies the following relational expression.

Here, k is the node order, λN (L (G)) is the maximum Laplacian eigenvalue of the graph G that satisfies the above, and λ2 (L (G)) is the graph G of the graph G that satisfies the above. The second minimum Laplacian eigenvalue, λ2 (L (G (V−1, E−k / 2))) is connected to the one node and the node from the graph G that satisfies the above The second minimum Laplacian eigenvalue of the graph from which (k / 2) links of the links are removed. On the computer,
In a graph G in which the communication network is composed of V nodes and E links, for different prime numbers p and prime numbers q satisfying the square remainder (p | q) = 1, the number of nodes is q + 1, and the node order A first construction procedure for constructing a connected Ramanujan graph where is p + 1;
When one or more of the nodes are removed from the Ramanujan graph, the Ramanujan graph does not become unconnected, and the node degree remains unchanged evenly before and after the node fails, and A second configuration procedure for configuring the graph G that is as large as possible within a range in which the second minimum Laplacian eigenvalue of the graph G satisfies the following relational expression.

Here, k is the node order, λN (L (G)) is the maximum Laplacian eigenvalue of the graph G that satisfies the above, and λ2 (L (G)) is the graph G of the graph G that satisfies the above. The second minimum Laplacian eigenvalue, λ2 (L (G (V−1, E−k / 2))) is connected to the one node and the node from the graph G that satisfies the above The second minimum Laplacian eigenvalue of the graph from which (k / 2) links of the links are removed.

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