CN108011784B - Dynamic optimization method for worst network connectivity - Google Patents

Abstract

The invention discloses a dynamic optimization method for worst network connectivity. The method comprises the following steps: calculating the network diameter to obtain a corresponding node pair set; counting the number of shortest paths with the length being the diameter to obtain a shortest path number set; reducing the node pair set, and judging whether to optimize according to the number of elements in the reduced set; selecting to execute or finish optimization according to the judgment result; and randomly selecting a plurality of node pairs in the set, building a new direct connection edge to form a new network, and re-executing the steps to realize dynamic optimization. The method of the invention optimizes the whole connectivity performance of the network dynamically from the worst situation by comparing the worst situation of the connectivity between the nodes, optimizing the node pairs and adding edges in a targeted manner. The invention can be widely applied to a large number of actual complex systems, such as Internet, traffic network and the like, and provides important reference basis for the optimization design of optical fiber routing layout, newly-added air routes, rail traffic and the like.

Description

Dynamic optimization method for worst network connectivity

Technical Field

The invention relates to a dynamic optimization method of network connectivity.

Background

Human life and production activities depend on a large number of complex systems, both natural and man-made, and for a given system, the connections and interaction patterns between the components can be represented by networks, the components of the system can be abstracted into vertices in the network, and the connections between the components can be abstracted into edges.

The overall connectivity of the network represents the transmission and service capabilities of the carrier to a greater extent, and therefore, the network is widely concerned in many application fields, such as the Internet, traffic networks, trade networks, and the like. Current considerations for network connectivity usually focus on the ratio relationship between the actual edge and all possible edges in both cases of intentional attack on nodes with high index (degree, betweenness, etc.) and random attack on all nodes (refer to the review on the Ecology letters published in Banks et al 2015).

For dynamic change of network connectivity, current research mostly focuses on network evolution generated by randomly adding edges or being dragged by external factors, and influences on overall robustness of the network after the evolution. For example, Foti et al (j.econ. dyn. control,2013) consider the change in robustness of a network in the face of random and deliberate attacks after adding edges, whereas the addition of edges in its trading network is affected by economic policies. The improvement of network connectivity is more considered on the statistical characteristics of edges and the connectivity among most nodes, and the increase of the edges has randomness. Current techniques lack proactive and targeted optimization and also lack the preferred way to add edges.

Disclosure of Invention

In order to solve the problem of targeted optimization of network connectivity, the invention evaluates the worst condition of connectivity between nodes by combining the robustness of the shortest path between the nodes, provides an optimal mode of edges to be added and realizes the dynamic optimization of the network connectivity.

In order to achieve the above technical object, the technical solution of the present invention is a method for dynamically optimizing the worst connectivity performance of a network, including the following steps,

the method comprises the following steps: calculating the network diameter and acquiring a node pair set corresponding to the diameter;

step two: counting the shortest path number between each node pair according to the node pair set obtained in the step one to obtain a shortest path number set corresponding to the node pair;

step three: according to the shortest path number set obtained in the step two, reducing the node pair set obtained in the step one to obtain node pairs under the condition of worst connectivity; judging whether to execute optimization according to the number of elements in the reduced set;

step four: according to the optimization execution situation in the judgment result obtained in the third step, randomly selecting a plurality of node pairs in the set obtained in the third step, and respectively adding a direct connection edge to generate a new network; and returning to the step one again, and circulating the steps to implement dynamic optimization.

In the method, the step one of calculating the network diameter and obtaining the node pair set corresponding to the diameter includes:

step 1: calculating the length of the shortest path between any two nodes in the network;

d_ij＝min{P¹ _i→j,P² _i→j,...,Pⁿ _i→j}

wherein d is_ijRepresenting the length of the shortest path from node i to node j (i < j), n is the statistical number of the shortest paths from node i to node j, P^k _i→jThe length value of the k path from the node i to the node j is represented, namely the number of edges passed by the k path; { P¹ _i→j,P² _i→j,...,Pⁿ _i→jIs the set of all path length values from node i to node j.

Step 2: calculating the diameter of the network;

calculating the maximum value of the shortest path length between any two nodes in the network according to the length of the shortest directed path between the two nodes obtained in the step 1 in the step one, namely the diameter:

D＝maxd_ij

wherein D represents the diameter of the network; maxd_ijRepresents the maximum value of the shortest path length between any two nodes.

And step 3: acquiring a node pair set corresponding to the diameter;

according to the diameter obtained in the step 2 in the step one, solving a node pair set corresponding to the diameter, wherein the step is as follows:

U＝{(i,j)|d_ij＝D}

wherein U represents a node pair set corresponding to the diameter, and (i, j) represents that the condition d is satisfied_ijPair D.

In the method, the step of obtaining the shortest path number set corresponding to the node pair in the step two is as follows:

according to the node pair set obtained in the step 3 in the step one, counting the shortest path number between each node pair to obtain a shortest path number set corresponding to the node pair, wherein the step is as follows:

V＝{M_ij|(i,j)∈U}

wherein M is_ijRepresenting a set of shortest path numbers corresponding to the node pair (i, j); v represents M_ijA collection of (a).

In the method, the step three of reducing the node pair set and judging whether to optimize is as follows:

step 1: acquiring node pairs under the situation of worst connectivity;

extracting node pairs with the shortest paths of only 1 according to the node pair set obtained in the step 3 in the step one, wherein the step is as follows:

U′＝{(i,j)|M_ij＝1}

where U' represents the set of node pairs after reduction of U, M_ij1 means that the shortest path between the node pair (i, j) is only 1; in this case, the minimum cut set of edges between node pairs is 1, and any node and edge on the path is destroyed, which results in an increase in the diameter of the network, so that the worst connectivity performance of the network is deteriorated.

Step 2: judging whether to optimize;

judging whether to optimize according to the node pair set obtained in the step 1 in the third step and the number of elements in the set, wherein the steps are as follows:

wherein (card (U') > 1)

The method is characterized in that the proposition that the number of elements in U is more than or equal to 1 is judged to be true or false; t represents that the proposition judgment result is true, namely entering the next optimization stage; f represents that the proposition judgment result is false, namely the optimization is terminated.

In the method, the step four of performing optimization includes:

optimizing according to the situation that step 2 in the step three needs to execute optimization, and the steps are as follows:

randomly selecting t node pairs (i, j) in U', and directly connecting one edge; wherein t < card (U'); here, the optimization implementer is given a certain degree of freedom to select the number of newly added edges for each optimization according to the actual situation.

The invention carries out active optimization on the network connectivity performance by comparing the worst situation of the connectivity among the nodes. Compared with the prior art, the addition of the edges is more targeted, the worst condition of the network connectivity is improved from the viewpoint of improving the robustness of the connection between the nodes, and the influence caused by deliberately attacking or randomly attacking some nodes under the worst condition is avoided, so that the worst condition is prevented from getting worse. The optimization process may result in the reduction of the network diameter, so that the purpose of optimizing the worst connectivity can also be achieved.

The dynamic optimization method is not only suitable for simple networks without directions and weights, but also suitable for directed networks and weighted networks. The condition of step one in the directed network is changed to i ≠ j. For the weighting network, the weight of the edge is converted into the path length for calculation. Therefore, the invention can be widely applied to a large number of actual complex systems, such as Internet, traffic network and other networks, and provides important reference basis for the implementation of optical fiber routing layout, newly-added air routes, tracks and the like.

The invention will be further explained with reference to the drawings.

Drawings

FIG. 1 is a flow chart of the present invention.

Fig. 2 is a schematic diagram of a complex network according to the present invention.

FIG. 3 is a schematic diagram of the optimization performed by the present invention.

Detailed Description

Referring to fig. 1, fig. 1 is a flow chart of the present invention. The following illustrates a specific embodiment of the present invention.

Example 1: dynamic optimization of worst-case connectivity performance of random networks

1) Obtaining complex networks

In this embodiment, a random complex network with 50 nodes is denoted as DN. The network is denoted as DN ═ (V, E), where V is the set of nodes and E is the set of edges between the nodes. The nodes contained in V are { V1, V2, …, V50 }. In the network, nodes v1-v9, v1-v18, v1-v20, v1-v34, v3-v6, v5-v12, v5-v16, v5-v37, v37-v 37, v37-v 37, v-v 37, v37-v 37, v-v 37, v31-v34, v32-v3, v32-v6, v32-v36, v32-v41, v33-v47, v34-v7, v7-v 7, v7-v 7, v7-v 7, v7-v 7, v7-v 7-v, An undirected and unweighted edge exists among v43-v47, v43-v48, v44-v39, v45-v38, v46-v28, v46-v33, v47-v14, v47-v34, v47-v45, v48-v42, v49-v44, v50-v3, v50-v30 and v50-v43, and other nodes are not connected.

Fig. 2 is a schematic diagram of a complex network obtained according to connection between nodes in embodiment 1 of the present invention.

2) Computing network diameter and corresponding node pair set

The network diameter is defined as the maximum value of the shortest path length between any two reachable nodes in the network. First, the length of the shortest path between any two nodes in the network is calculated. Taking the length calculation of the shortest path between the node pairs (v1, v2) as an example, the shortest paths existing between them are v1 to v38 to v2, wherein the shortest path includes 2 edges, and therefore has a length of 2. And sequentially calculating the length of the shortest path between any other two nodes, and then solving the maximum value in all length values. The maximum value of 6 is obtained, i.e. the diameter of the network. The corresponding node pair sets are { (v2, v10), (v3, v10), (v4, v10), (v10, v25), (v10, v49), (v25, v29) }.

2) Counting the shortest path number with the length as the diameter to obtain the shortest path number set

The number of shortest paths between the node pairs (v2, v10) is first determined, the node v2 walks along the edges, the number of walking edges is limited to 6, and the node v10 is reached, so that 3 shortest paths can be obtained. The number of shortest paths between the node pairs (v3, v10), (v4, v10), (v10, v25), (v10, v49), (v25, v29) is 6,1,6, and 20, respectively. The set of shortest path numbers obtained is {6,6,1,6,20 }.

3) Reducing node pairs and determining elements in a set

The node pairs with the shortest path number of only 1 are extracted, and the sets of { (v2, v10), (v3, v10), (v4, v10), (v10, v25), (v10, v49), (v25, v29) } are reduced to { (v10, v25) }. Judging the number of elements in the set, wherein the number of the elements exceeds 1, and executing the next optimization.

4) Randomly selecting a plurality of node pairs in the set, and newly building a direct connection edge

Here, only one element remains in the reduced set, so that 1 node pair is extracted, namely (v10, v25), and an undirected unweighted edge v10-v25 is added to the network to form a new network.

Fig. 3 is a schematic diagram of optimization performed in embodiment 1 of the present invention.

5) New round of optimization

And calculating the diameter of the new network and the corresponding node pair, obtaining the diameter as 6, and setting the corresponding node pair as { (v10, v49) }. The set of shortest path numbers obtained is {9 }. And extracting the node pairs with the shortest path number of only 1 to obtain an empty set. And judging the number of the elements in the set to be less than 1, thus finishing the optimization.

The above is an example analysis of the dynamic optimization of the worst performance of the network.

Claims (5)

1. A method for dynamically optimizing the worst-case network connectivity performance includes the following steps,

the method comprises the following steps: calculating the network diameter and acquiring a node pair set corresponding to the diameter;

step two: counting the shortest path number between each node pair according to the node pair set obtained in the step one to obtain a shortest path number set corresponding to the node pair;

step three: according to the shortest path number set obtained in the step two, reducing the node pair set obtained in the step one to obtain node pairs under the condition of worst connectivity; judging whether to execute optimization according to the number of elements in the reduced set;

step four: according to the situation of executing optimization in the judgment result obtained in the third step, randomly selecting a plurality of node pairs in the set obtained in the third step, and respectively adding a direct connection edge to generate a new network; and then returning to the step one again, and circulating the steps to realize dynamic optimization.

2. The method of claim 1, wherein the step of calculating the diameter of the network and obtaining the set of node pairs corresponding to the diameter in the first step comprises:

step 1: calculating the length of the shortest path between any two nodes in the network;

d_ij＝min{P¹ _i→j,P² _i→j,...,Pⁿ _i→j}

wherein d is_ijRepresenting nodes i to j (i)<j) Length of shortest path, n is the number statistic of shortest paths from node i to node j, P^k _i→jThe length value of the k path from the node i to the node j is represented, namely the number of edges passed by the k path; { P¹ _i→j,P² _i→j,...,Pⁿ _i→jThe set of all path length values from node i to node j;

step 2: calculating the diameter of the network;

calculating the maximum value of the shortest path length between any two nodes in the network according to the length of the shortest directed path between the two nodes obtained in the step 1 in the step one, namely the diameter:

D＝maxd_ij

wherein D represents the diameter of the network; maxd_ijThe maximum value of the shortest path length between any two nodes is represented;

and step 3: acquiring a node pair set corresponding to the diameter;

according to the diameter obtained in the step 2 in the step one, solving a node pair set corresponding to the diameter, wherein the step is as follows:

U＝{ (i，j)|d_ij＝D}

wherein U represents a set of node pairs corresponding to a diameter, ((ii) ((iii)), (i, j) indicates that the condition d is satisfied_ijPair D.

3. The method of claim 2, wherein the step of obtaining the shortest path number set corresponding to the node pair in the step two comprises:

according to the node pair set obtained in the step 3 in the step one, counting the shortest path number between each node pair to obtain a shortest path number set corresponding to the node pair, wherein the step is as follows:

V＝{ M_ij|(i，j)∈U}

wherein M is_ijRepresenting a set of shortest path numbers corresponding to the node pair (i, j); v represents M_ijA collection of (a).

4. The method of claim 3, wherein the step of reducing the set of nodes and determining whether to perform optimization in the third step comprises:

step 1: acquiring node pairs under the situation of worst connectivity;

extracting node pairs with the shortest paths of only 1 according to the node pair set obtained in the step 3 in the step one, wherein the step is as follows:

U＇＝{(i，j)|M_ij＝1}

wherein U' represents the node pair set after reduction of U, M_ij1 means that the shortest path between the node pair (i, j) is only 1; in this case, the minimum cut set of the edges between the node pairs is 1, and any node and edge on the broken path will cause the increase of the network diameter, so that the worst connectivity performance of the network is deteriorated;

step 2: judging whether to optimize;

judging whether to optimize according to the node pair set obtained in the step 1 in the third step and the number of elements in the set, wherein the steps are as follows:

wherein

The method is characterized in that the proposition that the number of elements in U' is more than or equal to 1 is judged to be true or false; t represents that the proposition judgment result is true, namely entering the next optimization stage; f represents that the proposition judgment result is false, namely the optimization is terminated.

5. The method of claim 4, wherein the step four of performing the optimization comprises:

optimizing according to the situation that step 2 in the step three needs to execute optimization, and the steps are as follows:

randomly selecting t node pairs (i, j) in U', and directly connecting one edge; wherein t < card (U'); here, the optimization implementer is given a certain degree of freedom to select the number of newly added edges for each optimization according to the actual situation.

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