A method for improving controllability of a symmetric network by time segmentation

Abstract

The application provides a method for improving controllability of a symmetric network by time segmentation, and relates to the technical field of network performance optimization. The method adopts a time segmentation method to divide a symmetric network into a symmetric time-varying network composed of multiple snapshots, and realizes the structural controllability of the whole network through the control of each snapshot. Given a symmetric network, the black and white coloring is performed through a breadth-first search algorithm for classification; the maximum matching of edges in the symmetric network is found; the undirected cactus is constructed according to the unmatched nodes and the matched edges; the network graph composed of each undirected cactus is used to construct a snapshot, and the number of snapshots is the same as the number of undirected cacti. When a symmetric network is divided into a symmetric time-varying network composed of multiple snapshots, the time segmentation method can effectively reduce the number of drive nodes in the symmetric network and improve the network controllability.

Description

Technical Field

[0001] This invention relates to the field of network performance optimization technology, and in particular to a method for improving the controllability of symmetric networks by utilizing time segmentation. Background Technology

[0002] The set of driving nodes in a real-world network system is determined by the network structure, and it's generally difficult to increase the number of driving nodes in a real network system. Taking the nervous system as an example, driving nodes represent sensory neurons, and their number and function are predetermined by human mechanisms; adding driving nodes cannot improve the controllability of the nervous system. Therefore, it's necessary to develop optimization strategies for the network structure without increasing the number of driving nodes to improve the controllability of the network system. Wang et al. reduced the number of driving nodes required to make the network controllable by adding edges. Zhang et al. studied three problems related to minimum-cost structural perturbations: adding edges, deleting edges, and deleting inputs, to make the network system structurally controllable or uncontrollable. Chanekar et al. found that adding edges or modifying edge weights can improve the controllability of the network. However, these methods are not applicable to all types of networks; the structure of some real-world networks is difficult to change. For example, in protein networks, the interactions between different proteins represent edges in the network, which are predetermined by biological mechanisms. Therefore, new methods are needed to improve the controllability of networks. Cui et al. proposed a time-segmentation scheme to transform a static network into a time-varying network composed of a set of time segments, thereby improving the controllability of the original network by leveraging the advantages of time-varying networks. This scheme, which improves the controllability of time-varying networks through time segmentation, considers the case where all parameters in the network are arbitrary and independent.

[0003] In real-world network systems, the parameters of many network structures are interdependent and mutually constrained. For example, symmetric networks are networks whose parameters have symmetric constraints and are prevalent in real-world networks. Examples include structural brain networks reconstructed from diffuse MRI data and small-signal network-preserving models of power networks. However, the problem of how to improve the controllability of symmetric networks through time segmentation without increasing the number of driving nodes remains unsolved. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to address the shortcomings of the prior art by providing a method for improving the controllability of symmetric networks using time segmentation, thereby improving the controllability of symmetric networks through time segmentation.

[0005] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is: a method for improving the controllability of symmetric networks by using time segmentation, which divides a symmetric network into a symmetric time-varying network composed of multiple snapshots, and realizes the structural controllability of the entire network by controlling each snapshot;

[0006] Preferably, the method employs a time-segmentation algorithm to divide a symmetric network into a symmetric time-varying network composed of multiple snapshots. Specifically, the method is as follows:

[0007] Step 1: Undirected Graph Classification; Given a symmetric network G(A), classify it by coloring it black and white using a breadth-first search algorithm; First, randomly select a node and color it black, and color all its neighboring nodes with the opposite color; repeat the coloring steps until all nodes are colored; if there exists a node in the symmetric network G(A) whose neighboring nodes have the same color as the node itself, then there is an odd cycle in the symmetric network G(A), denoted as G1; if there is no odd cycle in the symmetric network, it is denoted as G2;

[0008] Step 2: Find the maximum matching of edges in the symmetric network;

[0009] (1) Maximum edge matching in G1: First, treat each odd cycle in the symmetric network as a node to form a new network. Then, the network is found using the Hungarian algorithm. Find the maximum matching of edges in the middle; then expand each odd cycle and find the matching of edges in the odd cycle. The union of the matching of edges in G1 and the matching of edges in odd cycles is the maximum matching of edges in G1.

[0010] (2) Maximum matching of edges in G2: The maximum matching of edges in the G2 network is found directly by using the Hungarian algorithm to determine the matching edges and unmatched nodes;

[0011] Step 3: Construct an undirected cactus; After finding the maximum matching in the symmetric network, construct an undirected cactus based on the unmatched nodes and matched edges, in the following two cases:

[0012] (1) If there are unmatched nodes in G(A), let the number be n, and label the unmatched nodes as u1, u2...u n Then, n disjoint undirected cacti are needed to cover all nodes in the network. The specific process of constructing undirected cacti is as follows: In the unmatched node set u1, u2...u n Choose any node u m Let u1,...,u be the root node of the state path of n undirected cacti, and let the remaining unmatched nodes u1,...,u be the root node of the path. m-1 ,u m+1 ,...u nEach node is represented as a node on the state path of an n-1 undirected cactus, and the remaining matching edges are represented as cycles, to construct n disjoint undirected cacti.

[0013] (2) If there are no unmatched nodes in G(A), then only one undirected cactus is needed to cover all nodes in the network; therefore, an undirected cactus can be constructed by selecting any matched edge as the state path and the other matched edges as the cycle.

[0014] Step 4: Construct snapshots; Construct a snapshot of the network graph consisting of each undirected cactus, with the number of snapshots being the same as the number of undirected cacti.

[0015] The beneficial effects of adopting the above technical solution are as follows: This invention provides a method for improving the controllability of symmetric networks using time segmentation. Addressing the structural controllability problem of symmetric networks, it divides the symmetric network into a symmetric time-varying network composed of a set of snapshots using a time segmentation method. By studying the controllability of the symmetric time-varying network, the controllability of the symmetric network itself can be studied. When dividing a symmetric network into a symmetric time-varying network composed of multiple snapshots, the time segmentation method can effectively reduce the number of driving nodes in the symmetric network and improve network controllability. Attached Figure Description

[0016] Figure 1 A time-varying network diagram provided for an embodiment of the present invention;

[0017] Figure 2 A flowchart of the time segmentation algorithm provided in an embodiment of the present invention.

[0018] Figure 3 A symmetric network diagram provided for embodiments of the present invention;

[0019] Figure 4 Odd cycle graphs in symmetric networks provided in embodiments of the present invention;

[0020] Figure 5 A new network provided for embodiments of the present invention picture;

[0021] Figure 6 A network with a driver node is provided for an embodiment of the present invention, wherein (a) is snapshot 1, (b) is snapshot 2, and (c) is snapshot 3;

[0022] Figure 7 The symmetric network (1 driving node) with multiple driving nodes provided in the embodiments of the present invention is shown in (a) as a symmetric network, (b) as snapshot 1, (c) as snapshot 2, (d) as snapshot 3, and (e) as snapshot 4.

[0023] Figure 8The present invention provides a symmetric network with multiple driving nodes (2 driving nodes), wherein (a) is snapshot 1 and (b) is snapshot 2. Detailed Implementation

[0024] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples. The following examples are for illustrative purposes only and are not intended to limit the scope of the invention.

[0025] In this embodiment, a method for improving the controllability of symmetric networks using time segmentation is proposed. The method employs a time segmentation algorithm to divide a symmetric network into a symmetric time-varying network composed of multiple snapshots, and achieves the structural controllability of the entire network by controlling each snapshot.

[0026] In this embodiment, as Figure 1 The image shows snapshots (static network) of a time-varying network with four nodes, obtained at different time intervals. Figure 1 (a) shows the changes in the connections between the four nodes of the time-varying network over time. Each straight line corresponds to one node. Assume t i With t i+1 The time intervals (i = 1, 2, ..., 7) are all Δt. If two nodes are connected at a certain moment, they are connected by a line. Dividing the time-varying network by the time interval Δt, we obtain four snapshots, and the connection details are as follows. Figure 1 As shown in (b), the time-varying network is also divided by a time interval of 2Δt, resulting in two snapshots, as shown. Figure 1 As shown in (c), the time-varying network is divided into segments with a time interval of 4Δt, resulting in a snapshot, as follows. Figure 1 As shown in (d).

[0027] For example Figure 1 In the time-varying network shown, assuming node 1 is a fixed driving node, then B = (1 0 0 0). T When dividing the time-varying network by a time interval Δt ( Figure 1 b), corresponding to the 4 snapshots G(A) i (,B), i=1,2,…,4, where the adjacency matrices are respectively

[0028]

[0029] Its controllable subspace is Since dim(Ω1) = 2, the time-varying network has an uncontrollable structure. When the time-varying network is partitioned with a time interval of 2Δt ( Figure 1 c), corresponding to the two snapshots G(A) i (,B), i=1,2,…,4, where the adjacency matrices are respectively

[0030]

[0031] Its controllable subspace is Since dim(Ω²) = 4, the time-varying network is structurally controllable. However, when the time interval is sufficiently long, their corresponding static networks are identical. Figure 1 d).

[0032] As the examples above show, the ability to divide a time-varying network into snapshots at different time intervals affects the network's controllability. Therefore, for the same time-varying network, how should snapshots be divided to ensure controllability, and what is the minimum number of snapshots that can be used? This invention provides a method for snapshot division and the optimal number of snapshots for symmetric time-varying networks when the driving nodes remain fixed.

[0033]

[0034] In this embodiment, a time-segmentation algorithm is used to divide a symmetric network into a symmetric time-varying network composed of multiple snapshots, such as... Figure 2 As shown, the specific method is as follows:

[0035] Step 1: Undirected Graph Classification; Given a symmetric network G(A), classify it by coloring it black and white using the breadth-first search (BFS) algorithm; First, randomly select a node and color it black, and color all its neighboring nodes the opposite color (white); repeat the coloring steps until all nodes are colored (at this point, all nodes are either black or white); if there exists a node in the symmetric network G(A) whose neighboring nodes have the same color as itself (both are black or white), then there is an odd cycle in the symmetric network G(A), denoted as G1; if there is no odd cycle in the symmetric network, it is denoted as G2;

[0036] Step 2: Find the maximum matching of edges in the symmetric network;

[0037] (1) Maximum edge matching in G1: First, treat each odd cycle in the network as a node to form a new network. Then, the network is found using the Hungarian algorithm. Find the maximum matching of edges in the middle; then expand each odd cycle and find the matching of edges in the odd cycle. The union of the matching of edges in G1 and the matching of edges in odd cycles is the maximum matching of edges in G1.

[0038] (2) Maximum matching of edges in G2: The maximum matching of edges in the G2 network is found directly by using the Hungarian algorithm to determine the matching edges and unmatched nodes;

[0039] Step 3: Construct an undirected cactus; After finding the maximum matching in the network, construct an undirected cactus based on the unmatched nodes and matched edges, in the following two cases:

[0040] (1) If there are unmatched nodes in G(A), let the number be n, and label the unmatched nodes as u1, u2...u n Then, n disjoint undirected cacti are needed to cover all nodes in the network. The specific process of constructing undirected cacti is as follows: In the unmatched node set u1, u2...u n Choose any node u m Let u1,...,u be the root node of the state path of n undirected cacti, and let the remaining unmatched nodes u1,...,u be the root node of the path. m-1 ,u m+1 ,...u n Each node is represented as a node on the state path of an n-1 undirected cactus, and the remaining matching edges are represented as cycles, to construct n disjoint undirected cacti.

[0041] (2) If there are no unmatched nodes in G(A), then only one undirected cactus is needed to cover all nodes in the network; therefore, an undirected cactus can be constructed by selecting any matched edge as the state path and the other matched edges as the cycle.

[0042] Step 4: Construct snapshots; Construct a snapshot of the network graph consisting of each undirected cactus, with the number of snapshots being the same as the number of undirected cacti.

[0043] A symmetric network can be partitioned into a symmetric time-varying network consisting of multiple snapshots. If a symmetric network G(A) contains n unmatched nodes, it can be partitioned into a symmetric time-varying network consisting of n snapshots with one identical node i, where each snapshot is an undirected cactus rooted at node i. In this case, only one external input needs to be applied to node i in the symmetric time-varying network for the network to be structurally controllable. When the symmetric network is partitioned into n² snapshots using a time-segmentation algorithm, each snapshot contains... An undirected cactus can also be implemented using a time-segmentation algorithm for symmetric networks.

[0044] The algorithm for time segmentation of symmetric networks using undirected cacti can be divided into n stages. Each stage searches for an augmenting path starting from the unmatched node, traversing the entire graph at most once, and ensuring that each node is contained within at most n odd cycles. Therefore, the total time complexity of the algorithm is O(n log n). 3 ), where n is the total number of nodes in the symmetric network.

[0045] For a time-varying network, after dividing it into snapshots composed of multiple static networks, how do we determine whether the time-varying network is controllable? This embodiment first provides a criterion for judging the controllability of a symmetric time-varying network.

[0046] Theorem 1: If a symmetric time-varying network (A, B) satisfies the following two conditions:

[0047] (1) Each snapshot G m (A m A network of reachable nodes in (m=1,...,M) can be spanned by an undirected cactus;

[0048] (2) In the time-varying network (A,B), every node is reachable;

[0049] Then the symmetric time-varying network (A,B) structure is controllable.

[0050] Proof: First, as can be seen from condition (2), every node in the symmetric time-varying network (A,B) is reachable, that is, every node can be covered by at least one undirected cactus.

[0051] Suppose that the time-varying network (A,B) composed of snapshots contains an extension, and according to the definition of extension, we have the following formula:

[0052]

[0053] Where, P1∈R p×k There are no more than p-1 non-zero terms. Therefore, there exists a matrix pair (A) for the m-th snapshot. m B) can also be written in the form of the above equation. In snapshot m, the system matrix A corresponding to the unreachable node is... m Since all rows and columns in the array are zero, after removing these zero rows and columns, there exists (A m submatrix of B) It can still be written in the form of the above formula ( Let G represent the network of nodes reachable in snapshot m. From condition (1), we know that each snapshot G... m (A m The network formed by reachable nodes in G can be spanned by an undirected cactus, therefore G m (A m B) is structurally controllable and does not contain expansion, which contradicts the previous assumption that networks composed of snapshots do not contain expansion.

[0054] Note 1: The controllability of a time-varying network can be determined by calculating the dimension of the controllable subspace Ω of the entire time-varying network (e.g., ...). Figure 1If the dimension of the controllable subspace is the same as the number of nodes in the time-varying network (i.e., dim(Ω) = N), then the time-varying network is controllable; if the dimension of the controllable subspace is less than the number of nodes, then the time-varying network is uncontrollable. However, this method requires listing the system matrix and input matrix for each snapshot, which is computationally intensive for networks with a large number of nodes and complex structures. Assuming the symmetric time-varying network under study consists of N nodes and contains M snapshots, then it is necessary to calculate the controllable subspace... The controllability of a network is determined by the dimension of M, where A and B are both N×N matrices. When N and M are large, the calculation becomes cumbersome. Therefore, this embodiment starts from the network topology and studies the conditions for structural controllability of a symmetric time-varying network system (A,B), allowing for the determination of network structural controllability without computation.

[0055] Note 2: For the snapshot partitioning scheme of directed networks, the time-varying network composed of snapshots is structurally controllable, but this method is not applicable to the symmetric network studied in this invention. First, a breadth-first search algorithm is used for black-and-white coloring (coloring rule: the initial node is white, the nodes directly connected to it are colored black, then the adjacent nodes of the black nodes are colored white, and so on until all nodes are colored), transforming the network into a bipartite graph. The Kuhn-Munkres algorithm is then used to solve for the weighted maximum matching, thus achieving effective partitioning. However, this method cannot convert symmetric networks with odd cycles into bipartite graphs. Therefore, this method cannot be used for time segmentation of symmetric networks and cannot improve the controllability of symmetric networks. In summary, this embodiment proposes Theorem 1, which can extend the relevant conclusions in directed networks to symmetric networks, avoiding the above-mentioned problems.

[0056] The time-segmentation method can obtain different numbers of snapshots for a time-varying network by segmenting it in different ways. The fewer the number of snapshots, the lower the cost of controlling the network, and the better the segmentation method. Based on this, the following Theorem 2 is proposed in this embodiment of the invention.

[0057] Theorem 2: For a symmetric network (A, B), there exists a time segmentation method that splits it into a set of snapshots. Furthermore, the symmetric time-varying network composed of these snapshots is structurally controllable. If a set of snapshots G can be found... m (A m Given a network (A, B) where m = 1, 2, ..., M is composed of undirected cacti, and this set of snapshots covers every node in the symmetric network (A, B), then the number of snapshots M is minimized (i.e., ..., M). ).

[0058] Proof: Given a symmetric network (A, B), different snapshot groups can be obtained by splitting it according to a certain time segmentation method. Assuming that the snapshot G obtained from the splitm (A m When all nodes (m = 1, 2, ..., M) are composed of undirected cacti, the number of snapshots obtained is M. When each snapshot is composed of undirected cacti and these undirected cacti can cover all nodes in the network, according to Theorem 1, the symmetric time-varying network composed of these snapshots is structurally controllable.

[0059] Assume the number of snapshots M is not the minimum, i.e. Then at this point, at least Only one undirected cactus can cover all nodes in the network. Therefore, there must exist at least one node v in the network that is not in the preceding nodes. In an undirected cactus, otherwise the network would expand, causing the network structure to become uncontrollable. It is known that a symmetric time-varying network system composed of snapshots is structurally controllable, therefore there is no expansion in the network, and node v is in the first... In an undirected cactus. This contradicts the assumption, thus proving the point.

[0060] Note 3: When a static, uncontrollable symmetric network is divided into multiple snapshots, and each snapshot is an undirected cactus, it is not required that each edge of the symmetric network appears only once in these undirected cacti. Therefore, this time segmentation method can only guarantee that the number of snapshots after segmentation is uniquely determined, while the combination of these snapshots is not unique.

[0061] Among different time segmentation methods, segmenting the symmetric network according to an undirected cactus structure yields the fewest snapshots. Based on this, Theorem 3 further investigates the relationship between the number of driving nodes and the number of snapshots in a symmetric time-varying network after segmenting the time according to an undirected cactus structure.

[0062] Theorem 3 Consider a static symmetric network. Let n be the minimum number of driving nodes required to make the network structure controllable. Divide the static symmetric network into n² snapshots over time. If each snapshot has an undirected cactus structure and the number of driving nodes required to make each snapshot structure controllable is n¹, then (n, n1, n2 are all positive integers).

[0063] Proof: Divide the symmetric network into time segments according to an undirected cactus structure, forming a symmetric time-varying network composed of n snapshots. According to Theorem 1, each snapshot can be spanned by an undirected cactus; therefore, the entire symmetric time-varying network requires n driving nodes to achieve structural controllability. However, since nodes in the symmetric network are ubiquitous, the partitioning ensures that each snapshot has the same driving node. In this case, the symmetric time-varying network only requires one driving node and n snapshots to achieve structural controllability. Therefore, compared to the minimum number of driving nodes n required for structural controllability in the original static symmetric network, the number of driving nodes satisfies...

[0064] Assuming that after time segmentation, each snapshot is formed by n1 undirected cacti, then according to the structural controllability of symmetric networks, the number of driving nodes required in each snapshot is n1. After time segmentation, there are n2 snapshots, so the entire network requires n1 × n2 driving nodes and n2 snapshots to achieve structural controllability. During time segmentation, it can be guaranteed that each snapshot has n1 identical driving nodes. Therefore, the symmetric time-varying network only needs n1 driving nodes and n2 snapshots to achieve structural controllability. Thus, compared to the minimum number of driving nodes n required for structural controllability in the original static symmetric network, the number of driving nodes satisfies...

[0065] Note 4: For some real-world network systems, according to the conclusion of Theorem 3, the number of driver nodes can be selected based on specific circumstances, and the number of snapshots can be designed to achieve network structure control. For example, if a network system can only have one driver node (n1 = 1), then the number of snapshots can be designed as n2 = n; if a network system has n1 driver nodes, then the number of snapshots can be designed as follows:

[0066] Based on the above three theorems, this embodiment verifies the method of the present invention on a symmetric network with different driving nodes:

[0067] (1) A symmetric network with one driving node

[0068] Consider a symmetric network G(A), such as Figure 3 As shown, the network contains 27 nodes and 65 edges. The aforementioned time-segmentation algorithm is applied to this network for snapshot partitioning. First, based on step 1, the odd cycles of the network are determined using the node coloring method (selecting node 1 as the initial colored node, which is black), as shown... Figure 4 As shown, there are 4 odd cycles, each represented by a different dashed or solid line, denoted as G. 11 G 12 G 13 G 14 Then, following step 2, the odd cycle in G1 is processed, and G... 11 G 12 Consider it as a point (G) 12 The points obtained after processing are contained in G. 11 (in Chinese), G 13 G 14 Treat each point as a single point to form a new network. like Figure 5 As shown, and found The maximum matching of the middle edges is (17,19)(11,27)(12,13)(23,25). The next step is to... 11 G 12 G 13 G 14 Expand each side separately and find the matching of edges in each odd cycle, where G 11 The matching edges are (2,9)(5,6); G 12 The matching edges are (3,7)(8,18); G 13 The matching edge of G is (16,22); 14 The matching edges are (20,21), (4,15), and (10,14). Therefore, the unmatched nodes in G(A) are nodes 1, 24, and 26. Based on the number of unmatched nodes in G(A), it is determined that three disjoint undirected cacti are needed to cover all nodes in the network. Node 1 from the unmatched node set {1,24,26} is selected as the root node of the state path of the three undirected cacti. The remaining unmatched nodes 24 and 26 are then used as nodes on the state paths of two undirected cacti, respectively. The remaining matching edges are treated as cycles to construct the three disjoint undirected cacti. Finally, a snapshot is constructed, as shown below. Figure 6 As shown, G(A) is divided into three snapshots, each consisting of an undirected cactus. In each snapshot, hollow nodes represent reachable nodes, and solid lines represent edges in that snapshot.

[0069] Based on the structural controllability of symmetric networks, the above-mentioned symmetric network is structurally uncontrollable when only node 1 is used as the driving node; three driving nodes are required to achieve structural controllability. By using a time-segmentation method, this network only requires one driving node (selecting node 1 as the driving node) and three snapshots to achieve structural controllability, thus reducing the number of driving nodes.

[0070] (2) Symmetric networks with multiple driving nodes

[0071] In this embodiment, a symmetric network with multiple driving nodes can be divided into segments such as... using a time-segmentation algorithm. Figure 7 The four snapshots shown each time select node 1 as the driving node. Similarly, according to Theorem 3, we can obtain... Figure 8 The two snapshots shown are used to select node 1 and node 26 as the driving nodes in each snapshot.

[0072] Based on the structural controllability of symmetric networks, this symmetric network requires four driving nodes to achieve structural control (nodes 1, 16, 21, and 26 can be selected as driving nodes). When only node 1 is a driving node, or when both node 1 and 26 are driving nodes, the aforementioned symmetric network is structurally uncontrollable. By using a time-segmentation method, this network only requires one driving node (node ​​1 selected as the driving node) and four time snapshots, or two driving nodes (nodes 1 and 26 selected as driving nodes) and two snapshots to achieve structural controllability. Therefore, the time-segmentation method can reduce the number of driving nodes in the network system, lower control costs, and also improve the network's controllability.

[0073] (3) Model network and real network

[0074] Finally, the method of this invention was applied to model networks and real networks for analysis. This embodiment selected ER random networks, BA scale-free networks, Twitter retweet networks, animal social networks, protein networks, and urban local traffic networks. The number of nodes generated in the ER networks were N1 = 15 and N2 = 100, with connection probabilities P1 = 0.2 and P2 = 0.02. In the ER network 2 with 100 nodes, 12 isolated nodes were generated. Since this invention studies symmetric networks excluding isolated nodes, the actual number of nodes was 88. The BA network started with 10 nodes, introducing one new node and two edges each time, and generating scale-free networks with 50 and 100 nodes respectively, following the principle of prioritizing connections to nodes with higher degrees. In the Twitter retweet network, nodes represent Twitter users, and edges represent retweet relationships (forwards) between different users. In the animal social network, nodes represent individuals (birds) in the network, and edges represent different individuals using the same nesting chamber for roosting or nesting at any given time. In a protein network, nodes represent different proteins, and edges represent interactions between these proteins at any given time. In a local urban traffic network, nodes represent intersections, and edges represent the different roads connected by these intersections.

[0075] Based on the basic data of each network, the time segmentation algorithm proposed in this invention is used to obtain the number of driving nodes and the number of snapshots for each network, as shown in Table 1.

[0076] Table 1. Changes in the number of driving nodes in the model network and the real network after time segmentation algorithm.

[0077]

[0078] Table 1 lists the number of driving nodes and snapshots required by the time-segmentation algorithm for the model network and the real network. A comparison reveals that the product of the number of driving nodes and the number of snapshots for each snapshot in these networks is equal to the minimum number of driving nodes required by the original symmetric network. When the original network has only m driving nodes (m is any value other than the minimum number of driving nodes required by the original symmetric network), the original network is structurally uncontrollable under the influence of the driving nodes. Using the time-segmentation method, the original network only needs m driving nodes and corresponding snapshots to achieve structural controllability. Therefore, the time-segmentation method reduces the number of driving nodes required to achieve network controllability. Thus, based on the time-segmentation algorithm of this invention, an appropriate number of driving nodes can be selected in the real network according to the actual situation.

[0079] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope defined by the claims of the present invention.

Claims

1. A method for improving the controllability of symmetric networks using time segmentation, characterized in that: A symmetric network is divided into a symmetric time-varying network consisting of multiple snapshots, and the structural controllability of the entire network is achieved by controlling each snapshot. The symmetric time-varying network can be a Twitter retweet network, an animal social network, a protein network, or an urban local traffic network. In a Twitter retweet network, nodes represent Twitter users, and edges represent retweet relationships between different users. In an animal social network, nodes represent individuals, and edges represent different individuals using the same nesting chamber for resting or nesting at any given time. In a protein network, nodes represent different proteins, and edges represent interactions between different proteins at any given time. In an urban local traffic network, nodes represent intersections, and edges represent different roads connected by the intersections. The method employs a time-segmentation algorithm to divide a symmetric network into a symmetric time-varying network composed of multiple snapshots. The specific method is as follows: Step 1: Undirected graph classification; given a symmetric network The classification is performed using a breadth-first search algorithm with black and white coloring. Step 2: Find the maximum matching of edges in the symmetric network; Step 3: Construct an undirected cactus; after finding the maximum matching in the symmetric network, construct an undirected cactus based on the unmatched nodes and matched edges; The construction of undirected cacti can be divided into the following two cases: (1) If There are unmatched nodes in the array, the number of which is denoted as n, and these unmatched nodes are marked as... Then it is necessary Only a set of disjoint, undirected cacti can cover all nodes in the network. The specific process of constructing an undirected cactus is as follows: In the set of unmatched nodes... Choose any node As Find the root node of the state path of an undirected cactus, and let the remaining unmatched nodes... As respectively The nodes on the state path of an undirected cactus are used to construct the cycle, with the remaining matching edges forming a loop. A pair of non-intersecting, undirected cacti; (2) If If there are no unmatched nodes, then only one undirected cactus is needed to cover all nodes in the network; therefore, an undirected cactus can be constructed by selecting any matched edge as the state path and the other matched edges as the cycle. Step 4: Construct snapshots; Construct a snapshot of the network graph consisting of each undirected cactus, with the number of snapshots being the same as the number of undirected cacti.

2. The method for improving the controllability of symmetric networks using time segmentation according to claim 1, characterized in that: The specific method for step 1 is as follows: First, randomly select a node and color it black, then color all its neighboring nodes with the opposite color; repeat the coloring process until all nodes have been colored; if it is a symmetric network If a node in a network has neighboring nodes that are the same color as the node itself, then the network is symmetric. There exists a singular cycle, denoted as If there are no odd cycles in a symmetric network, it is denoted as... .

3. The method for improving the controllability of symmetric networks using time segmentation according to claim 2, characterized in that: The specific method for step 2 is as follows: (1) Maximum matching of edges: First, treat each odd cycle in the symmetric network as a node to form a new network. Then, the network is found using the Hungarian algorithm. Find the maximum matching of edges in the middle; then expand each odd cycle and find the matching of edges in the odd cycle. The union of the matching of edges in the middle and the matching of edges in the odd cycle is: Maximum matching of edges in the middle; (2) Maximum matching of middle edges: The network directly uses the Hungarian algorithm to find the maximum matching and determine the matching edges and unmatched nodes.

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