CN110781352B - Method for optimizing topological structure to realize network structure controllability at lowest cost - Google Patents

Method for optimizing topological structure to realize network structure controllability at lowest cost Download PDF

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CN110781352B
CN110781352B CN201910976273.2A CN201910976273A CN110781352B CN 110781352 B CN110781352 B CN 110781352B CN 201910976273 A CN201910976273 A CN 201910976273A CN 110781352 B CN110781352 B CN 110781352B
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项林英
汪霄
陈飞
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Northeastern University Qinhuangdao Branch
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Abstract

The invention provides a method for optimizing a topological structure with the lowest cost to realize network structure controllability, and relates to the technical field of control and information. The invention provides a directed network with an uncontrollable structure, provides a simple and practical method for calculating the cost of the edge on the basis of obtaining all the optimal edge adding configuration schemes, thereby calculating the network cost of each configuration scheme, and selects the configuration scheme with the lowest cost by comparing the network costs in all the schemes, so that the structure controllability of the network can be realized by optimizing the topological structure with the lowest cost.

Description

Method for optimizing topological structure to realize network structure controllability at lowest cost
Technical Field
The invention relates to the technical field of control and information, in particular to a method for optimizing a topological structure to realize controllability of a network structure at the lowest cost.
Background
A complex network is a network structure that is composed of multiple nodes and intricate relationships between the nodes. Over the past decades, due to the rapid development and popularity of the internet, mankind has entered the network era: from the internet to biological networks, from power networks to transportation networks, from biological networks to neural networks, humans have lived in a world full of a wide variety of complex networks. The complex network theory is to abstract various real network systems, abstract individuals in the systems into nodes, abstract the relationship between the individuals into edges in the network, so as to research the commonality of various networks which look different from each other and the universal method for processing the networks, and provide guidance for the analysis and control of the real network.
In 2011, Liu et al made pioneering work in the controllability of complex networks. The method combines network science, control theory and statistical physics, researches the structure controllability problem of the large-scale weighted directed network, and establishes a complex network controllability analysis framework. For a directed network with an uncontrollable structure, the structure controllability of the network can be generally realized by optimizing a network topology structure. Wang et al propose to achieve the goal of controlling the entire network using only one driving node by perturbing the network structure. For a large complex network, the whole network is controlled by only one driving node, the structure disturbance of the network is overlarge, and the cost is overlarge. Chen et al studied the problem of least edge adding for a directed network, but did not give optimal choice for multiple optimal configuration schemes, i.e., did not consider the problem of network control cost. Zhang et al, who presents a network cost problem, gives a measure of edge cost, but does not provide a simple and effective calculation method to calculate the whole network cost.
The study on the controllability of the complex network not only has important theoretical significance for the study on the whole network science, but also has obvious practical significance. For example, in a social network, how to establish a connection between strangers greatly accelerates the information interaction speed, which is a typical controllability analysis problem; in a large-scale power network, how to optimize the topology structure of the network so as to control the power supply of the whole area with a minimum number of substations is a typical network controllability optimization problem; in a traffic network, how to optimize traffic routes between cities greatly reduces travel time of people, and the traffic network is a typical network optimization problem.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a method for realizing network structure controllability by optimizing a topological structure with the lowest cost, a directed network with an uncontrollable structure is given, the cost of edges is calculated by a simple and practical method on the basis of obtaining all optimal edge adding configuration schemes, so that the network cost of each configuration scheme is calculated, and the network cost in all the schemes is compared, and then the configuration scheme with the lowest cost is selected, so that the network structure controllability can be realized by optimizing the topological structure with the lowest cost.
A method for realizing network structure controllability by optimizing a topological structure with the lowest cost comprises the following steps:
step 1: given an uncontrollable direction of a structureNetwork
Figure BDA0002233739680000021
Where U is the set of input nodes, EX,XAs a set of edges between state nodes, EU,XConstructing the directed network structure topological graph for the edge set between the state node and the input node;
step 2: obtaining the optimal edge adding configuration required by network structure controllability by using a minimum edge adding algorithm, establishing a model by using the optimal edge adding configuration as an objective function and using the structure controllability of the network after edge adding as a constraint condition, wherein the objective function is as follows:
Figure BDA0002233739680000022
constraint conditions are as follows: system for controlling a power supply
Figure BDA0002233739680000023
Is structurally controllable, wherein,
Figure BDA0002233739680000024
is the optimal edge-adding configuration and is,
Figure BDA0002233739680000025
for added edges, X is a set of state nodes,
Figure BDA0002233739680000026
adding a matrix for a feasible edge;
and step 3: utilizing network analysis software to introduce a specific directed network, selecting a network characteristic parameter-betweenness, calculating the betweenness of the nodes, and expressing the node load by using the betweenness;
and 4, step 4: determining the node capacity through a nonlinear relation existing between the node load and the node capacity;
and 5: measuring the cost of the nodes by using the node capacity so as to determine the cost of each node in the network;
step 6: comparing the capacities of two nodes on the edge in the network, and taking the larger node capacity as the cost of the edge;
and 7: computingAdding network cost of configuration schemes to all the optimal edges, determining a scheme with the lowest cost, realizing structural controllability of the network by adopting the scheme, calculating betweenness of nodes by utilizing network analysis software, taking the edge addition configuration scheme with the lowest cost as an objective function, adding configuration as a constraint condition by taking the added edge as the optimal edge, and establishing a model by using the objective function: min Cost (Net) ═ Σ Cost (l)ij)，
Figure BDA0002233739680000027
Constraint conditions are as follows:
Figure BDA0002233739680000028
where cost (Net) is the network cost, lijBeing connecting edges between state nodes, EX,XIs the set of edges between the state nodes;
the network cost for each configuration scheme is calculated by step 6, and the formula is as follows:
Cost(Net)＝ΣCost(lij),
where cost (net) represents the network cost.
The specific steps of the minimum edge filling algorithm in the step 2 are as follows:
step 2.1: classifying all nodes in the directed graph into a reachable node set and a non-reachable node set based on the reachability of the nodes;
step 2.2: calculating a maximum matching to obtain the number of key unreachable source connected pieces, and determining whether the node has an independent superior node and whether the node is used as the independent superior node;
step 2.3: adding edges to satisfy node reachability, the starting point of the edges being reachable nodes not serving as independent superior nodes, and the end point being a point in unreachable source SCCs without independent superior nodes;
step 2.4: for unreachable points which are not used as independent superior nodes in the network, a bridge edge set is added to ensure two conditions of controllable structure, namely, unreachable nodes do not exist in the network and expansion are both satisfied, so that an optimal edge addition configuration scheme is obtained.
In the step 3:
the node load is expressed by the betweenness of the nodes, and the formula is as follows:
Figure BDA0002233739680000031
wherein, CB(v) Denotes the betweenness, σ, of the node vst(v) Representing the number of s → t shortest paths, σ, through node vstRepresents the shortest path number of s → t.
In the step 4:
the nonlinear relation of the node load and the node capacity exists, and the formula is as follows:
Cap(v)＝CB(v)+β(CB(v))α,v＝1,2,...n,
wherein Cap (v) represents the capacity of the node v, and alpha is more than 0, and beta is more than 0.
In the step 5:
the node cost is measured by the node capacity, and the formula is as follows:
Cost(v)＝Cap(v)＝2CB(v),
where cost (v) represents the cost of node v.
In the step 6:
calculating the node capacity by the step 4, comparing the capacities of the two nodes on the edge, and taking the larger one as the capacity of the edge, namely the cost of the edge is as follows:
Cost(lij)＝max{Cap(vi),Cap(vj)}，
wherein v isiDenotes the ith node, vjDenotes the jth node, lijTo point from node i to an edge of node j, Cost (l)ij) Is represented byijThe cost of this edge.
Adopt the produced beneficial effect of above-mentioned technical scheme to lie in:
(1) the cost of the control network is calculated by defining the cost of the edge.
(2) The requirement on the network topological structure is simple, the applicability is strong, and the realization is easy.
(3) On the basis of determining the optimal edge adding configuration required by network controllability, the method can quickly calculate the network characteristic parameter-betweenness (node load) through pajek software, thereby calculating the network cost of each configuration scheme, and then selecting the configuration scheme with the lowest cost.
The invention introduces the cost problem of the network into the problem of realizing the controllability of the network structure, researches how to determine the lowest-cost optimized topological structure so as to realize the structure controllability of the network, provides a method for realizing the controllability of the network structure by the lowest-cost optimized topological structure, and simultaneously provides a simple and practical method for calculating the network cost of adding configuration schemes at different sides, and helps us to select the scheme with the lowest cost from all the optimal side adding schemes so as to realize the structure controllability of the network by the lowest-cost optimized topological structure.
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FIG. 1 is a flow chart of the present invention for implementing a lowest cost optimized topology to achieve network structure controllability;
FIG. 2 is a specific directed graph in an embodiment of the present invention;
fig. 3 is an original data diagram of node betweenness in a directed network corresponding to the first scheme in the embodiment of the present invention;
fig. 4 is an original data diagram of node betweenness in a directed network corresponding to the second scheme in the embodiment of the present invention.
Detailed Description
The following detailed description of embodiments of the present invention is provided in connection with the accompanying drawings and examples. The following examples are intended to illustrate the invention but are not intended to limit the scope of the invention.
A method for implementing network structure controllability by using a minimum cost optimized topology structure, as shown in fig. 1, includes the following steps:
step 1: given an uncontrollable directed network
Figure BDA0002233739680000041
Where U is the set of input nodes, EX,XAs edges between state nodesCollection, EU,XConstructing the directed network structure topological graph for the edge set between the state node and the input node;
in this embodiment, as shown in fig. 2, a directed network with an uncontrollable structure is given, where 1 input node and 8 state nodes are provided, and the number of connecting edges between the state nodes is 8;
step 2: obtaining the optimal edge adding configuration required by network structure controllability by using a minimum edge adding algorithm, establishing a model by using the optimal edge adding configuration as an objective function and using the structure controllability of the network after edge adding as a constraint condition, wherein the objective function is as follows:
Figure BDA0002233739680000042
constraint conditions are as follows: system for controlling a power supply
Figure BDA0002233739680000043
Is structurally controllable, wherein,
Figure BDA0002233739680000044
is the optimal edge-adding configuration and is,
Figure BDA0002233739680000045
for added edges, X is a set of state nodes,
Figure BDA0002233739680000046
adding a matrix for a feasible edge;
the minimum edge adding algorithm comprises the following specific steps:
step 2.1: classifying all nodes in the directed graph into a reachable node set R ═ x based on reachability of the nodes1,x2N ═ x for unreachable node set3,x4,x5,x6,x7,x8}；
Step 2.2: calculating a maximum matching to obtain the number of key unreachable source connected pieces (ZUSCC is 3), and determining a node x4、x6、x7Whether or not there is an independent upper node, and node x2、x3、x5Whether to act as an independent upper node；
Step 2.3: adding edges to satisfy node reachability, the starting point of the edges being reachable nodes not serving as independent superior nodes, and the end point being a point in unreachable source SCCs without independent superior nodes;
step 2.4: for unreachable points which are not used as independent superior nodes in the network, a bridge edge set is added to ensure two conditions of controllable structure, namely that unreachable nodes do not exist in the network and expansion are both satisfied, so that an optimal edge addition configuration scheme is obtained:
the first scheme is as follows:
Figure BDA0002233739680000051
scheme II:
Figure BDA0002233739680000052
and step 3: according to the configuration scheme in step 2.4, using network analysis software, in this embodiment, pajek software is used to import the directed network corresponding to each scheme, and then network characteristic parameter-betweenness is selected to calculate the network node betweenness corresponding to each scheme. The calculated original data are shown in fig. 3 and 4;
the node load is expressed by the betweenness of the nodes, and the formula is as follows:
Figure BDA0002233739680000053
wherein, CB(v) Denotes the betweenness, σ, of the node vst(v) Representing the number of s → t shortest paths, σ, through node vstRepresents the shortest path number of s → t.
And 4, step 4: determining the node capacity through a nonlinear relation existing between the node load and the node capacity;
the nonlinear relation of the node load and the node capacity exists, and the formula is as follows:
Cap(v)＝CB(v)+β(CB(v))α,v＝1,2,...n,
wherein Cap (v) represents the capacity of the node v, and alpha is more than 0, and beta is more than 0. Since the node load and the node capacity are in positive correlation, in order to avoid loss of generality, the alpha and the beta are 1 (not influencing the final result), so that the node capacity is determined to be Cap (v) ═ 2CB(v)。
The network node capacity for each scheme is shown in table 1.
Table 1: network node capacity comparison table of two schemes
Figure BDA0002233739680000054
Figure BDA0002233739680000061
And 5: measuring the cost of the nodes by using the node capacity so as to determine the cost of each node in the network;
the node cost is measured by the node capacity, and the formula is as follows:
Cost(v)＝Cap(v),
where cost (v) represents the cost of node v.
Step 6: comparing the capacities of two nodes on the edge in the network, and taking the larger node capacity as the cost of the edge;
calculating the node capacity by the step 4, comparing the capacities of the two nodes on the edge, and taking the larger one as the capacity of the edge, namely the cost of the edge is as follows:
Cost(lij)＝max{Cap(vi),Cap(vj)}，
wherein v isiDenotes the ith node, vjDenotes the jth node, lijTo point from node i to an edge of node j, Cost (l)ij) Is represented byijThe cost of this edge.
The cost of all edges in the network is shown in table 2.
TABLE 2 comparison of cost for all edges in two scheme networks
Figure BDA0002233739680000062
Figure BDA0002233739680000071
And 7: calculating the network cost of all the optimal edge adding configuration schemes, determining a scheme with the lowest cost, realizing the structural controllability of the network by adopting the scheme, calculating the betweenness of the nodes by utilizing network analysis software, using pajek software in the embodiment, taking the edge adding configuration scheme with the lowest cost as an objective function, taking the added edge as the optimal edge adding configuration as a constraint condition, and establishing a model, wherein the objective function is as follows: min Cost (Net) ═ Σ Cost (l)ij)，
Figure BDA0002233739680000072
Constraint conditions are as follows:
Figure BDA0002233739680000073
where cost (Net) is the network cost, lijBeing connecting edges between state nodes, EX,XIs the set of edges between the state nodes.
The network cost for each configuration scheme is calculated by step 6, and the formula is as follows:
Cost(Net)＝∑Cost(lij),
where cost (net) represents the network cost.
The network costs are shown in table 3.
TABLE 3 two schemes network cost comparison table
Figure BDA0002233739680000074
As can be seen from table 3, the network cost of the first solution is the lowest, so that the first solution is selected as the final edge addition configuration solution, and thus the structure controllability of the network can be realized by optimizing the topology structure with the lowest cost.
All steps are now complete. The invention researches a method for optimizing a topological structure with the lowest cost to realize network structure controllability. The cost index of the edge proposed by the method well measures the importance degree of the edge in the network, and a scheme with the lowest cost is determined through calculation. In the whole calculation process, pajek software is used for calculating node betweenness in the directed graph. The software is simple to operate, can quickly and effectively solve the problem of node load, and greatly improves the operation efficiency.
Finally, it should be noted that: the above examples are only intended to illustrate the technical solution of the present invention, but not to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those skilled in the art; the technical solutions described in the foregoing embodiments may still be modified, or some or all of the technical features may be equivalently replaced; such modifications and substitutions do not depart from the spirit and scope of the corresponding technical solutions as defined in the appended claims.

Claims (1)

1. A method for realizing network structure controllability by optimizing a topological structure with the lowest cost is characterized in that: the method comprises the following steps:
step 1: given an uncontrollable directed network
Figure FDA0003439812340000011
Figure FDA0003439812340000012
Where U is the set of input nodes, EX,XAs a set of edges between state nodes, EU,XConstructing the directed network structure topological graph for the edge set between the state node and the input node;
step 2: obtaining the optimal edge adding configuration required by network structure controllability by using a minimum edge adding algorithm, establishing a model by using the optimal edge adding configuration as an objective function and using the structure controllability of the network after edge adding as a constraint condition, wherein the objective function is as follows:
Figure FDA0003439812340000013
constraint conditions are as follows: network
Figure FDA0003439812340000014
Is structurally controllable, wherein,
Figure FDA0003439812340000015
is the optimal edge-adding configuration and is,
Figure FDA0003439812340000016
for added edges, X is a set of state nodes,
Figure FDA0003439812340000017
adding a matrix for a feasible edge;
the specific steps of the minimum edge adding algorithm in the step 2 are as follows:
step 2.1: classifying all nodes in the directed graph into a reachable node set and a non-reachable node set based on the reachability of the nodes;
step 2.2: calculating a maximum matching to obtain the number of key unreachable source connected pieces, and determining whether the node has an independent superior node and whether the node is used as the independent superior node;
step 2.3: adding edges to satisfy node reachability, the starting point of the edges being reachable nodes not serving as independent superior nodes, and the end point being a point in unreachable source SCCs without independent superior nodes;
step 2.4: for unreachable points which are not used as independent superior nodes in the network, a bridge edge set is added to ensure two conditions of controllable structure, namely, unreachable nodes and expansion do not exist in the network, so that an optimal edge adding configuration scheme is obtained;
and step 3: utilizing network analysis software to introduce a specific directed network, selecting a network characteristic parameter-betweenness, calculating the betweenness of the nodes, and expressing the node load by using the betweenness;
in the step 3:
the node load is expressed by the betweenness of the nodes, and the formula is as follows:
Figure FDA0003439812340000018
wherein, CB(v) Denotes the betweenness, σ, of the node vst(v) Representing the number of s → t shortest paths, σ, through node vstThe number of shortest paths representing s → t;
and 4, step 4: determining the node capacity through a nonlinear relation existing between the node load and the node capacity;
in the step 4:
the nonlinear relation of the node load and the node capacity exists, and the formula is as follows:
Cap(v)＝CB(v)+β(CB(v))α,v＝1,2,...n,
wherein Cap (v) represents the capacity of node v, CB(v) Denotes the betweenness of the node v, α>0,β>0；
And 5: measuring the cost of the nodes by using the node capacity so as to determine the cost of each node in the network;
in the step 5:
the node cost is measured by the node capacity, and the formula is as follows:
Cost(v)＝Cap(v),
wherein cost (v) represents the cost of the node v, and Cap (v) represents the capacity of the node v;
step 6: comparing the capacities of two nodes on the edge in the network, and taking the larger node capacity as the cost of the edge;
in the step 6:
calculating the node capacity by step 4, comparing the capacity of two nodes on the edge, and taking the larger one as the capacity of the edge, namely the cost of the edge is
Cost(lij)＝max{Cap(vi),Cap(vj)}，
Wherein v isiDenotes the ith node, vjDenotes the jth node, lijTo point from node i to an edge of node j, Cost (l)ij) Is represented byijThe cost of this edge;
and 7: calculating the network cost of all the optimal edge adding configuration schemes, determining a scheme with the lowest cost, realizing the structural controllability of the network by adopting the scheme, calculating the betweenness of the nodes by utilizing network analysis software, taking the edge adding configuration scheme with the lowest cost as an objective function, taking the added edge as the optimal edge adding configuration as a constraint condition, establishing a model, and using the objective function: min Cost (Net) ═ Σ Cost (l)ij)，
Figure FDA0003439812340000021
Constraint conditions are as follows:
Figure FDA0003439812340000022
where cost (Net) is the network cost, lijBeing connecting edges between state nodes, EX,XIs the set of edges between the state nodes;
wherein the network cost for each configuration scheme is calculated by step 6, and the formula is as follows:
Cost(Net)＝∑Cost(lij),
where Cost (Net) represents the network Cost, Cost (l)ij) Is represented byijThe cost of this edge.
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