CN108847986A

CN108847986A - A kind of control method for double-layer network

Info

Publication number: CN108847986A
Application number: CN201810689888.2A
Authority: CN
Inventors: 张锡哲; 周鑫
Original assignee: Northeastern University China
Current assignee: Northeastern University China
Priority date: 2018-06-28
Filing date: 2018-06-28
Publication date: 2018-11-20

Abstract

The present invention provides a kind of control method for double-layer network.Method includes：Obtain the first layer network and the maximum matching side collection of the second layer network, simulation node collection difference in double-layer network to be processed, judge that the maximum matching of the first layer network any of is matched with the maximum of the second layer network and whether belongs to perfect matching, if perfect matching, the minimum driving node collection of double-layer network is then obtained, and terminates operation.If not being perfect matching, the reachable set AC that interlocks is obtained₁(x), it meanwhile, obtains node y and replaces node x, while updating the matching of network maximum.This method can be by finding a simulation node set (MDS), allow to control any one layer of double-layer network, current driving node is replaced by the node in the staggeredly reachable set with driving node, it is effectively reduced the quantity of the total driving node of network, improves the control efficiency of multitiered network.

Description

A kind of control method for double-layer network

Technical field

The present invention relates to network technique fields, and in particular to a kind of control method for double-layer network.

Background technique

In actual life many things and its between relationship can be modeled as network, such as the interpersonal pass in social system It is net, scientist's coorporative network, epidemic transmission net and the network of communication lines, neuron net, gene regulatory network and albumen in the ecosystem Matter interactive network and Food web, telephone network, internet and WWW etc. in science technology system.To this increasingly complicated network The structure of system and behavioural analysis are to understand the premise of its inherent law.Guarantee the normal operation of these systems, it is therefore necessary to Study the control of complex network.For example, in transportation network, to avoid traffic congestion as far as possible, if will by increase road or Road width is widened to slow down traffic jam phenomenon, being operated to which road or crossing can just play a role？In Food web In, species are affected by environment and become extinct and affect the structure of food chain, how to guarantee the sustainability of the ecosystem？Because of spy In net, applying control to those nodes can guarantee the stability of network, adjust the communication of which node to maintain network just Normal communications status？In social networks, which node is selected to carry out information publication or regulation, so that the influence model that control message is propagated It encloses？These can all be analyzed by the research controlled complex network.

Many systems of real world are all interactions to form multilayered structure.For example, city traffic network is that collection is public Hand over the multilayered structure being integrated with subway；One physiological function, which is also required to several biosystem Collaboration, to be completed.Therefore, The research of multitiered network has important practical significance.

In order to control complex network, the part of nodes input control signal into network is needed, the company on side between node is passed through It connects, all nodes of network is driven to reach desired state, these nodes for being used to input control signal are known as driving node.It is multiple Miscellaneous network-based control, which refers to, enables complex network in the limited time control node input control signals several in network It is interior to reach any expecting state from original state.In order to fully control the state of all nodes of network, required least input Node set is known as simulation node collection (MDS).

It is directed to the control problem of multitiered network at present, first method is calculated separately each based on structural controllability theory The MDS of layer network, then takes union.Since the MDS of each layer network is not uniquely, the control obtained in this way is entirely more The driving node set of layer network is not optimal, therefore it is higher to control cost.

Another control method is proposed based on structural controllability theory.Assuming that the MDS of each layer network be it is identical, i.e., Find any one layer that a MDS allows to control multitiered network.According to structural controllability theory, the MDS for finding network can To be realized by the maximum matching for calculating figure.

Therefore how to go out to meet the minimum drive of constraint condition and maximum matched control double-layer network by BP equation calculation Dynamic node number N_DThe problem of as current urgent need to resolve.

Summary of the invention

(1) technical problems to be solved

For complex network control problem, the present invention proposes a kind of control method for double-layer network, can find control Simulation node set required for all nodes of double-layer network processed.Relative to the previous work for successively calculating driving node collection Make, the present invention can find less driving node, to control double-layer network, to improve the efficiency of control.

(2) technical solution

In order to achieve the above object, the main technical schemes that the present invention uses include：

A kind of control method for double-layer network, including：

S1, for double-layer network G (V, E) to be processed, obtain first layer network G in G (V, E)₁Maximum matching Simulation node collection MDS₁, obtain second layer network G in G (V, E)₂Maximum matchingSimulation node collection MDS₂；Its In, V is node set, node i ∈ V, i=1,2 ..., N, the G₁With the G₂Corresponding identical physical node；

S2, judge first layer network G₁Maximum matchingSecond layer network G₂Maximum matchingAny of Whether perfect matching is belonged to；

If S3,For perfect matching, then the simulation node collection MDS=MDS of the double-layer network G₂；

IfFor perfect matching, then the simulation node collection MDS=MDS of the double-layer network G₁；

IfIt is perfect matching, then the simulation node collection MDS={ i } of the double-layer network G, i ∈ {1,2,…,N}。

Optionally, the method also includes：

If S4,WithIt is not perfect matching, then obtains MDS₁And MDS₂Intersection sameSet=MDS₁∩MDS₂； And by the node in sameSet from MDS₁And MDS₂It is removed in set, obtains a new MDS₁ ⁽¹⁾And MDS₂ ⁽¹⁾Collection；

S5, MDS is obtained₁ ⁽¹⁾In each node x staggeredly reachable set AC₁(x)；

Obtain MDS₂ ⁽¹⁾In meet condition y₁∈MDS₂ ⁽¹⁾And y₁∈AC₁(x) node y₁；

At least one node y if it exists₁, then make MDS₁ ⁽¹⁾Remove corresponding x₁Element obtains MDS₁ ⁽²⁾, make MDS₂ ⁽¹⁾Removal y₁Element obtains MDS₂ ⁽²⁾, increase sameSet by y₁Element obtains sameSet⁽¹⁾, and execute update (G₁,x₁,y₁)；

Meanwhile obtaining MDS₂ ⁽¹⁾In meet condition y₂∈MDS₂ ⁽¹⁾And y₂Staggeredly reachable set AC₂(y₂) and AC₁(x) friendship Collection setA is not empty node y₂；

At least one node y if it exists₂, then make MDS₂ ⁽²⁾Remove corresponding x₂Element obtains MDS₁ ⁽³⁾, make MDS₂ ⁽²⁾Removal y₂Element obtains MDS₂ ⁽³⁾, an element z in setA is taken, sameSet is made⁽¹⁾Increase z element, obtains sameSet⁽²⁾, and hold Row update (G₁,x₂,z)、update(G₂,y₂,z)；

S6, MDS is being traversed₂ ⁽¹⁾In after each node, obtain MDS₁ ⁽⁴⁾、MDS₂ ⁽⁴⁾And sameSet⁽³⁾Collection；

Judge MDS₁ ⁽⁴⁾And MDS₂ ⁽⁴⁾It whether is sky, if MDS₁ ⁽⁴⁾And MDS₂ ⁽⁴⁾Middle there are a set for sky, then will be non- Null set and sameSet⁽³⁾Simulation node collection MDS of the union as double-layer network G；

Wherein, update (Gk, i, j) indicates to update network G_KMaximum matching, i.e., for the j's from node i to node All sides on zigzag path, by network G_KMatching when becoming non-matching, former network G_KIt is non-matching while become matching while, k Take 1 or 2.

If S7, MDS₁ ⁽⁴⁾And MDS₂ ⁽⁴⁾It is nonempty set, then obtains MDS₂ ⁽⁴⁾In each node b staggeredly reachable set AC₂ (b)；

Obtain MDS₁ ⁽⁴⁾In meet condition a₁∈MDS₁ ⁽⁴⁾And a₁∈AC₂(b) node a₁；

At least one node a if it exists₁, then make MDS₁ ⁽⁴⁾Remove a₁Element obtains MDS₁ ⁽⁵⁾, make MDS₂ ⁽⁴⁾It removes corresponding b₁Element obtains MDS₂ ⁽⁵⁾, make sameSet⁽³⁾Increase a₁Element obtains sameSet⁽⁴⁾, and execute update (G₂,b₁,a₁)；

Meanwhile obtaining MDS₁ ⁽⁴⁾In meet condition a2 ∈ MDS₁ ⁽⁴⁾And a₂Staggeredly reachable set AC₁(a₂) and AC₂(b) friendship Collection setB is not empty node a₂；

At least one node a if it exists₂, then make MDS₁ ⁽⁵⁾Remove a₂Element obtains MDS₁ ⁽⁶⁾, make MDS₂ ⁽⁵⁾It removes corresponding b₂Element obtains MDS₂ ⁽⁶⁾, an element c in setB is taken, sameSet is made⁽⁴⁾Increase c element, obtains sameSet⁽⁵⁾, and hold Row update (G₁,a₂,c)、update(G₂,b₂,c)；

S8, MDS is being traversed₂ ⁽⁴⁾In after each node, obtain MDS₁ ⁽⁷⁾、MDS₂ ⁽⁷⁾And sameSet⁽⁶⁾Collection；

It is repeated in a manner of step S5 and step S7, until MDS₁ ⁽⁷⁾And MDS₂ ⁽⁷⁾At least one of for sky, or sameSet⁽⁶⁾In element no longer change, then the simulation node of double-layer network G integrates MDS as MDS₁ ⁽⁷⁾、MDS₂ ⁽⁷⁾With sameSet⁽⁶⁾Union.

Optionally, before the step S5, the method also includes：

S4a, each node q in sameSet is performed the following operations：

Q is obtained in G₁In staggeredly reachable set AC₁(q), G₂In staggeredly reachable set AC₂(q)；

Acquisition meets condition m ∈ AC₁(q)∩MDS₂ ⁽¹⁾And n ∈ AC₂(q)∩MDS₁ ⁽¹⁾Node m, n；

At least one node m and there are at least one node n if it exists, then make MDS₁ ⁽¹⁾N element is removed, obtains one New MDS₁ ⁽¹⁾, make MDS₂ ⁽¹⁾M element is removed, the MDS of a update is obtained₂ ⁽¹⁾, make sameSet removal q element and increase m, n Element, obtains the sameSet of a update, and executes update (G₁, q, m), update (G₂,q,n)；

Correspondingly, the MDS in step S5₁ ⁽¹⁾、MDS₂ ⁽¹⁾It is the MDS updated in step S4a with sameSet₁And MDS₂、 sameSet；

Correspondingly, if the MDS updated₁And MDS₂Middle there are one is empty set, then by nonempty set and sameSet Simulation node collection MDS of the union as double-layer network G.

Optionally, the method also includes：

Judge whether there is at least one node y₁, meet y₁∈MDS₂ ⁽¹⁾And y₁∈AC₁(x)；

If it exists, step S5a is executed, otherwise, executes step S5b；

S5a, if it exists at least one node y₁, then make MDS₁ ⁽¹⁾Corresponding x element is removed, MDS is obtained₁ ⁽²⁾, make MDS₂ ⁽¹⁾ Remove y₁Element obtains MDS₂ ⁽²⁾, increase sameSet by y₁Element obtains sameSet⁽¹⁾, and execute update (G₁,x,y₁)；

S5b, judge MDS₂ ⁽¹⁾In whether there is at least one node y₂, meet y₂∈MDS₂ ⁽¹⁾And y₂Staggeredly reachable set AC₂(y₂) and AC₁(x) intersection setA is not empty；

At least one node y if it exists₂, then make MDS₂ ⁽¹⁾Corresponding x element is removed, MDS is obtained₁ ^(2)*, make MDS₂ ⁽¹⁾Removal y₂Element obtains MDS₂ ^(2)*, an element z in setA is taken, increases sameSet by z element, obtains sameSet^(1)*, and hold Row update (G₁,x,z)、update(G₂,y₂,z)；

S6, MDS is being traversed₂ ⁽¹⁾In after each node, obtain MDS₁ ⁽³⁾、MDS₂ ⁽³⁾And sameSet⁽²⁾Collection；

At this time, judge MDS₁ ⁽³⁾And MDS₂ ⁽³⁾It whether is sky, if MDS₁ ⁽³⁾And MDS₂ ⁽³⁾Middle there are one is empty set, Then by nonempty set and sameSet⁽²⁾Simulation node collection MDS of the union as double-layer network G；

Wherein, update (G_k, i, j) and it indicates to update network G_KMaximum matching, i.e. the friendship for the j from node i to node All sides on wrong path, by former network G_KMatching when becoming non-matching, former network G_KIt is non-matching while become matching while, k Take 1 or 2.

If S7, MDS₁ ⁽³⁾And MDS₂ ⁽³⁾It is nonempty set, then obtains MDS₂ ⁽³⁾In each node b staggeredly reachable set AC₂ (b)；

Judge whether there is node a₁, meet a₁∈MDS₁ ⁽³⁾And a₁∈AC₂(b)；

If it exists, step S7a is executed, otherwise, executes step S7b；

S7a, if it exists at least one node a₁, then make MDS₁ ⁽³⁾Remove a₁Element obtains MDS₁ ⁽⁴⁾, make MDS₂ ⁽³⁾Removal pair The b element answered, obtains MDS₂ ⁽⁴⁾, make sameSet⁽²⁾Increase a₁Element obtains sameSet⁽³⁾, and execute update (G₂,b, a₁)；

S8, MDS is being traversed₂ ⁽³⁾In after each node, obtain MDS₁ ⁽⁵⁾、MDS₂ ⁽⁵⁾And sameSet⁽⁴⁾Collection；

It is repeated in a manner of step S5 and step S7, until MDS₁ ⁽⁵⁾And MDS₂ ⁽⁵⁾At least one of for sky, or sameSet⁽⁴⁾In element no longer change, then the simulation node of double-layer network G integrates MDS as MDS₁ ⁽⁵⁾、MDS₂ ⁽⁵⁾With sameSet⁽⁴⁾Union.

S7b, judge MDS₁ ⁽³⁾In whether there is at least one node a₂, meet a₂∈MDS₁ ⁽³⁾And a₂Staggeredly reachable set AC₁(a₂) and AC₂(b) intersection setB is not empty；

At least one node a if it exists₂, then make MDS₁ ⁽³⁾Remove a₂Element obtains MDS₁ ^(4)*, make MDS₂ ⁽³⁾Removal corresponds to B element, obtain MDS₂ ^(4)*, an element c in setB is taken, sameSet is made⁽³⁾Increase c element, obtains sameSet^(3)*, and Execute update (G₁,a₂,c)、update(G₂,b,c)；

In traversal MDS₂ ⁽³⁾In after each node, obtain newest simulation node collection MDS₁ ⁽⁵⁾、MDS₂ ⁽⁵⁾And sameSet⁽⁴⁾Collection；

S8 ', it is repeated in a manner of step S5 and step S7, until MDS₁ ⁽⁵⁾And MDS₂ ⁽⁵⁾At least one of for sky, or sameSet⁽⁴⁾In element no longer change, then the simulation node of double-layer network G integrates MDS as MDS₁ ⁽⁵⁾、MDS₂ ⁽⁵⁾With sameSet₍₄₎Union.

Optionally, before the step S5, the method also includes：

S4a, each node q in sameSet is performed the following operations：

On the other hand, it the present invention also provides a kind of electronic equipment, including memory, processor, bus and is stored in On reservoir and the computer program that can run on a processor, the processor realize any of the above one when executing described program The step of.

(3) beneficial effect

The present invention has practical practical significance in the application that double-layer network controls, by finding a simulation section Point set (MDS) allows to control any one layer of double-layer network, passes through the node in the staggeredly reachable set with driving node Current driving node is replaced, the quantity of the total driving node of network is effectively reduced, improves the control efficiency of multitiered network, Meanwhile the present invention also provides effective measurement means for the controllability of multitiered network.

Detailed description of the invention

The drawings described herein are used to provide a further understanding of the present invention, constitutes a part of the invention, this hair Bright illustrative embodiments and their description are used to explain the present invention, and are not constituted improper limitations of the present invention.In the accompanying drawings：

Fig. 1 is a kind of control method flow diagram for double-layer network of the embodiment of the present invention one；

Fig. 2 is the schematic diagram that scientist provided by one embodiment of the present invention quotes collaboration network interactive network；

Fig. 3 is double-layer network of embodiment of the present invention schematic diagram；

Fig. 4 is a kind of control method flow diagram for double-layer network of the embodiment of the present invention two；

Fig. 5 is that a kind of control method for double-layer network of the embodiment of the present invention three calculates schematic diagram；

Fig. 6 is that three double-layer network initial maximum of the embodiment of the present invention matches schematic diagram；

Fig. 7 is that three double-layer network of the embodiment of the present invention executes maximum matching schematic diagram after operation；

Fig. 8 is that three double-layer network of the embodiment of the present invention executes maximum matching schematic diagram after operation；

Fig. 9 is a kind of calculating schematic diagram of the control method for double-layer network of the embodiment of the present invention four；

Figure 10 is that four double-layer network initial maximum of the embodiment of the present invention matches schematic diagram；

Figure 11 is that four double-layer network of the embodiment of the present invention executes maximum matching schematic diagram after operation；

Figure 12 is a kind of schematic diagram of bipartite graph matching provided in an embodiment of the present invention；

Figure 13 is present invention verifying example double-layer network controllability schematic diagram.

Specific embodiment

In order to preferably explain the present invention, in order to understand, with reference to the accompanying drawing, by specific embodiment, to this hair It is bright to be described in detail.

What the present invention was handled is the control problem of multitiered network, that is, finds out a MDS and allow to control entire Multilayer Network Any one layer of network.

The present invention is directed to this kind of network-based control problem, proposes a kind of identification multitiered network simulation node set Method can be found out by iterating to calculate the driving node of each layer network and fully control all node states of double-layer network most Small driving node set.

Consider network G (V, E), wherein V is node set, and E is line set.Matching collection when collection M is one independent, i.e., While while between not shared node.For a node, if it is the terminal on a matching side, which is matching section Point；It otherwise, is non-matching node.The most matching side collection of number of edges is referred to as a maximum matching of figure, is denoted as M*.If in figure All nodes are all matched nodes, then the maximum matching in the figure is known as perfect matching.

The size of the maximum matching side collection of network G (V, E) is denoted as | M* |, then minimum input theorem can be described as： If there are a perfect matchings in network G (V, E), then minimum input number of nodes NI needed for network G (V, E) or most is controlled Small input number of nodes ND is 1；Otherwise, equal to the quantity for matching corresponding non-matching node maximum in network.I.e.：

N_I=N_D=max N- | M^*|, 1 }

That is, not being 0 network for non-matching number of nodes, input number of nodes is non-matching number of nodes, and non- Matched node is a kind of situation of input node.For the network for the perfect matching that non-matching number of nodes is 0, input number of nodes It is 1, any node may be defined as input node in network.

In order to fully control a network, the minimum input needed is determined by the maximum matching of network.Matching section Point is controlled by its predecessor node, and non-matched node must then be controlled by an external input, therefore non-matched node is The input node of network.Once being inputted for each input node, the node in network has the forerunner of oneself, I.e. network is fully controllable.

It is given below about maximum matched several concepts.

Define 1：

The corresponding bipartite graph B=(V of G (V, E)⁺,V^-, E), V⁺Indicate upper node, V^-Indicate lower part node, the side v in G_i →v_jCorresponding to the side (v in B_i ⁺, v_j ^-).In a subgraph M of B, not depending in any two concentrated for M is same A vertex, then claiming M is a matching of bipartite graph.

The maximum matching of bipartite graph, is in given bipartite graph B=(V⁺,V^-, E) all matchings in, wherein including The most matching of number of edges find out.If in a matching, certain side is associated in each vertex and figure in figure, then claim This matching is exact matching, also referred to as perfect matching.

As shown in figure 12, Figure 12 (a) be a bipartite graph, overstriking while indicate matching while, therefore Figure 12 (b) indicate this two One matching of component, but its non-maximum matching, Figure 12 (c) are a maximum matching of bipartite graph, and Figure 12 (d) expression one is complete Complete to match, each vertex in figure is matched.

Define 2：

Non-matching point：Not belonging to any one the associated node in the matching side of M in bipartite graph is non-matching point, lower part Non-matching point in node is driving node.

Zigzag path：If there are such paths in B, on the path matching while and it is non-matching while be alternately present, that This path is known as zigzag path.Augmenting path is the zigzag path that endpoint is non-matching point.

Staggeredly reachable set：If there are zigzag paths between lower part node a and b, claim node a staggeredly up to node b.Node a All staggeredly reachable nodes be known as the staggeredly reachable set of a, be denoted as AC (a).In Figure 12 (b), node v4's is staggeredly reachable Integrate as AC (v4)={ v2, v3 }；In Figure 12 (c), the staggeredly reachable set of node v4 is AC (v4)={ v1 }.

Adjacent theorem：

(1) for arbitrary MDS, the staggeredly reachable set of each input node includes be likely to become driving node Node；

(2) if a node is not belonging to the staggeredly reachable set of any input node, which is centainly not belonging to any Minimum input node collection, i.e., will not be as the input node of network G.

According to the staggeredly connectivity of adjacent theorem and node, the node j in the staggeredly reachable set of driving node i can use Replace current driving node, maximum matching changes therewith, i.e., for the institute on from node i to the zigzag path of node j There is side, need to be by original matching when becoming non-matching, original is non-matching when becoming matching.By constantly replacing each layer Some initial driving nodes of network, so that the MDS of double-layer network finally converges to a minimum value, here it is this patents Core concept.

Embodiment one

Consider an oriented double-layer network G (V, E), double-layer network to be processed is the node group of a physical topological structure At tool there are two logical topological structure network, it includes first layer network Gs₁(V,E₁) and second layer network G₂(V,E₂)。G₁ And G₂The physical node set V of network is identical, node i ∈ V, i=1,2 ..., N, but the interaction relationship between node is not Together, that is, there is Different Logic topological structure, E₁、E₂, E be respectively G₁、G₂, G network line set, E₁≠E₂, and E=E₁+E₂。

Therefore, the characteristics of Multi-Layered Network Model that the present embodiment considers is that same group of physics section is shared on different types of even side Point.Specifically, scientist quotes collaboration network schematic diagram as shown in Figure 2, figure interior joint indicates scientist, and same group node expression is drawn With all scientists in collaboration network, solid line indicates that the adduction relationship between scientist, Fig. 2 (b) dotted line represent section in Fig. 2 (a) Cooperative relationship between scholar.The multitiered network that can be thus one by the network abstraction.

The present invention is based on structural controllability theories can be found out completely by iterating to calculate the driving node of each layer network The smallest driving node set (MDS) for controlling all node states of double-layer network, allows to control any of double-layer network One layer.

Method as shown in Figure 1, a kind of double-layer network control method of the present embodiment include：

S1, for double-layer network G (V, E) to be processed, use Hopcroft-Karp algorithm to obtain first layer in G (V, E) Network G₁Maximum matchingSimulation node collection MDS₁, obtain second layer network G in G (V, E)₂Maximum matching Simulation node collection MDS₂；

Oriented double-layer network as shown in Figure 3, every layer includes 5 nodes (N=5), solid line while and when dotted line indicate between node Different types of interaction relationship.In every layer network, overstriking when indicating matching, the side of non-overstriking indicates not match Side indicates driving node with the node that dotted line marks, therefore, the MDS of the double-layer network₁={ Isosorbide-5-Nitrae }, MDS₂={ Isosorbide-5-Nitrae }.

If it is, executing step S3, otherwise, step S4 is executed.

This when, the simulation node collection of double-layer network is obtained, operation terminates.

Specifically, oriented double-layer network as shown in Figure 3, the MDS={ Isosorbide-5-Nitrae } of double-layer network

If S4,WithIt is not perfect matching, then obtains MDS₁And MDS₂Intersection sameSet=MDS₁∩MDS₂； And by the node in sameSet from MDS₁And MDS₂It is removed in set, obtains a new MDS₁ ⁽¹⁾And MDS₂ ⁽¹⁾Collection.

I.e.：

SameSet=MDS₁∩MDS₂, MDS₁ ⁽¹⁾=MDS₁- sameSet, MDS₂ ⁽¹⁾=MDS₂-sameSet。

Note that during the execution of the algorithm, the simulation node collection of the first layer network is MDS always₁With sameSet's Union；Similarly, the simulation node collection of the second layer network is MDS always₂With the union of sameSet.

I.e.：

MDS₁ ⁽²⁾=MDS₁ ⁽¹⁾-{x₁, MDS₂ ⁽²⁾=MDS₂ ⁽¹⁾-{y₁, sameSet⁽¹⁾=sameSet ∪ { y₁, update (G₁,x₁,y₁)

I.e.：

MDS₁ ⁽³⁾=MDS₁ ⁽²⁾-{x₂, MDS₂ ⁽³⁾=MDS₂ ⁽²⁾-{y₂, sameSet⁽²⁾=sameSet⁽¹⁾∪ { z }, update(G₁,x₂,z)、update(G₂,y₂,z)；

Wherein, update (G_k, i, j) and it indicates to update network G_KMaximum matching, i.e. the friendship for the j from node i to node All sides on wrong path, by former network G_KMatching when becoming non-matching, former network G_KIt is non-matching while become matching while, k Take 1 or 2.Node i belongs to network G_KInitial driving node concentrate element, node j belong to node i staggeredly reachable set member Element belongs to G_kNode, the staggeredly reachable set of the node is network G with update_kMaximum matching it is determining.

The core concept of this step is that the node in the staggeredly reachable set for look for the driving node of the first layer network is worked as to replace Preceding driving node, so that MDS₁As much as possible with MDS₂It is overlapped.

At this point, having obtained the simulation node collection of double-layer network, operation terminates.

If MDS₁ ⁽³⁾And MDS₂ ⁽³⁾It is nonempty set, then continues to execute step S7, operation continues.

I.e.：

MDS₁ ⁽⁵⁾=MDS₁ ⁽⁴⁾-{a₁, MDS₂ ⁽⁵⁾=MDS₂ ⁽⁴⁾-{b₁, sameSet⁽⁴⁾=sameSet⁽³⁾∪{a₁, update(G₂,b₁,a₁)、

Meanwhile obtaining MDS₁ ⁽⁴⁾In meet condition a₂∈MDS₁ ⁽⁴⁾And a₂Staggeredly reachable set AC₁(a₂) and AC₂(b) friendship Collection setB is not empty node a₂；

I.e.：

MDS₁ ⁽⁶⁾=MDS₁ ⁽⁵⁾-{a₂, MDS₂ ⁽⁶⁾=MDS₂ ⁽⁵⁾-{b₂, sameSet⁽⁴⁾=sameSet⁽³⁾∪ { c }, update(G₁,a₂,c)、update(G₂,b₂,c)。

It is similar with the method for step S5, the core concept of this step be look for the second layer network driving node it is staggeredly reachable The node of concentration replaces current driving node, so that MDS₂As much as possible with MDS₁It is overlapped.

During specific implementation, after executing step S4, before the step S5, above-mentioned method shown in FIG. 1 is also Include the following steps：

S4a, each node q in sameSet is performed the following operations：

Correspondingly, the MDS in step S5₁ ⁽¹⁾、MDS₂ ⁽¹⁾It is the MDS updated in step S4a with sameSet₁And MDS₂、 sameSet。

The purpose of this step is the redundant node excluded in sameSet set, finds out the minimum driving of control multitiered network Node number.

Correspondingly, MDS is traversed in step S8₂ ⁽⁴⁾In after each node, if obtaining MDS₁ ⁽⁷⁾、MDS₂ ⁽⁷⁾It is non-empty Set；

It is then repeated in a manner of step S4a, step S5 and step S7, until MDS₁ ⁽⁷⁾And MDS₂ ⁽⁷⁾At least one of be Sky or sameSet⁽⁶⁾In element no longer change, then the simulation node of double-layer network G integrates MDS as MDS₁ ⁽⁷⁾、MDS₂ ⁽⁷⁾ And sameSet⁽⁶⁾Union.

Embodiment two

Optionally, Fig. 4 gives a kind of double-layer network control method of the present embodiment, and the method for the present embodiment includes：

If S4,WithIt is not perfect matching, then obtains MDS₁And MDS₂Intersection sameSet=MDS₁∩MDS₂； And by the node in sameSet from MDS₁And MDS₂It is removed in set, obtains a new MDS₁ ⁽¹⁾And MDS₂ ⁽¹⁾Collection；MDS₁ ⁽¹⁾= MDS₁- sameSet, MDS₂ ⁽¹⁾=MDS₂-sameSet；

If it exists, step S5a is executed, otherwise, executes step S5b；

If it exists, step S7a is executed, otherwise, executes step S7b；

Optionally, in above-mentioned the present embodiment the method, after executing S7, if it is decided that result be it is no, can execute with Lower step：

In traversal MDS₂ ⁽³⁾In after each node, obtain newest simulation node collection MDS₁ ⁽⁵⁾、MDS₂ ⁽⁵⁾And sameSet⁽⁴⁾Collection is in traversal MDS₂ ⁽³⁾In after each node, obtain newest simulation node collection MDS₁ ⁽⁵⁾、MDS₂ ⁽⁵⁾ And sameSet⁽⁴⁾Collection；

S8 ', it is repeated in a manner of step S5 and step S7, until MDS₁ ⁽⁵⁾And MDS₂ ⁽⁵⁾At least one of for sky, or sameSet⁽⁴⁾In element no longer change, then the simulation node of double-layer network G integrates MDS as MDS₁ ⁽⁵⁾、MDS₂ ⁽⁵⁾With sameSet⁽⁴⁾Union.

During specific implementation, after executing step S4, before the step S5, above-mentioned method shown in Fig. 4 is also Include the following steps：

S4a, each node q in sameSet is performed the following operations：

Correspondingly, MDS is traversed in step S8₂ ⁽³⁾In after each node, if obtaining MDS₁ ⁽⁵⁾、MDS₂ ⁽⁵⁾It is non-empty Set；

Embodiment three

Method in order to better understand the present invention, for the double-layer network shown in Fig. 6, the present embodiment provides a use The detailed calculating process of double-layer network control method operation, specifically, calculating process is as shown in figure 5, include：

101, it obtainsMDS₁、MDS₂。

Fig. 6 (a), Fig. 6 (c) are respectively the first layer and the second layer of double-layer network, and Fig. 6 (b), Fig. 6 (d) are each layer network pair The bipartite graph answered, overstriking side indicate the initial maximum matching of each layer network, and dotted line node indicates corresponding initial driving node.

MDS₁={ 3,4 }, MDS₂={ 1,2 }

102, judgeAny of whether belong to perfect matching.

If so, executing step 103, terminate operation, if not, executing step 104, continues operation.

103, MDS is exported, is terminated.

IfIt is all exact matching, then MDS={ i }, i ∈ { 1,2 ..., N }；

If onlyFor exact matching, then MDS=MDS₂；

If onlyFor exact matching, then MDS=MDS₁。

Because two-tier network does not all exactly match, step 104 is executed；

104, sameSet, MDS are obtained₁ ⁽¹⁾、MDS₂ ⁽¹⁾。

Obtain MDS₁And MDS₂Intersection sameSet=MDS₁∩MDS₂；And by the node in sameSet from MDS₁And MDS₂ It is removed in set, obtains a new MDS₁ ⁽¹⁾And MDS₂ ⁽¹⁾Collection.

I.e.：

SameSet=MDS₁∩MDS₂, MDS₁ ⁽¹⁾=MDS₁- sameSet, MDS₂ ⁽¹⁾=MDS₂-sameSet

In the present embodiment, MDS₁={ 3,4 }, MDS₂={ 1,2 },

Therefore, after executing this step,MDS₁ ⁽¹⁾={ 3,4 }, MDS₂ ⁽¹⁾={ 1,2 }

105, MDS is obtained₁ ⁽¹⁾In each node x staggeredly reachable set AC₁(x)；

As shown in fig. 6, MDS₁In include 2 nodes：X=3 is taken first, acquires AC₁(3)={ 5 }, next take x=4, ask Obtain AC₁(4)={ 1 }；

105a, MDS is obtained₂ ⁽¹⁾In meet condition y₁∈MDS₂ ⁽¹⁾And y₁∈AC₁(x) node y₁；

At least one node y if it exists₁,

(1) make MDS₁ ⁽¹⁾Remove corresponding x₁Element obtains MDS₁ ⁽²⁾；

(2) make MDS₂ ⁽¹⁾Remove y₁Element obtains MDS₂ ⁽²⁾；

(3) increase sameSet by y₁Element obtains sameSet⁽¹⁾；

(4) update (G is executed₁,x₁,y₁)。

At this point, MDS₁ ⁽¹⁾={ 3,4 }, MDS₂ ⁽¹⁾={ 1,2 }, it is seen then that MDS₂ ⁽¹⁾It is middle that there are a node y₁=1 meets y₁∈ AC₁(x), so,

(1) make MDS₁ ⁽¹⁾Remove corresponding x₁Element { 4 }, obtains MDS₁ ⁽²⁾={ 3 }；

(2) make MDS₂ ⁽¹⁾Remove y₁Element { 1 }, obtains MDS₂ ⁽²⁾={ 2 }；

(3) increase sameSet by y₁Element { 1 }, obtains sameSet⁽¹⁾={ 1 }

As shown in Fig. 7 (a), Fig. 7 (b), with node y₁(y₁=1) x is replaced₁(x₁=4), G₁The maximum matching of network is further It is updated to { 1,3 }, as shown in Fig. 7 (c), Fig. 7 (d), G₂The maximum matching of network is further updated to { 1,2 }.

(4) update (G is executed₁,x₁,y₁), update G₁Maximum matching in network.

As shown in Fig. 7 (b), for G₁In network from node 4 to the zigzag path of node 1 on all sides, by former G₁Network In matching when becoming non-matching, by former G₁It is non-matching when becoming matching in network.

105b, judge MDS₂ ⁽¹⁾In whether there is at least one node y₂, and y₂Staggeredly reachable set AC₂(y₂) and AC₁(x) Intersection setA be not empty；

At least one node y if it exists₂, then

(1) make MDS₁ ⁽²⁾Remove corresponding x₂Element obtains MDS₁ ⁽³⁾；

(2) make MDS₂ ⁽²⁾Remove y₂Element obtains MDS₂ ⁽³⁾；

(3) an element z in setA is taken, sameSet is made⁽¹⁾Increase z element, obtains sameSet⁽²⁾；

(4) and update (G is executed₁,x₂,z)、update(G₂,y₂,z)。

At this point, MDS₁ ⁽²⁾={ 3 }, MDS₂ ⁽²⁾={ 2 }, so, MDS₂ ⁽¹⁾There are a node y₂=2, interlock reachable set AC₂(y₂)=AC₂(2)={ 5 }, and setA=AC₂(y₂)∩AC₁(x)={ 5 }, setA are not empty；

(1) make MDS₁ ⁽²⁾Remove corresponding x₂Element { 3 }, obtains

(2) make MDS₂ ⁽²⁾Remove y₂Element { 2 }, obtains

(3) an element z in setA is taken, sameSet is made⁽¹⁾Increase z element { 5 }, obtains sameSet⁽²⁾={ 1,5 }

As shown in figure 8, replacing node x with node z (z=5)₂(x₂=3) and node y₂(y₂=2) at this point, each layer network Maximum matching is further updated to { 1,5 } and { 1,5 },

(4) update (G is executed₁,x₂, z),

As shown in Fig. 8 (b), for G₁In network from node 3 to the zigzag path of node 5 on all sides, by former G₁Network In matching when becoming non-matching, by former G₁It is non-matching when becoming matching in network.

Execute update (G₂,y₂, z),

As shown in Fig. 8 (d), for G₂In network from node 2 to the zigzag path of node 5 on all sides, by former G₁Network In matching when becoming non-matching, by former G₁It is non-matching when becoming matching in network.

106, in traversal MDS₂ ⁽¹⁾In after each node, obtain MDS₁ ⁽⁴⁾、MDS₂ ⁽⁴⁾And sameSet⁽³⁾Collection；

Judge MDS₁ ⁽⁴⁾And MDS₂ ⁽⁴⁾It whether is empty.

If MDS₁ ⁽⁴⁾And MDS₂ ⁽⁴⁾Middle there are a set for sky, then by nonempty set and sameSet⁽³⁾Union conduct The simulation node collection MDS of double-layer network G.

At this time

Therefore, the simulation node set MDS=sameSet of double-layer network shown in Fig. 6 is obtained⁽³⁾={ 1,5 }.

Example IV

For the double-layer network shown in Figure 10, the present embodiment provides one using the detailed of double-layer network control method operation Calculating process, specifically, calculating process is as shown in figure 9, include：

201, it obtainsMDS₁、MDS₂。

Figure 10 (a), Figure 10 (c) are respectively the first layer and the second layer of double-layer network, and Figure 10 (b), Figure 10 (d) are each layer net The corresponding bipartite graph of network, overstriking side indicate the initial maximum matching of each layer network, and dotted line node indicates corresponding initial driving section Point, i.e., when initial, MDS₁={ Isosorbide-5-Nitrae }, MDS₂={ 2,4,5 },.

202 judgementsAny of whether belong to perfect matching.

If so, executing step 103, terminate operation, if not, executing step 204, continues operation.

Because two-tier network does not all exactly match, step 204 is executed

204, sameSet, MDS are obtained₁ ⁽¹⁾、MDS₂ ⁽¹⁾。

At this point, sameSet={ 4 }, MDS₁ ⁽¹⁾={ 1 }, MDS₂ ⁽²⁾={ 2,5 }.

204a, each node q in sameSet is performed the following operations：

Double-layer network as shown in Figure 10, for the node q=4 in sameSet, its interlocking in the first and second layer network Reachable set is respectively AC₁(4)={ 2 }, AC₂(4)={ 1 }；

Because there are node m=2 to meet m ∈ AC₁(4)∩MDS₂, and there are node n=1 to meet n ∈ AC₂(4)∩MDS₁,

As shown in figure 11, by MDS₁ ⁽¹⁾It removes n element { 1 }, obtains a new MDS₁ ⁽¹⁾, update

Make MDS₂ ⁽¹⁾It removes m element { 2 }, obtains the MDS of a update₂ ⁽¹⁾={ 5 }

Make sameSet removal q element and increase m, n element, obtains the sameSet, the sameSet=of update of a update { 1,2 }.

The maximum matching of each layer network is updated to { 1,2 } and { 1,2,5 } respectively at this time.

As shown in figure 11, update (G is executed₁, q, m), update (G₂, q, n),

Because at this momentSo the final three simulation node collection for gathering the double-layer network asked and obtained are MDS={ 1,2,5 }.

In the present embodiment, the redundant node in sameSet set can be excluded, is executed directly down

Calculating process example as shown in Figure 9,1 is not by this operation as a result, 2 be the knot using this operation method Fruit.

If not passing through the operation of step 204a, because of MDS₁And MDS₂In the staggeredly reachable set of node be all empty, institute With no node it can replace them so that the driving node of two-tier network is further identical, final three set are asked and obtained The simulation node of double-layer network integrates as MDS={ 1,2,4,5 }, and the simulation node obtained by the present embodiment method Integrate as MDS={ 1,2,5 }, so, the redundant node in sameSet set can be excluded by this method.

Verify example

For the validity of further verification method, this verifying example generates N=1000, average in-degree<k_in>Equal to average Out-degree<k_out>Two groups of ER networks, enabling it is respectively the first layer G of double-layer network₁With second layer G₂。

The measurement of network controllabilityIt is with network average degree<k>(<k>=2<k_in>=2<k_out>) variation And the case where changing, is as shown in figure 13.

The controllability of round scatterplot expression double-layer network in Figure 13, the controllability of the first layer network of rectangular scatterplot expression, three Angular scatterplot indicates the controllability of the second layer network.

On the other hand, the present embodiment also provides a kind of electronic equipment, including memory, processor, bus and is stored in On memory and the computer program that can run on a processor, the processor realize any of the above one when executing described program The step of item.

In addition, the embodiment of the present invention also provides a kind of computer storage medium, it is stored thereon with computer program, the journey It realizes when sequence is executed by processor such as the step of above method any one.

Finally it should be noted that：Above-described embodiments are merely to illustrate the technical scheme, rather than to it Limitation；Although the present invention is described in detail referring to the foregoing embodiments, those skilled in the art should understand that： It can still modify to technical solution documented by previous embodiment, or to part of or all technical features into Row equivalent replacement；And these modifications or substitutions, it does not separate the essence of the corresponding technical solution various embodiments of the present invention technical side The range of case.

Claims

1. a kind of control method for double-layer network, which is characterized in that including：

S1, for double-layer network G (V, E) to be processed, obtain the maximum matching of first layer network G 1 in G (V, E)It is minimum Driving node collection MDS₁, obtain second layer network G in G (V, E)₂Maximum matchingSimulation node collection MDS₂；Wherein, V For node set, node i ∈ V, i=1,2 ..., N, the G₁With the G₂Corresponding identical physical node；

S2, judge first layer network G₁Maximum matchingSecond layer network G₂Maximum matchingAny of whether belong to In perfect matching；

IfIt is perfect matching, then the simulation node collection MDS={ i } of the double-layer network G, i ∈ 1, 2,…,N}。

2. the method according to claim 1, wherein the double-layer network is the node of a physical topological structure There are two the networks of logical topological structure for the tool of composition.

3. according to the method described in claim 2, it is characterized in that, the method also includes：

If S4,WithIt is not perfect matching, then obtains MDS₁And MDS₂Intersection sameSet=MDS₁∩MDS₂；And it will Node in sameSet is from MDS₁And MDS₂It is removed in set, obtains a new MDS₁ ⁽¹⁾And MDS₂ ⁽¹⁾Collection；

At least one node y if it exists₁, then make MDS₁ ⁽¹⁾Remove corresponding x₁Element obtains MDS₁ ⁽²⁾, make MDS₂ ⁽¹⁾Remove y₁Member Element obtains MDS₂ ⁽²⁾, increase sameSet by y₁Element obtains sameSet⁽¹⁾, and execute update (G₁,x₁,y₁)；

Meanwhile obtaining MDS₂ ⁽¹⁾In meet condition y₂∈MDS₂ ⁽¹⁾And y₂Staggeredly reachable set AC₂(y₂) and AC₁(x) intersection setA It is not empty node y₂；

At least one node y if it exists₂, then make MDS₂ ⁽²⁾Remove corresponding x₂Element obtains MDS₁ ⁽³⁾, make MDS₂ ⁽²⁾Remove y₂Member Element obtains MDS₂ ⁽³⁾, an element z in setA is taken, sameSet is made⁽¹⁾Increase z element, obtains sameSet⁽²⁾, and execute update(G₁,x₂,z)、update(G₂,y₂,z)；

Judge MDS₁ ⁽⁴⁾And MDS₂ ⁽⁴⁾It whether is sky, if MDS₁ ⁽⁴⁾And MDS₂ ⁽⁴⁾Middle there are a set for sky, then by nonvoid set Conjunction and sameSet⁽³⁾Simulation node collection MDS of the union as double-layer network G；

Wherein, update (G_k, i, j) and it indicates to update network G_KMaximum matching, i.e., for the staggeredly road of the j from node i to node All sides on diameter, by network G_KMatching when becoming non-matching, former network G_KIt is non-matching while become matching while, k take 1 or 2。

4. according to the method described in claim 3, it is characterized in that, the method also includes：

If S7, MDS₁ ⁽⁴⁾And MDS₂ ⁽⁴⁾It is nonempty set, then obtains MDS₂ ⁽⁴⁾In each node b staggeredly reachable set AC₂(b)；

At least one node a if it exists₁, then make MDS₁ ⁽⁴⁾Remove a₁Element obtains MDS₁ ⁽⁵⁾, make MDS₂ ⁽⁴⁾Remove corresponding b₁Member Element obtains MDS₂ ⁽⁵⁾, make sameSet⁽³⁾Increase a₁Element obtains sameSet⁽⁴⁾, and execute update (G₂,b₁,a₁)；

Meanwhile obtaining MDS₁ ⁽⁴⁾In meet condition a₂∈MDS₁ ⁽⁴⁾And a₂Staggeredly reachable set AC₁(a₂) and AC₂(b) intersection setB It is not empty node a₂；

At least one node a if it exists₂, then make MDS₁ ⁽⁵⁾Remove a₂Element obtains MDS₁ ⁽⁶⁾, make MDS₂ ⁽⁵⁾Remove corresponding b₂Member Element obtains MDS₂ ⁽⁶⁾, an element c in setB is taken, sameSet is made⁽⁴⁾Increase c element, obtains sameSet⁽⁵⁾, and execute update(G₁,a₂,c)、update(G₂,b₂,c)；

It is repeated in a manner of step S5 and step S7, until MDS₁ ⁽⁷⁾And MDS₂ ⁽⁷⁾At least one of for empty or sameSet⁽⁶⁾In element no longer change, then the simulation node of double-layer network G integrates MDS as MDS₁ ⁽⁷⁾、MDS₂ ⁽⁷⁾And sameSet⁽⁶⁾'s Union.

5. according to the method described in claim 3, it is characterized in that, before the step S5, the method is also after step S4 Including：

S4a, each node q in sameSet is performed the following operations：

At least one node m and there are at least one node n if it exists, then make MDS₁ ⁽¹⁾Remove n element, obtain one it is new MDS₁ ⁽¹⁾, make MDS₂ ⁽¹⁾M element is removed, the MDS of a update is obtained₂ ⁽¹⁾, make sameSet removal q element and increase m, n element, The sameSet of a update is obtained, and executes update (G₁, q, m), update (G₂,q,n)；

Correspondingly, if the MDS updated₁And MDS₂Middle there are a set for sky, then by the union of nonempty set and sameSet Simulation node collection MDS as double-layer network G.

6. according to the method described in claim 2, it is characterized in that, the method also includes：

If it exists, step S5a is executed, otherwise, executes step S5b；

S5a, if it exists at least one node y₁, then make MDS₁ ⁽¹⁾Corresponding x element is removed, MDS is obtained₁ ⁽²⁾, make MDS₂ ⁽¹⁾Removal y₁Element obtains MDS₂ ⁽²⁾, increase sameSet by y₁Element obtains sameSet⁽¹⁾, and execute update (G₁,x,y₁)；

S5b, judge MDS₂ ⁽¹⁾In whether there is at least one node y₂, meet y₂∈MDS₂ ⁽¹⁾And y₂Staggeredly reachable set AC₂(y₂) With AC₁(x) intersection setA is not empty；

At least one node y if it exists₂, then make MDS₂ ⁽¹⁾Corresponding x element is removed, MDS is obtained₁ ^(2)*, make MDS₂ ⁽¹⁾Remove y₂Member Element obtains MDS₂ ^(2)*, an element z in setA is taken, increases sameSet by z element, obtains sameSet^(1)*, and execute update(G₁,x,z)、update(G₂,y₂,z)；

At this time, judge MDS₁ ⁽³⁾And MDS₂ ⁽³⁾It whether is sky, if MDS₁ ⁽³⁾And MDS₂ ⁽³⁾Middle there are a set for sky, then will be non- Null set and sameSet⁽²⁾Simulation node collection MDS of the union as double-layer network G；

Wherein, update (G_k, i, j) and it indicates to update network G_KMaximum matching, i.e., for the staggeredly road of the j from node i to node All sides on diameter, by former network G_KMatching when becoming non-matching, former network G_KIt is non-matching while become matching while, k takes 1 Or 2.

7. according to the method described in claim 6, it is characterized in that, the method also includes：

If S7, MDS₁ ⁽³⁾And MDS₂ ⁽³⁾It is nonempty set, then obtains MDS₂ ⁽³⁾In each node b staggeredly reachable set AC₂(b)；

If it exists, step S7a is executed, otherwise, executes step S7b；

S7a, if it exists at least one node a₁, then make MDS₁ ⁽³⁾Remove a₁Element obtains MDS₁ ⁽⁴⁾, make MDS₂ ⁽³⁾It removes corresponding B element, obtains MDS₂ ⁽⁴⁾, make sameSet⁽²⁾Increase a₁Element obtains sameSet⁽³⁾, and execute update (G₂,b,a₁)；

It is repeated in a manner of step S5 and step S7, until MDS₁ ⁽⁵⁾And MDS₂ ⁽⁵⁾At least one of for empty or sameSet⁽⁴⁾In element no longer change, then the simulation node of double-layer network G integrates MDS as MDS₁ ⁽⁵⁾、MDS₂ ⁽⁵⁾And sameSet⁽⁴⁾'s Union.

8. the method according to the description of claim 7 is characterized in that the method also includes：

S7b, judge MDS₁ ⁽³⁾In whether there is at least one node a₂, meet a₂∈MDS₁ ⁽³⁾And a₂Staggeredly reachable set AC₁(a₂) With AC₂(b) intersection setB is not empty；

At least one node a if it exists₂, then make MDS₁ ⁽³⁾Remove a₂Element obtains MDS₁ ^(4)*, make MDS₂ ⁽³⁾Remove corresponding b member Element obtains MDS₂ ^(4)*, an element c in setB is taken, sameSet is made⁽³⁾Increase c element, obtains sameSet^(3)*, and execute update(G₁,a₂,c)、update(G₂,b,c)；

9. according to the method described in claim 6, it is characterized in that, before the step S5, the method is also after step S4 Including：

S4a, each node q in sameSet is performed the following operations：

10. a kind of electronic equipment, which is characterized in that on a memory and can be including memory, processor, bus and storage The computer program run on processor, the processor are realized when executing described program such as claim 1-9 any one Step.