CN102857525A - Community Discovery Method Based on Random Walk Strategy - Google Patents

Abstract

The invention discloses a community discovery method based on a random walk strategy, which mainly comprises three parts of contents, namely network initialization, random walk and community tendency analysis. The invention is characterized in that the following problems of the existing community discovery method are solved: 1) the community structure of the network under a certain single level can be obtained, and the community division condition of the network under multiple levels cannot be completely given; 2) dividing networks with overlapping community structures is not free and the quality of the obtained communities is not very high; 3) nodes with multiple identities in overlapping communities are not analyzed quantitatively. In conclusion, the invention can not only discover the overlapping communities in the network, but also discover the community structure of the network at different levels. And in the method, the introduced concept of community tendency makes the quantitative analysis of the overlapped communities possible.

Description

Community discovery method based on the random walk strategy

Technical field

The present invention relates to the complex network field, relate in particular to discovery technique and its implementation of community structure in a kind of complex network.

Background technology

The community discovery technology is a basic research in the complex network.About the research of community structure, on the whole, from the module level, community structure usually is the height non-trivial, and this is so that we always are difficult to find reasonable method to come that network is carried out community divides.Mainly contain the reason of following two aspects.

The first, because the community in the network is nested: some less communities form a slightly large community, and these slightly large communities continue to form a larger community, finally make whole network present a large community, such as the institutional framework of school.Because community's module of every one deck can both embody the specific function of system in the network, so the level form in the institutional framework is very effective, this also requires us to seek a kind of method and finds community structure under all levels, and not only is confined to the community structure under a certain single level.Although traditional hierarchy clustering method can produce the level of division with comparalive ease, the quality of subregion but is not very clear.Although the modularization degree that Newman and Girvan proposes is exactly to weigh a kind of instrument of dividing quality, it can only judge that the community under certain independent level divides quality.Until in recent years, just there was at leisure the scholar to begin to pay close attention to the problem that significant community divides under the many levels.

Second Problem is exactly that some nodes usually are under the jurisdiction of a plurality of communities, thereby forms overlapping community.For example, people also have hobby etc. according to family, friend, occupation, can be a member in a plurality of different social groups.During network when dividing this node and belong to a plurality of community with conventional method, it is unable to do what one wishes to tend to seem, also can reduce the quality of the community that obtains simultaneously.And this has also hidden some important information, often causes dividing incorrect.

In addition, existing community discovery method usually is to analyze the network with a specific character.When having simultaneously this two specific character in the network, we often just are difficult to provide satisfied result.

Based on above-mentioned these problems, we have proposed the community discovery method based on the random walk strategy, the overlapping community of the method in not only can discovering network, and the multilayered structure in can discovering network.And we have also introduced the tendentious concept of community, and we can be analyzed quantitatively to overlapping community.

Summary of the invention

In view of this, main purpose of the present invention is to provide a kind of new community discovery technology---based on the community discovery method of random walk strategy, and its overlapping community in not only can discovering network, and the multilayered structure in can discovering network.And can analyze quantitatively overlapping community.

Here, we have introduced the agent set, and each the element individuality in the set is that each agent represents a node in the corresponding network, and can move in a two-dimensional space.In discovery procedure, each agent can be in its all of its neighbor point, but selects randomly a corresponding agent of abutment points as the destination with a changeable probability, moves to the coordinate points at agent place, this destination from oneself coordinate points.If node belongs to same community, then the tightness value between them should be higher, between the agent that namely these nodes are corresponding should be closely to link to each other.Thereby these agent that move will little by little form some groups by randomly movement.And these agent that closely link to each other are gathered in the same coordinate points of two-dimensional space the most at last.

The present invention is achieved by the following technical solutions, mainly comprises following three some works:

A, netinit;

B, random walk;

Community's tendentiousness of C, analysis node.

Wherein the netinit of A part comprises following three steps:

A1, give the corresponding the mobile agent of each node definition in the network;

A2, according to the attribute of each node of tightness matrix M initialization;

A3, each node forms an independently community when initial.

A given network G, it has n summit and m bar limit.The adjacency matrix of G is

Element a in the matrix _IjRelationship strength between representation node i and the j.

Wherein in the steps A 1, we distinguish agent and its corresponding node with subscript.For example, i ₀The agent that representation node i is corresponding.

Wherein in the steps A 2, the tightness matrix can utilize adjacency matrix

Obtain.

The wherein random walk of B part is for each agent i ₀, it comprises following steps:

B1, calculating i ₀The attractability value CA ' of community;

B2, calculating i ₀The at random movement probability of random walk;

B3, agent i ₀Begin to move toward other places according to the coordinate points from own place of movement probability at random.

Wherein among the step B2, agent i ₀To c (j ₀) the at random movement probability that moves of the coordinate points at place can obtain by normalization community attractability.

Wherein community's tendentiousness of the analysis node of C part comprises following two steps:

Under C1, the current community structure of calculating, community's tendentiousness value of each node;

C2, calculate the Shannon quotient of each node according to community's tendentiousness value of each node.

Community discovery method based on the random walk strategy provided by the invention, it at first carries out the A part, and this network is carried out initialization; Then carry out the B part.Execute after the B part, if the community structure in the current network no longer changes, carry out so the C part, otherwise continue to carry out the B part.It is worthy of note, the value of multi-scale parameters t should be set first before carrying out the A part.By changing the t value, can obtain the community structure of network under all levels.

In technical scheme provided by the present invention, to divide by the community that adopts a kind of strategy based on random walk to carry out network, it has following characteristics:

At first, the at random movement probability of agent is relevant with the community's attractability between agent, and this probability is variable in the process of random walk, and is namely adaptive.

Next is that the community discovery method based on the random walk strategy that the present invention proposes is the probabilistic method of a community discovery aspect.It, still has an opportunity to leave this incorrect community and move to correct community in the later stage based on the agent of random walk when node is divided into incorrect community in early days because of carelessness.

Moreover be that we can analyze the node characteristic that has multiple identities in the community structure quantitatively by the frequency of statistics node in different communities.

Description of drawings

Fig. 1 is the application of the present invention in an artificial network of eight nodes;

Fig. 2 (a) is community's tendentiousness distribution comparison diagram that all nodes belong to respectively the i of community, ii, iii and iv in the artificial network of H 13-4 of verifying among the present invention under four community structures;

Fig. 2 (b) is the Shannon quotient figure of the artificial network of H 13-4 each node when four community structures of verifying among the present invention;

Fig. 2 (c) is the Shannon quotient figure of the artificial network of H 13-4 each node when 16 community structure of verifying among the present invention;

Fig. 3 (a) is the division figure as a result of true community network Zachary ' the s Karate club network verified among the present invention;

Fig. 3 (b) is the Shannon quotient figure of true community network Zachary ' s Karate club's network each node under 2 community structures of verifying among the present invention;

Fig. 4 is the cumulative distribution function graph of a relation of the gained community size of true social network scientist's collaborative network of verifying among the present invention.

Embodiment

Core concept of the present invention is to utilize the part of network to explore to seek the natural community that each node belongs to.

For making the purpose, technical solutions and advantages of the present invention clearer, the below tells about respectively the implementation details of netinit, random walk and node community this three some work of tendentiousness of summary of the invention part.

(1) netinit

A given network G, it has n summit and m bar limit.The adjacency matrix of G is n * n matrix A _{N * n}, the element a in the matrix _IjRelationship strength between representation node i and the j.Especially, in without weight graph, if i is connected with j (namely having the limit to connect) then a _IjValue is 1, otherwise value is 0.If weighted graph then refers to a _IjValue be weights between i and the j.Element value on the M diagonal is undefined, for convenient, is made as 0 here.

During netinit, in steps A 1, distinguish agent and its corresponding node with subscript here.For example, i ₀The agent that representation node i is corresponding.

In steps A 2, the tightness matrix M can obtain by following mode:

When non-directed graph G is when having no right to net, node i and node j in the Given Graph, if they are abutment points (two points that have the limit to link to each other), then the tightness value between them increases a _Ij, namely 1.A common abutment points k (being called public abutment points) is arranged between i and j, and the tightness between i and the j can increase by 1 again; And between i and j, a plurality of public abutment points being arranged, the tightness value between i and the j then increases corresponding time.Final mathematic(al) representation is as follows: $m_{\mathrm{ij}}=m_{\mathrm{ij}}^0+1+1*|\mathrm{\τ}\left(i\right)\∩\mathrm{\τ}\left(j\right)|=1+|\mathrm{\τ}\left(i\right)\∩\mathrm{\τ}\left(j\right)|.$ Wherein, m _IjThe value of element after as calculated among the M,

The element value of matrix M when initial.τ (i) is the set of the abutment points of summit i.| x| is the element number among the set x.

When non-directed graph G is weighted network, similarly, node i and j in the Given Graph, they are abutment points, the tightness value between them still increases a at this moment _IjBut when i and j have public abutment points k, the tightness value between i and the j will increase

Rather than a _IjWhen public abutment points between i and the j has when a plurality of, the value that the tightness value between i and the j increases then is

Final mathematic(al) representation is as follows:

$m_{\mathrm{ij}}=m_{\mathrm{ij}}^0+a_{\mathrm{ij}}+\underset{k\∈\mathrm{\τ}\left(i\right)\∩\mathrm{\τ}\left(j\right)}{\mathrm{\Σ}}(a_{\mathrm{ij}}*\sqrt{a_{\mathrm{ik}}*a_{\mathrm{jk}}})$

$=a_{\mathrm{ij}}(1+\underset{k\∈\mathrm{\τ}\left(i\right)\∩\mathrm{\τ}\left(j\right)}{\mathrm{\Σ}}\sqrt{a_{\mathrm{ik}}*a_{\mathrm{jk}}})$

In steps A 3, each node in the network oneself forms an independently community, always total n community when namely initial.The residing coordinate of agent corresponding to each node this moment also is different.Be that n node has when initial n community, the agent of their correspondences altogether also to be in n different coordinate points.

(2) random walk

In step B1, suppose agent j ₀To agent i ₀Attractability be a _Ij(a _IjThe element in the adjacency matrix of current network), c (j then ₀) to agent i ₀Attractability be defined as ${\mathrm{CA}}^{\′}(i_0,c\left(j_0\right))={\left(\frac{\underset{s\∈\mathrm{\τ}\left(i\right)\∩C\left(j\right)}{\mathrm{\Σ}}{m}_{\mathrm{is}}\underset{p,q\∈\mathrm{\τ}\left(i\right)\∩C\left(j\right)}{\mathrm{\Σ}}{m}_{\mathrm{pq}}}{\underset{p,q\∈C\left(j\right)}{\mathrm{\Σ}}{m}_{\mathrm{pq}}}\right)}^t,$ I wherein ₀, j ₀Difference representation node i and agent corresponding to node j, c (j ₀) be and agent j ₀The set of identical all agent in coordinate points position, C (j) is c (j ₀) in the set of all its corresponding nodes of agent, t, t＞0th, multi-scale parameters (being also referred to as resolution parameter), m _IjThe element in the tightness matrix M, record be tightness value between node i and node j.

In step B2, agent i ₀To c (j ₀) the at random movement probability that moves of the coordinate points at place can obtain by normalization community attractability, mathematical description is

I wherein ₀, j ₀Difference representation node i and agent corresponding to node j, c (j ₀) be and agent j ₀The set of identical all agent in coordinate points position, CA ' (i ₀, c (j ₀)) be c (j ₀) to agent i ₀Community's attractability, Γ (i) is the set that the corresponding agent of abutment points of node i consists of.

In step B3, agent i ₀Actual will the movement toward which coordinate points depended on the at random movement probability value that it to which coordinate points moves.Here agent i ₀Select past at random that coordinate points of movement probability value maximum to move.

(3) node community tendentiousness

In step C1, given node i and the final collection Z of community that finds, node i belongs to the Com of community _x, Com _xCommunity's tendentiousness of ∈ Z is that node i is at the Com of community _xIn frequency, formula is described as $\mathrm{CT}(i,{\mathrm{Com}}_x)=\frac{s(i,{\mathrm{Com}}_x)}{{\mathrm{\Σ}}_{{\mathrm{Com}}_k\∈Z}s(i,{\mathrm{Com}}_k)}.$

In step C2, consider a common node i, its Shannon merchant (a kind of about probabilistic index, as to be designated as H (i)) is defined as $H\left(i\right)=-\underset{{\mathrm{Com}}_k\∈Z}{\mathrm{\Σ}}\mathrm{CT}(i,{\mathrm{Com}}_k){\log }_b\mathrm{CT}(i,{\mathrm{Com}}_k),$ Wherein b is the radix of logarithm, CT (i, Com _x) be that node i belongs to the Com of community _x, Com _xCommunity's tendentiousness of ∈ Z.

It should be noted that technical scheme provided by the invention before each the realization, should at first arrange the value of multi-scale parameters t.By constantly changing the t value, finally can obtain the community structure of network under all levels.

Accompanying drawing 1 is technical scheme provided by the invention in the artificial application of having no right in the network of eight node.Situation when the subgraph of accompanying drawing 1 (a) has been described netinit.Each node oneself forms a community, and 8 different communities are arranged when namely initial.Subgraph (b) has been described agentA ₀Select at random agentB ₀The coordinate points at place is the destination, then leaves and locates to move to B ₀The process of coordinate points.And subgraph (c) is the community structure (two communities) that finally marks off.

Accompanying drawing 2 (a) is technical scheme provided by the invention in that H 13-4 is artificial when having no right to use in the network, and all nodes belong to respectively community's tendentiousness distribution comparison diagram of the i of community, ii, iii and iv under four community structures.The artificial network of H 13-4 is uniform (each degree of node is identical) at degree, and two predefined hierarchical structures are arranged, the introducing on limit relies in two nodes whether belong to one group between the node: nodes is 256, company's limit number between each node and inner community is 13, and the company's limit number between outside community is 4, and and network in company's limit number between other any one nodes be 1.When accompanying drawing 2 (a) has provided multi-scale parameters t=0.4, the comparison diagram that community's tendentiousness of all nodes distributes.2 (a) can find out with reference to the accompanying drawings, and all nodes have been divided into four communities.Node 1 belongs to same community to node 64, and node 65 belongs to same community to node 128, and node 129 belongs to same community to node 192, and final node 193 to 256 belongs to same community.And community's tendentiousness value of node is: CT (i, i)＞0.624, i ∈ [1,64], CT (i, ii)＞0.623, i ∈ [65,128], CT (i, iii)＞0.644, i ∈ [129,192] and CT (i, iv)＞0.612, i ∈ [193,256].When multi-scale parameters t greater than 0.4 the time, have 16 communities and form, the connection of these 16 little community inside very closely (than before in 4 communities that divide the contact of community inside more tight).By the CT value under each yardstick of standardization, " ratio of correct partitioning site " all is 100%.

In order further to analyze the characteristic of the H 13-4 network described in the accompanying drawing 2 (a), accompanying drawing 2 (b) and accompanying drawing 2 (c) have provided respectively the values of ambiguity of H 13-4 network each node when four communities and 16 community, i.e. Shannon quotient.Contrast this two Shannon quotient figure, and " company's limit number of each node and inner community is 13, and with company's limit number of outside community be 4 " this structure is so that the Shannon quotient in inner community can be lower.

The division that accompanying drawing 3 (a) obtains when to be technical scheme provided by the invention use in true community network Zachary ' s Karate club network is figure as a result.Node in Zachary ' the s Karate club network represents the clubbite, the social interaction between the line-up of delegates of limit, and it includes 34 members and 78 limits altogether, and the limit represents the relation between the clubbite.When multi-scale parameters t=0.6, Zachary ' sKarate club network is divided into 2 communities, represents with square and circle respectively in the drawings.But when t becomes large, and even network can little by little be divided into 4 communities of 3 communities.4 community structures are in the drawings with different gray scale signs.Resulting 3 community structures of technical scheme provided by the invention have two kinds of situations.The first situation is that node 1,2,3,4,8,9,12,13,14,18,20,22 belongs to same community, node 5,6,7,11,17 belongs to same community, and node 10,15,16,19,21,23,24,25,26,27,28,29,30,31,32,33,34 belongs to same community; The second situation is that node 1,2,3,4,5,6,7,8,11,12,13,14,17,18,20,22,31 belongs to same community, 25,26,29,32 belong to same community, and 9,10,15,16,19,21,23,24,27,28,30,33,34 belong to same community.

Accompanying drawing 3 (b) is that technical scheme provided by the invention is when using in Zachary ' the sKarate club network described in the accompanying drawing 3 (a), the Shannon quotient figure of each node under resulting 2 community structures, the multi-scale parameters t=0.6 of this moment.

Accompanying drawing 4 is that technical scheme provided by the invention is applied to live network---the cumulative distribution function graph of a relation about community's size that obtains during scientist's collaborative network.Scientist's collaborative network obtains in the BibTeX bibliography at first; Summit in the network represents the author; If cooperated at least one piece of paper or works between two authors, then there is the limit to connect between them, the value on limit represents the number of cooperation paper together or works; 7343 summits and 11898 limits are arranged in the network diagram after the simplification.Four subgraphs in the accompanying drawing 4 are that multi-scale parameters t is respectively 1,2,3 and 4 o'clock cumulative distribution table.In each subgraph, cumulative distribution all is similar to the 3 rank power rates of obeying and distributes.

In a word, the above is preferred embodiment of the present invention only, is not for limiting protection scope of the present invention.Within the spirit and principles in the present invention all, any modification of doing, be equal to replacement, improvement etc., all should be included within protection scope of the present invention.

Claims (10)

1. the community structure discovery technique in the complex network is characterized in that this technology not only can be divided overlapping community, and can obtain the community structure of network under at all levels.

2. community structure discovery technique according to claim 1 is characterized in that, this technology is based on the part of network and explores to seek the natural community that each node belongs to.

3. according to claim 1 to the described community structure discovery technique of 2 any one, it is characterized in that, realize that this Technology Need introduces one group of agent set, each element individuality in the set represents a node in the corresponding network, these agent can move in a two-dimensional space, in mobile process, will little by little form some groups, and be gathered in the most at last the same coordinate points of two-dimensional space simultaneously.

4. according to claim 1 to the described community structure discovery technique of 2 any one, it is characterized in that, realize this Technology Need introducing community attractability, the tendentious concept of community, and comprise community's tendentiousness three partial contents of netinit, random walk and analysis node.

5. the random walk described in according to claim 4 is characterized in that, it comprises and calculates community's attractability, calculates at random that movement probability and agent move three steps.

6. according to claim 4 to the described community of 5 any one attractability, it is characterized in that, suppose agent j ₀To agent i ₀Attractability be a _Ij(a _IjThe element in the current network adjacency matrix), c (j then ₀) to agent i ₀Community's attractability be defined as ${\mathrm{CA}}^{\′}(i_0,c\left(j_0\right))={\left(\frac{\underset{s\∈\mathrm{\τ}\left(i\right)\∩C\left(j\right)}{\mathrm{\Σ}}{m}_{\mathrm{is}}\underset{p,q\∈\mathrm{\τ}\left(i\right)\∩C\left(j\right)}{\mathrm{\Σ}}{m}_{\mathrm{pq}}}{\underset{p,q\∈C\left(j\right)}{\mathrm{\Σ}}{m}_{\mathrm{pq}}}\right)}^t,$ I wherein ₀, j ₀Difference representation node i and agent corresponding to node j, c (j ₀) refer to the j with agent ₀The set of identical all agent in coordinate points position, C (j) refers to c (j ₀) in the set of node in the corresponding network of all agent representatives, t, t＞0th, multi-scale parameters (being also referred to as resolution parameter), m _IjThe element in the tightness matrix M, record be tightness value between node i and node j.

7. according to claim 4 to the described community of 6 any one attractability, it is characterized in that, when utilizing this community's attractability to come that network divided community, can access the community structure of network under many levels.

8. according to claim 4 to the defined community of 6 any one attractability, it is characterized in that normalization c (j ₀) to agent i ₀Community's attractability can obtain agent i ₀Move to c (j ₀) the probability of coordinate position, i.e. at random movement probability described in the claim 5, and normalized mathematical formulae is

I wherein ₀, j ₀Difference representation node i and agent corresponding to node j, c (j ₀) refer to the j with agent ₀The set of identical all agent in coordinate points position, CA ' (i ₀, c (j ₀)) be c (j ₀) to agent i ₀Community's attractability, Γ (i) is the set that the corresponding agent of abutment points of node i consists of.

9. community according to claim 4 tendentiousness is characterized in that, utilizes this community's tendentiousness can analyze quantitatively the node characteristic that has multiple identities in the overlapping community.

10. community according to claim 4 tendentiousness is characterized in that, given node i and the final collection Z of community that finds, and node i belongs to the Com of community _x, Com _xCommunity's tendentiousness of ∈ Z is that node i is at the Com of community _xIn frequency, formula is described as

S (i, Com wherein _x) be that node i is divided into the Com of community _xIn number of times.

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