CN103020267A - Complex network community structure mining method based on triangular cluster multi-label transmission - Google Patents

Abstract

The invention discloses a complex network community structure mining method based on triangular cluster multi-label transmission, which includes the steps of: searching the mutually disjointed triangular clusters in the network; setting different labels for the nodes of different triangular clusters to be used as an initial state for the transmission of the labels; synchronously updating the transmission of the labels of all the nodes in the network according to a multi-label renewal rule; comparing the updated label sets with the previously updated label sets; continuing to update the transmission of the labels if the labels are still being transmitted; if the labels oscillate, eliminating the oscillation and then continuing to update; if the labels no longer change again, stopping updating to obtain the overlapped community structures in the network. The complex network community structure mining method is lower in time complexity and is applicable for large-scale complex networks. In addition, the method can well detect the overlapped sections of the network community structures, has good robustness and is higher in detection accuracy.

Description

The complex network community structure method for digging of propagating based on label more than the triangle bunch

Technical field

What the present invention relates to is the method in a kind of complex network field, specifically a kind of complex network community structure method for digging of propagating based on label more than the triangle bunch.

Background technology

Complex network is a kind of abstraction form of complication system in the real world, node in the complex network represents the individuality in the complication system, the connection between the node then in the representative system between the individuality according to a kind of relation of certain regular self-assembling formation or arteface.At present, complex network has been widely used in characterizing the various complication systems such as electric power networks, communication network, internet, social networks.

Community structure is an important topological property of complex network, and whole complex network is comprised of several community structures, and the node of each community structure inside connects very tight, and the connection between the community structure is then relatively sparse.The community structure of complex network is corresponding to the functional unit that has common feature in the reality or organizations, for example community structure is exactly that the number of site of common topic is discussed in the internet, and the community structure in social networks is exactly a group that has people's composition of common interest hobby.Therefore, come the characteristic of phase-split network and function to be of great practical significance by the community structure of excavating in the complex network.

The community structure of complex network is excavated the character that needs to satisfy two keys:

The first, complexity is low.The scale of complex network is often very huge, can comprise several thousand even up to ten thousand nodes, and under the network of this scale, if the complexity of method is higher, carrying out so time overhead that community structure excavates will be very large;

The second, effective overlay structure testing mechanism.In the actual complex network, community structure ubiquity overlapping phenomenon, namely some nodes in the complex network can belong to a plurality of community structures simultaneously, and this just requires the community structure method for digging can detect the lap of community structure in the complex network.

Find by literature search, M.E.J.Newman and M.Girvan have proposed a kind of community mining method based on shortest path in article " Community structure in social and biological networks[J] " (" community structure in social and bio-networks ") (Proc.Natl.Acad.Sci.USA99,7821-7826 (2001)) (PNAS).The method obtains the shortest path table of whole network at first by the shortest path between any two points in the computational grid; Then utilize number of times that every limit of this table statistics passed through by shortest path as the weights on this limit, and remove the limit of weights maximum in the network, recomputate afterwards the weights on each bar limit in the whole network; Repeat above step, until whole network is divided into rational community structure.The accuracy of detection of this scheme is not high, and the method complexity is higher, and can't detect and have overlapping community structure.

Find through retrieval again, the people such as Filippo Radicchi and Claudio Castellano have proposed a kind of community mining scheme based on the Local network topology structure in article " Defining and identifying communities in networks[J] " (" community structure in definition and the recognition network ") (Proc.Natl.Acad.Sci.USA101,2658-2663 (2004)) (PNAS).Its method is: local topology characteristics of community structure in the Analysis of Complex network at first, and with the triangular structure in the network as the most basic structure of Web Community, the quantity of the triangular structure of every limit institute subordinate is as the weights on this limit in the statistics network; Then, remove the limit of weights maximum in the network, recomputate the weights on each bar limit in the whole network; Repeat above step, until whole network is divided into rational community structure.This scheme has been considered the local features of network topology structure, and accuracy of detection has a certain upgrade, but still can't solve these two problems of detection of complexity height and community structure lap.

Find through retrieval again, the people such as Gergely Palla and Imre Derenyi have proposed a community structure method for digging based on complete subgraph in article " Uncovering the overlapping community structure of complex networks in nature and society[J] " (" excavate in nature and the community network and have overlapping community structure ") (Nature435 (7043), 814-818 (2005)) (nature).For each node in the complex network, at first calculate all complete subgraphs that comprise this node, construct the overlapping matrix of these complete subgraphs; Then filter according to the connectivity pair overlapping matrix, thereby obtain having overlapping community structure.Have overlapping community structure although the method can be excavated, because complexity is too high, do not have actual using value.

Find finally by retrieval, the people such as U.N.Raghavan and R.Albert has proposed a kind of label transmission method that community structure is excavated that is applied in article " Near linear time algorithm to detect community structures in large-scale networks[J] " (being applied to a kind of method near linear time complexity that community structure is excavated in the large scale network) (Phys.Rev.E76,036106 (2007)) (physical comment).Distribute an independently label at first for during this scheme initialization each node in the complex network; Afterwards when each iteration all with the tag update of each node for occupying the label of ratio maximum in its neighbor node, stop iteration when the label of all nodes in the network all no longer changes, the node that have same label this moment just belongs to same community structure.Thus, just can mark off the community structure of complex network.The method complexity is very low, has to approach linear time complexity, goes for large-scale complex network.But the robustness of the method is very poor, and accuracy of detection is not high, and can't detect and have overlapping community structure.

Summary of the invention

The object of the invention is to for above-mentioned the deficiencies in the prior art, propose a complex network community structure method for digging of propagating based on label more than the triangle bunch.Its main thought is at first to seek the core of community structure as the initial state of label propagation, adopts afterwards many tag updates rules in the renewal of label, guarantees the detection effect of community structure lap.

The present invention is achieved through the following technical solutions:

A kind of complex network community mining method of propagating based on label more than the triangle bunch, its characteristics are: the method may further comprise the steps: mutually disjoint triangle bunch in the search network; The initial state of propagating as label for the mutually different label of Node configuration in the different triangles bunch; According to many tag update rules, for all nodes in the network, carry out synchronously the propagation of label and upgrade; The label array that obtains after upgrading and the label array that before renewal obtains are compared; If label is still being propagated, then proceed to upgrade; If label produces vibration, proceed again behind the Processing for removing that then vibrates to upgrade; If label no longer changes, then stop to upgrade, obtain the overlapping community structure that has in the network.

The concrete steps of the method are as follows:

1. give an empty label for each node of complex network G to be measured;

2. calculate every limit e by following formula _IjTriangle bunch coefficient T C (e _Ij):

TC(e _ij)＝|N(v _i)∩N(v _j)|

Wherein: N (v _i) be node v _iThe set of neighbor node, ∩ is the operational symbol that seeks common ground, || be the operational symbol of asking cardinal of the set, i is 1,2,3 ..., N;

3. search out the limit of triangle bunch coefficient maximum;

4. all nodes in the triangle on this limit bunch are given a radix that is different from other labels in the network and are 1 label; Afterwards these nodes are deleted from network, are obtained new network G ',

5. judge whether to exist triangle bunch coefficient greater than 0 limit, exist, then return step 2., otherwise will this moment in the network label of all nodes save as a label array L ⁽⁰⁾, 6. the original state as label is propagated enters step;

6. the propagation that utilizes following formula to carry out synchronously label is upgraded:

$L_i^{\left(t\right)}=\arg \underset{l}{\max }\underset{v_j\∈N\left(v_i\right)}{\mathrm{\Σ}}\mathrm{\δ}(L_j^{(t-1)},l)$

$\mathrm{\δ}(L_j^{(t-1)},l)=\left\{\begin{array}{cc}1,& l\∈L_j^{(t-1)}\\ 0,& l\∉L_j^{(t-1)}\end{array}\right.$

Wherein: Node v _iLabel after carrying out upgrading for the t time, t is the positive integer more than 1, N (v _i) be node v _iThe set of neighbor node,

Node v _iNeighbor node v _jLabel after the t-1 time renewal,

The operational symbol of asking the set of element l, the l that tries to achieve so that behind the max defined function obtain maximal value.When upgrading for the t time, the label array L that utilizes the t-1 time renewal to obtain ^(t-1), for the rule of all nodes among the complex network G according to many tag updates, carry out synchronously the propagation of label and upgrade, and be saved to new label array L ^(t)

7. compare the label array: upgrade the label array L that obtains with the t time ^(t)With upgrade before the label array L obtain for t-1 time ⁽¹⁾, L ⁽²⁾..., L ^(t-1)Compare,

When t time label array is all not identical, then return step 6.;

Work as L ⁽¹⁾, L ⁽²⁾..., L ^(t-2)Middle existence and L ^(t)Identical label array then enters and shows to upgrade and vibration occurred, then carries out step 8.;

As label array L ^(t-1)And L ^(t)Identical, then enter step 9.

8. Processing for removing vibrates: supposition label array L ⁽¹⁾, L ⁽²⁾..., L ^(t-2)In have L ^(k)And L ^(t)Identical, then at label array L ^(k+1)..., L ^(t)In search out all and upgrade the node that change occured label from being updated to the t time for the k+1 time, at L ^(t)It is 1 label that middle label with these nodes replaces to respectively the radix that is different from other labels in the network; Then return step 6.;

9. obtain the overlapping community structure of having of network, the node that has the same label element in the network belongs to same community structure, and the label radix is the lap of community structure greater than 1 node.

Technical solution of the present invention is explained as follows:

1. calculate the original state that label is propagated:

For complex network G=(V, E) to be measured, wherein V={v _i| i=1,2 ..., N} is the node set of complex network G, v _iRepresent i node, N is the quantity of Node Contraction in Complex Networks,

The limit set of complex network G, e _Ij=(v _i, v _j) expression network in connected node v _iAnd v _jThe limit, be each the node v in the network G _i(i=1,2 ..., N) give an empty label.At first, calculate every limit e _Ij(i, j=1,2 ..., bunch coefficient of triangle N); Then, search out the limit of triangle bunch coefficient maximum, it is 1 label that all nodes in the triangle on this limit bunch are reset a radix that is different from other labels in the network; Afterwards these nodes are deleted from network, obtain new network G '=(V', E ');

Repeat said process, until do not exist triangle bunch coefficient greater than 0 limit in the network, the label of all nodes in this moment network is saved as a label array

As the original state of label propagation, wherein

The label of expression node vi when label is propagated original state;

2. the propagation of label is upgraded:

When upgrading for the t time, the label array L that utilizes the t-1 time renewal to obtain ^(t-1), for the rule of all nodes among the complex network G=(V, E) according to many tag updates, carry out synchronously the propagation of label and upgrade, and be saved to new label array $L^{\left(t\right)}=[L_1^t,L_2^t,...,L_i^t,...,L_N^t],$ Wherein Expression node v _iLabel after the t time renewal;

3, the label array is relatively:

Upgrade the label array L that obtains with the t time ^(t)With upgrade before the label array L obtain for t-1 time ⁽¹⁾, L ⁽²⁾..., L ^(t-1)Compare,

When t time label array is all not identical, then continue to carry out according to step 2 the propagation renewal of label;

Work as L ⁽¹⁾, L ⁽²⁾..., L ^(t-2)Middle existence and L ^(t)Identical label array then shows to upgrade vibration to have occurred, and the Processing for removing that must vibrate continues to carry out according to step 2 the propagation renewal of label afterwards again;

As label array L ^(t-1)And L ^(t)Identical, then show network G=(V, E) label of all nodes all no longer changes in, the propagation that stops label this moment is upgraded, the node that has the same label element in the network namely belongs to same community structure, and the label radix is the lap of community structure greater than 1 node.

The original state that described calculating label is propagated.For complex network G=(V, E) to be measured, wherein V={v _i| i=1,2 ..., N is the node set of complex network G, and vi represents i node, and N is the quantity of Node Contraction in Complex Networks, E={e _Ij=(v _i, v _j) | i, j=1,2 ..., N} is the limit set of complex network G, e _Ij=(v _i, v _j) expression network in connected node v _iAnd v _jThe limit, be each the node v in the network G _i(i=1,2 ... N, give an empty label.At first, calculate every limit e _Ij(i, j=1,2 ..., bunch coefficient of triangle N); Then, search out the limit of triangle bunch coefficient maximum, it is 1 label that all nodes in the triangle on this limit bunch are reset a radix that is different from other labels in the network; Afterwards these nodes are deleted from network, obtain new network G '=(V', E').Repeat said process, until do not exist triangle bunch coefficient greater than 0 limit in the network.The label of all nodes in this moment network is saved as a label array

As the original state of label propagation, wherein

Expression node v _iLabel when label is propagated original state.Be specially:

Described limit e _IjTriangle bunch, refer to comprise in the network limit e _IjThe topological structure that forms of all triangles.

Described limit e _IjTriangle bunch coefficient, refer to comprise in the network limit e _IjAll leg-of-mutton quantity, its computing formula is as follows:

TC(e _ij)＝|N(v _i)∩N(v _j)|

Wherein, TC (e _Ij) be limit e _IjTriangle bunch coefficient, N (v _i) be node v _iThe set of neighbor node, ∩ is the operational symbol that seeks common ground, || be the operational symbol of asking cardinal of the set.

Described node v _iNeighbor node, refer in the network and node v _iBetween the node of fillet is arranged.

Described cardinality of a set, the number of element in referring to gather.

Described node v _iLabel, be one the set L _i={ l _Ik| k=1,2 ..., K}, its element l _IkBe used for characterizing node v _iThe numbering of affiliated community structure, K is cardinality of a set.If the radix of this label is 1, then show node v _iOnly belong to a community structure; If the radix of this label is greater than 1, node v then _iBelong to simultaneously a plurality of community structures.

Described empty label refers to gather the label for empty set, and namely radix is 0 label.The label that node has is empty label, illustrates that then this node does not belong to any community structure.

Described label array is the array of a 1 * N, is used for preserving the label of all nodes of network.Wherein, N is the quantity of nodes,

The label that all nodes have when being illustrated in label propagation original state,

Node v when propagating original state for label _iThe label that has;

Represent the label that rear all nodes of the t time renewal have,

Be the t time renewal posterior nodal point v _iThe label that has.

The propagation of described label is upgraded.When upgrading for the t time, the label array L that utilizes the t-1 time renewal to obtain ^(t-1), for the rule of all nodes among the complex network G=(V, E) according to many tag updates, carry out synchronously the propagation of label and upgrade, and be saved to new label array

Wherein

Expression node v _iLabel after the t time renewal.Be specially:

Described many tag updates rule is:

$L_i^{\left(t\right)}=\arg \underset{l}{\max }\underset{v_j\∈N\left(v_i\right)}{\mathrm{\Σ}}\mathrm{\δ}(L_j^{(t-1)},l)$

$\mathrm{\δ}(L_j^{(t-1)},l)=\left\{\begin{array}{cc}1,& l\∈L_j^{(t-1)}\\ 0,& l\∉L_j^{(t-1)}\end{array}\right.$

Wherein, Node v _iLabel after carrying out upgrading for the t time, t is the positive integer more than 1, N (v _i) be node v _iThe set of neighbor node,

Node v _iNeighbor node v _jLabel after the t-1 time renewal,

The operational symbol of asking the set of element l, the l that tries to achieve so that behind the max defined function obtain maximal value.

The described propagation of carrying out synchronously label is upgraded, and refers to renewal arbitrary node v the t time _iLabel

Can only be by upgrading the label array L that obtains t-1 time before ⁽¹⁾, L ⁽²⁾..., L ^(t-1)Calculate, and other any node v that obtained with the t time renewal _j(the label of j ≠ i)

Irrelevant.

Described label array relatively.Upgrade the label array L that obtains with the t time ^(t)With upgrade before the label array L obtain for t-1 time ⁽¹⁾, L ⁽²⁾..., L ^(t-1)Compare.If t time label array is all not identical, then continue to carry out according to step 2 the propagation renewal of label; If L ⁽¹⁾, L ⁽²⁾..., L ^(t-2)Middle existence and L ^(t)Identical label array then shows to upgrade vibration to have occurred, and the Processing for removing that must vibrate continues to carry out according to step 2 the propagation renewal of label afterwards again; If label array L ^(t-1)And L ^(t)Identical, then show network G=(V, E) label of all nodes all no longer changes in, the propagation that stops label this moment is upgraded, the node that has the same label element in the network namely belongs to same community structure, and the label radix is the lap of community structure greater than 1 node.Be specially:

Described vibration is eliminated, and the steps include: to suppose label array L ⁽¹⁾, L ⁽²⁾..., L ^(t-2)In have L ^(k)And L ^(t)Identical, then at label array L ^(k+1)..., L ^(t)In search out all and upgrade the node that change occured label from being updated to the t time for the k+1 time, at L ^(t)It is 1 label that middle label with these nodes replaces to respectively the radix that is different from other labels in the network.

The present invention has improved performance and effect that label is propagated from three aspects.At first, the core texture of complex network community structure is triangle bunch.The present invention marks off disjoint triangle bunch from network, and the initial state of distributing different labels to propagate as label, take full advantage of the topological property of complex network, Primary Construction the basic configuration of community structure, be in a ratio of that all nodes distribute mutually different label as the method for initial state very large optimization to be arranged in the network.Secondly, the present invention has introduced many tag update rules, so that propagating, the label of each node upgrades the label information that takes full advantage of neighbor node, so that the local message of node can not lose, compare with the update rule of selecting at random optimum label, many tag updates rule has improved robustness and accuracy that label is propagated greatly.At last, adopt the vibration Processing for removing to the label of vibration node again assignment, solved preferably the oscillation problem under the synchronous update mode, thoroughly eliminated the randomness defective of introducing owing to the update sequence of node in the asynchronous refresh mode, and can by parallel computation, improve counting yield.

Description of drawings

Fig. 1 is method flow diagram of the present invention.

Fig. 2 is the original state synoptic diagram that the label of this method under karate club network propagated.

Fig. 3 be this method under karate club network, obtain have an overlapping community structure synoptic diagram.

Embodiment

The below elaborates to embodiments of the invention, and present embodiment is implemented under take technical solution of the present invention as prerequisite, provided detailed embodiment and concrete operating process, but protection scope of the present invention is not limited to following embodiment.

Present embodiment adopts the classical data set Zachary karate club network in the community network, and this network comprises 34 nodes and 78 limits.Comprise that specifically step is as follows:

1. give an empty label for each node of karate club network to be measured;

2. calculate the triangle bunch coefficient T C (e of every limit ej by following formula _Ij):

TC(e _ij)＝|N(v _i)∩N(v _j)|

Wherein: N (v _i) be node v _iThe set of neighbor node, ∩ is the operational symbol that seeks common ground, || be the operational symbol of asking cardinal of the set, i is 1,2,3 ..., N;

3. search out the limit of triangle bunch coefficient maximum;

4. all nodes in the triangle on this limit bunch are given a radix that is different from other labels in the network and are 1 label; Afterwards these nodes are deleted from network, are obtained new network G ',

5. judge whether to exist triangle bunch coefficient greater than 0 limit, exist, then return step 2., otherwise will this moment in the network label of all nodes save as a label array L ⁽⁰⁾, 6. the original state as label is propagated enters step;

5. the label that is obtained by step is propagated the label array L of original state ⁽⁰⁾Comprise altogether 3 kinds of different non-NULL label l ₁, l ₂And l ₃, corresponding to 3 mutually disjoint triangles that exist in the karate club network bunch, as shown in Figure 2, wherein each label represents with a circle, and the node in the different circles has different non-NULL labels, and blank node represents to have the node of sky label.

6. the propagation that utilizes following formula to carry out synchronously label is upgraded:

$L_i^{\left(t\right)}=\arg \underset{l}{\max }\underset{v_j\∈N\left(v_i\right)}{\mathrm{\Σ}}\mathrm{\δ}(L_j^{(t-1)},l)$

$\mathrm{\δ}(L_j^{(t-1)},l)=\left\{\begin{array}{cc}1,& l\∈L_j^{(t-1)}\\ 0,& l\∉L_j^{(t-1)}\end{array}\right.$

Wherein:

Node v _iLabel after carrying out upgrading for the t time, t is the positive integer more than 1, N (v _i) be node v _iThe set of neighbor node,

Node v _iNeighbor node v _jLabel after the t-1 time renewal,

The operational symbol of asking the set of element l, the l that tries to achieve so that behind the max defined function obtain maximal value, when the t time is upgraded, the label array L that utilizes the t-1 time renewal to obtain ^(t-1), for the rule of all nodes among the complex network G according to many tag updates, carry out synchronously the propagation of label and upgrade, and be saved to new label array L ^(t)

7. compare the label array: upgrade the label array L that obtains with the t time ^(t)With upgrade before the label array L obtain for t-1 time ⁽¹⁾, L ⁽²⁾..., L ^(t-1)Compare,

When t time label array is all not identical, then return step 6.;

Work as L ⁽¹⁾, L ⁽²⁾..., L ^(t-2)Middle existence and L ^(t)Identical label array then enters and shows to upgrade and vibration occurred, then carries out step 8.;

As label array L ^(t-1)And L ^(t)Identical, then enter step 9.

8. Processing for removing vibrates: supposition label array L ⁽¹⁾, L ⁽²⁾..., L ^(t-2)In have L ^(k)And L ^(t)Identical, then at label array L ^(k+1)..., L ^(t)In search out all and upgrade the node that change occured label from being updated to the t time for the k+1 time, at L ^(t)It is 1 label that middle label with these nodes replaces to respectively the radix that is different from other labels in the network; Then return step 6.;

9. obtain the overlapping community structure of having of network, the node that has the same label element in the network belongs to same community structure, and the label radix is the lap of community structure greater than 1 node.

Utilize this method, the label array L that after the 2nd renewal, obtains ⁽²⁾With the label array L that obtains after the 1st renewal ⁽¹⁾Identical, upgrade so stop the propagation of label, obtain the overlapping community structure that has of karate club network, as shown in Figure 3.Can find out, co-exist in 3 community structure C in the karate club network ₁, C ₂And C ₃, represent with a circle that respectively the node in the circle is the member node of this community structure; Node 5, node 10 and node 11 are the lap of community structure, represent with the node of black; Community structure C ₁And C ₂Own node 5 and node 11 together, the C of community ₂And C ₃Own node 10 together.

More than the typical network data collection is used in experiment, and method flow of the present invention is had been described in detail.The background knowledge of this experimental result and data set is basically identical, is divided the node ratio that enters correct community structure and has reached 95.59%.In addition, modularity is a very important index parameter that is used for weighing the community structure quality, the modularity of the community structure that obtains under karate club network of this method is 0.3912 as calculated, be in close proximity to theoretic maximal value 0.4020, this has also verified accuracy and the validity of this method.

Claims (2)

1. complex network community mining method of propagating based on label more than the triangle bunch, it is characterized in that: the method may further comprise the steps: mutually disjoint triangle bunch in the search network; The initial state of propagating as label for the mutually different label of Node configuration in the different triangles bunch; According to many tag update rules, for all nodes in the network, carry out synchronously the propagation of label and upgrade; The label array that obtains after upgrading and the label array that before renewal obtains are compared; If label is still being propagated, then proceed to upgrade; If label produces vibration, proceed again behind the Processing for removing that then vibrates to upgrade; If label no longer changes, then stop to upgrade, obtain the overlapping community structure that has in the network.

2. the complex network community mining method of propagating based on label more than the triangle bunch according to claim 1, it is characterized in that: the concrete steps of the method are as follows:

1. give an empty label for each node of complex network G to be measured;

2. calculate every limit e by following formula _IjTriangle bunch coefficient T C (e _Ij):

TC(e _ij)＝|N(v _i)∩N(v _j)|

Wherein: N (v _i) be node v _iThe set of neighbor node, ∩ is the operational symbol that seeks common ground, || be the operational symbol of asking cardinal of the set, i is 1,2,3 ..., N;

3. search out the limit of triangle bunch coefficient maximum;

4. all nodes in the triangle on this limit bunch are given a radix that is different from other labels in the network and are 1 label; Afterwards these nodes are deleted from network, are obtained new network G ',

5. judge whether to exist triangle bunch coefficient greater than 0 limit, exist, then return step 2., otherwise will this moment in the network label of all nodes save as a label array L ⁽⁰⁾, 6. the original state as label is propagated enters step;

6. the propagation that utilizes following formula to carry out synchronously label is upgraded:

$L_i^{\left(t\right)}=\arg \underset{l}{\max }\underset{v_j\∈N\left(v_i\right)}{\mathrm{\Σ}}\mathrm{\δ}(L_j^{(t-1)},l)$

$\mathrm{\δ}(L_j^{(t-1)},l)=\left\{\begin{array}{cc}1,& l\∈L_j^{(t-1)}\\ 0,& l\∉L_j^{(t-1)}\end{array}\right.$

Wherein:

Node v _iLabel after carrying out upgrading for the t time, t is the positive integer more than 1, N (v _i) be node v _iThe set of neighbor node, Node v _iNeighbor node v _jLabel after the t-1 time renewal, || be the operational symbol of asking cardinal of the set,

The operational symbol of asking the set of element l, the l that tries to achieve so that behind the max defined function obtain maximal value, when the t time is upgraded, the label array L that utilizes the t-1 time renewal to obtain ^(t-1), for the rule of all nodes among the complex network G according to many tag updates, carry out synchronously the propagation of label and upgrade, and be saved to new label array L ^(t)

7. compare the label array: upgrade the label array L that obtains with the t time ^(t)With upgrade before the label array L obtain for t-1 time ⁽¹⁾, L ⁽²⁾..., L ^(t-1)Compare,

When t time label array is all not identical, then return step 6.;

Work as L ⁽¹⁾, L ⁽²⁾..., L ^(t-2)Middle existence and L ^(t)Identical label array then enters and shows to upgrade and vibration occurred, then carries out step 8.;

As label array L ^(t-1)And L ^(t)Identical, then enter step 9.

8. Processing for removing vibrates: supposition label array L ⁽¹⁾, L ⁽²⁾..., L ^(t-2)In have L ^(k)And L ^(t)Identical, then at label array L ^(k+1)..., L ^(t)In search out all and upgrade the node that change occured label from being updated to the t time for the k+1 time, at L ^(t)It is 1 label that middle label with these nodes replaces to respectively the radix that is different from other labels in the network; Then return step 6.;

9. obtain the overlapping community structure of having of network, the node that has the same label element in the network belongs to same community structure, and the label radix is the lap of community structure greater than 1 node.

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