CN103425868A - Complex network community detection method based on fractal feature - Google Patents

Abstract

A complex network community detection method based on a fractal feature includes two phases of offline static complex network community detection processing and online dynamic complex network community detection processing. Multi-scale features of a complex network are fully utilized, a renormalization process is used as a bridge for connecting different scales, the purpose of using previous community structure information to detect a new community structure is achieved, incremental community detection is achieved, new attempts for researching dynamic complex network community structures are conducted, and satisfying results are obtained.

Description

Complex network community discovery method based on fractal characteristic

Technical field

The invention belongs to Web Community's discover method technical field, relate to a kind of complex network community discovery method based on fractal characteristic.

Background technology

Many complexity that exist in real world and huge system can be described with network, we are referred to as complex network.Complex network is the abstract of complication system, and the individuality in complication system is nodes, and the limit between node is according to certain relation of certain regular self-assembling formation or arteface between individuality.Comprising various types of complex networks in real world, the network formed as interlinked between the page in community network, technical network, biological networks, network, paper coauthorship network, reference citation network etc.In these real worlds, a large amount of complex networks is to be combined by many dissimilar nodes, the connection wherein existed between identical type node is many, and the connection of dissimilar node is relatively less, this specific character of complex network is called community structure.

Research finds, the complex network in reality has three large features: 1. worldlet (Small-world), although refer to that the scale of complex network may be very large, wherein the shortest path between any two nodes is smaller.Small-world network has little Path length (characteristic path length) and high convergence factor (clustering coefficient) simultaneously.2. scaleless property (Scale-free), refer to that the degree distribution of node in complex network is obeyed or approximate obedience power law (power law) distribution.3. self-similarity (Self-similarity), refer to that complex network and the part of self have approximate similarity, namely has fractal (Fractal) feature.In order to explore the design feature of complex network, and then the function of understanding complex network, people conduct extensive research the community structure of complex network, have proposed numerous community discovery methods, mainly are divided into four kinds of methods: condensing method, splitting method, optimization method and analogy method.In numerous methods, these four kinds of methods are not independently, and a kind of method may embody multiple thought simultaneously.

Fractal characteristic is the feature that in reality, complex network has, and according to research, in having many networks of community structure, the most stable is exactly the network with fractal characteristic.The fractal characteristic that confirms complex network can be adjusted the structure of network, so being applied to community discovery, the fractal characteristic of complex network has broad prospects, and be a kind of thinking of novelty.

Although analysis and theoretical research that many scholars have carried out the fractal characteristic of complex network at present.But its achievement and method are analyzed and confirm the fractal characteristic of complex network, there is no to propose the complex network community discovery method based on fractal characteristic.The problem of its existence is, complex network essence is exactly the nonlinear system of a complexity, since fractal characteristic is an important character in real network, itself and complex network community structure have anything to contact so; Can be to the research of the community discovery aspect of complex network helpful etc.Trace it to its cause, not paying attention to fractal characteristic can not only disclose in the nonlinear system in order and unordered unification, and can fail the characteristics such as utilization of unifying better of determinacy and randomness, cause fractal feature only to be confined to theory and analysis for applying the complex network community discovery method, fail to propose practical solution complex network community discovery method.

Summary of the invention

The object of the present invention is to provide a kind of complex network community discovery method based on fractal characteristic, solve the promptness problem existed in existing complex network community discovery method, meet the promptness requirement of the dynamic change of complex network topologies.

Technical scheme of the present invention is, the complex network community discovery method based on fractal characteristic comprises two stages of processing of the community discovery of the processing of community discovery of static complex network of off-line and online Research Dynamic Complex Networks.

Characteristics of the present invention also are:

First stage: the processing of the community discovery of the static complex network of off-line, its process is as follows:

The topological structure G=(V, E) of step 1, input complex network; Complex network uses non-directed graph G=(V, E) to mean, V and E are respectively the set on node and limit;

Step 2, initialization;

The distance of step 3, calculating neighborhood of nodes, renormalization is apart from minimum node;

Step 4, renewal network;

Step 5, Output rusults.

Above-mentioned steps 2 is specially: calculate the degree of each node in V, node n _iDegree be designated as deg _i, mean to be connected in network the number on the limit of node; Weight w is set in every in E limit _IjIf, n _iAnd n _jIt is 1 that a degree is arranged, and making its weights is 0, and other situation weights are 1.Ream weight positizing number of times k=0, and make d=0, d _k=0, C (d) _k=0;

Above-mentioned steps 3 is specially: if in complex network, all original point is all by the renormalization mistake, skips to step 4 and upgrade network, otherwise continue this step; Calculate the distance of all neighborhood of nodes according to formula (1):

$\left|\right|n_i-n_j\left|\right|=\frac{1}{|{\mathrm{deg}}_i-{\mathrm{deg}}_j|+1}\·\min \{{\mathrm{deg}}_i,{\mathrm{deg}}_j\}\·w_{\mathrm{ij}},---\left(1\right)$

Wherein, deg _i, deg _jMean respectively node n _i, n _jDegree, min{x, y} means to get a less value.W _IjExpression is to limit (n _i, n _j) weights.Get the minimum value d of all limits middle distance, the end points on the limit that is d by all distances carries out renormalization.What here renormalization adopted is the principle of processing at first at most, first processes and has the point that bee-line limit number is maximum, first the adjacent the shortest related point in limit of this point is carried out to renormalization, is about to it and merges into a point.Repeat this process until distance is all processed for the limit of d.Calculate distance:

d _k＝d _k-1+d （2）

With correlation function C (d) _k:

${C\left(d\right)}_k=\frac{1}{N^2}{\mathrm{\Σ}}_{i,j=1}^{N}H(d-||n_i-n_j|\left|\right)---\left(3\right)$

Wherein, the number that N is node in network, n _i, n _jFor the node in network.H (y) is step function (step function),

$H(y)\left\{\begin{array}{cc}1& y\&GreaterEqual;0\\ 0& y<0\end{array}\right.---\left(4\right)$

Make k=k+1.

Above-mentioned steps 4 is specially: after the renormalization of step 2, and the weight w on every limit _IjBy following strategy, calculate: at first making all weights is all 1; If n now _iAnd n _jBefore process previous step renormalization, there is s limit to be connected, w _Ij=w _Ij/ s; If n _iAnd n _jThe degree that has one is 1, and the point of its representative has m, and it puts renormalization, w by m _Ij=w _Ij* m.

Above-mentioned steps 5 is: the draw fractal dimension of estimating complex network the value that provides its correlation dimension;

The correlation dimension of network is defined as follows:

$D_C=\underset{d\&RightArrow;0}{\lim }\frac{\ln C\left(d\right)}{\ln d}---\left(5\right)$

1) K the different { D of community _i(i=1 ... K), wherein

For $\&ForAll;i\&NotEqual;j,D_i\&cap;D_j=\mathrm{\&phi;};$

2) a series of (C (d), the complex network fractal dimension Ds of d) putting right value and estimating thus _c.

Subordinate phase: the processing of the community discovery of online Research Dynamic Complex Networks:

The community discovery of Research Dynamic Complex Networks refers on the static network basis, calculates the network of the new growth (evolution) changed along with the time; Implementation procedure is described below:

Step 1, input complex network mean that with non-directed graph G=(V, E) V and E are respectively the set on node and limit;

Step 2, renewal network topology structure;

Step 3, removal do not have influential limit to change to the complex network community structure;

Step 4, adjustment yardstick;

Step 5, result output.

Above-mentioned input complex network comprises:

1) t complex network G (t)=(V (t), E (t)) constantly;

2) community structure { D of t moment complex network _i(t) } (i=1 ... K), wherein

For $\&ForAll;i\&NotEqual;j,D_i\left(t\right)\&cap;D_j\left(t\right)=\mathrm{\&phi;};$

Minor increment d when 3) each community is as a node between the node of complex network _m

4) be carved into the t+1 variation on limit constantly during t: Δ E ⁺With Δ E ^-, Δ E ⁺With Δ E ^-Mean respectively the new limit produced and the limit of extinction.

Above-mentioned renewal network topology structure, be carved into complex network during from t the t+1 variation on limit constantly and be updated to t complex network G (t) constantly, thereby obtain the topological structure of G (t+1), for subsequent calculations apart from the time use.

Above-mentioned removal does not have influential limit to change to the complex network community structure, from Δ E ⁺Two end points of middle removal are at the same D of community _i(t) limit in, from Δ E ^-Two end points of middle removal are at two D of community _iAnd D (t) _j(t) limit in, wherein i ≠ j.

Above-mentioned adjustment yardstick, for Δ E ⁺With Δ E ^-In the remaining related { D of community of element also _{I (p)}(t) } (p=1 ... q), node and the limit of each inside, community are condensed by top static network method, meet following one of two things and just stop: in coacervation process, the distance of each node is not less than d for the first time _mAll nodes in community are condensed into a node.

Adjust yardstick again, the result of upper step is carried out to the community discovery process based on renormalization, suitable community structure is all crossed or obtain to the original point newly added until all by renormalization.Also a kind of possible situation is exactly, and the part that community splits into is combined with other community.Here just need first to consider in practice division, then the part after division is considered to situation about merging together with remaining community.Processing for this situation is exactly first to divide and then merge.

The above results output, comprise t+1 constantly complex network community structure and to the estimated value of complex network correlation dimension:

(1) community structure { D of t+1 moment complex network _i(t+1) } (i=1 ... L), wherein $\underset{i=1}{\overset{L}{\&cup;}}{D}_i(t+1)=V(t+1),$ For $\&ForAll;i\&NotEqual;j,D_i(t+1)\&cap;D_j(t+1)=\mathrm{\&phi;}$ The point that wherein V (t+1) deducts extinction by V (t) is added the point of new generation and forms;

(2) a series of (C (d), the complex network fractal dimension Ds of d) putting right value and estimating thus _c.

The present invention has following beneficial effect:

1, to take full advantage of the fractal characteristic of complex network in reality be the community discovery service in the present invention.Using renormalization and inverse process thereof as the instrument that changes yardstick, the community structure of dynamic network is changed and is studied, proposed the increment type dynamic network community discovery method based on dimensional variation, and confirmed its feasibility on real network.Characteristics of the present invention are exactly the Analysis On Multi-scale Features that takes full advantage of complex network, with renormalization process as contact the bridge between different scale, thereby reaching community structure information before is that new community structure is found purpose used, realized the community discovery of increment type, for the community structure of research trends complex network has been done new trial, and obtained satisfied result.

2, the present invention adopts fractal principle, the community discovery method of dynamic increment formula is proposed, be intended to solve complex network community discovery method, the topologies change rule that discloses complex network and function restructuring by the fractal characteristic technology, better solve the community discovery methods such as WWW, transportation network, Scientific Cooperation network, electric power networks, human relation network, cell neural network and infectious disease network.Especially provide new approaches for the network information security.

The accompanying drawing explanation

Fig. 1 the present invention is based in the complex network community discovery method of fractal characteristic for estimating the log-log plot of complex network correlation dimension;

Fig. 2 is the schematic diagram that the present invention is based on network renormalization in the complex network community discovery method of fractal characteristic;

Fig. 3 the present invention is based on the different conditions figure of complex network in coacervation process in the complex network community discovery method of fractal characteristic;

Fig. 4 the present invention is based on the schematic diagram that in the complex network community discovery method of fractal characteristic, complex network develops.

Embodiment

Below in conjunction with the drawings and specific embodiments, the present invention is described in detail.

Complex network community discovery method based on fractal characteristic comprises two stages of processing of the community discovery of the processing of community discovery of static complex network of off-line and online Research Dynamic Complex Networks:

First stage: the processing of the community discovery of the static complex network of off-line, as reference Fig. 3, step is as follows:

The topological structure G=(V, E) of step 1, input complex network; Complex network uses non-directed graph G=(V, E) to mean, V and E are respectively the set on node and limit;

Step 2, initialization.Calculate the degree of each node in V, node n _iDegree be designated as deg _i, mean to be connected in network the number on the limit of node; Weight w is set in every in E limit _IjIf, n _iAnd n _jIt is 1 that a degree is arranged, and making its weights is 0, and other situation weights are 1.Ream weight positizing number of times k=0, and make d=0, d _k=0, C (d) _k=0;

The distance of step 3, calculating neighborhood of nodes, renormalization is apart from minimum node.Renormalization (renormalization) is a physical concept, and its definition is in order to overcome the divergence difficulty in quantum field theory, makes the theoretical a kind of smooth theoretical treatment method of calculating.Its essence is to change slightly looking (coarsening) degree in observation, analyzes quantitatively whereby the variation of physical quantity to disclose certain rule.In complex network, constantly the point that is less than certain distance is regarded as to a point with regard to being equivalent to, until whole complex network all becomes a point.If original point (refer to the point of not crossed by renormalization, there is no the point of merged mistake) all in complex network, all by the renormalization mistake, skip to step 4, otherwise continue this step.Calculate the distance of all neighborhood of nodes according to formula (1):

$\left|\right|n_i-n_j\left|\right|=\frac{1}{|{\mathrm{deg}}_i-{\mathrm{deg}}_j|+1}\&CenterDot;\min \{{\mathrm{deg}}_i,{\mathrm{deg}}_j\}\&CenterDot;w_{\mathrm{ij}},---\left(1\right)$

Wherein, deg _i, deg _jMean respectively node n _i, n _jDegree, min{x, y} means to get a less value.W _IjExpression is to limit (n _i, n _j) weights.Get the minimum value d of all limits middle distance, the end points on the limit that is d by all distances carries out renormalization.What here renormalization adopted is the principle of processing at first at most, first processes and has the point that bee-line limit number is maximum, first the adjacent the shortest related point in limit of this point is carried out to renormalization, is about to it and merges into a point.Repeat this process until distance is all processed for the limit of d.Calculate distance:

d _k＝d _k-1+d （2）

With correlation function C (d) _k:

${C\left(d\right)}_k=\frac{1}{N^2}{\mathrm{\&Sigma;}}_{i,j=1}^{N}H(d-||n_i-n_j|\left|\right)---\left(3\right)$

Wherein, the number that N is node in network, n _i, n _jFor the node in network.H (y) is step function (step function),

$H(y)\left\{\begin{array}{cc}1& y\&GreaterEqual;0\\ 0& y<0\end{array}\right.---\left(4\right)$

Make k=k+1;

Step 4, renewal network.After the renormalization of step 2, variation has all occurred in the node number of network and the number on limit, so the weights on the degree of each node and each limit all need to recalculate.The size of degree can be calculated according to the number on limit, the weight w on every limit _IjNeed to calculate by following strategy: at first making all weights is all 1; If n now _iAnd n _jBefore process previous step renormalization, there is s limit to be connected, w _Ij=w _Ij/ s; If n _iAnd n _jThe degree that has one is 1, and the point of its representative has m, and it puts renormalization, w by m _Ij=w _Ij* m;

Step 5, Output rusults.Community's result of the complex network that output marks off, the draw fractal dimension of estimating complex network the value that provides its correlation dimension.The correlation dimension of network is defined as follows:

$D_C=\underset{d\&RightArrow;0}{\lim }\frac{\ln C\left(d\right)}{\ln d}---\left(5\right)$

Correlation dimension is convenient directly to be measured from experiment, is widely used, and it was suggested in nineteen eighty-three.In community discovery method of the present invention, using the parameter of using correlation dimension as the complex network fractal characteristic, i.e. fractal dimension.From formula (5), can see, with meter box counting dimension similar correlation dimension also can be with (C (d), d) the fitting a straight line slope under the double-log system is estimated.Specifically the result of whole method has following two:

1) K the different { D of community _i(i=1 ... K), wherein

For $\&ForAll;i\&NotEqual;j,D_i\&cap;D_j=\mathrm{\&phi;};$

It is 2) a series of that (C (d), d) put right value, and the complex network fractal dimension D estimated according to the method shown in accompanying drawing 1 thus _c.

The method is the condensing method of a community discovery, if from different level height, its result is showed, can obtain different community structures, and this also understands that the complex network community structure has level on the other hand, as shown in Figure 4.

Subordinate phase: the processing of the community discovery of online Research Dynamic Complex Networks; The community discovery of Research Dynamic Complex Networks refers on the static network basis, calculates the network of the new growth (evolution) changed along with the time; Implementation procedure is described below:

Step 1, input complex network mean that with non-directed graph G=(V, E) V and E are respectively the set on node and limit; Have 4 inputs:

1) t complex network G (t)=(V (t), E (t)) constantly;

2) community structure { D of t moment complex network _i(t) } (i=1 ... K), wherein

For $\&ForAll;i\&NotEqual;j,D_i\left(t\right)\&cap;D_j\left(t\right)=\mathrm{\&phi;};$

Minor increment d when 3) each community is as a node between the node of complex network _m

4) be carved into the t+1 variation on limit constantly during t: Δ E ⁺With Δ E ^-(meaning respectively the new limit produced and the limit of extinction).

Step 2, renewal network topology structure.Complex network is carved into to the t+1 variation on limit constantly during from t and is updated to t complex network G (t) constantly, thereby obtain the topological structure of G (t+1), for subsequent calculations apart from the time use.

Step 3, removal do not have influential limit to change to the complex network community structure.From Δ E ⁺Two end points of middle removal are at the same D of community _i(t) limit in, from Δ E ^-Two end points of middle removal are at two D of community _iAnd D (t) _j(t) limit in, wherein i ≠ j.The community of complex network is exactly the network subgraph large and little with outside Connection Density with inner Connection Density.There are following two kinds so hold the intelligible variation that can not impact Web Community: the limit between the limit in the increase community between node and minimizing community between node.These two kinds of modes can make community more condense, so can not produce the variation of community.

Step 4, adjustment yardstick, go deep into inside, community and condensed.For Δ E ⁺With Δ E ^-In the remaining related { D of community of element also _{I (p)}(t) } (p=1 ... q), node and the limit of each inside, community are condensed by top static network method, meet following one of two things and just stop: in coacervation process, the distance of each node is not less than d for the first time _mAll nodes in community are condensed into a node.Here the computing method of distance still adopt the calculating means of formula (1).The related community of this step only need to consider the contact between its inner node, does not need to consider the contact between community, so be independently for the calculating of each community, can use parallel method to improve processing speed.The result of this step will be condensed into the former community divided by not to complex network, the network that the piece that the node newly added and division community form is mixed to form, and the minor increment of this network is d _m, wherein these three kinds of elements differ and have established a capital, and certainly pass through this step, and three kinds of elements above-mentioned all have been expressed as node.The complex network that may make certain community that division occurs develops.A kind of situation is to wither away in the limit of inside, community, can make the density of inside, community reduce, thereby may cause this community to produce division; The second situation is that to only have an end points be the node of this inside, community the new limit that produces, can increase contacting of the inner node in community and outside, thereby also likely cause this community to produce, divides.

Step 5, again adjust yardstick, the result of upper step is carried out to the community discovery process based on renormalization, suitable community structure is all crossed or obtain to the original point newly added until all by renormalization.The complex network that may make two or more communities produce merging develops.The community here also can mean newly-increased original point, because from different yardsticks, community is also the node under certain yardstick.A plurality of communities merge and can further be decomposed into two communities and first merge and merge with other community, thereby only need to consider that the situation that two communities merge is just passable.The situation of the complex network evolution that may make two communities merge is exactly to make the contact of two community's points closeer, has namely increased two intercommunal limits newly.Also a kind of possible situation is exactly, and the part that community splits into is combined with other community.Here just need first to consider in practice division, then the part after division is considered to situation about merging together with remaining community.Processing for this situation is exactly first to divide and then merge.

Step 6, result output.It is the t+1 community structure of complex network constantly that Output rusults mainly contains two one, and another is the estimated value to the complex network correlation dimension:

(1) community structure { D of t+1 moment complex network _i(t+1) } (i=1 ... L), wherein $\underset{i=1}{\overset{L}{\&cup;}}{D}_i(t+1)=V(t+1),$ For $\&ForAll;i\&NotEqual;j,D_i(t+1)\&cap;D_j(t+1)=\mathrm{\&phi;}$ The point that wherein V (t+1) deducts extinction by V (t) is added the point of new generation and forms.

(2) a series of (C (d), the complex network fractal dimension Ds of d) putting right value and estimating thus _c.

The present invention can obtain complex network t+1 community structure constantly, can obtain t+1 minor increment constantly during end, and t+1 complex network topologies figure constantly, if be carved into the variation delta E on the limit of t+2 network constantly when t+1 has been arranged ⁺With Δ E ^-, just can continue to obtain t+2 complex network community structure constantly, continuing is exactly the implication place of increment type, the state in a moment after the state by last the time obtains with variable quantity.

The invention provides a kind of complex network community discovery method based on fractal characteristic, utilize fractal characteristic and the feature such as multiple dimensioned of actual complex network to serve community discovery, be a new thinking, examples of implementation have also confirmed validity and the feasibility of the method.

In embodiment, select Zachary karate club network, it is a network between all members of karate club in school of the U.S..This network is comprised of 34 nodes and 78 limits, and node represents each member in club, and limit means the social bond of member outside club.Wayne Zachary spent two years observed and studied this club at 20 century 70s, and afterwards due to inside, club has produced division, has become two clubs.Because Wayne Zachary has carried out detailed investigation and research to this club, so this network just becomes a highest network of research community discovery problem frequency of utilization.Use this network to be verified complex network community discovery method of the present invention.Its result is as shown in table 1:

Arithmetic accuracy on table 1Zachary karate club network

No matter from accuracy rate, recall rate or, from the statistics of F-value, the present invention has desirable result, because only have a node to be divided in wrong community for whole network, so degree of accuracy of the present invention or more satisfactory.

Second embodiment is the increment type dynamic network community discovery method based on dimensional variation for dynamic network for checking, the example of choosing is U.S. NCAA(National Collegiate Athletic Association) the rugby network is the rugby conventional competition code of 2000 racing seasons of American college physical culture league matches of collection.In this network, node means the team of university of each competition, with the name of university, means, has 115 universities; Limit between node means the conventional competition between school team, has 613 matches between these school teams.In reality, these teams are divided into 11 different playing areas, each playing area is comprised of 8 to 12 teams, match between the team of same playing area is higher than the match frequency between the team of different playing area, and the every team of the match in playing area is on average 7, and the every team of the match between playing area is on average 4.But it should be noted that it is inhomogeneous that match between playing area distributes, for the team that belongs to different playing area, the match between near distance is more than the match between distance.These known community structures make this network be usually used in the detection to community discovery method.The present invention divides this network for two parts, and first is 76 nodes and 426 limits therebetween wherein, these nodes and limit main composition 8 playing areas in 11 playing areas, use static method to obtain the community structure of these points.Second portion is remaining node and limit in network, and the variation by it with limit means, i.e. Δ E ⁺With Δ E ^-, then in conjunction with the division result of 76 nodes of first, and minor increment (being 8.9344 in experiment).So just formed the input of delta algorithm in the 3rd chapter, result is as shown in table 2:

Arithmetic accuracy on table 2NACC rugby network

As can be seen from the table, although team's number in the playing area had in discover method of the present invention is not too accurate, but in most playing area, team still can well identify, the effect of the present invention on this real data collection is still more gratifying on the whole, average rate of accuracy reached is to 89%, and recall rate has also reached 95%.

Claims (9)

1. the complex network community discovery method based on fractal characteristic is characterized in that: two stages of processing that comprise the community discovery of the processing of community discovery of static complex network of off-line and online Research Dynamic Complex Networks.

2. the complex network community discovery method based on fractal characteristic as claimed in claim 1 is characterized in that: the first stage, and the processing of the community discovery of the static complex network of off-line, its process is as follows:

The topological structure G=(V, E) of step 1, input complex network; Complex network uses non-directed graph G=(V, E) to mean, V and E are respectively the set on node and limit;

Step 2, initialization;

The distance of step 3, calculating neighborhood of nodes, renormalization is apart from minimum node;

Step 4, renewal network;

Step 5, Output rusults.

3. the complex network community discovery method based on fractal characteristic as claimed in claim 2 is characterized in that:

Step 2 is specially, and calculates the degree of each node in V, node n _iDegree be designated as deg _i, mean to be connected in network the number on the limit of node; Weight w is set in every in E limit _IjIf, n _iAnd n _jIt is 1 that a degree is arranged, and making its weights is 0, and other situation weights are 1; Ream weight positizing number of times k=0, and make d=0, d _k=0, C (d) _k=0;

Step 3 is specially: if in complex network, all original point is all by the renormalization mistake, skips to step 4 and upgrade network, otherwise continue this step; Calculate the distance of all neighborhood of nodes according to formula (1):

$\left|\right|n_i-n_j\left|\right|=\frac{1}{|{\mathrm{deg}}_i-{\mathrm{deg}}_j|+1}\&CenterDot;\min \{{\mathrm{deg}}_i,{\mathrm{deg}}_j\}\&CenterDot;w_{\mathrm{ij}},---\left(1\right)$

Wherein, deg _i, deg _jMean respectively node n _i, n _jDegree, min{x, y} means to get a less value; w _IjExpression is to limit (n _i, n _j) weights; Get the minimum value d of all limits middle distance, the end points on the limit that is d by all distances carries out renormalization; What here renormalization adopted is the principle of processing at first at most, first processes and has the point that bee-line limit number is maximum, first the adjacent the shortest related point in limit of this point is carried out to renormalization, is about to it and merges into a point; Repeat this process until distance is all processed for the limit of d; Calculate distance:

d _k＝d _k-1+d （2）

With correlation function C (d) _k:

${C\left(d\right)}_k=\frac{1}{N^2}{\mathrm{\&Sigma;}}_{i,j=1}^{N}H(d-||n_i-n_j|\left|\right)---\left(3\right)$

Wherein, the number that N is node in network, n _i, n _jFor the node in network; H (y) is step function (step function),

$H\left(y\right)=\left\{\begin{array}{cc}1& y\&GreaterEqual;0\\ 0& y<0\end{array}\right.---\left(4\right)$

Make k=k+1;

Step 4 is specially: after the renormalization of step 2, and the weight w on every limit _IjBy following strategy, calculate: at first making all weights is all 1; If n now _iAnd n _jBefore process previous step renormalization, there is s limit to be connected, w _Ij=w _Ij/ s; If n _iAnd n _jThe degree that has one is 1, and the point of its representative has m, and it puts renormalization, w by m _Ij=w _Ij* m;

Step 5 is: the draw fractal dimension of estimating complex network the value that provides its correlation dimension; The correlation dimension of network is defined as follows:

$D_C=\underset{d\&RightArrow;0}{\lim }\frac{\ln C\left(d\right)}{\ln d}---\left(5\right)$

1) K the different { D of community _i(i=1 ... K), wherein

For $\&ForAll;i\&NotEqual;j,D_i\&cap;D_j=\mathrm{\&phi;};$

2) a series of (C (d), the complex network fractal dimension Ds of d) putting right value and estimating thus _c.

4. the complex network community discovery method based on fractal characteristic as described as claim 1-3 any one, it is characterized in that: subordinate phase, the processing of the community discovery of online Research Dynamic Complex Networks: the community discovery of Research Dynamic Complex Networks refers on the static network basis, calculates the network of the new growth changed along with the time; Implementation procedure is as follows:

Step 1, input complex network mean that with non-directed graph G=(V, E) V and E are respectively the set on node and limit;

Step 2, renewal network topology structure;

Step 3, removal do not have influential limit to change to the complex network community structure;

Step 4, adjustment yardstick;

Step 5, result output.

5. the complex network community discovery method based on fractal characteristic as claimed in claim 4, it is characterized in that: described input complex network comprises:

1) t complex network G (t)=(V (t), E (t)) constantly;

2) community structure { D of t moment complex network _i(t) } (i=1 ... K), wherein

For $\&ForAll;i\&NotEqual;j,D_i\left(t\right)\&cap;D_j\left(t\right)=\mathrm{\&phi;};$

Minor increment d when 3) each community is as a node between the node of complex network _m

4) be carved into the t+1 variation on limit constantly during t: Δ E ⁺With Δ E ^-, Δ E ⁺With Δ E ^-Mean respectively the new limit produced and the limit of extinction.

6. the complex network community discovery method based on fractal characteristic as claimed in claim 4, it is characterized in that: described renewal network topology structure, complex network being carved into to the t+1 variation on limit constantly during from t is updated to t complex network G (t) constantly, thereby obtain the topological structure of G (t+1), for subsequent calculations apart from the time use.

7. the complex network community discovery method based on fractal characteristic as claimed in claim 4 is characterized in that: described removal does not have influential limit to change to the complex network community structure, from Δ E ⁺Two end points of middle removal are at the same D of community _i(t) limit in, from Δ E ^-Two end points of middle removal are at two D of community _iAnd D (t) _j(t) limit in, wherein i ≠ j.

8. the complex network community discovery method based on fractal characteristic as claimed in claim 4 is characterized in that: described adjustment yardstick, and for Δ E ⁺With Δ E ^-In the remaining related { D of community of element also _{I (p)}(t) } (p=1 ... q), node and the limit of each inside, community are condensed by top static network method, meet following one of two things and just stop: in coacervation process, the distance of each node is not less than d for the first time _mAll nodes in community are condensed into a node.

9. the complex network community discovery method based on fractal characteristic as claimed in claim 4 is characterized in that: described result output, comprise t+1 constantly complex network community structure and to the estimated value of complex network correlation dimension:

(1) community structure { D of t+1 moment complex network _i(t+1) } (i=1 ... L), wherein $\underset{i=1}{\overset{L}{\&cup;}}{D}_i(t+1)=V(t+1),$ For $\&ForAll;i\&NotEqual;j,D_i(t+1)\&cap;D_j(t+1)=\mathrm{\&phi;}$ The point that wherein V (t+1) deducts extinction by V (t) is added the point of new generation and forms;

(2) a series of (C (d), the complex network fractal dimension Ds of d) putting right value and estimating thus _c.

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