CN103425868B - Complex network community discovery method based on fractal characteristic - Google Patents

Abstract

Complex network community discovery method based on fractal characteristic, including the processing and two stages of process of community discovery of online Research Dynamic Complex Networks of community discovery of the static complex network of off-line.The present invention makes full use of the Analysis On Multi-scale Features of complex network, by renormalization process as the bridge between contact different scale, thus reaching community structure information before is the purpose used by new community structure finds, achieve the community discovery of increment type, community structure for research trends complex network has done new trial, and has obtained satisfied result.

Description

Complex network community discovery method based on fractal characteristic

Technical field

The invention belongs to network community discovery method technical field, relate to a kind of complex network community based on fractal characteristic Discovery method.

Background technology

Present in real world many complicated and huge system can describe with network, we term it complex web Network.Complex network is the abstract of complication system, and the individuality in complication system is nodes, and the limit between node is then individual Between according to certain rule self-assembling formation or certain relation of arteface.Real world comprises various types of complex web Network, the network formed as interlinked between the page in community network, technical network, biological networks, network, paper are collaborateed Network, reference citation network etc..In these real worlds, substantial amounts of complex network is by many different types of combination of nodes Forming, the connection existed between the most identical type node is the most, and the connection of dissimilar node is relatively fewer, multiple This characteristic of miscellaneous network is referred to as community structure.

Research finds, the complex network in reality has three big features: 1. worldlet (Small-world), although referring to The scale of complex network may be very big, but the shortest path between any two of which node is smaller.Small-word networks Network has little characteristic path length (characteristic path length) and high convergence factor simultaneously (clustering coefficient).2. scaleless property (Scale-free), refers to the degree distribution of node in complex network Obey or approximation obeys power law (power law) distribution.3. self-similarity (Self-similarity), refers to complex network With the similarity that the local of self has approximation, namely there is fractal (Fractal) feature.In order to explore the knot of complex network Structure feature, and then understand the function of complex network, the community structure of complex network is conducted extensive research by people, it is proposed that Numerous community discovery methods, are broadly divided into four kinds of methods: condensing method, splitting method, optimization method and analogy method.Numerous In method, these four method is not independent, and a kind of method may embody multiple thought simultaneously.

Fractal characteristic is the feature that in reality, complex network has, and root is it was found that have the many of community structure In network, the most stable of network exactly with fractal characteristic.Confirm that the structure of network can be entered by the fractal characteristic of complex network Row sum-equal matrix, has broad prospects so the fractal characteristic of complex network is applied to community discovery, is the thinking of a kind of novelty.

Although the analysis that the fractal characteristic of complex network has been carried out by many scholars at present and theoretical research.But its achievement And method is only that the fractal characteristic to complex network is analyzed and confirms, do not have to propose complex network based on fractal characteristic Community discovery method.It there is problems of, and complex network essence is exactly a complicated nonlinear system, since fractal characteristic It is an important properties in real network, then it has anything to contact with complex network community structure；Can be to complex web Research in terms of the community discovery of network is helpful etc..Trace it to its cause, do not pay attention to fractal characteristic and can not only disclose non-linear In system in order with unordered unification, and can fail the features such as utilization of preferably unifying by definitiveness Yu randomness, cause Fractal feature is used for applying complex network community discovery method to be limited only to theory and analysis, fails to propose practical solution multiple Miscellaneous network community discovery method.

Summary of the invention

It is an object of the invention to provide a kind of complex network community discovery method based on fractal characteristic, solve existing multiple Promptness problem present in miscellaneous network community discovery method, the promptness of the dynamically change meeting complex network topologies is wanted Ask.

The technical scheme is that, complex network community discovery method based on fractal characteristic, including the static state of off-line The process of the community discovery of complex network and two stages of process of the community discovery of online Research Dynamic Complex Networks.

The feature of the present invention also resides in:

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

Step 1, topological structure G=(V, E) of input complex network；Complex network non-directed graph G=(V, E) represents, V With the set that E is respectively node and limit；

Step 2, initialization；

Step 3, the distance of calculating neighborhood of nodes, the node that renormalization distance is minimum；

Step 4, renewal network；

Step 5, output result.

Above-mentioned steps 2 is particularly as follows: calculate the degree of each node, node n in V_iDegree be designated as deg_i, represent in network and connect Number to the limit of node；Each edge in E is set weight w_ijIf, n_iAnd n_jHaving a degree is 1, then making its weights is 0, its Its situation weights is 1.Make renormalization number of times k=0, and make d=0, d_k=0, C (d)_k=0；

If above-mentioned steps 3 is particularly as follows: all of original point is the most by renormalization mistake in complex network, then skip to step 4 Update network, otherwise continue this step；The distance of all neighborhood of nodes is calculated 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_jRepresent node n respectively_i, n_jDegree, min{x, y} represent and take a less value.w_ijRepresent To limit (n_i,n_j) weights.Take minima d of distance in all limits, the end points on the limit that all distances are d is carried out renormalization.? Here renormalization uses the principle at most processed at first, the most first processes and has the point that beeline limit number is most, first by this The point involved by adjacent the shortest limit of point carries out renormalization, is i.e. merged into a point.Repeat this process until distance is d Limit the most processed.Computed range:

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, N is the number of node, n in network_i, n_jFor the node in network.H (y) is jump function (step Function), i.e.

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

Make k=k+1.

Above-mentioned steps 4 particularly as follows: after the renormalization of step 2, the weight w of each edge_ijThen based on following strategy comes Calculate: first making all of weights is all 1；If now n_iAnd n_jS limit is had to be connected before previous step renormalization, then w_ij= w_ij/s；If n_iAnd n_jThe degree having one is 1, and its representative point has m, and i.e. it is by m some renormalization, then w_ij= w_ij*m。

Above-mentioned steps 5 is: the draw fractal dimension estimating complex network the value providing 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 different community { D_i(i=1 ... K), whereinFor $\&ForAll;i\&NotEqual;j,D_i\&cap;D_j=\mathrm{\&phi;};$

2) a series of (C (d), d) to value and the complex network fractal dimension D that thus estimates_c。

Second stage: the process of the community discovery of online Research Dynamic Complex Networks:

The community discovery of Research Dynamic Complex Networks refers on the basis of static network, calculates the new growth changed over time The network of (evolution)；Realize process prescription as follows:

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

Step 2, renewal network topology structure；

Complex network community structure is not had influential limit to change by step 3, removal；

Step 4, adjustment yardstick；

Step 5, result export.

Above-mentioned input complex network, including:

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

2) community structure { D of t complex network_i(t) } (i=1 ... K), whereinFor $\&ForAll;i\&NotEqual;j,D_i\left(t\right)\&cap;D_j\left(t\right)=\mathrm{\&phi;};$

3) each community is as minimum range d between the node of complex network during a node_m；

4) t is to the change on t+1 moment limit: Δ E⁺With Δ E^-, Δ E⁺With Δ E^-Represent newly generated limit and extinction respectively Limit.

Above-mentioned renewal network topology structure, updates complex network to t from the change of t to t+1 moment limit On complex network G (t), thus obtain the topological structure of G (t+1), use when subsequent calculations distance.

Complex network community structure is not had influential limit to change by above-mentioned removal, i.e. from Δ E⁺Two end points of middle removal are same One community D_iT the limit in (), from Δ E^-Two end points of middle removal are at Liang Ge community D_i(t) and D_jLimit in (t), wherein i ≠ j.

Above-mentioned adjustment yardstick, for Δ E⁺With Δ E^-In community { D involved by still remaining element_i(p)(t) } (p=1 ... Q), the node within each community and limit static network method above are condensed, meet one of following two situation Just stop: in coacervation process, the distance of each node is for the first time not less than d_m；All nodes in community are condensed into a node.

Adjust yardstick again, the result of upper step carried out community discovery process based on renormalization, until all newly add former Initial point is all crossed by renormalization or obtains suitable community structure.A kind of possible situation is exactly, and community splits into Part is combined with other community.Here it is accomplished by the most first considering division, then the part after division and remaining society District considers situation about merging together.Process for this situation is exactly first to divide then to remerge.

The above results exports, including the community structure of t+1 moment complex network and the estimation to complex network correlation dimension Value:

(1) community structure { D of t+1 moment complex network_i(t+1) } (i=1 ... L), wherein $\underset{i=1}{\overset{L}{\&cup;}}{D}_i(t+1)=V(t+1),$ Right In $\&ForAll;i\&NotEqual;j,D_i(t+1)\&cap;D_j(t+1)=\mathrm{\&phi;}$ Wherein V (t+1) is deducted the point of extinction by V (t) and adds newly generated point and formed；

(2) a series of (C (d), d) to value and the complex network fractal dimension D that thus estimates_c。

There is advantages that

1, during the present invention makes full use of reality, the fractal characteristic of complex network is community discovery service.With renormalization and inverse Process is studied as the instrument of change yardstick, the community structure change to dynamic network, it is proposed that based on dimensional variation Increment type dynamic network community discovery method, and in real network, confirm its feasibility.The feature of the present invention is filled exactly Divide the Analysis On Multi-scale Features utilizing complex network, by renormalization process as the bridge between contact different scale, thus reach it Front community structure information is the purpose used by new community structure finds, it is achieved that the community discovery of increment type, dynamic for research The community structure of state complex network has done new trial, and has obtained satisfied result.

2, the present invention uses fractal principle, proposes the community discovery method of dynamic increment formula, it is intended to by fractal characteristic skill Art solves complex network community discovery method, the topologies change rule disclosing complex network and function integrity, more preferably solves WWW, transportation network, Scientific Cooperation network, electric power networks, human relation network, cell neural network and infectious disease network etc. Community discovery method.Especially provide new approaches for the network information security.

Accompanying drawing explanation

Fig. 1 is for estimating complex network correlation dimension in present invention complex network based on fractal characteristic community discovery method The double logarithmic chart of number；

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

Fig. 3 be in present invention complex network based on fractal characteristic community discovery method complex network in coacervation process Different conditions figure；

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

Detailed description of the invention

The present invention is described in detail with detailed description of the invention below in conjunction with the accompanying drawings.

Complex network community discovery method of based on fractal characteristic, including the community discovery of the static complex network of off-line Two stages of process of the community discovery of process and online Research Dynamic Complex Networks:

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

Step 1, topological structure G=(V, E) of input complex network；Complex network non-directed graph G=(V, E) represents, V With the set that E is respectively node and limit；

Step 2, initialization.Calculate the degree of each node, node n in V_iDegree be designated as deg_i, represent and network be connected to knot The number on the limit of point；Each edge in E is set weight w_ijIf, n_iAnd n_jHaving a degree is 1, then making its weights is 0, other feelings Condition weights are 1.Make renormalization number of times k=0, and make d=0, d_k=0, C (d)_k=0；

Step 3, the distance of calculating neighborhood of nodes, the node that renormalization distance is minimum.Renormalization (renormalization) Being a physical concept, its definition is to overcome the divergence difficulty in quantum field theory, makes Theoretical Calculation be entered smoothly A kind of theoretical treatment method of row.Its essence is to change thick regarding in observation to change (coarsening) degree, the most quantitatively Analyze the change of physical quantity to disclose certain rule.In complex network, it is equivalent to constantly the point less than certain distance Regard a point as, until whole complex network all becomes a point.If all of original point (by weight is just referring to not in complex network The point changed, does not i.e. have the point of merged mistake) the most by renormalization mistake, then skip to step 4, otherwise continue this step.Foundation Formula (1) calculates the distance of all neighborhood of nodes:

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

Wherein, deg_i, deg_jRepresent node n respectively_i, n_jDegree, min{x, y} represent and take a less value.w_ijRepresent To limit (n_i,n_j) weights.Take minima d of distance in all limits, the end points on the limit that all distances are d is carried out renormalization.? Here renormalization uses the principle at most processed at first, the most first processes and has the point that beeline limit number is most, first by this The point involved by adjacent the shortest limit of point carries out renormalization, is i.e. merged into a point.Repeat this process until distance is d Limit the most processed.Computed range:

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, N is the number of node, n in network_i, n_jFor the node in network.H (y) is jump function (step Function), i.e.

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

Make k=k+1；

Step 4, renewal network.After the renormalization of step 2, the node number of network and the number on limit all there occurs change Changing, therefore the degree of each node and the weights on each bar limit are required for recalculating.The size of degree can be according to the number meter on limit Calculate, the weight w of each edge_ijThen need to calculate by following strategy: first making all of weights is all 1；If now n_iAnd n_jS limit is had to be connected before previous step renormalization, then w_ij=w_ij/s；If n_iAnd n_jThe degree having one is 1, and its generation The point of table has m, and i.e. it is by m some renormalization, then w_ij=w_ij*m；

Step 5, output result.Community's result of the complex network that output marks off, draws and estimates the fractal of complex network Dimension also provides the value of 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.The present invention's In community discovery method, the correlation dimension parameter as complex network fractal characteristic, i.e. fractal dimension will be used.Can from formula (5) To see, correlation dimension similar with box-counting dimension can also be with (C (d), d) the fitting a straight line slope under double-log system is estimated Meter.The result of the most whole method has a following two:

1) K different community { D_i(i=1 ... K), whereinFor $\&ForAll;i\&NotEqual;j,D_i\&cap;D_j=\mathrm{\&phi;};$

2) a series of (C (d), d) to value, and the complex network estimated thus in accordance with the method shown in accompanying drawing 1 divides Shape dimension D_c。

The method is the condensing method of a community discovery, if its result is shown from different level height, Can obtain different community structures, this illustrates that complex network community structure has level the most from another point of view, such as accompanying drawing 4 institute Show.

Second stage: the process of the community discovery of online Research Dynamic Complex Networks；The community discovery of Research Dynamic Complex Networks refers to On the basis of static network, calculate the network of the new growth (evolution) changed over time；Realize process prescription as follows:

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

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

2) community structure { D of t complex network_i(t) } (i=1 ... K), whereinFor $\&ForAll;i\&NotEqual;j,D_i\left(t\right)\&cap;D_j\left(t\right)=\mathrm{\&phi;};$

3) each community is as minimum range d between the node of complex network during a node_m；

4) t is to the change on t+1 moment limit: Δ E⁺With Δ E^-(representing the limit of newly generated limit and extinction respectively).

Step 2, renewal network topology structure.Complex network is updated t from t to the change on t+1 moment limit Complex network G (t) on, thus obtain the topological structure of G (t+1), use when subsequent calculations distance.

Complex network community structure is not had influential limit to change by step 3, removal.I.e. from Δ E⁺Two end points of middle removal exist Same community D_iT the limit in (), from Δ E^-Two end points of middle removal are at Liang Ge community D_i(t) and D_jLimit in (t), wherein i ≠ j.The community of complex network is exactly the network subgraph big and little with external connection density with internal Connection Density.So it is easy to understand The change that Web Community will not be impacted have a following two: increase the limit between community's interior knot and reducing and tie between community Limit between point.Both modes Hui Shi community more condenses, so the change of community will not be produced.

Step 4, adjustment yardstick, go deep into being condensed inside community.For Δ E⁺With Δ E^-In involved by still remaining element Community { D_i(p)(t) } (node within each community and limit static network method above are coagulated by p=1 ... q) Poly-, meet one of following two situation and just stop: in coacervation process, the distance of each node is for the first time not less than d_m；Institute in community Node is had to be condensed into a node.Here the computational methods of distance still use the calculating means of formula (1).Involved by this step Community have only to consider the contact between its internal node, be not required to consider the contact between community, so for each community Calculating be independent, it is possible to use parallel method improves processing speed.The result of this step will be condensed into complex network Network that the block formed by the former community not divided, the node being newly added and division community is mixed to form, this network Minimum range be d_m, wherein these three element differs and has established a capital, and of course passes through this step, three kinds of elements above-mentioned all by It is expressed as node.Certain community may be made to occur the complex network of division to develop.A kind of situation is to wither away in the limit within community, The density within community can be made to reduce, consequently, it is possible to cause this community to produce division；The second situation is the new limit produced Only one of which end points is the node within this community, can increase contacting of community's internal node and outside, thus be also possible to lead Cause this community and produce division.

Step 5, again adjust yardstick, the result of upper step is carried out community discovery process based on renormalization, until all The original point newly added all is crossed by renormalization or obtains suitable community structure.Two or more community may be made to produce merging Complex network develops.Here community can also represent newly-increased original point, because in terms of different yardsticks, community is also certain Node under yardstick.Multiple communities merge and can be further broken into Liang Ge community and first merge and merge with other community, thus Have only to consider that the situation that Liang Ge community merges is the most permissible.The situation that the complex network that possible Shi Liangge community merges develops is just It is so that the contact of Liang Ge community point is closeer, has namely increased two intercommunal limits newly.A kind of possible situation is exactly, The part that one community splits into is combined with other community.Here it is accomplished by the most first considering division, then after division Part consider situation about merging together with remaining community.Process for this situation is exactly first to divide then to remerge.

Step 6, result export.Output result mainly has two one to be the community structure of t+1 moment complex network, another Individual is the estimated value to complex network correlation dimension:

(1) community structure { D of t+1 moment complex network_i(t+1) } (i=1 ... L), wherein $\underset{i=1}{\overset{L}{\&cup;}}{D}_i(t+1)=V(t+1),$ Right In $\&ForAll;i\&NotEqual;j,D_i(t+1)\&cap;D_j(t+1)=\mathrm{\&phi;}$ Wherein V (t+1) is deducted the point of extinction by V (t) and adds newly generated point and formed.

(2) a series of (C (d), d) to value and the complex network fractal dimension D that thus estimates_c。

The present invention can obtain the community structure in complex network t+1 moment, at the end of can obtain the narrow spacing in t+1 moment From, and the complex network topologies figure in t+1 moment, if there being t+1 moment variation delta E to the limit of the network in t+2 moment⁺With ΔE^-, it is possible to continuing to obtain the complex network community structure in t+2 moment, continuing is exactly the implication place of increment type, by State time previous and variable quantity obtain the state of later moment in time.

The invention provides a kind of complex network community discovery method based on fractal characteristic, utilize actual complex network Fractal characteristic and the feature such as multiple dimensioned serve community discovery, are new thinkings, and examples of implementation also demonstrate that the method Effectiveness and feasibility.

In embodiment, selecting Zachary karate club network, it is the karate club in one school of the U.S. A network between all members.This network is made up of 34 nodes and 78 limits, and node represents each one-tenth in club Member, while represent member's social connection outside club.Wayne Zachary spent two years to see at 20 century 70s Examining and study this club, later due to inside, club creates division, becomes Liang Ge club.Due to Wayne Zachary has carried out detailed investigation and research to this club, so this network just becomes research community and sends out Existing problem uses the network that frequency is the highest.Use this network that complex network community discovery method of the present invention is tested Card.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 point of view of the statistical result of F-value, the present invention has preferable result, because of For only one of which node for whole network be divided into mistake community in, so the degree of accuracy of the present invention still than Comparatively ideal.

Second embodiment is used to verify the increment type dynamic network community based on dimensional variation for dynamic network Discovery method, the example chosen is U.S. NCAA(National Collegiate Athletic Association) rugby Network, is the rugby conventional competition code of American college sports league 2000 racing season collected.In the network, node Represent each team of university taken in competition, represent with the name of university, have 115 universities；Limit between node represent school team it Between conventional competition, have 613 matches between these school teams.In reality, these teams are divided into 11 different playing areas, often Individual playing area is made up of 8 to 12 teams, and the match between same playing area team is than the match frequency between different playing area teams Wanting height, the every team of match in playing area is averagely 7, and the every team of match between playing area is averagely 4.It would be appreciated that match Interval match is unevenly distributed, and for belonging to the team in different playing area, the match between near is than distance Match between Yuan wants many.Community structure known to these makes this network be usually used in the detection to community discovery method.This This network has been divided into two parts by invention, and Part I is 76 nodes therein and 426 limits therebetween, these nodes and limit 8 playing areas in 11 playing areas of main composition, use static method to obtain the community structure of these points.Part II is network In remaining node and limit, it is represented with the change on limit, i.e. Δ E⁺With Δ E^-, then in conjunction with 76 nodes of Part I Division result, and minimum range (being 8.9344 in an experiment).Constitute the input of delta algorithm in the 3rd chapter, knot Fruit 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 having in the discovery method of the present invention is less accurate, but Being in most playing area, team still can well identify, and the present invention is on this real data collection on the whole Effect still ratio more satisfactory, average rate of accuracy reached is to 89%, and recall rate has also reached 95%.

Claims (7)

1. complex network community discovery method based on fractal characteristic, it is characterised in that: include the static complex network of off-line The process of community discovery and two stages of process of the community discovery of online Research Dynamic Complex Networks, wherein, first stage off-line The process of the community discovery of static complex network, its process is as follows:

Step 1, topological structure G=(V, E) of input complex network；Complex network non-directed graph G=(V, E) represents, V and E It is respectively node and the set on limit；

Step 2, initialization, specifically, calculate the degree of each node, node n in V_iDegree be designated as deg_i, represent in network and connect Number to the limit of node；Each edge in E is set weight w_ijIf, n_iAnd n_jHaving a degree is 1, then making its weights is 0, its Its situation weights is 1；Make renormalization number of times k=0, and make d=0, d_k=0, C (d)_k=0, wherein, d be between community two node Small distance, for the first time d=0, d_kIt is the superposition distance for community two node, d_k=0 refers in the case of k=0, d_kIt is assigned to 0 value, C (d)_kIt is the correlation function of community two node, C (d) in the case of k=0 time_kIt is assigned to 0 value；

Step 3, the distance of calculating neighborhood of nodes, the node that renormalization distance is minimum, if particularly as follows: all of former in complex network Initial point the most by renormalization mistake, then skips to step 4 and updates network, otherwise continue this step；Calculate all adjacent according to formula (1) The distance of node:

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

Wherein, deg_i, deg_jRepresent node n respectively_i, n_jDegree, min{x, y} represent and take a less value；w_ijIt is represented to limit (n_i,n_j) weights；Take minima d of distance in all limits, the end points on the limit that all distances are d is carried out renormalization；Here Renormalization uses the principle at most processed at first, the most first processes and has the point that beeline limit number is most, first this is put Point involved by the shortest adjacent limit carries out renormalization, is i.e. merged into a point；Repeat the limit that this process is d until distance The most processed；Computed range:

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, N is the number of node, n in network_i, n_jFor the node in network；H (y) is jump function (step Function), i.e.

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

Make k=k+1；

Step 4, renewal network, particularly as follows: after the renormalization of step 2, the weight w of each edge_ijThen come by following strategy Calculate: first making all of weights is all 1；If now n_iAnd n_jS limit is had to be connected before previous step renormalization, then w_ij= w_ij/s；If n_iAnd n_jThe degree having one is 1, and its representative point has m, and i.e. it is by m some renormalization, then w_ij= w_ij*m；

Step 5, output result: the draw fractal dimension estimating complex network the value providing its correlation dimension；The association of network Dimension is defined as follows:

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

1) K different community { D_i(i=1 ... K), whereinFor

2) a series of (C (d), d) to value and the complex network fractal dimension D that thus estimates_c。

2. complex network community discovery method based on fractal characteristic as claimed in claim 1, it is characterised in that: second-order Section, the process of the community discovery of online Research Dynamic Complex Networks: the community discovery of Research Dynamic Complex Networks refers on static network basis On, calculate the network of the new growth changed over time；Realize process as follows:

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

Step 2, renewal network topology structure；

Complex network community structure is not had influential limit to change by step 3, removal；

Step 4, adjustment yardstick；

Step 5, result export.

3. complex network community discovery method based on fractal characteristic as claimed in claim 2, it is characterised in that: described input Complex network, including:

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

2) community structure { D of t complex network_i(t) } (i=1 ... K), whereinFor

3) each community is as minimum range d between the node of complex network during a node_m；

4) t is to the change on t+1 moment limit: Δ E⁺With Δ E^-, Δ E⁺With Δ E^-Represent the limit of newly generated limit and extinction respectively.

4. complex network community discovery method based on fractal characteristic as claimed in claim 2, it is characterised in that: described renewal Network topology structure, updates complex network to complex network G (t) of t from the change of t to t+1 moment limit, from And obtain the topological structure of G (t+1), use when subsequent calculations distance.

5. complex network community discovery method based on fractal characteristic as claimed in claim 2, it is characterised in that: described removal Influential limit is not had to change complex network community structure, i.e. from Δ E⁺Two end points of middle removal are at same community D_iIn (t) Limit, from Δ E^-Two end points of middle removal are at Liang Ge community D_i(t) and D_jLimit in (t), wherein i ≠ j.

6. complex network community discovery method based on fractal characteristic as claimed in claim 2, it is characterised in that: described adjustment Yardstick, for Δ E⁺With Δ E^-In community { D involved by still remaining element_i(p)(t) } (p=1 ... q), inside each community Node and limit be condensed by above static network method, meet one of following two situation and just stop: in coacervation process The distance of each node is for the first time not less than d_m；All nodes in community are condensed into a node.

7. complex network community discovery method based on fractal characteristic as claimed in claim 2, it is characterised in that: described result Output, including the community structure of t+1 moment complex network and the estimated value to complex network correlation dimension:

(1) community structure { D of t+1 moment complex network_i(t+1) } (i=1 ... L), whereinForWherein V (t+1) is deducted the point of extinction by V (t) and adds newly generated point and formed；

(2) a series of (C (d), d) to value and the complex network fractal dimension D that thus estimates_c。