Combined method to detect communities in graphs of interacting objects Текст научной статьи по специальности «Математика»

Combined method

to detect communities in graphs

of interacting objects1

Sofiya Y. Lobanova

BSc Program Student

Tikhonov Moscow Institute of Electronics and Mathematics National Research University Higher School of Economics Address: 20, Myasnitskaya Street, Moscow, 101000, Russian Federation E-mail: [email protected]

Alexander A. Chepovskiy

Associate Professor,School of Applied Mathematics Tikhonov Moscow Institute of Electronics and Mathematics National Research University Higher School of Economics Address: 20, Myasnitskaya Street, Moscow, 101000, Russian Federation E-mail: [email protected]

Abstract

In this paper, we propose and implement a method for detecting intersecting and nested communities in graphs of interacting objects of different natures. For this, two classical algorithms are taken: a hierarchical agglomerate and one based on the search for k-cliques. The combined algorithm presented is based on their consistent application. In addition, parametric options are developed that are responsible for actions with communities whose sizes are smaller than the given k, and also with single vertices. Varying these parameters allows us to take into account differences in the topology of the original graph and thus to correct the algorithm.

The testing was carried out on real data, including on a group of graphs of a social network, and the qualitative content of the resulting partition was investigated. To assess the differences between the integrated method and the classical algorithms of community detections, a common measure of similarity was used. As a result, it is clearly shown that the resulting partitions are significantly different. We found that for the approach proposed in the article the index of the numerical characteristic of the partitioning accuracy, modularity, can be lower than the corresponding value for other approaches. At the same time, the result of an integrated method is often more informative due to intersections and nested community structure.

A visualization of the partition obtained for one of the examples by an integrated method at the first and last steps is presented. Along with the successfully found set of parameters of the integrated method for small communities and cut off vertices in the case of social networks, some shortcomings of the proposed model are noted. Proposals are made to develop this approach by using a set of parametric algorithms.

Key words: graph, graph analysis, community detection, graph structure, social network analysis, big data.

Citation: Lobanova S.Y., Chepovskiy A.A. (2017) Combined method to detect communities in graphs of interacting objects. Business Informatics, no. 4 (42), pp. 64-73. DOI: 10.17323/1998-0663.2017.4.64.73.

1 This work was supported by the Russian Foundation for Basic Research (projects No. 16-29-09546 and No. 16-07-00641)

Introduction

Today, a wide variety of real-world systems of many fields can be represented as complex networks or graphs, namely, sets of vertices and edges with nontrivial topological internals. Examples of such networks run from technology [1] to biological fields [2] and social [3] studies.

The notion of community is quite general and associated with object classification. Depending on the context, it may refer to the module, the class, the group, the cluster, etc. This notion is of great value for solving the community detection task for the Internet [4]. The issue is topical for the implementation of on-the-edge search engines, data filtration and further automated sorting.

Typical networks to study are popular social network services. Then the user accounts are taken for the nodes, and the connections between them are determined by the formal notions of "friendship", "subscription", etc., so that we obtain a graph of interacting objects. Such networks have the structure of implicit communities — groups of people united by some additional feature, for example: classmates, subscribers to a number of public persons or thematic publics, criminal elements.

The problem of community detection in graphs of interacting objects is of great value to network analysis. In the general case, this task is NP-complete, so the paper proposes to use a method with a number of terminal options since it is required to deal with graphs from various fields [5, 6]. These options are designed to possibly refine the resulting community structure and are based on the initial graph topology and corresponding field.

1. Community detection methods

The well-known GN algorithm proposed by Girvan and Newman in 2002 [7, 8] is among the pioneers of divisive algorithms. Initially all the graph vertices share one community.

Each step of the GN algorithm starts with the edge betweenness [9] calculation for every edge in the graph [8] and then it removes those with the highest score. The algorithm repeats this procedure, and it splits the network into disconnected parts, which in their turn go through the same process. The loop breaks when there are no more edges in the graph or when the split reaches modularity's [10] maxima.

Modularity is a numerical characteristic of the correctness of a partition. Currently, this is the most popular measure of the quality of the partition, and it is used directly or indirectly in the study of complex networks. As a rule, a closer to real partitioning has larger values of modularity.

Modularity Q is defined on an existing community structure of a graph:

2m

A,-

k¡kj 2m

öfarCj), (1)

where m the number of edges; A - the adjacency matrix element; k. - the degree of the vertex i;

c. - the index of the class to which the vertex

i

belongs;

[1, c^Cj.

The edge betweenness for a given edge is the fraction of the shortest paths passing this edge among all shortest paths:

aM

cß{e)= X

s*t ,veV

(2)

where e — the given edge; V — the set of graph vertices; ast — the number of the shortest paths between the vertices s and i;

— the number of the shortest paths between the vertices s and t passing through the edge e.

The calculation of edge betweenness CB(e) is computationally expensive, as was mentioned by Girvan and Newman [1, 8]. So in the case of studying large networks, it is better to apply different techniques [1].

The fast hierarchical approach was proposed by Blondel et al. in 2008 [11] for the general case of weighted graphs. The algorithm is agglomerative, so it combines current vertices to new supervertices (communities) step by step. The increase of modularity for such split must be maximal as well. At the end of each iteration, a new hierarchy level is reached. The loop breaks if there is no increase of modularity. The method is limited more by the processing memory than the time.

The resulting community structure may not make sense since the approach is based on the locals of vertices. That is why it is not always obvious if some intermediate split meets the reality. Moreover, the outcomes depend on the vertices order at the first step [12].

The aforementioned approaches mean that communities are not intersecting with each other. Furthermore, in real-world systems vertices are rarely belong to single community [13]. As a consequence, the resulting partition may not accurately describe the actual network.

Further on, for CPM we denote one of the well-known methods. In their work, Palla et al. [14, 15] presented a novel approach for detecting the overlapping community structure in complex networks.

In this method, finding the structure of overlapping communities is related to the problem of finding k-cliques in a network (complete subgraphs of size k). Given the clique size k, this algorithm first finds all k-cliques within the network. Then it associates two k-cliques if they share k - 1 nodes (Figure 1). Each community is made over a maximum set of adjacent k-cliques [15].

To measure the quality of such partitions, there are many modifications of classical mod-

Fig. 1. Determined overlapping communities with the CPM algorithm for 4-cliques

ularity. In this paper, we consider the variant proposed in [13]:

1

Qm 2m ^^

ceC i,jsc

4

kj^ 2m

1

0,Oj

(3)

where C= {cv c2, ..., c^} — the set of overlapping communities;

Qj — the number of communities to which the vertex i belongs.

2. Description of the Combined method

In this paper, we consider a combination of the hierarchical Blondel algorithm with the CPM method of network partition into overlapping communities. By combination we mean a consistent application of the two methods. The idea is that at first our network is divided into large non-intersecting communities, and then we find smaller intersecting communities within them. Hereafter, we call this algorithm the "Combined method" (CM).

However, if Blondel's method identifies a finite community that does not split into parts in terms of a real network, then it is no longer useful to run CPM on it. Therefore, we apply the CPM algorithm to each separate c community identified by the Blondel method only if the percentage of vertices with high connection cen-trality does not exceed a predetermined value

9

0, cB (v) < ä

(4)

where nC — number of vertices in c community;

cB(v) — connection centrality of the vertex v;

a and p(nc) — algorithm parameters, 0<a<l;0</J<l.

The relatively large connection centrality value characterizes the vertex as possibly connecting different communities. If the vertex is removed, the graph can be divided into several components. A similar consideration is used in the Girvan—Newman algorithm [1, 8]. The presence of a sufficient number of such peaks in the community may indicate that it has not been finalized completely, and further fragmentation by the Blondel algorithm is required. Therefore, the condition of applying CPM depends both on the value of connection centrality and on the percentage of vertices which have this value higher than some threshold a. The threshold values for cB(v) and the vertex percentage are determined by the input parameters of algorithm a and ¡5, and thus they are able to influence the final partition.

As a result of using CPM, vertices that are not included in cliques are cut off, and there is a problem of assigning them to the community. The solution is a system of two new parameters t , and m , of the Combined

1 opt opt

method. Depending on their values, not only the cut off vertices are redistributed, but also small communities. This allows us to partially influence the resulting structure, and in some cases — to avoid combining graph sheets or small communities into large, meaningless communities. This characteristic of the partition, which can be called "picking up junk", is intrinsic to many modularity optimization methods, including the Blondel algorithm

[11, 16, 17].

To accurately define and illustrate the operation of the parameters t and mopt, let

us consider CR the set of

communities identified by the Blondel algorithm, where rc is their number. For convenience, we assume that if the individual community CR^CR does not satisfy the condition of applying the CPM, then the set CCPMi = CRt duplicates it. Otherwise, CCPM. is defined as the set of communities obtained as a result of the CPM inside C,

CPM,'

Ccpm, ~ i CCPM ,CCPM ,...,CCPM ?,

1 I M '2 lmc¡ J

(5)

where mct number of communities in CR e CR that were identified by CPM algorithm.

Let us fix the result of CPM partition as Cf and denote the community structure after redistribution with the parameter i as C'f. The parameters / and mgpt are required when the algorithm starts, along with the size of the clique k and the limitations a and /?. The community structure Cf is first transformed according to the value of the parameter i , and then — according the value of the parameter m t.

opt

The first parameter / is responsible for the distribution of the cut off vertices of the CPM application and may have the following values (Table 1).

Table 1.

Description of the operation of iopt parameter

None Cut off vertices that do not belong to any community

He Each cut off vertex is assigned to an individual (single) community within the community in which it was before applying CPM

i2r Each cut off vertex is assigned to an individual (single) community that does not intersect with others

ia2e A group of all cut off vertices that were in the same community prior to CPM application forms a community within it

ia2r All the cut off vertices form one community that does not intersect with others

The second parameter mopt works with small communities. The community is considered small if the number of vertices in it does not exceed the size of the clique k — the input parameter of the algorithm. This restriction is chosen because k is the minimum community size that can be indicated using the CPM method. The mgpt parameter can have the following values (Table 2).

Table 2.

Description of the operation of mojtt parameter

None The structure of communities does not change

mdel Small communities disband, but if the vertex belongs to other communities, then it remains in them. Otherwise, it will not belong to any community

m2c All small communities that are in a common community (i.e., enclosed in it) are united into one common community that does not have enclosed communities

m2r All small communities form one common community that does not intersect with others, even if the vertices belonged to other communities

Each of the transformations is done once, so when new small communities (possibly for m2r) appear, they will not be merged with the new common community. The resulting structure of C'f communities network is formed from Cf in accordance with the principles shown in Tables 1 and 2.

Separately, we should consider a combination of parameters with the values "i2c / None". In this case, when a small community is identified, it often turns out that all its members are also identified as individual communities, and large communities are divided into one larger part and several individual parts. To avoid a bunch of communities in the final structure, the authors use a modification: in the case described for small communities, individual communities disband, so only the smallest

community remains. For a large community, it remains itself and the individual communities remain within it (that is, we remove the part of the internal partition except of those individual communities). Figure 2 shows a possible result of the CM operation within one of the communities obtained in the first step. No similar result can be obtained by the separate application of Blondel and CPM algorithms. Meanwhile, in a number of cases in the graph analysis such a partition would be extremely informative.

Fig. 2. Possible final partition within one of the communities obtained in the first step of CM

3. Tests

The results of the CM operation on different types of graphs were evaluated and compared with the separate results of the Blondel and CPM algorithms. In addition, comparison was made with the results of some classical algorithms: GN, quick greedy optimization of the modularity FG [18] and the CONGA algorithm (Cluster Overlap Newman-Girvan Optimized algorithm) [19]. We considered dependence of the CM results on the parameters k, a, P, i , m The size of the clique k is considered in the following range: k e [3, 10]. For each partition, the value of Qis calculated, and for the partitions obtained by CM, their similarity measure a> to other partitions must be calculated.

Collins and Dent [20, 21] proposed the a> (Omega Index) for estimating the similarity of two partitions, based on the number of pairs of vertices adjacent in the same number of communities. Let C1 and C2 be different partitions,K, K2 — corresponding number of communities. Let tj(C) be the set of pairs of vertices occurring together in the C partition exactly j times. Then co is defined as follows:

a>(C',C")= A 'V (6)

where cou

(C'.C)

— observed value, d)e (C',C") — expected value:

0).

0).

1 min(iQ ,K2)

'u(c>c") = è I Mc')n^(c")|

m

j=o

* tmn(Á1>£2)

m

j=o

Value of a>(C',C") = 1 in the case of absolute equality of two partitions.

The authors studied the results of applying CM and selected classical algorithms on more than 70 graphs of interacting objects of various nature, such as: social networks, scientists citation networks, interactions of proteins,

infrastructure networks. In a number of cases, for graphs from one group and with similar topological parameters, the most optimal values of the correcting parameters for CM have been identified. The software realization is written in Python 3.5, standard and open libraries, including classical algorithms, were used to work with graphs. Semi-automatic test blocks are implemented to find the best values for Qm parameter values. Visualization of the partitions obtained for the graphs is provided by third-party software.

Let us consider the results on ego-graphs obtained from the social network VKontakte (Table 3) in more detail. The ego-graphs in this case are graphs formed by a width search starting from a single vertex with a depth of 1 and adding all the edges between the vertices previously obtained. At the same time, as a starting point we took vertices with a degree in the range from about 100 to 500. This made it possible to obtain not only graphs with different topological properties, but ones which are also convenient for qualitative analysis of their sizes. The main objectives of this work are to study the content of the resulting partitions and to find the most suitable values of CM parameters.

Table 3.

Social network graphs for testing

n m avk dst avntr avclu maxcl bet clo

G 105 792 15.1 0.1451 38.04 0.712 15 44.5 0.5434

g2 120 1226 20.4 0.1717 52.39 0.668 14 49.3 0.5507

G3 158 1352 17.1 0.1090 44.41 0.767 17 69.9 0.5309

G4 158 1596 20.2 0.1287 63.80 0.638 21 68.4 0.5374

187 1873 20.0 0.1077 51.11 0.674 16 83.0 0.5308

G 229 1726 15.1 0.0661 26.67 0.633 14 106.5 0.5184

G7 439 4143 18.9 0.0431 41.47 0.602 16 209.6 0.5117

g8 461 4299 18.7 0.0405 49.36 0.665 17 220.7 0.5110

Notation: n - number of vertices; m - number of edges; avk- average degree of vertices; dst- connection density; avntr - average number of triangles; avclu - average cluster coefficient; maxcl - size of a maximal clique; bet - averaged betweenness centrality; clo - averaged closeness.

Table 4.

Test results

s(CM) Q KA ov s(BL) Q BL ov s(CP) Q CP ov s(GN) Q GN ov s(FG) Q FG ov s(CN) Q CN ov

G, 38 0.213 4 0.320 4 0.167 17 0.250 3 0.315 3 0.245

G 21 0.367 4 0.385 4 - 0.013 20 0.329 4 0.371 4 0.136

G. 28 0.650 4 0.614 5 0.390 15 0.581 5 0.581 5 0.494

G4 39 0.327 4 0.342 3 0.016 47 0.175 3 0.313 4 0.100

G, 36 0.048 5 0.369 3 - 0.008 27 0.250 5 0.325 110 0.011

G 51 0.224 5 0.465 4 - 0.029 22 0.408 6 0.393 4 0.273

g7 90 0.501 7 0.572 8 - 0.149 54 0.536 8 0.509 5 0.304

G8 99 0.349 7 0.569 11 -0.272 53 0.514 6 0.552 8 0.195

Notation: s( CM) - number of communities identified by CM; s(BL) - number of communities identified by the Blon-del algorithm; s(CP) - number of communities identified by CPM; s(GN) - number of communities identified by GN; s(FG) - number of communities identified by FG; s(CN) - number of communities identified by CONGA; Each of Q£ - value Q for the partition, obtained by A algorithm.

In most cases, CM identified, as expected, a greater number of communities on these graphs (Table 4). For this group of graphs representing ego-graphs of the same social network, the most effective parameter option for the content of the partitions was the i2c/m2r and k = 4. A qualitative expert analysis of the composition of the groups singled out by the algorithms examined, especially of small communities of several vertices, showed the greatest efficiency especially for CM. For example, the Blondel, CPM and FG algorithms, as expected, left no large communities. And the smallest communities (including 1 element) identified by the GN algorithm for each of the graphs were not all meaningfully justified, and in some cases, taking into account the edges of the graph, they should be combined with the others.

It should be noted that the value of Q™ was not always higher than of other similar algorithms due to the fact that CM calculates another local extremum of the given quality measure, which is partially confirmed by the calculated values of the similarity measure m of these partitions (Table 5). As you can see from the tables, not only the number of the

communities obtained by CM and the other community algorithms differs greatly, but so do the sets of vertices combined into groups. After all, due to the peculiarities of the considered classical algorithms, some types of partitions cannot be obtained from their application, even theoretically. For example, the situation in Figure 2 for CPM cannot be obtained. All combined, this shows that the partition obtained as a result of CM was essentially different from the others.

The partition obtained within the CM is hierarchical, with intersecting and nested communities, which gives an opportunity for more complete understanding of the structure of the investigated graph and better corresponds to the actual system of interaction of objects of some nature, in particular, users of social networks. As an illustration, let us consider the example for G3 in more detail.

At the first step, the CM identifies four communities in the graph G3: CR, CRi, CR, with the sizes 44, 41, 37, 36, respectively (Figure 3). This graph is very revealing, because the communities identified by the Blondel algorithm are of the similar size. Here CR is a typical

Table 5.

The values of on of the results of partitioning the graphs by CM and other algorithms

co(BL) w(CPM) rn(GN) o(FG) œ(CONGA)

G, 0.0811 -0.0478 0.0209 0.0729 0.0226

G2 -0.0125 0.0139 0.0152 -0.0087 0.0655

G3 0.0160 -0.0166 -0.0017 0.0043 0.0149

G4 0.0459 -0.0077 -0.0098 0.0047 -0.0147

G5 0.0651 -0.0058 0.0232 0.0133 -0.0055

G6 0.0915 0.0003 0.0004 0.0159 0.0088

G7 0.0452 0.0009 0.0061 0.0020 0.0066

g8 0.0152 0.0137 -0.0070 0.0096 0.0079

Notation: o(BL) - for partitioning a graph with CM and the Blondel algorithm; m(CPM) - for partitioning a graph with CM and CPM algorithm; m(GN)- for partitioning a graph with CM and GN algorithm; m(FG)-for partitioning a graph with CM and FG algorithm; co(CONGA) - for partitioning a graph with CM and CONGA algorithm.

"junk" community containing the original vertex of the ego-graph, leaf vertices and small communities. The other three communities are more meaningful, they are user groups, most of which have no connections with representatives of the other two communities.

Then, after applying the following CM steps on the graph Gy the four communities are transformed into 28 as follows. Within the three meaningful dense communities, minimal

Fig. 3. The four communities identified by the Blondel algorithm in the graph G3

changes occur. Thus, the algorithm found one vertex less connected to the others inside the C„, for which, according to the i2c/None options, a separate community inside CRwas selected, and the remaining 40 vertices were put in the second internal community for CR.. The community CR remained completely unchanged, and for CR three vertices were found. They were separated into their individual internal communities, like in the CR2 case, meanwhile creating a large remainder — a community of 33 vertices .

The main changes occurred at the second step of the CM within CR. Thus, in this community 4 others were identified. They intersected along the original vertex of the ego-graph. In addition, similar to the situation with the other two upper-level communities, 14 individual communities within the original community were identified (Figure 4).

In other ego-graph GA with the same number of vertices, but with a different structure (Table 3), 39 communities were identified after applying CM to it. And initially, at the first step, 4 large groups were also identified (here

Fig. 4. Communities obtained by CM

I CRi | = 50, | CK% | = 44, | | = 44, | CRt | = 20) inside of which already at the second step a search for intersecting nested communities is made. In this case, CR and CR did not change, but within the Q and CR 28 and 7 nested communities with intersections of different sizes were allocated, respectively.

It should be noted that according to the users of social networks whose ego-graphs were analyzed, other partitions obtained by the compared methods carried less information in themselves. However the Q score was higher than for CM in a number of cases (Table 4). In particular, the results of algorithms that are components of CM were recognized as less informative and, for example, for GN the partitioning was partially superfluous.

Conclusion

A new algorithm, called by the authors the "Combined method" (CM), has been invented and presented to effectively iden-

tificate partitions of implicit communities in the graphs of interacting objects of various natures. We apply CM and well-known classical algorithms on the set of graphs from different industries which were taken from open sources. We studied the graph partitions into communities so obtained. Comparison of the modularity value Qm on the resultant partitions and the similarity measure co between the result of CM and the remaining algorithms showed that in most cases there are significant differences in the resulting sets of communities. In addition, for a set of graphs from the social network, a meaningful analysis of the partitions obtained was carried out, and it showed that the result of CM with the options selected on this set more accurately identifies groups of interactions between users.

The proposed algorithm was run on graphs with different parameters. During the testing, it was found that the use of the parameters of the restrictions a and /? on these graphs does not improve the quality of the resulting partition, since a result different from the results of applying the Blondel and CM algorithm (with no constraints) was achieved only in the case a > 0 and /? = 0. Further investigation was carried out at the values of the parameters a, = 0 and fi = 1. Thus, the CPM method was run on all the Blondel communities generated by the algorithm.

The research showed that the most suitable values of additional parameters for social networks are i2c/None within the CM designations. At the same time, it cannot be stated that for a graph of interacting objects of a different nature this combination will be more effective than the others. Thus, the authors consider further studies on the design and implementation of parametric algorithms for the partition of communities. We also consider a possible use of automated tools to select suitable options and parameter values based on topological features and the original nature of graphs. ■

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