An interaction-based method for detecting overlapping community structure in real-world networks

Abstract

A central theme of network analysis, these days, is the detection of community structure as it offers a coarse-grained view of the network at hand. A more interesting and challenging task in network analysis involves the detection of overlapping community structure due to its wide-spread applications in synthesising and interpreting the data arising from social, biological and other diverse fields. Certain real-world networks possess a large number of nodes whose memberships are spread through multiple groups. This phenomenon called community structure with pervasive overlaps has been addressed partially by the development of a few well-known algorithms. In this paper, we presented an algorithm called Interaction Coefficient-based Local Community Detection (IC-LCD) that not only uncovers the community structures with pervasive overlaps but do so efficiently. The algorithm extracted communities through a local expansion strategy which underlie the notion of interaction coefficient. We evaluated the performance of IC-LCD on different parameters such as speed, accuracy and stability on a number of synthetic and real-world networks, and compared the results with well-known baseline algorithms, namely DEMON, OSLOM, SLPA and COPRA. The results give a clear indication that IC-LCD gives competitive performance with the chosen baseline algorithms in uncovering the community structures with pervasive overlaps. The time complexity of IC-LCD is $\mathcal {O}(nc_{\max })$, where n is the number of nodes, and $c_{\max }$ is the maximum size of a community detected in a network.

Appendix

Here we shall illustrate the concepts of node and edge interaction coefficients, with the help of examples. Then we shall see how the seed expansion takes place using these coefficients.

1.1 Appendix A.1: Illustration of node and edge interaction coefficients

Consider the graph given below (Fig. 5).

Fig. 5

Expansion of the seeds $u_0$ and $v_0$

Take $C_1 = \lbrace u_0 \rbrace$, and $C_2 = \lbrace v_0 \rbrace$. Let us expand $C_1$, and $C_2$ using the node interaction coefficient $\xi _{\text {node}}$, taking $\xi _0 = 0.5$. Note that $N_{C_1} = \lbrace u_1, u_2, u_3, u_4 \rbrace$ and $N_{C_2} = \lbrace v_1, v_2, v_3, v_4 \rbrace$. For each $1 \le i \le 4$, we have

$$\begin{aligned} \xi _\text {node}(u_i, C_1) = \frac{1}{5} < \xi _0. \end{aligned}$$

This means $C_1$ would not expand. However, for each $1 \le i \le 4$ we have

$$\begin{aligned} \xi _\text {node}(v_i, C_2) = \frac{1}{2} = \xi _0, \end{aligned}$$

which means $C_2$ can expand to all its neighbours, and becomes $C_2 = \lbrace v_0, v_1, v_2, v_3, v_4 \rbrace$. So, $N_{C_2} = \lbrace v_5, v_6, \ldots , v_{12} \rbrace$. Now for each $5 \le i \le 12$, we have

$$\begin{aligned} \xi _\text {node}(v_i, C_2) = \frac{1}{4} < \xi _0, \end{aligned}$$

which means $C_2$ cannot be expanded further. The case we have considered is specific. But, it captures the two important types of seeds which are—highly clustered, and lowly clustered. The same strategy, such as the one based on node interaction coefficient, will not work for the expansion of both the kinds of seeds.

Fig. 6

A network with community structure

Therefore, we have introduced the concept of edge interaction coefficient $\xi _{{\text {edge}}}$. An edge $e_{uv}$ essentially interacts with a subgraph C through its endpoints u and v. To arrive at a formula for $\xi _\text {edge}(e_{uv}, C)$, we use the following assumption: If both u and v have more neighbours in C, then $e_{uv}$ interacts with C highly. So, look at the quantity

$$\begin{aligned} \min \bigl \lbrace \left| N_u \cap V_C \right| , \left| N_v \cap V_C \right| \bigr \rbrace . \end{aligned}$$

To normalise it we can divide it by the minimum or the maximum of the degrees of u and v. Moreover, we wish $\xi _\text {edge}(e_{uv}, C)$ to be highest when $N_u \subseteq V_C \backslash \lbrace v \rbrace$, $N_v \subseteq V_C \backslash \lbrace u \rbrace$, and $d_u = d_v$. Keeping, all these requirements, we get Eq. (2). It is apparent that $0 \le \xi _{\text {edge}} \le 1$. It can be seen that $\xi _\text {edge}(e_{uv}, C) = 1$ iff $d_u = d_v$ and $\left| N_u \cap V_C\right| = \left| N_v \cap V_C \right|$. In the denominator of Eq. (2) we have taken $\max \lbrace d_u, d_v \rbrace$ instead of $\min \lbrace d_u, d_v \rbrace$. To see why let us look at the case given in the picture below.

The node u has 3 neighbours in C, and 6 neighbours outside C. So, $\xi _\text {node}(u, C) = 1/3$ which is much smaller than the threshold $\xi _0$. Consequently, u must not join C in any case. However, v has 5 neighbours in C and just 2 neighbours outside C. So, v would surely join C. Now let us compute the node interaction coefficient of u with $C \cup \lbrace v \rbrace$. We have

$$\begin{aligned} \xi _\text {node}(u, C \cup \lbrace v \rbrace ) = \frac{4}{9} < \xi _0 \end{aligned}$$

Thus u would not join $C \cup \lbrace v \rbrace$ too. Consider, now the case when $\min \lbrace d_u, d_v \rbrace$ is the numerator in Eq. (2). Then

$$\begin{aligned} \xi _\text {edge}(e_{uv}, C) = \frac{1+3}{7} = \frac{4}{7} > \xi _0 \end{aligned}$$

In this case u joins C. Thus $\min \lbrace d_u, d_v \rbrace$ is not an appropriate choice for the denominator in Eq. (2).

1.2 Appendix A.2: Illustration of seed expansion phase

Table 6 Before augmentation step, $V_{\text {new}} = \varnothing$ for $C = \lbrace 30, 31 \rbrace$ in Fig. 6

Table 7 Before augmentation step, $V_{\text {new}} = \varnothing$ for $C = \lbrace 1,2,3 \rbrace$ in Fig. 6

We illustrate the GET-NEW-NODES() procedure through examples. Note that we do not specify any criterion for selecting seeds, so any node may serve as a seed. Then it may well happen that certain seeds, especially the low degree nodes, stop expanding after growing to few nodes, or do not expand at all. Let us consider a few examples assuming that $\xi _0 = 0.5$ and $n_{\min } = 4$.

Example 1

Consider the graph given in Fig. 6.

Let $C = \lbrace 30, 31 \rbrace$. Then $N_C = \lbrace 25, 26, 29 \rbrace$. In order to compute $V_{\text {new}}$, the steps followed before the augmentation step are listed in Table 6. No node of $N_C$ is added to $V_{\text {new}}$, leaving $V_{\text {new}}$ empty. On the other hand, during the augmentation step, we find that $\xi _\text {node}(25, C \cup N_C) = 1/2, \xi _\text {node}(26, C \cup N_C) = 3/4$ and $\xi _\text {node}(29, C \cup N_C) = 3/5$, which makes $V_{\text {new}} = \lbrace 25, 26, 29 \rbrace$.

Example 2

This time consider the subgraph $C = \lbrace 1,2,3 \rbrace$ in the graph given in Fig. 6. Here $N_C = \lbrace 4,5,14,22,23 \rbrace$. Then before the augmentation step $V_{\text {new}}$ remains empty as shown in Table 7. However, in this case even the augmentation step does not help, as $\xi _\text {node}(u, C \cup N_C) < \xi _0$ for all $u \in N_C$, and so, $V_{\text {new}} = \varnothing$. Thus the subgraph C is not expandable to a full community. Such groups of nodes are likely to join multiple communities and form the basis for pervasive overlaps.