Disjoint and Non-Disjoint Community Detection with Control of Overlaps Between Communities

Abstract

Overlapping community detection has become an important challenge in networks analysis that motivates researchers to propose community detection methods that best fit existing complex and non-disjoint structures in real-world networks such as social, scientific and collaborative networks. Existing overlapping community detection methods usually build large overlaps between communities, larger than expected, and do not allow users to interact with the system to regulate this size, except those allowing to include hard constraints. To solve these issues, we propose a novel non-disjoint community detection method, referred to as CDCO, which easily allows users to interact with the system and regulate overlaps between communities based on existing relationships between nodes in the network. In the same way that allowing to analysts to control the number of communities or the minimal number of actors in the community, CDCO allows to regulate overlaps using an $\alpha$ parameter which can favor or penalize overlaps. The regulation of overlaps is introduced in the objective criterion and optimized iteratively during the community detection process. Extensive experiments, conducted on both simulated and real-world networks having different sizes of overlaps, show the importance of the regulation of overlaps when a non-disjoint partitioning of the network is needed and show that CDCO outperforms existing conventional methods in terms of both F-measure and NMI.

Appendix A

Given a network containing 8 connected nodes as described in Fig. 6. We assume we want to organize the network into 2 communities $C_1$ and $C_2$ using the proposed CDCO method. We considered in this illustrative example that nodes $v_2$ and $v_4$ are initialized as the community attractor of $C_1$ and $C_2$ respectively. We give in the following the different steps of the proposed method to decide the assignment of node $v_8$ to $C_1$, to $C_2$ or to both $C_1$ and $C_2$. We give a step by step execution of the proposed method using two cases : the first case when $\alpha =0$ which leads to a non-disjoint assignment of node $v_8$ while in the second case we increase the value of $\alpha =2$ to reduce the overlaps between the two communities. We show how we can easily adjust the size of overlaps.

Fig. 6

Results of the proposed method on an illustrative example. a A network containing eight nodes. We considered a partitioning of the network into two communities $\{C_1, C_2\}$ where $\breve{a}_{1}=v_2$ and $\breve{a}_{2}=v_4$ b results of CDCO with $\alpha =0$, c results of CDCO with $\alpha =2$

A.1 First Case with $\alpha =0$

• 1. Evaluate the degree of connectivity between $v_8$ and each community using Eq. (3):

  $$\begin{aligned}&{\text {Conn}}(v_8, C_1)= {\text {Conn}}(v_8,v_2)= \frac{1}{3+2-1}= \frac{1}{4} = 0.25 \\&{\text {Conn}}(v_8, C_2)= {\text {Conn}}(v_8,v_4)= \frac{2}{3+2-2}=\frac{2}{3}= 0.66 \end{aligned}$$

• 2. Evaluate the degree of connectivity between $v_8$ and the combination of the community using Eq. (4):

  $$\begin{aligned} {\text {Conn}}(v_8, (C_1C_2) )= {\text {Conn}}(v_8,(v_2\cup v_4) )= \frac{3}{3+4-3}=\frac{3}{4} =0.75 \end{aligned}$$

  We can show from these results that the degree of connectivity of the node $v_8$ is maximal if assigned to both $C_1$ and $C_2$. However, to decide its final assignment we must evaluate the local error of node $v_8$ using Eq. (7) for each of these alternatives and take the alternative with the minimal error.

• 3. Evaluate the local error of $v_8$ when assigned to the nearest community ($C_2$) using Eq. (7):

  $$\left( {1 + \frac{2}{3}} \right)^{0} \left( {1 - \frac{2}{3}} \right) = 0.33$$

• 4. Evaluate the local error of $v_8$ when assigned to the first and to the second community ($C_2C_1$) using Eq. (7):

  $$\begin{aligned} \left( 2+\frac{3}{4}\right) ^0 \left( 1-\frac{3}{4}\right) = 0.25 \end{aligned}$$

We show now that the minimal error is obtained when $v_8$ is assigned to both ($C_2C_1$). Therefore, to minimize the objective criterion (Eq. 5), $v_8$ must be assigned to the first and to the second community. These steps must be repeated for each node $v_i$ in the network to build the partitioning matrix C. We report in Fig. 6b the obtained partitioning on this illustrative example by using the proposed method with $\alpha =0$.

A.2 Second Case: Reduce Overlaps by Using $\alpha =2$

• 1. The first and the second steps described in the first case still valid for this case. We will compute now local error with $\alpha =2$

• 2. Evaluate the local error of $v_8$ when assigned to the nearest community ($C_2$) using Eq. (7):

  $$\begin{aligned} \left( 1+\frac{2}{3}\right) ^2 \left( 1-\frac{2}{3}\right) = 0.9 \end{aligned}$$

• 3. Evaluate the local error of $v_8$ when assigned to the first and the second community ($C_2C_1$) using Eq. (7):

  $$\begin{aligned} \left( 2+\frac{3}{4}\right) ^2 \left( 1-\frac{3}{4}\right) = 1.8 \end{aligned}$$

We show now that the minimal error is obtained when $v_8$ is only assigned to the second community ($C_2$). Therefore, to minimize the objective criterion (Eq. 5), $v_8$ must be assigned to the second community ($C_2$). These steps must be repeated for each node $v_i$ in the network to build the partitioning matrix C. We report in Fig. 6c the obtained partitioning on this illustrative example by using the proposed method with $\alpha =2$.