Overlapping community detection algorithms using Modularity and the cosine

TL;DR

Two overlapping network community detection algorithms are presented that build on the two-step approach, using the extended modularity and cosine function, using the extended modularity and cosine function.

Abstract

The issue of network community detection has been extensively studied across many fields. Most community detection methods assume that nodes belong to only one community. However, in many cases, nodes can belong to multiple communities simultaneously.This paper presents two overlapping network community detection algorithms that build on the two-step approach, using the extended modularity and cosine function. The applicability of our algorithms extends to both undirected and directed graph structures. To demonstrate the feasibility and effectiveness of these algorithms, we conducted experiments using real data.

Figures (15)

• Figure 1: Applying the Parameterized Modularity Overlap Algorithm for undirected graphs with $\theta=1$, we obtained three communities are $C_1=\{1,2,3,4,5\}$, $C_2=\{5,6,7,8,9,10\}$, and $C_3=\{10,11,12,13,14,15,16\}$. Vertex $5$ belongs to communities $C_1$ and $C_2$. Vertex $10$ belongs to both communities $C_2$ and $C_3$.
• Figure 2: Applying the Cosine Overlap Algorithm with $\theta=0.6$, we obtained three communities $C1=\{4, 5, 11, 14, 15, 16, 21, 22, 27, 19\}$, $C2=\{1, 2, 6, 7, 9, 10, 13, 17, 19, 23\}$ and $C3=\{0, 3, 8, 12, 18, 20, 24, 25, 26, 28, 29, 5, 7, 22\}$. Vertex $19$ belongs to communities $C_1$ and $C_2$. Vertex $7$ belongs to both communities $C_2$ and $C_3$, and vertices $5$, $22$ belongs to both communities $C_1$ and $C_3$.
• Figure 3: Applying the Parameterized Modularity Overlap Algorithm for directed graphs with $\theta=1$, we obtained two communities $C_1=\{1,2,3,4,10\}$ and $C_2=\{5,6,7,8,9,10\}$. Vertex $10$ belongs to communities $C_1$ and $C_2$.
• Figure 4: This chart illustrates the maximum modularity obtained in Experiment 1 for undirected graphs; we experimented on ten randomly generated graphs using the LFR benchmark with all the parameters taken with a uniform distribution in the following corresponding intervals: $N \in [400;500]$, $on \in [60;80]$, $om \in [2;5]$.
• Figure 5: This chart illustrates the maximum ONMI obtained in Experiment 1 for undirected graphs; we experimented on ten randomly generated graphs using the LFR benchmark with all the parameters taken with a uniform distribution in the following corresponding intervals: $N \in [400;500]$, $on \in [60;80]$, $om \in [2;5]$.

Theorems & Definitions (5)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Definition 3.1