Unit Disk Graph-Based Node Similarity Index for Complex Network Analysis

Natarajan Meghanathan

doi:10.1155/2019/6871874

Abstract

We seek to quantify the extent of similarity among nodes in a complex network with respect to two or more node-level metrics (like centrality metrics). In this pursuit, we propose the following unit disk graph-based approach: we first normalize the values for the node-level metrics (using the sum of the squares approach) and construct a unit disk graph of the network in a coordinate system based on the normalized values of the node-level metrics. There exists an edge between two vertices in the unit disk graph if the Euclidean distance between the two vertices in the normalized coordinate system is within a threshold value (ranging from 0 tok, where k is the number of node-level metrics considered). We run a binary search algorithm to determine the minimum value for the threshold distance that would yield a connected unit disk graph of the vertices. We refer to “1 − (minimum threshold distance/k)” as the node similarity index (NSI; ranging from 0 to 1) for the complex network with respect to the k node-level metrics considered. We evaluate the NSI values for a suite of 60 real-world networks with respect to both neighborhood-based centrality metrics (degree centrality and eigenvector centrality) and shortest path-based centrality metrics (betweenness centrality and closeness centrality).

Highlights

The weights assigned to nodes (a.k.a. vertices) in a complex network are either topology-based or domain-based or a combination of both
For ease of presentation and visualization, we show the distribution of the vertices in the example graph using two dimensions at a time: the neighborhood-based degree centrality (DEG) and eigenvector centrality (EVC) metrics and the shortest path-based betweenness centrality (BWC) and closeness centrality (CLC) metrics
The division of the minimum threshold distance by √k would negate the impact of the number of node-level metrics considered for similarity assessment and capture the impact of the actual node-level metrics considered in their entirety

Summary

Introduction

The weights assigned to nodes (a.k.a. vertices) in a complex network are either topology-based or domain-based or a combination of both. The rest of the paper is organized as follows: Section 2 describes the proposed procedure to construct the unit disk graph of the vertices based on a coordinate system comprising of the normalized values for the node-level metrics as well as explains the use of the binary search algorithm to determine the minimum threshold distance value that is required to obtain a connected unit disk graph; the section analyzes the time complexity and memory space requirements of the binary search algorithm as well as illustrates the whole process using a toy network of eight vertices. The memory requirements of the algorithm is O(V2), where V is the number of vertices in the real-world network graph GR as well

Overview of the Real-World Networks

Node Similarity Index of the Real-World Networks

Literature Network

Related Work

Findings

Conclusions