Edge Importance in Complex Networks

TL;DR

This paper compares two methods for estimating edge importance in complex networks, focusing on their computational performance and potential applications in network simplification.

Contribution

It introduces and evaluates two novel approaches—partial derivatives of communicability and Perron sensitivity—for assessing edge importance in medium to large networks.

Findings

Partial derivatives effectively identify important edges.

Perron sensitivity provides a complementary measure.

Both methods are computationally feasible for large networks.

Abstract

Complex networks are made up of vertices and edges. The latter connect the vertices. There are several ways to measure the importance of the vertices, e.g., by counting the number of edges that start or end at each vertex, or by using the subgraph centrality of the vertices. It is more difficult to assess the importance of the edges. One approach is to consider the line graph associated with the given network and determine the importance of the vertices of the line graph, but this is fairly complicated except for small networks. This paper compares two approaches to estimate the importance of edges of medium-sized to large networks. One approach computes partial derivatives of the total communicability of the weights of the edges, where a partial derivative of large magnitude indicates that the corresponding edge may be important. Our second approach computes the Perron sensitivity of the edges. A high sensitivity signals that the edge may be important. The performance of these methods and some computational aspects are discussed. Applications of interest include to determine whether a network can be replaced by a network with fewer edges with about the same communicability.