Weak ties and how to find them

Abstract

This paper aims to obtain a better characterization of “weak ties”, edges between communities in a network. Community detection has been, for the last decade, one of the major topics in network science, with applications ranging from e-commerce in social networks or web page characterizations, up to control and engineering problems. There are many methods, which characterize, how well a network is split into communities or clusters, and a set of methods, based on a wide range set of principles, have been designed to find such communities. In a network, where nodes and edges are characterized only by their topological properties, communities are characterized only by being densely internally connected by edges, while edges between communities are scarce. Usually, the most convenient method how to find the in-between edges is to use the community detection methods to find the communities and from them the edges between them. However, the sole characterization of these edges, “weak ties”, is missing. When such characterization is mentioned, typically it is based on high value of edge betweenness, i.e. number of shortest paths going through the edge. While this is mostly valid for networks characterized by a high modularity value, it is very often misleading. Search for a community detection resulting in maximum modularity value is an NP complete problem, and since the selection of weak ties characterizes the communities, their selection is an NP complete problem as well. One can only hope to find a good heuristic. In our paper, we show, to what extent the high edge betweenness characterization is misleading and we design a method, which better characterizes the “weak tie” edges.