Abstract
Identification of community structures in complex network is of crucial importance for understanding the system’s function, organization, robustness and security. Here, we present a novel Ollivier-Ricci curvature (ORC) inspired approach to community identification in complex networks. We demonstrate that the intrinsic geometric underpinning of the ORC offers a natural approach to discover inherent community structures within a network based on interaction among entities. We develop an ORC-based community identification algorithm based on the idea of sequential removal of negatively curved edges symptomatic of high interactions (e.g., traffic, attraction). To illustrate and compare the performance with other community identification methods, we examine the ORC-based algorithm with stochastic block model artificial networks and real-world examples ranging from social to drug-drug interaction networks. The ORC-based algorithm is able to identify communities with either better or comparable performance accuracy and to discover finer hierarchical structures of the network. This opens new geometric avenues for analysis of complex networks dynamics.
Highlights
In this work, we propose a novel geometric approach in network community identification by using the Ollivier-Ricci curvature[5] (ORC) concept
The Ollivier-Ricci curvature (ORC) captures two fundamental properties of the structure of complex networks: First, the ORC associated with each edge of the network encodes its shortest path characteristics[6]
The ORC provides information about the frequency of triangles, characterized by the clustering coefficient, within a neighborhood of two adjacent vertices[12,13]. Starting from these premises, in this work, we aim to address the following questions: Can the ORC help us discover the underlying hierarchical functional characteristics of a complex network? Can the ORC curvature provide algorithmic hints towards solving the hard problem of community identification?
Summary
We propose a novel geometric approach in network community identification by using the Ollivier-Ricci curvature[5] (ORC) concept. The ORC captures the notion of network flows of shortest paths via the Wasserstein’s distance formulation wherein a negatively curved edge is a “bottleneck”, along which traffic is intense in a scheme that minimizes the “cost” of transferring “commodities”, while, positively curved edges contribute to transport of “commodities” along with many other edges. Negatively curved edges could be interpreted as “bridges” between communities and cutting them would isolate the network flow between communities. In this context, a “community” is defined as a robust transport of information within the community. Robust means that if some edges are cut information is still going to flow. The ORC is recently being applied as a tool in various research areas such as in wireless networking[6,7], quantum computation[8,9] as well as robustness analysis of complex networks[10,11]
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