Local multiresolution order in community detection

Community detection algorithms attempt to find the best clusters of nodes in an arbitrary complex network. Multi-scale ("multiresolution") community detection extends the problem to identify the best network scale(s) for these clusters. The latter task is generally accomplished by analyzing community stability simultaneously for all clusters in the network. In the current work, we extend this general approach to define local multiresolution methods, which enable the extraction of well-defined local communities even if the global community structure is vaguely defined in an average sense. Toward this end, we propose measures analogous to variation of information and normalized mutual information that are used to quantitatively identify the best resolution(s) at the community level based on correlations between clusters in independently-solved systems. We demonstrate our method on two constructed networks as well as a real network and draw inferences about local community strength. Our approach is independent of the applied community detection algorithm save for the inherent requirement that the method be able to identify communities across different network scales, with appropriate changes to account for how different resolutions are evaluated or defined in a particular community detection method. It should, in principle, easily adapt to alternative community comparison measures.Comment: 19 pages, 11 figure

Physical Models in Community Detection with Applications to Identifying Structure in Complex Amorphous Systems

We present an exceptionally accurate spin-glass-type Potts model for the graph theoretic problem of community detection. With a simple algorithm, we find that our approach is exceptionally accurate, robust to the effects of noise, and competitive with the best currently available algorithms in terms of speed and the size of solvable systems. Being a local measure of community structure, our Potts model is free from a resolution limit that hinders community solutions for some popular community detection models. It further remains a local measure on weighted and directed graphs. We apply our community detection method to accurately and quantitatively evaluate the multi-scale: multiresolution ) structure of a graph. Our multiresolution algorithm calculates correlations among multiple copies: replicas ) of the same graph over a range of resolutions. Significant multiresolution structures are identified by strongly correlated replicas. The average normalized mutual information and variation of information give a quantitative estimate of the best resolutions and indicate the relative strength of the structures in the graph. We further investigate a phase transition effect in community detection, and we elaborate on its relation to analogous physical phase transitions. Finally, we apply our community detection methods to ascertain the most natural complex amorphous structures in two model glasses in an unbiased manner. We construct a model graph for the physical systems using the potential energy to generate weighted edge relationships for all pairs of atoms. We then solve for the communities within the model network and associate the best communities with the natural structures in the physical systems

Heavy Tails, Generalized Coding, and Optimal Web Layout

This paper considers Web layout design in the spirit of source coding for data compression and rate distortion theory, with the aim of minimizing the average size of files downloaded during Web browsing sessions. The novel aspect here is that the object of design is layout rather than codeword selection, and is subject to navigability constraints. This produces statistics for file transfers that are heavy tailed, completely unlike standard Shannon theory, and provides a natural and plausible explanation for the origin of observed power laws in Web traffic. We introduce a series of theoretical and simulation models for optimal Web layout design with varying levels of analytic tractability and realism with respect to modeling of structure, hyperlinks, and user behavior. All models produce power laws which are striking both for their consistency with each other and with observed data, and their robustness to modeling assumptions. These results suggest that heavy tails are a permanent and ubiquitous feature of Internet traffic, and not an artifice of current applications or user behavior. They also suggest new ways of thinking about protocol design that combines insights from information and control theory with traditional networking

Towards Online Multiresolution Community Detection in Large-Scale Networks

The investigation of community structure in networks has aroused great interest in multiple disciplines. One of the challenges is to find local communities from a starting vertex in a network without global information about the entire network. Many existing methods tend to be accurate depending on a priori assumptions of network properties and predefined parameters. In this paper, we introduce a new quality function of local community and present a fast local expansion algorithm for uncovering communities in large-scale networks. The proposed algorithm can detect multiresolution community from a source vertex or communities covering the whole network. Experimental results show that the proposed algorithm is efficient and well-behaved in both real-world and synthetic networks