Stability of graph communities across time scales

Abstract

27/11/12 mebDear Professor Barahona, Re: Stability of graph communities across time scales Flow graphs: Interweaving dynamics and structure A dynamical model of lipoprotein metabolism Thank you for your submissions to Spiral .Could you please advice on the version of the paper you uploaded using the arXiv source. I have not been able to assess if this is an author version peer-reviewed or is it an author version non peer reviewed. Spiral digital repository only accept peer-reviewed papers. 29/11/12 meb. Peer reviewed version OK to Pub.The complexity of biological, social, and engineering networks makes it desirable to find natural partitions into clusters ( or communities) that can provide insight into the structure of the overall system and even act as simplified functional descriptions. Although methods for community detection abound, there is a lack of consensus on how to quantify and rank the quality of partitions. We introduce here the stability of a partition, a measure of its quality as a community structure based on the clustered autocovariance of a dynamic Markov process taking place on the network. Because the stability has an intrinsic dependence on time scales of the graph, it allows us to compare and rank partitions at each time and also to establish the time spans over which partitions are optimal. Hence the Markov time acts effectively as an intrinsic resolution parameter that establishes a hierarchy of increasingly coarser communities. Our dynamical definition provides a unifying framework for several standard partitioning measures: modularity and normalized cut size can be interpreted as one-step time measures, whereas Fiedler’s spectral clustering emerges at long times. We apply our method to characterize the relevance of partitions over time for constructive and real networks, including hierarchical graphs and social networks, and use it to obtain reduced descriptions for atomic-level protein structures over different time scales