Communities in Networks

We survey some of the concepts, methods, and applications of community detection, which has become an increasingly important area of network science. To help ease newcomers into the field, we provide a guide to available methodology and open problems, and discuss why scientists from diverse backgrounds are interested in these problems. As a running theme, we emphasize the connections of community detection to problems in statistical physics and computational optimization.Comment: survey/review article on community structure in networks; published version is available at http://people.maths.ox.ac.uk/~porterm/papers/comnotices.pd

Recent Advances in Graph Partitioning

We survey recent trends in practical algorithms for balanced graph partitioning together with applications and future research directions

Communication Network Design: Balancing Modularity and Mixing via Optimal Graph Spectra

By leveraging information technologies, organizations now have the ability to design their communication networks and crowdsourcing platforms to pursue various performance goals, but existing research on network design does not account for the specific features of social networks, such as the notion of teams. We fill this gap by demonstrating how desirable aspects of organizational structure can be mapped parsimoniously onto the spectrum of the graph Laplacian allowing the specification of structural objectives and build on recent advances in non-convex programming to optimize them. This design framework is general, but we focus here on the problem of creating graphs that balance high modularity and low mixing time, and show how "liaisons" rather than brokers maximize this objective

Quadratically-Regularized Optimal Transport on Graphs

Optimal transportation provides a means of lifting distances between points on a geometric domain to distances between signals over the domain, expressed as probability distributions. On a graph, transportation problems can be used to express challenging tasks involving matching supply to demand with minimal shipment expense; in discrete language, these become minimum-cost network flow problems. Regularization typically is needed to ensure uniqueness for the linear ground distance case and to improve optimization convergence; state-of-the-art techniques employ entropic regularization on the transportation matrix. In this paper, we explore a quadratic alternative to entropic regularization for transport over a graph. We theoretically analyze the behavior of quadratically-regularized graph transport, characterizing how regularization affects the structure of flows in the regime of small but nonzero regularization. We further exploit elegant second-order structure in the dual of this problem to derive an easily-implemented Newton-type optimization algorithm.Comment: 27 page