Tensor Spectral Clustering for Partitioning Higher-order Network Structures

Spectral graph theory-based methods represent an important class of tools for studying the structure of networks. Spectral methods are based on a first-order Markov chain derived from a random walk on the graph and thus they cannot take advantage of important higher-order network substructures such as triangles, cycles, and feed-forward loops. Here we propose a Tensor Spectral Clustering (TSC) algorithm that allows for modeling higher-order network structures in a graph partitioning framework. Our TSC algorithm allows the user to specify which higher-order network structures (cycles, feed-forward loops, etc.) should be preserved by the network clustering. Higher-order network structures of interest are represented using a tensor, which we then partition by developing a multilinear spectral method. Our framework can be applied to discovering layered flows in networks as well as graph anomaly detection, which we illustrate on synthetic networks. In directed networks, a higher-order structure of particular interest is the directed 3-cycle, which captures feedback loops in networks. We demonstrate that our TSC algorithm produces large partitions that cut fewer directed 3-cycles than standard spectral clustering algorithms.Comment: SDM 201

Minimum Cuts in Near-Linear Time

We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a ``semi-duality'' between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized algorithm that finds a minimum cut in an m-edge, n-vertex graph with high probability in O(m log^3 n) time. We also give a simpler randomized algorithm that finds all minimum cuts with high probability in O(n^2 log n) time. This variant has an optimal RNC parallelization. Both variants improve on the previous best time bound of O(n^2 log^3 n). Other applications of the tree-packing approach are new, nearly tight bounds on the number of near minimum cuts a graph may have and a new data structure for representing them in a space-efficient manner

A min-cut approach to functional regionalization, with a case study of the Italian local labour market areas

In several economical, statistical and geographical applications, a territory must be subdivided into functional regions. Such regions are not fixed and politically delimited, but should be identified by analyzing the interactions among all its constituent localities. This is a very delicate and important task, that often turns out to be computationally difficult. In this work we propose an innovative approach to this problem based on the solution of minimum cut problems over an undirected graph called here transitions graph. The proposed procedure guarantees that the obtained regions satisfy all the statistical conditions required when considering this type of problems. Results on real-world instances show the effectiveness of the proposed approach