Centrality and mode detection in dynamic contact graphs; a joint diagonalisation approach

Abstract

This paper presents a technique for analysis of dynamic contact networks aimed at extracting periods of time during which the network changes behaviour. The technique is based on tracking the eigenvectors of the contact network in time (efficiently) using a technique called Joint Diagonalisation (JD). Repeated application of JD then shows that real-world networks naturally break into several modes of operation which are time dependent and in one real-world case, even periodic. This shows that a view of real-world contact networks as realisations from a single underlying static graph is mistaken. However, the analysis also shows that a small finite set of underlying static graphs can approximate the dynamic contact graphs studied. We also provide the means by which these underlying approximate graphs can be constructed. Core to the approach is the analysis of spanning trees constructed on the contact network. These trees are the routes a broadcast would take given a random starting location and we find that these propagation paths (in terms of their eigenvector decompositions) cluster into a small subset of modes which surprisingly correspond to clusters in time. The net result is that a dynamic network may be approximated as a (small finite) set of static graphs. Most interestingly the MIT dataset shows a periodic behaviour which allows us to know in advance which mode the network will be in. This has obvious consequences as individuals in the network take differing roles in differing modes. Finally, we demonstrate the technique by constructing a synthetic network with an 4 underlying modes of operation; creating synthetics contacts and then used JD to extract the original underlying modes.