Temporal Ordered Clustering in Dynamic Networks

Krzysztof Turowski, Jithin K. Sreedharan, Wojciech Szpankowski

Abstract: Given a single snapshot of a dynamic network in which nodes arrived at distinct time instants along with edges, we aim at inferring a partial order σ between the node pairs such that u <σ v indicates node u arrived earlier than node v in the graph. The inferred partial order can be deduced to a natural clustering of the nodes into K ordered clusters C1 ≺ ⋯ ≺ CK such that for i < j, nodes in cluster Ci joined the network before nodes in cluster Cj, with K being a data-driven parameter and not known upfront. We first formulate our problem for a general dynamic graph, and propose an integer programming framework that finds the optimal partial order, achieving the best precision (i.e., fraction of successfully ordered node pairs) for a fixed density (i.e., fraction of comparable node pairs). We then design algorithms to find temporal ordered clusters that efficiently approximate the optimal solution. To illustrate our techniques, we apply our methods to the vertex copying model (also known as the duplication-divergence model).