Temporal Ordered Clustering in Dynamic Networks
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In temporal ordered clustering, given a single snapshot of a dynamic network, we aim at partitioning its nodes into K ordered clusters C_1 ≺...≺ C_K such that for i<j nodes in cluster C_i arrive to the dynamic graph before nodes in cluster C_j. The problem of inferring evolution of a dynamic network is of considerable significance in many applications ranging from predicting the age of proteins to track the expansion of fake news in online social networks. We first formulate our problem for a general dynamic graph, and propose an integer programming framework that finds the optimal partial order, which describes the clusters achieving the best precision (i.e., fraction of successfully ordered node pairs in the partial order) for a particular density (i.e., fraction of comparable node pairs in the partial order). We provide a method to solve a linear programming relaxation of the original optimization using importance sampling on a Markov chain corresponding to the graph evolution. Inspired by this solution, we design unsupervised and supervised algorithms to find temporal ordered clusters. Finally, we instantiate our model to the duplication-divergence model (also known as the vertex copying model) which turns out to present a real challenge when compared to other network models, as explained in the paper. We validate the proposed algorithms tailored to the duplication-divergence model on synthetic data and various real-world networks.
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