Temporal Ordered Clustering in Dynamic Networks: Unsupervised and Semi-Supervised Learning Algorithms

In
temporal ordered clustering
, given a single snapshot of a dynamic network in which nodes arrive at distinct time instants, we aim at partitioning its nodes into
$K$
ordered clusters
$\mathcal {C}_1 \prec \cdots \prec \mathcal {C}_K$
such that for
$i< j$
, nodes in cluster
$\mathcal {C}_i$
arrived before nodes in cluster
$\mathcal {C}_j$
, with
$K$
being a data-driven parameter and not known upfront. Such a problem is of considerable significance in many applications ranging from tracking the expansion of fake news to mapping the spread of information. We first formulate our problem for a general dynamic graph, and propose an integer programming framework that finds the optimal clustering, represented as a strict partial order set, 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 develop a sequential importance procedure and design unsupervised and semi-supervised algorithms to find temporal ordered clusters that efficiently approximate the optimal solution. To illustrate the techniques, we apply our methods to the vertex copying (duplication-divergence) model which exhibits some edge-case challenges in inferring the clusters as compared to other network models. Finally, we validate the performance of the proposed algorithms on synthetic and real-world networks.