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

## TL;DR

This paper introduces unsupervised and semi-supervised algorithms for temporal ordered clustering in dynamic networks, effectively inferring the chronological order of node clusters with applications in information spread analysis.

## Contribution

It formulates the problem as an integer programming task and proposes efficient algorithms to approximate the optimal temporal clustering in dynamic graphs.

## Key findings

- Algorithms achieve high precision in ordering node pairs.
- Effective in synthetic and real-world network datasets.
- Addresses challenges in inferring order in complex network models.

## Abstract

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.