On Detection and Structural Reconstruction of Small-World Random Networks

TL;DR

This paper investigates the detection and reconstruction of small-world networks, specifically the Watts-Strogatz model, analyzing the computational and statistical challenges across different parameter regimes.

Contribution

It introduces methods tailored to various difficulty regions in detecting and reconstructing the Watts-Strogatz small-world network structure.

Findings

Identifies phase transitions in detectability and reconstructability.

Proposes algorithms for different regions of the parameter space.

Provides theoretical analysis of the problem's complexity.

Abstract

In this paper, we study detection and fast reconstruction of the celebrated Watts-Strogatz (WS) small-world random graph model \citep{watts1998collective} which aims to describe real-world complex networks that exhibit both high clustering and short average length properties. The WS model with neighborhood size $k$ and rewiring probability probability $\beta$ can be viewed as a continuous interpolation between a deterministic ring lattice graph and the Erd\H{o}s-R\'{e}nyi random graph. We study both the computational and statistical aspects of detecting the deterministic ring lattice structure (or local geographical links, strong ties) in the presence of random connections (or long range links, weak ties), and for its recovery. The phase diagram in terms of $(k,\beta)$ is partitioned into several regions according to the difficulty of the problem. We propose distinct methods for the various regions.