On Detection and Structural Reconstruction of Small-World Random Networks

In this paper, we study detection and fast reconstruction of the celebrated Watts-Strogatz (WS) small-world random graph model <xref ref-type="bibr" rid="ref29">[29]</xref> which aims to describe real-world complex networks that exhibit both high clustering and short average length properties. The WS model with neighborhood size <inline-formula> <tex-math notation="LaTeX">$k$</tex-math></inline-formula> and rewiring probability probability <inline-formula> <tex-math notation="LaTeX">$\beta$</tex-math><alternatives><inline-graphic xlink:href="liang-ieq2-2703102.gif"/> </alternatives></inline-formula> can be viewed as a continuous interpolation between a deterministic ring lattice graph and the Erdős-Rényi random graph. We study the computational and statistical aspects of detection and recovery of the deterministic ring lattice structure (strong ties) in the presence of random connections (weak ties). The phase diagram in terms of <inline-formula><tex-math notation="LaTeX">$(k,\beta)$</tex-math><alternatives> <inline-graphic xlink:href="liang-ieq3-2703102.gif"/></alternatives></inline-formula> is shown to consist of several regions according to the difficulty of the problem. We propose distinct methods for these regions.

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