Spectral partitioning of time-varying networks with unobserved edges

We discuss a variant of `blind’ community detection, in which we aim to partition an unobserved network from the observation of a (dynamical) graph signal defined on the network. We consider a scenario where our observed graph signals are obtained by filtering white noise input, and the underlying network is different for every observation. In this fashion, the filtered graph signals can be interpreted as defined on a time-varying network. We model each of the underlying network realizations as generated by an independent draw from a latent stochastic blockmodel (SBM). To infer the partition of the latent SBM, we propose a simple spectral algorithm for which we provide a theoretical analysis and establish consistency guarantees for the recovery. We illustrate our results using numerical experiments on synthetic and real data, highlighting the efficacy of our approach.

💡 Research Summary

The paper tackles the problem of uncovering community structure in a network that is never directly observed. Instead of trying to infer the exact adjacency matrix, the authors assume that at each of m time instances a graph signal y(ℓ) is generated by filtering white‑noise w(ℓ) through a graph filter H(L(ℓ)) where L(ℓ) is the Laplacian of a graph drawn independently from a stochastic block model (SBM) with fixed affinity matrix Ω. The underlying graph therefore changes from sample to sample, but the SBM parameters remain the same. The goal is to recover the latent partition of the SBM using only the collection {y(ℓ)}ₗ₌₁^m.

Algorithm 1 is remarkably simple: compute the empirical covariance Ĉ_y = (1/m)∑ₗ y(ℓ)y(ℓ)ᵀ, perform an eigen‑decomposition, keep the top k eigenvectors, and run k‑means on their rows to obtain a partition of the n nodes. No adjacency information is required, and the number of communities k is assumed known (it could be estimated from the spectrum).

The theoretical contribution proceeds in three steps. Proposition 1 shows that, under mild conditions (bounded spectral norm of the filter and bounded moments of the noise), the empirical covariance converges to its expectation C_y with a rate O((log log n)² · n^{½−2/q}/√m). This result holds for any filter and does not yet exploit the SBM structure. Proposition 2 then derives the exact form of C_y when the latent graphs follow a planted‑partition SBM with two equal‑size blocks. The covariance can be written as
  C_y = (c₃−c₁)I + G