Bobak Nazer

Aditya Gangrade, Praveen Venkatesh, Bobak Nazer et al.

We propose and analyze the problems of \textit{community goodness-of-fit and two-sample testing} for stochastic block models (SBM), where changes arise due to modification in community memberships of nodes. Motivated by practical applications, we consider the challenging sparse regime, where expected node degrees are constant, and the inter-community mean degree ($b$) scales proportionally to intra-community mean degree ($a$). Prior work has sharply characterized partial or full community recovery in terms of a "signal-to-noise ratio" ($\mathrm{SNR}$) based on $a$ and $b$. For both problems, we propose computationally-efficient tests that can succeed far beyond the regime where recovery of community membership is even possible. Overall, for large changes, $s \gg \sqrt{n}$, we need only $\mathrm{SNR}= O(1)$ whereas a naïve test based on community recovery with $O(s)$ errors requires $\mathrm{SNR}= Θ(\log n)$. Conversely, in the small change regime, $s \ll \sqrt{n}$, via an information-theoretic lower bound, we show that, surprisingly, no algorithm can do better than the naïve algorithm that first estimates the community up to $O(s)$ errors and then detects changes. We validate these phenomena numerically on SBMs and on real-world datasets as well as Markov Random Fields where we only observe node data rather than the existence of links.

Aditya Gangrade, Bobak Nazer, Venkatesh Saligrama

The change detection problem is to determine if the Markov network structures of two Markov random fields differ from one another given two sets of samples drawn from the respective underlying distributions. We study the trade-off between the sample sizes and the reliability of change detection, measured as a minimax risk, for the important cases of the Ising models and the Gaussian Markov random fields restricted to the models which have network structures with $p$ nodes and degree at most $d$, and obtain information-theoretic lower bounds for reliable change detection over these models. We show that for the Ising model, $Ω\left(\frac{d^2}{(\log d)^2}\log p\right)$ samples are required from each dataset to detect even the sparsest possible changes, and that for the Gaussian, $Ω\left( γ^{-2} \log(p)\right)$ samples are required from each dataset to detect change, where $γ$ is the smallest ratio of off-diagonal to diagonal terms in the precision matrices of the distributions. These bounds are compared to the corresponding results in structure learning, and closely match them under mild conditions on the model parameters. Thus, our change detection bounds inherit partial tightness from the structure learning schemes in previous literature, demonstrating that in certain parameter regimes, the naive structure learning based approach to change detection is minimax optimal up to constant factors.