Learning Bollobás-Riordan Graphs Under Partial Observability

Abstract

This work examines the problem of learning the topology of a network (graph learning) from the signals produced at a subset of the network nodes (partial observability). This challenging problem was recently tackled assuming that the topology is drawn according to an Erdős-Rényi model, for which it was shown that graph learning under partial observability is achievable, exploiting in particular homogeneity across nodes and independence across edges. However, several real-world networks do not match the optimistic assumptions of homogeneity/independence, for example, high heterogeneity is often observed between very connected nodes (hubs) and scarcely connected peripheral nodes. Random graphs with preferential attachment were conceived to overcome these issues. In this work, we discover that, over first-order vector autoregressive systems with a stable Laplacian combination matrix, graph learning is achievable under partial observability, when the network topology is drawn according to a popular preferential attachment model known as the Bollobás-Riordan model.

I Can Read Your Mind: Control Mechanism Secrecy of Networked Dynamical Systems under Inference Attacks

“…Under specific conditions, it is possible to obtain an (asymptotically) accurate estimate of the local topology by estimator (46). Recent works [68], [81], [82] have made some progress in deriving the conditions, for example, i) the topology is in symmetric Erdős-Rényi random graph form with vanishing connection probability, and ii) the ratio of the observable nodes to all nodes converges to constant as the network goes to infinity. These methods cannot infer directed topology where the specified node conditions are not available.…”

“…Corollary 3 (see [97]). Consider the system (82). If the added noises satisfy the exact convergence condition lim k→∞ k−1 l=0 W k−l−1 θ(l) = 0, then the noise-adding algorithm cannot guarantee the the differential privacy of the initial state.…”

Topology Inference on Partially Observable Mobile Robotic Networks under Formation Control

“…The works [24], [53], [54] have explored the conditions of using the truncated estimator to approximate the ground truth 1 . Nevertheless, these conditions are not consistent with our problem setting, and ŴFF is far away from the ground truth from basic linear algebra, i.e., More precisely, let V F = V\V F and the formation dynamics (3) can be divided into…”

“…Despite the asymptotic boundedness of the OLS estimator (54), the proposed method nevertheless can be used as the basis for inferring the local topology when a finite number of observations are available.…”

“…In[24],[53],[54], the conditions of using estimator(25) are summarized as: i) the topology is in symmetric Erdős-Rényi random graph form with vanishing connection probability, and ii) the ratio of the observable nodes to all nodes converges to constant as the size of the network goes to infinity.…”