Estimating the Topology of Preferential Attachment Graphs Under Partial Observability

This work addresses the problem of learning the topology of a network from the signals emitted by the network nodes. These signals are generated over time through a linear diffusion process, where neighboring nodes exchange messages according to the underlying network topology, and aggregate them according to a certain combination matrix. We consider the demanding setting of graph learning under partial observability, where signals are available only from a limited fraction of nodes, and we want to establish whether the topology of these nodes can be estimated faithfully, despite the presence of possibly many latent nodes. Recent results examined this problem when the network topology is generated according to an Erdős-Rényi random model. However, Erdős-Rényi graphs feature homogeneity across nodes and independence across edges, while, over several real-world networks, significant heterogeneity is observed between very connected "hubs" and scarcely connected peripheral nodes, and the edge construction process entails significant dependencies across edges. Preferential attachment random graphs were conceived primarily to fill these gaps. We tackle the problem of graph learning over preferential attachment graphs by focusing on the following setting: first-order vector autoregressive models equipped with a stable Laplacian combination matrix, and a network topology drawn according to the popular Bollobás-Riordan preferential attachment model. The main result established in this work is that a combination-matrix estimator known as Granger estimator achieves graph learning under partial observability.