Learning Erdős-Rényi Graphs under Partial Observations: Concentration or Sparsity?

This work examines the problem of graph learning over a diffusion network when data can be collected from a limited portion of the network (partial observability). While most works in the literature rely on a degree of sparsity to provide guarantees of consistent graph recovery, our analysis moves away from this condition and includes the demanding setting of dense connectivity. We ascertain that suitable estimators of the combination matrix (i.e., the matrix that quantifies the pairwise interaction between nodes) possess an identifiability gap that enables the discrimination between connected and disconnected nodes. Fundamental conditions are established under which the subgraph of monitored nodes can be recovered, with high probability as the network size increases, through universal clustering algorithms. This claim is proved for three matrix estimators: i) the Granger estimator that adapts to the partial observability setting the solution that is optimal under full observability ; ii) the one-lag correlation matrix; and iii) the residual estimator based on the difference between two consecutive time samples. Comparison among the estimators is performed through illustrative examples that reveal how estimators that are not optimal in the full observability regime can outperform the Granger estimator in the partial observability regime. The analysis reveals that the fundamental property enabling consistent graph learning is the statistical concentration of node degrees, rather than the sparsity of connections.