Signed Graph Learning with Hidden Nodes

Signed graphs, which are characterized by both positive and negative edge weights, have recently attracted significant attention in the field of graph signal processing (GSP). Existing works on signed graph learning typically assume that all graph nodes are available. However, in some specific applications, only a subset of nodes can be observed while the remaining nodes stay hidden. To address this challenge, we propose a novel method for identifying signed graph that accounts for hidden nodes, termed \textit{signed graph learning with hidden nodes under column-sparsity regularization} (SGL-HNCS). Our method is based on the assumption that graph signals are smooth over signed graphs, i.e., signal values of two nodes connected by positive (negative) edges are similar (dissimilar). Rooted in this prior assumption, the topology inference of a signed graph is formulated as a constrained optimization problem with column-sparsity regularization, where the goal is to reconstruct the signed graph Laplacian matrix without disregarding the influence of hidden nodes. We solve the constrained optimization problem using a tailored block coordinate descent (BCD) approach. Experimental results using synthetic data and real-world data demonstrate the efficiency of the proposed SGL-HNCS method.