A Random Dot Product Graph Model for Weighted and Directed Networks

In its most basic form, the Random Dot Product Graph (RDPG) model assigns a low-dimensional vector to each vertex, and postulates that an edge between any two nodes exists with probability given by the inner product of said vectors. Recently, this latent position model has been extended to account for weighted graphs (the so-called Weighted (W-)RDPG), now embedding each node with a sequence of vectors. For a given node pair, the inner product between the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k-\text{th}$</tex> elements of the respective sequences specifies the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k-\text{th}$</tex> moment of the edge weight's distribution. However, graphs adhering to this nonparametric W-RDPG model are constrained to be undirected and homophilic (i.e., the adjacency matrix must be positive semidefinite in expectation). In this work, we extend the model's expressivity by proposing a variant for directed graphs, which may also include heterophilic nodes. To this end, we endow each vertex with two sequences, respectively modeling the node's incoming and outgoing connectivity behavior. We propose an embedding algorithm to estimate the latent nodal sequences from an observed adjacency matrix, and also discuss graph generation when the latent positions are given. The effectiveness of the novel weighted and directed (WD-)RDPG model is illustrated via several test cases, including both synthetic and real-life networks.