DIGRAC: Digraph Clustering Based on Flow Imbalance

Abstract

Node clustering is a powerful tool in the analysis of networks. Here, we introduce a graph neural network framework with a novel scalable Directed Mixed Path Aggregation (DIMPA) scheme to obtain node embeddings for directed networks in a self-supervised manner, including a novel probabilistic imbalance loss. The method is end-to-end in combining embedding generation and clustering without an intermediate step. In contrast to standard approaches in the literature, in this paper, directionality is not treated as a nuisance, but rather contains the main signal. In particular, we leverage the recently introduced cut flow imbalance measure, which is tightly related to directionality; cut flow imbalance is optimized without resorting to spectral methods or cluster labels. Experimental results on synthetic data, in the form of directed stochastic block models and real-world data at different scales, demonstrate that our method attains state-of-the-art results on directed graph clustering, for a wide range of noise and sparsity levels and graph structures.Preprint. Under review.

MSGNN: A Spectral Graph Neural Network Based on a Novel Magnetic Signed Laplacian

“…Importantly, unlike the other directed graph Laplacians mentioned here, the magnetic Laplacian is a complex, Hermitian matrix rather than a real, symmetric matrix. We also note [5], which constructs a GNN for node clustering on directed graphs based on flow imbalance. All of the above works are restricted to unsigned graphs, i.e., graphs with positive edge weights.…”

“…ij be a convolution matrix defined by (4) or (5). Given the ( − 1)-st layer hidden representation matrix X ( −1) , we define X ( ) columnwise by…”

“…In a signed graph, clusters can be formed by grouping together nodes with positive links and separating nodes with negative links (see [12]). In a directed graph, clusters can be determined by a directed flow on the network (see [5]). More generally, we can define clusters based on an underlying meta-graph, where meta-nodes, each of which corresponds to a cluster in the network, can be distinguished based on either signed or directional information (e.g., flow imbalance [5]).…”

“…For example, in the analysis of social networks, positive and negative edges could model friendship or enmity, and directional information could model the influence of one person on another [2,3]. Signed/directed networks also arise when analyzing time-series data with lead-lag relationships [4], detecting influential groups in social networks [5], and computing rankings from pairwise comparisons [6]. Additionally, signed and directed networks are a natural model for group conflict analysis [7], modeling the interaction network of the agents during a rumor spreading process [8], and maximizing positive influence while formulating opinions [9].…”

“…In addition to testing MSGNN on established data sets, we also devise a novel synthetic model which we call the Signed Directed Stochastic Block Model (SDSBM), which generalizes both the (undirected) Signed Stochastic Block Model from [12] and the (unsigned) Directed Stochastic Block Model from [5]. Analogous to these previous models, our SDSBM can be defined by a meta-graph structure and additional parameters describing density and noise levels.…”