Frustrated Random Walks: A Fast Method to Compute Node Distances on Hypergraphs
Enzhi Li, Scott Nickleach, Bilal Fadlallah
TL;DR
This work tackles the problem of computing node distances on hypergraphs by leveraging expected hitting times of random walks. It introduces frustrated random walks (FRW), which incorporate an acceptance probability to counteract the bias of simple random walks in heavily-weighted, scale-free hypergraphs, and presents a unified framework to compute hitting times for both SRW and FRW via a recurrence and generating-function approach. The authors derive explicit constructions for the SRW and FRW transition matrices, formulate the associated $B$ matrices, and show how to obtain the expected hitting times by solving a linear system with conjugate gradient methods, achieving near-linear time in sparse settings. Empirical results on real-world hypergraphs indicate that FRW produces more intuitive neighbor rankings than SRW, often approaching or matching the performance of DeepWalk while offering substantial speed advantages, especially when the number of target nodes is small. The work thus provides a fast, interpretable distance measure for hypergraphs with strong theoretical guarantees and practical applicability to large-scale, complex networks.
Abstract
A hypergraph is a generalization of a graph that arises naturally when attribute-sharing among entities is considered. Compared to graphs, hypergraphs have the distinct advantage that they contain explicit communities and are more convenient to manipulate. An open problem in hypergraph research is how to accurately and efficiently calculate node distances on hypergraphs. Estimating node distances enables us to find a node's nearest neighbors, which has important applications in such areas as recommender system, targeted advertising, etc. In this paper, we propose using expected hitting times of random walks to compute hypergraph node distances. We note that simple random walks (SRW) cannot accurately compute node distances on highly complex real-world hypergraphs, which motivates us to introduce frustrated random walks (FRW) for this task. We further benchmark our method against DeepWalk, and show that while the latter can achieve comparable results, FRW has a distinct computational advantage in cases where the number of targets is fairly small. For such cases, we show that FRW runs in significantly shorter time than DeepWalk. Finally, we analyze the time complexity of our method, and show that for large and sparse hypergraphs, the complexity is approximately linear, rendering it superior to the DeepWalk alternative.