Motif Cut Sparsifiers
Michael Kapralov, Mikhail Makarov, Sandeep Silwal, Christian Sohler,, Jakab Tardos

TL;DR
This paper introduces motif cut sparsifiers, a method to create sparse subgraphs that approximately preserve motif-based cut structures in graphs, enabling efficient analysis of complex network organization.
Contribution
It develops a polynomial-time algorithm for constructing motif cut sparsifiers with nearly linear edges, extending graph and hypergraph sparsification techniques to motifs.
Findings
Constructed sparse subgraphs with O(n/^2) edges
Preserves motif cut structure within 1+ factor
Provides lower bounds for motif-induced sparsification
Abstract
A motif is a frequently occurring subgraph of a given directed or undirected graph . Motifs capture higher order organizational structure of beyond edge relationships, and, therefore, have found wide applications such as in graph clustering, community detection, and analysis of biological and physical networks to name a few. In these applications, the cut structure of motifs plays a crucial role as vertices are partitioned into clusters by cuts whose conductance is based on the number of instances of a particular motif, as opposed to just the number of edges, crossing the cuts. In this paper, we introduce the concept of a motif cut sparsifier. We show that one can compute in polynomial time a sparse weighted subgraph with only edges such that for every cut, the weighted number of copies of crossing the cut in is within a …
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