Conductance Estimation in Digraphs: Submodular Transformation, Lov\'asz Extension and Dinkelbach Iteration

Abstract

Conventional spectral digraph partitioning methods typically symmetrize the adjacency matrix, thereby transforming the directed graph partitioning problem into an undirected one, where bipartitioning is commonly linked to minimizing graph conductance. However, such symmetrization approaches disregard the directional dependencies of edges in digraphs, failing to capture the inherent imbalance crucial to directed network modeling. Building on the parallels between digraph conductance and conductance under submodular transformations, we develop a generalized framework to derive their continuous formulations. By leveraging properties of the Lov\'asz extension, this framework addresses the fundamental asymmetry problem in digraph partitioning. We then formulate an equivalent fractional programming problem, relax it via a three-step Dinkelbach iteration procedure, and design the Directed Simple Iterative ($\mathbf{DSI}$) algorithm for estimating digraph conductance. The subproblem within $\mathbf{DSI}$ is analytically solvable, and the algorithm is guaranteed to converge provably to a binary local optimum. Extensive experiments on synthetic and real-world networks demonstrate that our $\mathbf{DSI}$ algorithm significantly outperforms several state-of-the-art methods in digraph conductance minimization.