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Fast Computation of Kemeny's Constant for Directed Graphs

Haisong Xia, Zhongzhi Zhang

TL;DR

Two novel approximation algorithms for estimating Kemeny's constant on directed graphs with theoretical error guarantees are proposed and extensive numerical experiments validate the superiority of these algorithms over baseline methods in terms of efficiency and accuracy.

Abstract

Kemeny's constant for random walks on a graph is defined as the mean hitting time from one node to another selected randomly according to the stationary distribution. It has found numerous applications and attracted considerable research interest. However, exact computation of Kemeny's constant requires matrix inversion, which scales poorly for large networks with millions of nodes. Existing approximation algorithms either leverage properties exclusive to undirected graphs or involve inefficient simulation, leaving room for further optimization. To address these limitations for directed graphs, we propose two novel approximation algorithms for estimating Kemeny's constant on directed graphs with theoretical error guarantees. Extensive numerical experiments on real-world networks validate the superiority of our algorithms over baseline methods in terms of efficiency and accuracy.

Fast Computation of Kemeny's Constant for Directed Graphs

TL;DR

Two novel approximation algorithms for estimating Kemeny's constant on directed graphs with theoretical error guarantees are proposed and extensive numerical experiments validate the superiority of these algorithms over baseline methods in terms of efficiency and accuracy.

Abstract

Kemeny's constant for random walks on a graph is defined as the mean hitting time from one node to another selected randomly according to the stationary distribution. It has found numerous applications and attracted considerable research interest. However, exact computation of Kemeny's constant requires matrix inversion, which scales poorly for large networks with millions of nodes. Existing approximation algorithms either leverage properties exclusive to undirected graphs or involve inefficient simulation, leaving room for further optimization. To address these limitations for directed graphs, we propose two novel approximation algorithms for estimating Kemeny's constant on directed graphs with theoretical error guarantees. Extensive numerical experiments on real-world networks validate the superiority of our algorithms over baseline methods in terms of efficiency and accuracy.
Paper Structure (29 sections, 14 theorems, 49 equations, 2 figures, 3 tables, 2 algorithms)

This paper contains 29 sections, 14 theorems, 49 equations, 2 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph and Random Walk
  4. Hitting Time and Kemeny's Constant
  5. Existing Methods
  6. Method based on truncated random walks
  7. Theoretical Results
  8. Algorithm Design
  9. Adaptive simulation from each node
  10. Iteration over node subset
  11. Algorithm Based on Directed Tree Sampling
  12. Numerical Experiments
  13. Results on Real-world Networks
  14. Efficiency
  15. Accuracy
  16. ...and 14 more sections

Key Result

lemma 1

Let $G=(V,E)$ be a digraph with transition matrix $\boldsymbol{P}$ and stationary distribution $\boldsymbol{\pi}$. The fundamental matrix $\boldsymbol{F}$ can be expressed as

Figures (2)

  • Figure 1: Running time of different approximate algorithms with varying error parameter $\epsilon$ on real-world networks: CAIDA (a), Higgs (b), Youtube (c), Pokec (d), Skitter (e) and Wikilink-fr (f).
  • Figure 2: Mean relative error of different approximate algorithms with varying error parameter $\epsilon$ on real-world networks: Sister cities (a), PGP (b), CAIDA (c), Wikilink-wa (d), Epinions (e) and EU Inst (f).

Theorems & Definitions (33)

  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • lemma 4
  • proof
  • theorem 1
  • proof
  • lemma 5: Hoeffding's inequality
  • ...and 23 more