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Power-Law Graphs Have Minimal Scaling of Kemeny Constant for Random Walks

Published: 20 April 2020 Publication History

Abstract

The mean hitting time from a node i to a node j selected randomly according to the stationary distribution of random walks is called the Kemeny constant, which has found various applications. It was proved that over all graphs with N vertices, complete graphs have the exact minimum Kemeny constant, growing linearly with N. Here we study numerically or analytically the Kemeny constant on many sparse real-world and model networks with scale-free small-world topology, and show that their Kemeny constant also behaves linearly with N. Thus, sparse networks with scale-free and small-world topology are favorable architectures with optimal scaling of Kemeny constant. We then present a theoretically guaranteed estimation algorithm, which approximates the Kemeny constant for a graph in nearly linear time with respect to the number of edges. Extensive numerical experiments on model and real networks show that our approximation algorithm is both efficient and accurate.

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  • (2024)Efficient Approximation of Kemeny's Constant for Large GraphsProceedings of the ACM on Management of Data10.1145/36549372:3(1-26)Online publication date: 30-May-2024
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  • (2024)Resistance Eccentricity in Graphs: Distribution, Computation and Optimization2024 IEEE 40th International Conference on Data Engineering (ICDE)10.1109/ICDE60146.2024.00315(4113-4126)Online publication date: 13-May-2024
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cover image ACM Conferences
WWW '20: Proceedings of The Web Conference 2020
April 2020
3143 pages
ISBN:9781450370233
DOI:10.1145/3366423
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 20 April 2020

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Author Tags

  1. Hitting time
  2. Kemeny constant
  3. Linear system solver
  4. Normalized Laplacian matrix
  5. Random projection
  6. Random walk
  7. Spectral graph theory

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WWW '20
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WWW '20: The Web Conference 2020
April 20 - 24, 2020
Taipei, Taiwan

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Cited By

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  • (2024)Efficient Approximation of Kemeny's Constant for Large GraphsProceedings of the ACM on Management of Data10.1145/36549372:3(1-26)Online publication date: 30-May-2024
  • (2024)Fast Computation of Kemeny's Constant for Directed GraphsProceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining10.1145/3637528.3671859(3472-3483)Online publication date: 25-Aug-2024
  • (2024)Resistance Eccentricity in Graphs: Distribution, Computation and Optimization2024 IEEE 40th International Conference on Data Engineering (ICDE)10.1109/ICDE60146.2024.00315(4113-4126)Online publication date: 13-May-2024
  • (2024)Resistance Distances in Directed Graphs: Definitions, Properties, and ApplicationsTheoretical Computer Science10.1016/j.tcs.2024.114700(114700)Online publication date: Jun-2024
  • (2024)On Kemeny's constant and stochastic complementLinear Algebra and its Applications10.1016/j.laa.2024.09.001Online publication date: Sep-2024
  • (2023)Optimal Scale-Free Small-World Graphs with Minimum Scaling of Cover TimeACM Transactions on Knowledge Discovery from Data10.1145/358369117:7(1-19)Online publication date: 6-Apr-2023
  • (2023)Measures and Optimization for Robustness and Vulnerability in Disconnected NetworksIEEE Transactions on Information Forensics and Security10.1109/TIFS.2023.327997918(3350-3362)Online publication date: 2023
  • (2021)An Efficient and Scalable Algorithm for Estimating Kemeny's Constant of a Markov Chain on Large GraphsProceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining10.1145/3447548.3467431(964-974)Online publication date: 14-Aug-2021
  • (2021)Kemeny’s constant and Kirchhoffian indices for conjoined highly symmetric graphsDiscrete Applied Mathematics10.1016/j.dam.2021.07.007302:C(215-220)Online publication date: 30-Oct-2021

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