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Linear Programming and Community Detection

Author

Listed:
  • Alberto Del Pia

    (Department of Industrial and Systems Engineering & Wisconsin Institute for Discovery, University of Wisconsin–Madison, Madison, Wisconsin 53715)

  • Aida Khajavirad

    (Department of Industrial & Systems Engineering, Lehigh University, Bethlehem, Pennsylvania 18015)

  • Dmitriy Kunisky

    (Department of Computer Science, Yale University, New Haven, Connecticut 06520)

Abstract

The problem of community detection with two equal-sized communities is closely related to the minimum graph bisection problem over certain random graph models. In the stochastic block model distribution over networks with community structure, a well-known semidefinite programming (SDP) relaxation of the minimum bisection problem recovers the underlying communities whenever possible. Motivated by their superior scalability, we study the theoretical performance of linear programming (LP) relaxations of the minimum bisection problem for the same random models. We show that, unlike the SDP relaxation that undergoes a phase transition in the logarithmic average degree regime, the LP relaxation fails in recovering the planted bisection with high probability in this regime. We show that the LP relaxation instead exhibits a transition from recovery to nonrecovery in the linear average degree regime. Finally, we present nonrecovery conditions for graphs with average degree strictly between linear and logarithmic.

Suggested Citation

  • Alberto Del Pia & Aida Khajavirad & Dmitriy Kunisky, 2023. "Linear Programming and Community Detection," Mathematics of Operations Research, INFORMS, vol. 48(2), pages 885-913, May.
    • Handle: RePEc:inm:ormoor:v:48:y:2023:i:2:p:885-913
    DOI: 10.1287/moor.2022.1282
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