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Step-By-Step Community Detection in Volume-Regular Graphs

Authors Luca Becchetti , Emilio Cruciani , Francesco Pasquale , Sara Rizzo

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Author Details

Luca Becchetti
  • Sapienza Università di Roma, Italy
Emilio Cruciani
  • Gran Sasso Science Institute, L'Aquila, Italy
Francesco Pasquale
  • Università di Roma "Tor Vergata", Italy
Sara Rizzo
  • Gran Sasso Science Institute, L'Aquila, Italy

Funding

  • Becchetti, Luca: Partially supported by ERC Advanced Grant 788893 AMDROMA "Algorithmic and Mechanism Design Research in Online Markets" and MIUR PRIN project ALGADIMAR "Algorithms, Games, and Digital Markets".
  • Pasquale, Francesco: Partially supported by the University of "Tor Vergata" under research program "Mission: Sustainability" project ISIDE (grant no. E81I18000110005).

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Luca Becchetti, Emilio Cruciani, Francesco Pasquale, and Sara Rizzo. Step-By-Step Community Detection in Volume-Regular Graphs. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 20:1-20:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ISAAC.2019.20

Abstract

Spectral techniques have proved amongst the most effective approaches to graph clustering. However, in general they require explicit computation of the main eigenvectors of a suitable matrix (usually the Laplacian matrix of the graph).
Recent work (e.g., Becchetti et al., SODA 2017) suggests that observing the temporal evolution of the power method applied to an initial random vector may, at least in some cases, provide enough information on the space spanned by the first two eigenvectors, so as to allow recovery of a hidden partition without explicit eigenvector computations. While the results of Becchetti et al. apply to perfectly balanced partitions and/or graphs that exhibit very strong forms of regularity, we extend their approach to graphs containing a hidden k partition and characterized by a milder form of volume-regularity. We show that the class of k-volume regular graphs is the largest class of undirected (possibly weighted) graphs whose transition matrix admits k "stepwise" eigenvectors (i.e., vectors that are constant over each set of the hidden partition). To obtain this result, we highlight a connection between volume regularity and lumpability of Markov chains. Moreover, we prove that if the stepwise eigenvectors are those associated to the first k eigenvalues and the gap between the k-th and the (k+1)-th eigenvalues is sufficiently large, the Averaging dynamics of Becchetti et al. recovers the underlying community structure of the graph in logarithmic time, with high probability.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • Community detection
  • Distributed algorithms
  • Dynamics
  • Markov chains
  • Spectral analysis

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