The Maximum Label Propagation Algorithm on Sparse Random Graphs

Authors Charlotte Knierim, Johannes Lengler, Pascal Pfister, Ulysse Schaller, Angelika Steger

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Author Details

Charlotte Knierim
  • ETH Zurich, Switzerland
Johannes Lengler
  • ETH Zurich, Switzerland
Pascal Pfister
  • ETH Zurich, Switzerland
Ulysse Schaller
  • ETH Zurich, Switzerland
Angelika Steger
  • ETH Zurich, Switzerland

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Charlotte Knierim, Johannes Lengler, Pascal Pfister, Ulysse Schaller, and Angelika Steger. The Maximum Label Propagation Algorithm on Sparse Random Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 58:1-58:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


In the Maximum Label Propagation Algorithm (Max-LPA), each vertex draws a distinct random label. In each subsequent round, each vertex updates its label to the label that is most frequent among its neighbours (including its own label), breaking ties towards the larger label. It is known that this algorithm can detect communities in random graphs with planted communities if the graphs are very dense, by converging to a different consensus for each community. In [Kothapalli et al., 2013] it was also conjectured that the same result still holds for sparse graphs if the degrees are at least C log n. We disprove this conjecture by showing that even for degrees n^epsilon, for some epsilon>0, the algorithm converges without reaching consensus. In fact, we show that the algorithm does not even reach almost consensus, but converges prematurely resulting in orders of magnitude more communities.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Random graphs
  • Theory of computation → Distributed algorithms
  • random graphs
  • distributed algorithms
  • label propagation algorithms
  • consensus
  • community detection


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