Skip to Main content Skip to Navigation
Journal articles

Step-by-step community detection in volume-regular graphs

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 (usuallythe Laplacian matrix of the graph). Recent work (e.g., Becchetti et al., SODA 2017) suggests that observing the temporal evolutionof the power method applied to an initial random vector may, at least in some cases, provide enoughinformation on the space spanned by the first two eigenvectors, so as to allow recovery of a hiddenpartition without explicit eigenvector computations. While the results of Becchetti et al. applyto perfectly balanced partitions and/or graphs that exhibit very strong forms of regularity, weextend their approach to graphs containing a hiddenkpartition and characterized by a milderform of volume-regularity. We show that the class ofk-volume regulargraphs is the largest class ofundirected (possibly weighted) graphs whose transition matrix admitsk“stepwise” eigenvectors (i.e.,vectors that are constant over each set of the hidden partition). To obtain this result, we highlight aconnection between volume regularity and lumpability of Markov chains. Moreover, we prove that ifthe stepwise eigenvectors are those associated to the firstkeigenvalues and the gap between thek-th and the (k+1)-th eigenvalues is sufficiently large, theAveragingdynamics of Becchetti et al.recovers the underlying community structure of the graph in logarithmic time, with high probabil
Complete list of metadata

https://hal.archives-ouvertes.fr/hal-03007156
Contributor : Emilio Cruciani Connect in order to contact the contributor
Submitted on : Friday, December 11, 2020 - 3:25:58 PM
Last modification on : Friday, January 8, 2021 - 11:22:06 AM
Long-term archiving on: : Friday, March 12, 2021 - 7:54:13 PM

File

1907.07149.pdf
Files produced by the author(s)

Identifiers

Collections

Citation

Luca Becchetti, Emilio Cruciani, Francesco Pasquale, Sara Rizzo. Step-by-step community detection in volume-regular graphs. Theoretical Computer Science, Elsevier, 2020, 847, pp.49-67. ⟨10.1016/j.tcs.2020.09.036⟩. ⟨hal-03007156⟩

Share

Metrics

Record views

64

Files downloads

55