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Large very dense subgraphs in a stream of edges

Published online by Cambridge University Press:  25 January 2022

Claire Mathieu  and
Michel de Rougemont Open the ORCID record for Michel de Rougemont [Opens in a new window]
Claire Mathieu
Affiliation:
CNRS and IRIF, Paris, France
Michel de Rougemont*
Affiliation:
University Paris II and IRIF, Paris, France
*
*Corresponding author. Email: mdr@irif.fr
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Abstract

We study the detection and the reconstruction of a large very dense subgraph in a social graph with n nodes and m edges given as a stream of edges, when the graph follows a power law degree distribution, in the regime when $m=O(n. \log n)$. A subgraph S is very dense if it has $\Omega(|S|^2)$ edges. We uniformly sample the edges with a Reservoir of size $k=O(\sqrt{n}.\log n)$. Our detection algorithm checks whether the Reservoir has a giant component. We show that if the graph contains a very dense subgraph of size $\Omega(\sqrt{n})$, then the detection algorithm is almost surely correct. On the other hand, a random graph that follows a power law degree distribution almost surely has no large very dense subgraph, and the detection algorithm is almost surely correct. We define a new model of random graphs which follow a power law degree distribution and have large very dense subgraphs. We then show that on this class of random graphs we can reconstruct a good approximation of the very dense subgraph with high probability. We generalize these results to dynamic graphs defined by sliding windows in a stream of edges.

Keywords

dense subgraphsclusteringstreamingprobabilistic analysisrandom graphsapproximation

Information

Type
Research Article
Information
Network Science , Volume 9 , Issue 4 , December 2021 , pp. 403 - 424
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

Action Editor: Ulrik Brandes

*

A preliminary version was presented at FODS 2020 (Foundations of Data Science) Conference

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