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Bond Percolation in Small-World Graphs with Power-Law Distribution

Authors Luca Becchetti, Andrea Clementi, Francesco Pasquale, Luca Trevisan, Isabella Ziccardi

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Author Details

Luca Becchetti
  • Sapienza University of Rome, Italy
Andrea Clementi
  • University of Rome Tor Vergata, Italy
Francesco Pasquale
  • University of Rome Tor Vergata, Italy
Luca Trevisan
  • Bocconi University, Milan, Italy
Isabella Ziccardi
  • Bocconi University, Milan, Italy

Funding

Luca Trevisan and Isabella Ziccardi: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 834861).
  • Becchetti, Luca: Supported by the ERC Advanced Grant 788893 AMDROMA, EC H2020RIA project SoBigData++ (871042), PNRR MUR project PE0000013-FAIR, PNRR MUR project IR0000013-SoBigData.it.
  • Clementi, Andrea: Partially supported by Spoke 1 “FutureHPC & BigData” of the Italian Research Center on High-Performance Computing, Big Data and Quantum Computing (ICSC) funded by MUR Missione 4 Componente 2 Investimento 1.4: Potenziamento strutture di ricerca e creazione di “campioni nazionali” di R&S (M4C2-19) - Next Generation EU (NGEU).

Cite As Get BibTex

Luca Becchetti, Andrea Clementi, Francesco Pasquale, Luca Trevisan, and Isabella Ziccardi. Bond Percolation in Small-World Graphs with Power-Law Distribution. In 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 257, pp. 3:1-3:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.SAND.2023.3

Abstract

Full-bond percolation with parameter p is the process in which, given a graph, for every edge independently, we keep the edge with probability p and delete it with probability 1-p. Bond percolation is studied in parallel computing and network science to understand the resilience of distributed systems to random link failure and the spread of information in networks through unreliable links. Moreover, the full-bond percolation is equivalent to the Reed-Frost process, a network version of SIR epidemic spreading.
We consider one-dimensional power-law small-world graphs with parameter α obtained as the union of a cycle with additional long-range random edges: each pair of nodes {u,v} at distance L on the cycle is connected by a long-range edge {u,v}, with probability proportional to 1/L^α. Our analysis determines three phases for the percolation subgraph G_p of the small-world graph, depending on the value of α.
- If α < 1, there is a p < 1 such that, with high probability, there are Ω(n) nodes that are reachable in G_p from one another in 𝒪(log n) hops; 
- If 1 < α < 2, there is a p < 1 such that, with high probability, there are Ω(n) nodes that are reachable in G_p from one another in log^{𝒪(1)}(n) hops; 
- If α > 2, for every p < 1, with high probability all connected components of G_p have size 𝒪(log n).

Subject Classification

ACM Subject Classification
  • Theory of computation → Random network models
Keywords
  • Information spreading
  • gossiping
  • epidemics
  • fault-tolerance
  • network self-organization and formation
  • complex systems
  • social and transportation networks

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