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Percolation in self-similar networks.

M Angeles Serrano¹, Dmitri Krioukov, Marián Boguñá

¹Departament de Química Física, Universitat de Barcelona, Barcelona, Spain.

Physical Review Letters

|March 17, 2011

Summary

This study proves that many self-similar networks, including scale-free and real-world networks, exhibit a zero percolation threshold. This finding simplifies understanding network connectivity without assuming treelike structures.

Area of Science:

Network Science
Statistical Physics
Graph Theory

Background:

Percolation theory studies the connectivity of random networks.
Self-similar networks, like scale-free networks, exhibit similar structures across different scales.
Understanding the percolation threshold is crucial for network robustness and function.

Purpose of the Study:

To demonstrate a simple proof for a zero percolation threshold in a broad class of self-similar networks.
To identify the key network property responsible for this phenomenon.
To extend the understanding of percolation beyond treelike network assumptions.

Main Methods:

Developing a general proof for percolation threshold in self-similar networks.
Analyzing the size of the giant component in these networks.

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Utilizing the hierarchical structure of nested subgraphs.

Main Results:

Graphs in a general class of self-similar networks possess a zero percolation threshold.
The proof applies to various network types, including random scale-free, growing scale-free, and real-world networks.
The derivation of the giant component size does not rely on the networks being treelike.

Conclusions:

The hierarchical structure with growing average degree is pivotal for zero percolation in self-similar networks.
This finding simplifies the analysis of connectivity in complex networks.
The results have implications for understanding the robustness and behavior of diverse real-world networks.