Exact formula for bond percolation on cliques.

Area of Science:

• Complex networks analysis
• Statistical physics
• Graph theory

Background:

• Understanding the behavior of complex networks is crucial in various scientific fields.
• Percolation theory models the formation of connected clusters in random systems.
• Networks composed of cliques present unique structural properties.

Purpose of the Study:

• To derive exact solutions for the size of the giant connected component in clique-based networks under bond percolation.
• To determine the percolation threshold of this network model analytically.
• To explore the applicability of the Erdős-Gallai theorem to configuration model networks with clique subgraphs.

Main Methods:

• Exact solution derivation for giant component size.
• Analysis of bond percolation on networks composed of cliques.
• Application of theoretical results to locate the percolation threshold.
• Examination of the Erdős-Gallai theorem for network graphicality.

Main Results:

• Exact solutions for the giant connected component size were obtained.
• The location of the percolation threshold was determined, with analytical solutions provided where feasible.
• The Erdős-Gallai theorem was investigated as a condition for graphicality in specific network configurations.

Conclusions:

• The derived solutions offer valuable insights into the structure and phase transitions of complex networks.
• The findings are applicable to diverse fields such as graph theory, epidemiology, and fragmentation theory.
• The study contributes to the analytical understanding of percolation phenomena in structured network models.