Extended-range percolation and complex networks

This PhD thesis investigates percolation in complex networks, focusing on a novel model called extended-range percolation. This model is studied on various complex structures, with an exact solution provided using generating functions and message-passing techniques. Additionally, a novel model for highly-clustered random graphs is presented and analyzed both numerically and analytically. The thesis begins by introducing basic concepts of percolation theory and provides a comprehensive review of critical properties on uncorrelated locally tree-like random graphs. It aims to bridge gaps in existing literature by offering a unified framework for understanding percolation's critical properties. The second part of the thesis discusses extended-range percolation, where inactive nodes do not disrupt connectivity within a defined interaction range R. An exact solution is presented using generating functions and mean-field-like recursive equations, revealing non-universal behavior in strongly heterogeneous networks. A message-passing framework is developed to analyze extended-range percolation on tree-like networks for arbitrary R, and its application to multiplex networks is explored, leading to novel percolation critical phenomena. In the third part, a model for highly-clustered random graphs is introduced by closing open triads in uncorrelated random graphs. This allows for the exact computation of topological properties and provides insights into how clustering affects percolation critical exponents. Finally, a brief summary of a parallel project on optimizing quantum communication networks is presented, demonstrating practical applications of theoretical findings. This thesis advances the understanding of percolation processes in complex networks through innovative models, contributing valuable insights into communication systems and other fields influenced by network theory. These findings pave the way for future research directions that could further explore the applications of percolation theory in various contexts.