Localization of nonbacktracking centrality on dense subgraphs of sparse networks

Area of Science:

• Network science
• Statistical physics

Background:

• Nonbacktracking centrality (NBC) is vital for modeling network processes like nonrecurrent epidemics.
• Understanding NBC localization in complex networks is crucial for predicting process dynamics.

Purpose of the Study:

• To investigate the localization phenomenon of nonbacktracking centrality (NBC) in infinite sparse networks containing a finite subgraph.
• To derive explicit expressions for NBC in localized states and analyze its decay patterns.

Main Methods:

• Analysis of the nonbacktracking matrix for composite networks (finite subgraph + infinite network).
• Derivation of explicit formulas for NBC under localized conditions.
• Mathematical analysis of NBC decay rates around the subgraph.

Main Results:

• The largest eigenvalue of the composite network's nonbacktracking matrix is determined by the subgraph and enclosing network eigenvalues.
• Explicit expressions for NBC are derived for nodes within the subgraph and its vicinity.
• NBC exhibits exponential decay around the finite subgraph, with a rate independent of the enclosing network's structure.

Conclusions:

• Nonbacktracking centrality localizes to finite subgraphs and their immediate surroundings.
• The derived formulas provide accurate predictions for NBC distribution, even in real-world networks.
• Findings offer valuable insights into the behavior of network-based processes.