Reduced Google matrix

TL;DR

This paper introduces a reduced Google matrix $G_R$ inspired by quantum scattering theory to analyze interactions within a selected subset of nodes in large directed networks, accounting for indirect links and effective interactions.

Contribution

The paper develops a novel reduced Google matrix $G_R$ framework, inspired by quantum physics, to analyze subset interactions within large directed networks.

Findings

$G_R$ captures effective interactions among subset nodes.

Method allows analysis of indirect links in large networks.

Provides numerical methods for computing $G_R$.

Abstract

Using parallels with the quantum scattering theory, developed for processes in nuclear and mesoscopic physics and quantum chaos, we construct a reduced Google matrix $G_R$ which describes the properties and interactions of a certain subset of selected nodes belonging to a much larger directed network. The matrix $G_R$ takes into account effective interactions between subset nodes by all their indirect links via the whole network. We argue that this approach gives new possibilities to analyze effective interactions in a group of nodes embedded in a large directed networks. Possible efficient numerical methods for the practical computation of $G_R$ are also described.