# Nonbacktracking expansion of finite graphs

## TL;DR

This paper introduces a nonbacktracking expansion of finite graphs, creating an infinite tree that accurately captures local structure, enabling exact solutions for cooperative models and improving understanding of percolation and graph destruction.

## Contribution

It constructs an infinite, locally treelike tree matching finite graph local structure, facilitating exact message passing solutions for cooperative models on finite graphs.

## Key findings

- Exact solutions for cooperative models on finite graphs using the nonbacktracking expansion.
- Analysis of the accuracy and limitations of message passing algorithms.
- Comparison of computational complexity between message passing and simulations.

## Abstract

Message passing equations yield a sharp percolation transition in finite graphs, as an artifact of the locally treelike approximation. For an arbitrary finite, connected, undirected graph we construct an infinite tree having the same local structural properties as this finite graph, when observed by a nonbacktracking walker. Formally excluding the boundary, this infinite tree is a generalization of the Bethe lattice. We indicate an infinite, locally treelike, random network whose local structure is exactly given by this infinite tree. Message passing equations for various cooperative models on this construction are the same as for the original finite graph, but here they provide the exact solutions of the corresponding cooperative problems. These solutions are good approximations to observables for the models on the original graph when it is sufficiently large and not strongly correlated. We show how to express these solutions in the critical region in terms of the principal eigenvector components of the nonbacktracking matrix. As representative examples we formulate the problems of the random and optimal destruction of a connected graph in terms of our construction, the nonbacktracking expansion. We analyze the limitations and the accuracy of the message passing algorithms for different classes of networks and compare the complexity of the message passing calculations to that of direct numerical simulations. Notably, in a range of important cases, simulations turn out to be more efficient computationally than the message passing.