# Relating Topological Determinants of Complex Networks to Their Spectral Properties: Structural and Dynamical Effects

## TL;DR

This paper establishes a physically grounded formula linking the largest eigenvalue of a network's adjacency matrix to specific subgraph structures, enhancing understanding of spectral properties and their impact on network dynamics.

## Contribution

It introduces a novel formula relating the largest eigenvalue to subgraph eigenvalues, validated across synthetic and real networks, and explores implications for network dynamics and heterogeneity.

## Key findings

- The formula accurately predicts the largest eigenvalue in various networks.
- Spectral properties differ significantly between static and preferential attachment networks.
- Implications for dynamics on networks are demonstrated.

## Abstract

The largest eigenvalue of a network's adjacency matrix and its associated principal eigenvector are key elements for determining the topological structure and the properties of dynamical processes mediated by it. We present a physically grounded expression relating the value of the largest eigenvalue of a given network to the largest eigenvalue of two network subgraphs, considered as isolated: The hub with its immediate neighbors and the densely connected set of nodes with maximum $K$-core index. We validate this formula showing that it predicts with good accuracy the largest eigenvalue of a large set of synthetic and real-world topologies. We also present evidence of the consequences of these findings for broad classes of dynamics taking place on the networks. As a byproduct, we reveal that the spectral properties of heterogeneous networks built according to the linear preferential attachment model are qualitatively different from those of their static counterparts.