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.
Highlights
The spectral properties of complex topologies [1] play a crucial role in our understanding of the structure and function of real networked systems
Particular interest in this case is placed on the study of the principal eigenvector ffig (PEV), defined as the eigenvector of the adjacency matrix with the largest eigenvalue ΛM (LEV)
The dependence of this threshold on the network topology is well approximated by the socalled quenched mean-field theory (QMF), predicting it to be equal to the inverse of the LEV, λc Inserting into this expression the LEV scaling form given by
Summary
The spectral properties of complex topologies [1] play a crucial role in our understanding of the structure and function of real networked systems. The LEV plays a pivotal role in the behavior of many dynamical systems on complex networks, such as epidemic spreading [7], synchronization of weakly coupled oscillators [8], weighted percolation on directed networks [9], models of genetic control [10], or the dynamics of excitable elements [11] In these kinds of dynamical processes, the LEV is related, through different analytical techniques, to the critical point at which a transition between different phases takes place: In terms of some generic control parameter λ, a critical point λc is found to be, in general, inversely proportional to the LEV ΛM.
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