The largest eigenvalue of the adjacency matrix of networks is a key quantity determining several important dynamical processes on complex networks. Based on this fact, we present a quantitative, objective characterization of the dynamical importance of network nodes and links in terms of their effect on the largest eigenvalue. We show how our characterization of the dynamical importance of node...

We show that nonconstant eigenfunctions of the p-Laplacian do not necessarily have an average value of 0, as they must when p = 2. This fact has implications for deriving a sharp variational characterization of the second eigenvalue for a general class of nonlinear eigenvalue problems.

Scaling is a commonly used technique for standard eigenvalue problems to improve the sensitivity of the eigenvalues. In this paper we investigate scaling for generalized and polynomial eigenvalue problems (PEPs) of arbitrary degree. It is shown that an optimal diagonal scaling of a PEP with respect to an eigenvalue can be described by the ratio of its normwise and componentwise condition number...

Newman’s measure for (dis)assortativity, the linear degree correlation coefficient ρD, is reformulated in terms of the total number Nk of walks in the graph with k hops. This reformulation allows us to derive a new formula from which a degree-preserving rewiring algorithm is deduced, that, in each rewiring step, either increases or decreases ρD conform our desired objective. Spectral metrics (e...

A signed graph is a simple with two types of edges: positive and negative. homomorphism from G to another H mapping \(\varphi : V(G) \rightarrow V(H)\) that preserves vertex adjacencies balance closed walks (the the parity number negative edges). The chromatic \(\chi _s(G)\) order smallest such there H.