Investigating stability of Laplacians on signed digraphs via eventual positivity

Abstract

Signed Laplacian matrices generally fail to be diagonally dominant and may fail to be stable. For both undirected and directed graphs, in this paper we present conditions guaranteeing the stability of signed Laplacians based on the property of eventual positivity, a Perron-Frobenius type of property for signed matrices. Our conditions are necessary and sufficient for undirected graphs, but only sufficient for digraphs, the gap between necessity and sufficiency being filled by matrices who have this Perron-Frobenius property on the right but not on the left side (i.e., on the transpose). An exception is given by weight balanced signed digraphs, where eventual positivity corresponds to positive semidefinitness of the symmetric part of the Laplacian. Analogous conditions are obtained for signed stochastic matrices.

Collective decision-making on networked systems in presence of antagonistic interactions

“…The following lemma is a collection of fundamental and well-known properties of the Laplacian matrix, see for instance [51,52,[55][56][57]. Intuitively, (i)÷(vi) follow directly from the definition of Laplacian, the Geršgorin's Theorem, and the Perron-Frobenius Theorem, and (vii) is shown in [57,Theorem 2]. (i) L1 = 0, i.e., 0 is always an eigenvalue of L with right eigenvector 1;…”

“…Something more can be added when a digraph is not only strongly connected but also weight-balanced, as shown for instance in the recent work [57]. When a digraph G is strongly connected then it does not have isolated nodes, which means that the in-degree matrix ∆ in is positive definite.…”

Extensions of Perron-Frobenius Theory

“…Definition 1.6. A matrix A ∈ R n,n possesses • the Perron-Frobenius property if it has a positive dominant eigenvalue λ 1 = ρ(A) > 0 and the corresponding eigenvector x (1) ≥ 0;…”

“…Later in 2017, Shi, Altafini and Baras [50] combined generalized Perron-Frobenius theory, graph theory and elementary algebraic recursion to define an algebraic-graphical method for signed networks. Recently in 2019, Altafini [1] extended his former research and studied the Laplacian matrix of the form (4.3) which represents the negative weights on the adjacency matrix. If the adjacency matrix A of the system (4.3) is eventually positive, the author applied [43,Theorem 3.3] to prove the stability of signed Laplacian matrix L.…”

“….} and the associated right and left Perron eigenvectors x(1) , y(1) are nonnegative.The matrix B 2l has double its nonzero eigenvalues (λ 2l 1 , λ 2l 2 , . .…”