Nonnegative Graphs Research Articles

Leader-Follower Bipartite Output Synchronization on Signed Digraphs Under Adversarial Factors via Data-Based Reinforcement Learning.

The optimal solution to the leader-follower bipartite output synchronization problem is proposed for heterogeneous multiagent systems (MASs) over signed digraphs in the presence of adversarial inputs in this article. For the MASs, the dynamics and dimensions of the followers are different. Distributed observers are first designed to estimate the leader's two-way state and output over signed digraphs. Then, the leader-follower bipartite output synchronization problem on signed graphs is translated into a conventional output distributed leader-follower problem over nonnegative graphs after the state transformation by using the information of followers and observers. The effect of adversarial inputs in sensors or actuators of agents is mitigated by designing the resilient H∞ controller. A data-based reinforcement learning (RL) algorithm is proposed to obtain the optimal control law, which implies that the dynamics of the followers is not required. Finally, a simulation example is given to verify the effectiveness of the proposed algorithm.

Symbolic Algorithms Computing Gram Congruences in the Coxeter Spectral Classification of Edge-bipartite Graphs, I. A Gram Classification

We continue the Coxeter spectral study of the category UBigrm of loop-free edgebipartite (signed) graphs ∆, with m ≥ 2 vertices, we started in [SIAM J. Discr. Math. 27(2013), 827-854] for corank r = 0 and r = 1. Here we study the class of all non-negative edge-bipartite graphs ∆ ∈ UBigrn+r of corank r ≥ 0, up to a pair of the Gram Z-congruences ∼Z and ≈Z, by means of the non-symmetric Gram matrix Ǧ∆ ∈ Mn+r(Z) of ∆, the symmetric Gram matrix G∆ := 1 2 [Ǧ∆ + Ǧ tr ∆ ] ∈ Mn+r(Z), the Coxeter matrix Cox∆ := −Ǧ∆ · Ǧ −tr ∆ ∈ Mn+r(Z) and its spectrum specc∆ ⊂ C, called the Coxeter spectrum of ∆. One of the aims in the study of the category UBigrn+r is to classify the equivalence classes of the non-negative edge-bipartite graphs in UBigrn+r with respect to each of the Gram congruences ∼Z and ≈Z. In particular, the Coxeter spectral analysis question, when the strong congruence ∆ ≈Z ∆′ holds (hence also ∆ ∼Z ∆′ holds), for a pair of connected non-negative graphs ∆,∆′ ∈ UBigrn+r such that specc∆ = specc∆′ , is studied in the paper. One of our main aims is an algorithmic description of a matrix B defining the Gram Z-congruences ∆ ≈Z ∆′ and ∆ ∼Z ∆′, that is, a Z-invertible matrix B ∈Mn+r(Z) such that Ǧ∆′ = B · Ǧ∆ ·B and G∆′ = B ·G∆ ·B, respectively. We show that, given a connected non-negative edge-bipartite graph ∆ in UBigrn+r of corank r ≥ 0 there exists a simply laced Dynkin diagram D, with n vertices, and a connected canonical r-vertex extension D := D of D of corank r (constructed in Section 2) such that ∆ ∼Z D. We also show that every matrix B defining the strong Gram Zcongruence ∆ ≈Z ∆′ in UBigrn+r has the form B = C∆ · B · C−1 ∆′ , where C∆, C∆′ ∈ Mn+r(Z) are fixed Z-invertible matrices defining the weak Gram congruences ∆ ∼Z D and ∆′ ∼Z D with an r-vertex extended graph D, respectively, and B ∈ Mn+r(Z) is Z-invertible matrix lying in the isotropy group Gl(n+r,Z)D of D. Moreover, each of the columns κ ∈ Z n+r of B is a root of ∆, Address for correspondence: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Torun, Poland. ∗Supported by Polish Research Grant NCN 2011/03/B/ST1/00824. Received November 2015; revised March 2016 20 D. Simson / Symbolic Algorithms Computing the Gram Congruence, I i.e., κ · Ǧ∆ · κ = 1. Algorithms constructing the set of all such matrices B are presented in case when r = 0. We essentially use our construction of a morsification reduction map φD : UBigrD → MorD that reduces (up to ≈Z) the study of the set UBigrD of all connected non-negative edgebipartite graphs ∆ in UBigrn+r such that ∆ ∼Z D to the study of Gl(n+r,Z)D-orbits in the set MorD ⊆ Gl(n+r,Z) of all matrix morsifications of the graph D.

Predictable Dynamics of Opinion Forming for Networks With Antagonistic Interactions

For communities of agents which are not necessarily cooperating, distributed processes of opinion forming are naturally represented by signed graphs, with positive edges representing friendly and cooperative interactions and negative edges the corresponding antagonistic counterpart. Unlike for nonnegative graphs, the outcome of a dynamical system evolving on a signed graph is not obvious and it is in general difficult to characterize, even when the dynamics are linear. In this paper, we identify a significant class of signed graphs for which the linear dynamics are however predictable and show many analogies with positive dynamical systems. These cases correspond to adjacency matrices that are eventually positive, for which the Perron–Frobenius property still holds and implies the existence of an invariant cone contained inside the positive orthant. As examples of applications, we determine cases in which it is possible to anticipate or impose unanimity of opinion in decision/voting processes even in presence of stubborn agents, and show how it is possible to extend the PageRank algorithm to include negative links.

Upper bounds for the second largest eigenvalue of symmetric nonnegative matrices

The paper suggests uppe bound on the second largest eigenvalue and the sum of two largest eigenvalues of symmetric nonnegative matrices and graphs. Conditions necessary and sufficient for some of the counds to be attained are established. Special attention is paid to the subclass of matrices with zero diagonal entries and with off-diagonal entries not exceeding unity, which obviously contains the adjacency matrices of undirected graphs. Bibliography: 8 titles.