Perron-Frobenius Theorem Research Articles

Some novel results of a two-community SIR model with asymmetric structure

In this paper, a two-community SIR model with an asymmetric structure is proposed. We provide a concise method to derive an explicit formula of the basic reproduction number ℛ0 and present a route conception for the explanation of ℛ0. Finally, we find a close relationship between ℛ0 and the final size by the Perron–Frobenius theorem and the Krasnoselskii fixed point theorem.

On Spectral Radius and Energy of a Graph with Self-Loops

The spectral radius of a square matrix is the maximum among absolute values of its eigenvalues. Suppose a square matrix is nonnegative; then, by Perron–Frobenius theory, it will be one among its eigenvalues. In this paper, Perron–Frobenius theory for adjacency matrix of graph with self-loops AGS will be explored. Specifically, it discusses the nontrivial existence of Perron–Frobenius eigenvalue and eigenvector pair in the matrix AGS−σnI, where σ denotes the number of self-loops. Also, Koolen–Moulton type bound for the energy of graph GS is explored. In addition, the existence of a graph with self-loops for every odd energy is proved.

Accurate Computations with Block Checkerboard Pattern Matrices

In this work, block checkerboard sign pattern matrices are introduced and analyzed. They satisfy the generalized Perron–Frobenius theorem. We study the case related to total positive matrices in order to guarantee bidiagonal decompositions and some linear algebra computations with high relative accuracy. A result on intervals of checkerboard matrices is included. Some numerical examples illustrate the theoretical results.

Quality assessment and community detection methods for anonymized mobility data in the Italian Covid context

We discuss how to assess the reliability of partial, anonymized mobility data and compare two different methods to identify spatial communities based on movements: Greedy Modularity Clustering (GMC) and the novel Critical Variable Selection (CVS). These capture different aspects of mobility: direct population fluxes (GMC) and the probability for individuals to move between two nodes (CVS). As a test case, we consider movements of Italians before and during the SARS-Cov2 pandemic, using Facebook users’ data and publicly available information from the Italian National Institute of Statistics (Istat) to construct daily mobility networks at the interprovincial level. Using the Perron-Frobenius (PF) theorem, we show how the mean stochastic network has a stationary population density state comparable with data from Istat, and how this ceases to be the case if even a moderate amount of pruning is applied to the network. We then identify the first two national lockdowns through temporal clustering of the mobility networks, define two representative graphs for the lockdown and non-lockdown conditions and perform optimal spatial community identification on both graphs using the GMC and CVS approaches. Despite the fundamental differences in the methods, the variation of information (VI) between them assesses that they return similar partitions of the Italian provincial networks in both situations. The information provided can be used to inform policy, for example, to define an optimal scale for lockdown measures. Our approach is general and can be applied to other countries or geographical scales.

Moderate deviation principle for guesswork

In this paper, we establish a moderate deviation principle for the logarithm of guesswork, which was initially studied by Massey to quantify the number of guesses needed to ascertain a discrete random variable. Our approach is based on an asymptotic analysis for Perron–Frobenius eigenvalue of some perturbed matrix and several moments estimates bounds for guesswork.

АСИМПТОТИЧНІ ВЛАСТИВОСТІ ВИПАДКОВИХ МАТРИЦЬ

The work explores the asymptotic properties of large-dimensional stochastic matrices N under the condition of independence of matrix elements or rows (columns). An analysis of the main properties of eigenvalues of stochastic matrices is conducted. The work is dedicated to investigating the asymptotic characteristics of random matrices under the absence of the second moment and also considers the presence of "heavy tails" in the corresponding transitions in the adjacency matrices of the respective graph. The main result of the work is formulated in terms of the transition matrix of a discrete Markov chain and its eigenvalues. In proving the theorem, a non-degenerate Markov chain is considered, describing a mathematical model of random processes that do not change over time with known transition probabilities between states and a corresponding stochastic matrix P, one of whose eigenvalues is unity, and all eigenvalues of the stochastic matrix do not exceed it in absolute value. The proof of this fact follows from the Perron–Frobenius theorem, which concerns the properties of positive matrices and their eigenvalues. Thus, the theorem considered manages to expand the class of random matrices A for which convergence of eigenvalues of the matrix can be applied under the conditions imposed on the elements of the adjacency matrix. Moreover, the imposed conditions are relaxed compared to classical results, where the existence of a finite second moment for the elements of the adjacency matrix is required. This result generalizes both classical results for the normal distribution and similar results of other authors. The obtained result can be used in graph clustering problems to choose the optimal number of clusters, namely, they can be applied to determine the optimal number of clusters in a Grid system, complex networks, in investigations of the structure of the web space, etc.

Event-triggered distributed Nash equilibrium seeking over directed graphs and its application to power management

In this paper, the problem of distributed discrete-time Nash equilibrium computation with event-triggered communication over directed graphs is studied. Each agent can only receive strategies from its neighbors through a directed and unbalanced graph, while its cost function may rely on the information of all agents. To this end, each agent needs to estimate the strategy profile. Meanwhile, an event-triggered communication mechanism is introduced, which can effectively compensate for network resource loss. Concretely, specific moments that each agent transmits its information are decided by the pre-designed event-triggered condition. In this setting, two discrete-time Nash equilibrium seeking algorithms with event-triggered communication are designed, respectively considering that the Perron-Frobenius eigenvector of the adjacency matrix associated the directed communication graph is known and unknown in advance. It is rigorously proved that both algorithms can linearly converge to the Nash equilibrium with a small error. Finally, a power management example in smart grids is presented to verify the performance of proposed algorithms.

On the Admissibility and Stability of Multiagent Nonlinear Interconnected Positive Systems With Heterogeneous Delays

Many multi-agent interconnected systems include typical nonlinearities which are highly sensitive to inevitable communication delays. This makes their analysis challenging and the generalization of results from linear interconnected systems theory to those nonlinear interconnected systems very limited. This paper deals with the analysis of Multi-Agent Nonlinear Interconnected Positive Systems (MANIPS). The main contributions of this work are two fold. Based on Perron-Frobenius theorem, we first study the “admissibility” property for MANIPS, and show that it is a fundamental requirement for this category of systems. Then, using admissibility/positivity properties and sequences of functions theory, we propose a suitable Lyapunov function candidate to conduct the analysis of the dynamical behavior of such systems. We show that the stability of MANIPS is reduced to the positiveness property (i.e. negative or positive definiteness) of a new specific matrix-valued function ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {Z}$</tex-math></inline-formula> ) that we derive in this paper. Furthermore, the obtained results generalize the existing theory. The quality of the results achieved are demonstrated through the applications of the developed theory on cells with multi-stage maturation process dynamical models.

Perron-Frobenius type theorem for nonnegative tubal matrices in the sense of t-product

This paper is focused on the study of Perron-Frobenius (PF) type theorem for nonnegative tubal matrices, where a tubal matrix is a matrix with every element being a tubal scalar (a vector in the usual sense). Such a matrix is often recognized as a third-order tensor. The product between tubal scalars, tubal vectors, and tubal matrices can be done by the powerful t-product. In this paper, we define nonnegative/positive/strongly positive tubal scalars/vectors/matrices, and establish several properties that are analogous to their matrix counterparts. In particular, we introduce irreducible tubal matrices and provide two equivalent characterizations. Then, the celebrated PF theorem is established on the nonnegative irreducible tubal matrices. We show that some conclusions of the PF theorem for nonnegative irreducible matrices, such as the existence of a positive eigenvector, can be generalized to the tubal matrix setting, while some are not. For those conclusions that can not be extended, weaker conclusions are proved. We also show that, if the nonnegative irreducible tubal matrix contains a strongly positive tubal scalar, then most conclusions of the matrix PF theorem hold.

Topological and random spread models with frozen symbols.

When a symbol or a type has been "frozen" (namely, a type of which an individual only produces one individual of the same type), its spread pattern will be changed and this change will affect the long-term behavior of the whole system. However, in a frozen system, the ξ-matrix and the offspring mean matrix are no longer primitive so that the Perron-Frobenius theorem cannot be applied directly when predicting the spread rates. In this paper, our goal is to characterize these key matrices and analyze the spread rate under more general settings both in the topological and random spread models with frozen symbols. More specifically, we propose an algorithm for explicitly computing the spread rate and relate the rate with the eigenvectors of the ξ-matrix or offspring mean matrix. In addition, we reveal that the growth of the population is exponential and that the composition of the population is asymptotically periodic. Furthermore, numerical experiments are provided as supporting evidence for the theory.

Approximate spectral decomposition of Fisher information matrix for simple ReLU networks

We argue the Fisher information matrix (FIM) of one hidden layer networks with the ReLU activation function. For a network, let W denote the d×p weight matrix from the d-dimensional input to the hidden layer consisting of p neurons, and v the p-dimensional weight vector from the hidden layer to the scalar output. We focus on the FIM of v, which we denote as I. Under certain conditions, we characterize the first three clusters of eigenvalues and eigenvectors of the FIM. Specifically, we show that the following approximately holds. (1) Since I is non-negative owing to the ReLU, the first eigenvalue is the Perron–Frobenius eigenvalue. (2) For the cluster of the next maximum values, the eigenspace is spanned by the row vectors of W. (3) The direct sum of the eigenspace of the first eigenvalue and that of the third cluster is spanned by the set of all the vectors obtained as the Hadamard product of any pair of the row vectors of W. We confirmed by numerical calculation that the above is approximately correct when the number of hidden nodes is about 10000.

A dynamic input-output model of the Tyumen Oblast south's development based on the von Neumann algorithm

Subject. This article deals with the problem of constructing a real trajectory of the region's development and comparing it with the stated one. Objectives. The article aims to develop an algorithm for calculating and analyzing a dynamic input-output model of the south of the Tyumen Oblast, including social and infrastructural subsystems, and households as an endogenous variable-aggregate. Methods. For the study, we used the methods of construction and analysis of dynamic models by W. Leontief and J. von Neumann. Indicators of balanced growth trajectories were estimated using the Perron–Frobenius theorem. Results. The article proposes an algorithm for the formation of an investment variable. The matrix of direct costs for each year and on average for the periods 2017–2019 and 2017–2020 was identified. Based on the available statistical data, von Neumann highways and model development trajectories were constructed. Conclusions. The shown mechanism for a possible comparison of actual and planned optimal development can help organize digital monitoring of the region's development and analyze deviations from the optimal forecast.

Statistical theory of asymmetric damage segregation in clonal cell populations.

Asymmetric damage segregation (ADS) is ubiquitous among unicellular organisms: After a mother cell divides, its two daughter cells receive sometimes slightly, sometimes strongly different fractions of damaged proteins accumulated in the mother cell. Previous studies demonstrated that ADS provides a selective advantage over symmetrically dividing cells by rejuvenating and perpetuating the population as a whole. In this work we focus on the statistical properties of damage in individual lineages and the overall damage distributions in growing populations for a variety of ADS models with different rules governing damage accumulation, segregation, and the lifetime dependence on damage. We show that for a large class of deterministic ADS rules the trajectories of damage along the lineages are chaotic, and the distributions of damage in cells born at a given time asymptotically becomes fractal. By exploiting the analogy of linear ADS models with the Iterated Function Systems known in chaos theory, we derive the Frobenius-Perron equation for the stationary damage density distribution and analytically compute the damage distribution moments and fractal dimensions. We also investigate nonlinear and stochastic variants of ADS models and show the robustness of the salient features of the damage distributions.

Some Upper Bounds of Maximum &amp;lt;I&amp;gt;E&amp;lt;/I&amp;gt;-Eigenvalues of Uniform Hypergraphs

A hypergraphs, as a generalization of a general graph, is often used as an effective tool to describe complex structures in discrete mathematics, computer science and other fields. Hypergraph theory and related parameters of hypergraph are important research topics in hypergraph theory. In particular, the problem of spectral extremum of graphs has been widely concerned. This problem originates from the problem proposed by Brualdi and Solheid in 1986. That is to find the upper and lower bounds of spectral radius of a given graph class and characterize the polar graph that reaches the upper and lower bounds. Let <I>H </I>be a uniform hypergraph. Let <I>A</I>(<I>H</I>) be the adjacency tensor of <I>H</I>. In this work, by using Perron-Frobenius theorem, Hölder’s inequality and inequality of arithmetic and geometric means, we establish some upper bounds for the maximum <I>E</I>-eigenvalue of a uniform hypergraph instead of the degrees of vertices and edge number of hypergraph H. In addition, we characterize the extremal hypergraphs that reach the upper bounds.

Learning by Population Genetics and Matrix Riccati Equation

A model of learning as a generalization of the Eigen's quasispecies model in population genetics is introduced. Eigen's model is considered as a matrix Riccati equation. The error catastrophe in the Eigen's model (when the purifying selection becomes ineffective) is discussed as the divergence of the Perron-Frobenius eigenvalue of the Riccati model in the limit of large matrices. A known estimate for the Perron-Frobenius eigenvalue provides an explanation for observed patterns of genomic evolution. We propose to consider the error catastrophe in Eigen's model as an analog of overfitting in learning theory; this gives a criterion for the presence of overfitting in learning.

Multidigraph Autocatalytic Set for Modelling Complex Systems

The motion of solid objects or even fluids can be described using mathematics. Wind movements, turbulence in the oceans, migration of birds, pandemic of diseases and all other phenomena or systems can be understood using mathematics, i.e., mathematical modelling. Some of the most common techniques used for mathematical modelling are Ordinary Differential Equation (ODE), Partial Differential Equation (PDE), Statistical Methods and Neural Network (NN). However, most of them require substantial amounts of data or an initial governing equation. Furthermore, if a system increases its complexity, namely, if the number and relation between its components increase, then the amount of data required and governing equations increase too. A graph is another well-established concept that is widely used in numerous applications in modelling some phenomena. It seldom requires data and closed form of relations. The advancement in the theory has led to the development of a new concept called autocatalytic set (ACS). In this paper, a new form of ACS, namely, multidigraph autocatalytic set (MACS) is introduced. It offers the freedom to model multi relations between components of a system once needed. The concept has produced some results in the form of theorems and in particular, its relation to the Perron–Frobenius theorem. The MACS Graph Algorithm (MACSGA) is then coded for dynamic modelling purposes. Finally, the MACSGA is implemented on the vector borne disease network system to exhibit MACS’s effectiveness and reliability. It successfully identified the two districts that were the main sources of the outbreak based on their reproduction number, R0.

The first eigenvector of a distance matrix is nearly constant

Let x1,…,xn be points in a metric space and define the distance matrix D∈Rn×n by Dij=d(xi,xj). The Perron-Frobenius Theorem implies that there is an eigenvector v∈Rn with non-negative entries associated to the largest eigenvalue. We prove that this eigenvector is nearly constant in the sense that the inner product with the constant vector 1∈Rn is large〈v,1〉≥12⋅‖v‖ℓ2⋅‖1‖ℓ2 and that each entry satisfies vi≥‖v‖ℓ2/4n. Both inequalities are sharp.

Dynamics of products of nonnegative matrices

The aim of this manuscript is to understand the dynamics of products of nonnegative matrices. We extend a well known consequence of the Perron-Frobenius theorem on the periodic points of a nonnegative matrix to products of finitely many nonnegative matrices associated to a word and later to products of nonnegative matrices associated to a word, possibly of infinite length.

System Monotonicity and Subspace Tracking: A Geometric Perspective of the Frisch–Shapiro Scheme

The Shapiro scheme, together with the closely related Frisch–Kalman scheme, has been an important approach to system identification and statistical analysis. A longstanding result on this scheme, known as the Shapiro theorem, is both informative and significant. This article imparts a geometric understanding to the Shapiro theorem and generalizes it to the asymmetric setting using the notion of cone-invariance. In particular, we establish the equivalence between two important properties of a real-valued square matrix—irreducible orthant-invariance and simplicity of its dominant eigenvalue under arbitrary diagonal perturbations. The result can be regarded as a converse Perron–Frobenius theorem. Furthermore, we investigate two applications of the proposed result in systems and control, namely, characterization of irreducibly orthant-monotone nonlinear systems and subspace tracking via decentralized control. We also extend the established result to accommodating polyhedral cones and obtain several insights.

Equilibrium states for non-transitive random open and closed dynamical systems

AbstractWe prove a random Ruelle–Perron–Frobenius theorem and the existence of relative equilibrium states for a class of random open and closed interval maps, without imposing transitivity requirements, such as mixing and covering conditions, which are prevalent in the literature. This theorem provides the existence and uniqueness of random conformal and invariant measures with exponential decay of correlations, and allows us to expand the class of examples of (random) dynamical systems amenable to multiplicative ergodic theory and the thermodynamic formalism. Applications include open and closed non-transitive random maps, and a connection between Lyapunov exponents and escape rates through random holes. We are also able to treat random intermittent maps with geometric potentials.