Perron Frobenius Research Articles

Chaotic fields out of equilibrium are observable independent

Chaotic dynamics is always characterized by swarms of unstable trajectories, unpredictable individually, and thus generally studied statistically. It is often the case that such phase-space densities relax exponentially fast to a limiting distribution, that rules the long-time average of every observable of interest. Before that asymptotic time scale, the statistics of chaos is generally believed to depend on both the initial conditions and the chosen observable. I show that this is not the case for a widely applicable class of models, that feature a phase-space (‘field’) distribution common to all pushed-forward or integrated observables, while the system is still relaxing towards statistical equilibrium or a stationary state. This universal profile is determined by both leading and first subleading eigenfunctions of the transport operator (Koopman or Perron–Frobenius) that maps phase-space densities forward or backward in time.

A Perron–Frobenius type result in Banach algebras via asymptotic closeness to a cone

For an element a of a Banach algebra (scaled to spectral radius 1) we prove that the spectral radius is contained in the spectrum, if the sequence of powers (ak)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(a^k)$$\\end{document} is asymptotically not too far from a normal cone.

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.

Upper bounds for the spectral radius of matrices having the Perron–Frobenius property

A new upper bound for the spectral radius of matrices having the Perron–Frobenius property is given by considering the position of positive entries. Some examples involving the largest zero of polynomials and the spectral radius of the iterative matrix for the Perron–Frobenius splitting are given to show the superiority of the theoretical result.

Finite dimensional Frobenius–Perron algebras

Motivated by the Frobenius–Perron (FP) theory of endofunctors of a [Formula: see text]-linear category, we introduce the notion of a FP algebra. This notion can be seen as a generalization of a fusion algebra, and is related to the PP theory of endofunctors of the categories of finite dimensional representations of Taft algebras. Our main result is to classify such algebras with small dimensions according to their FP dimension vectors. Moreover, we introduce the notion of a FP Hopf algebra which can be used to categorify a FP algebra via the Green ring of the Hopf algebra.

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.

Data-Driven Stochastic Optimal Control With Safety Constraints Using Linear Transfer Operators

We provide a data-driven framework for optimal control of a continuous-time control-affine stochastic dynamical system. The proposed framework relies on the linear operator theory involving Perron-Frobenius (P-F) and Koopman operators. Our first result involving the P-F operator provides a convex formulation to the optimal control problem in the dual space of densities. This convex formulation of the stochastic optimal control problem leads to an infinite-dimensional convex program. The finite-dimensional approximation of the convex program is obtained using a data-driven approximation of the linear operators. We provide rigorous error bounds and convergence rate for the data-driven approximation of the linear operators and comment on the convergence rate of the optimal control w.r.t. data size. Our second results demonstrate using the Koopman operator, dual to the P-F operator, for the stochastic optimal control design. We show that the Hamilton Jacobi Bellman (HJB) equation can be expressed using the Koopman operator. We provide an iterative procedure along the lines of a popular policy iteration algorithm based on the data-driven approximation of the Koopman operator for solving the HJB equation. The two formulations, namely the convex formulation involving the P-F operator and the Koopman-based formulation using the HJB equation, can be viewed as dual to each other where the duality follows due to the dual nature of the P-F and Koopman operators. Finally, we present examples to demonstrate the efficacy of the developed framework and verify the convergence rates for the operator and optimal control numerically.

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.

Ensemble forecasts in reproducing kernel Hilbert space family

A methodological framework for ensemble-based estimation and simulation of high dimensional dynamical systems such as the oceanic or atmospheric flows is proposed. To that end, the dynamical system is embedded in a family of reproducing kernel Hilbert spaces (RKHS) with kernel functions driven by the dynamics. In the RKHS family, the Koopman and Perron–Frobenius operators are unitary and uniformly continuous. This property warrants they can be expressed in exponential series of diagonalizable bounded evolution operators defined from their infinitesimal generators. Access to Lyapunov exponents and to exact ensemble based expressions of the tangent linear dynamics are directly available as well. The RKHS family enables us the devise of strikingly simple ensemble data assimilation methods for trajectory reconstructions in terms of constant-in-time linear combinations of trajectory samples. Such an embarrassingly simple strategy is made possible through a fully justified superposition principle ensuing from several fundamental theorems.

An innovative orthogonal matrix based on nonlinear chaotic system for compressive sensing

Compressed sensing exploits the signal’s sparsity by non-uniform sampling to achieve high-quality signal reconstruction at low sampling rates. This work aims to show the efficient performance of chaotic binary orthogonal matrices (CBOM) in compressed sensing. The nonlinear, high dimensional, irregular, and high complexity properties of chaotic systems can provide more diverse and efficient ways of sampling and reconstructing signals. The CBOM construction method proposed in this paper is divided into two steps, in the first step, the real-valued sequence of a one-dimensional chaotic map is binarised using the proposed Threshold-Matching Symbol Algorithm (TMSA) to obtain a chaotic binary sequence (CBS). The i.i.d properties of the CBS were proved using the Perron–Frobenius operator and the properties of the joint probability. The binarized CBS conditionally preserves the pseudo-random of chaotic sequences, as evidenced by the derivation of the well-distribution measure and the k-order correlation measure. In the second step, the binarised sequence CBS was split and orthogonalized to construct CBOM, which satisfies low storage and correlation. We prove that CBOM obeys the Restricted Isometric Condition (RIP). The orthogonalization of the matrix will further reduce matrix column correlation and improve the quality of the reconstruction. Numerical simulation results show that the proposed matrix has considerable sampling efficiency, comparable to Gaussian and partial Hadamard matrices, close to the theoretical limit. Meanwhile, the generation and reconstruction time of the proposed matrix is smaller than other matrices. Our framework covers partial one-dimensional chaotic maps, including Chebyshev, Tent, Logistic, and so on. We can easily apply this paradigm to various fields.

Theoretical Evaluation of the Reinjection Probability Density Function in Chaotic Intermittency

The traditional theory of chaotic intermittency developed for return maps hypothesizes a uniform density of reinjected points from the chaotic zone to the laminar one. In the past few years, we have described how the reinjection probability density function (RPD) can be generalized as a power law function. Here, we introduce a broad and general analytical approach to determine the RPD function and other statistical variables, such as the characteristic relation traditionally utilized to characterize the chaotic intermittency type. The proposed theoretical methodology is simple to implement and includes previous studies as particular cases. It is compared with numerical data, the M function methodology, and the Perron–Frobenius technique, showing high accuracy between them.

Hessian Distributed Ant Optimized Perron–Frobenius Eigen Centrality for Social Networks

Terabytes of data are now being handled by an increasing number of apps, and rapid user decision-making is hampered by data analysis. At the same time, there is a rise in interest in big data analysis for social networks at the moment. Thus, adopting distributed multi-agent-based technology in an optimum way is one of the solutions to effective big data analysis for social networks. Studying the development of a social network helps users gain an understanding of interactions and relationships and guides them in making decisions. In this study, a method called Hessian Distributed Ant Optimized and Perron–Frobenius Eigen Centrality (HDAO-PFEC) is developed to analyze large amounts of data (i.e., Big Data) in a computationally accurate and efficient manner. Designing an adaptable Multi-Agent System architecture for large data analysis is the primary goal of HDAO-PFEC. Initially, using a Hessian Mutual Distributed Ant Optimization MapReduce model, comparable user interest tweets are produced in a computationally efficient manner. Eigen Vector Centrality is a measure of a node’s importance in a network (i.e., a social network), which allows association with other significant nodes (i.e., users), allowing for a greater effect on social networks. With this goal in mind, a MapReduce methodology in the Hadoop platform using Big Data, which enables quick and ordered calculations, is used in a distributed computing method to estimate the Eigen Vector Centrality value for each social network member. Lastly, extensive investigative experimental learning demonstrates the HDAO-PFEC method’s use and accuracy as well as its time and overhead on the well-known sentiment 140 dataset.

A Perron–Frobenius analysis of wall-bounded turbulence

The Perron–Frobenius operator (PFO) is adapted from dynamical-system theory to the study of turbulent channel flow. It is shown that, as long as the analysis is restricted to the system attractor, the PFO can be used to differentiate causality and coherence from simple correlation without performing interventional experiments, and that the key difficulty remains the collection of enough data to populate the operator matrix. This is alleviated by limiting the analysis to two-dimensional projections of the phase space, and developing a series of indicators to choose the best parameter pairs from a large number of possibilities. The techniques thus developed are applied to the study of bursting in the inertial layer of the channel, with emphasis on the process by which bursts are reinitiated after they have decayed. Conditional averaging over phase-space trajectories suggested by the PFO shows, somewhat counter-intuitively, that a key ingredient for the burst recovery is the development of a low-shear region near the wall, overlaid by a lifted shear layer. This is confirmed by a computational experiment in which the control of the mean velocity profile by the turbulence fluctuations is artificially relaxed. The behaviour of the mean velocity profile is thus modified, but the association of low wall shear with the initiation of the bursts is maintained.

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.

Eventual cone invariance revisited

We consider finite-dimensional real vector spaces X ordered by a closed cone X+ with non-empty interior and study eventual nonnegativity of matrix semigroups (etA)t≥0 with respect to this cone.Our first contribution is the observation that, for general cones, one needs to distinguish between different notions of eventual nonnegativity: (i) uniform eventual nonnegativity means that etA maps X+ into X+ for all sufficiently large times t; (ii) individual eventual nonnegativity means that for each x∈X+ the vector etAx is in X+ for all t larger than an x-dependent time t0; and (iii) weak eventual nonnegativity means that for each x∈X+ and each functional x′ in the dual cone X+′ the value 〈x′,etAx〉 is in [0,∞) for all t larger than an x- and x′-dependent time t0. Until now, only the first of these notions has been studied in the literature. We demonstrate by examples that, somewhat surprisingly for finite-dimensional spaces, all three notions are different.Our second contribution is to show that typical Perron–Frobenius like properties remain valid under the weakest of the above notions.Third, we study a strengthened form of the above mentioned concepts, namely eventual positivity. We prove that the uniform, individual and weak versions of this property are – in contrast to the nonnegative case – equivalent, and that they can be characterized by spectral properties.

Convex Approach to Data-Driven Off-Road Navigation via Linear Transfer Operators

We consider the problem of optimal control design for navigation on off-road terrain. We use a traversability measure to characterize the difficulty of navigation on off-road terrain. The traversability measure captures terrain properties essential for navigation, such as elevation maps, roughness, slope, and texture. The terrain with the presence or absence of obstacles becomes a particular case of the proposed traversability measure. We provide a convex formulation to the off-road navigation problem by lifting the problem to the density space using the linear Perron-Frobenius (P-F) operator. The convex formulation leads to an infinite-dimensional optimal navigation problem for control synthesis. We construct the finite-dimensional approximation of the optimization problem using data. We use a computational framework based on the data-driven approximation of the Koopman operator. This makes the proposed approach data-driven and applicable to cases where an explicit system model is unavailable. Finally, we apply the proposed navigation framework with single integrator dynamics and Dubin's car model.

Vector-valued Ruelle operator for weakly contractive IFS and Dini matrix potentials

The (scalar) Ruelle operator theory is well-known both in fractal geometry and dynamical systems. In this paper we consider vector-valued Ruelle operators for weakly contractive iterative function systems associated with Dini matrix potentials. We generalize the result in the paper (J. Math. Anal. Appl. 299 341–56) to weakly contractive iterated function system. More exactly, our main theorem gives a sufficient condition for the vector-valued Ruelle operator with the Perron–Frobenius property.