Perron-Frobenius Theory Research Articles

Principal eigenvectors in hypergraph Turán problems

For a general class of hypergraph Turán problems with uniformity $r$, we investigate the principal eigenvector for the $p$-spectral radius (in the sense of Keevash-Lenz-Mubayi and Nikiforov) for the extremal graphs, showing in a strong sense that these eigenvectors have close to equal weight on each vertex (equivalently, showing that the principal ratio is close to $1$). We investigate the sharpness of our result; it is likely sharp for the Turán tetrahedron problem. In the course of this latter discussion, we establish a lower bound on the $p$-spectral radius of an arbitrary $r$-graph in terms of the degrees of the graph. This builds on earlier work of Cardoso-Trevisan, Li-Zhou-Bu, Cioabă-Gregory, and Zhang. The case $1 < p < r$ of our results leads to some subtleties connected to Nikiforov's notion of $k$-tightness, arising from the Perron-Frobenius theory for the $p$-spectral radius. We raise a conjecture about these issues and provide some preliminary evidence for our conjecture.

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.

An algorithm for calculating spectral radius of $ s $-index weakly positive tensors

<abstract><p>In this paper, we introduced $ s $-index weakly positive tensors and discussed the calculation of the spectral radius of this kind of nonnegative tensors. Using the diagonal similarity transformation of tensor and Perron-Frobenius theory of nonnegative tensor, the calculation method of the maximum $ H $-eigenvalue of $ s $-index weakly positive tensors was given. A variable parameter was introduced in each iteration of the algorithm, which is equivalent to a translation transformation of the tensor in each iteration to improve the calculation speed. At the same time, it was proved that the algorithm is linearly convergent for the calculation of the spectral radius of $ s $-index weakly positive tensors. The final numerical example shows the effectiveness of the algorithm.</p></abstract>

A unified approach to nonlinear Perron-Frobenius theory

Let f:R>0n→R>0n be an order-preserving and homogeneous function. We show that the set of eigenvectors of f in R>0n is nonempty and bounded in Hilbert's projective metric if and only if f satisfies a condition involving upper and lower Collatz-Wielandt numbers of readily computed auxiliary functions. This condition generalizes a test for the existence of eigenvectors using hypergraphs that was proved by Akian, Gaubert, and Hochart. We include several examples to show how the new condition can be combined with the hypergraph test to give a systematic approach to determine when homogeneous and order-preserving functions have eigenvectors in R>0n. We also observe that if the entries of f are real analytic functions on R>0n, then the set of eigenvectors of f in R>0n is nonempty and bounded in Hilbert's projective metric if and only if the eigenvector is unique, up to scaling.

Sum Rate and Max-Min Rate for Cellular-Enabled UAV Swarm Networks

This paper investigates the fundamental rate performances in the highly promising cellular-enabled unmanned aerial vehicle (UAV) swarm networks, which can provide ubiquitous wireless connectivity for supporting various Internet of things (IoT) applications. We first provide the formulations for the sum rate maximization and max-min rate, which are two nonlinear optimization problems subject to the constraints of UAV transmit power and antenna parameters at base station (BS). For the sum rate maximization problem, we propose an iterative algorithm to solve it utilizing the Karush–Kuhn–Tucker (KKT) condition. For the max-min rate problem, we transform it to an equivalent conditional eigenvalue problem based on the nonlinear Perron-Frobenius theory, and thus design an iterative algorithm to obtain the solution of such problem. Finally, numerical results are presented to indicate the effect of some key parameters on the rate performances in such networks.

Entrywise lower and upper bounds for the Perron vector

Let an irreducible nonnegative matrix A and a positive vector x be given. Assume αx≤Ax≤βx for some 0<α≤β∈R. Then, by Perron-Frobenius theory, α and β are lower and upper bounds for the Perron root of A. As for the Perron vector x⁎, only bounds for the ratio γ:=maxi,j⁡xi⁎/xj⁎ are known, but no error bounds against some given vector x. In this note we close this gap. For a given positive vector x and provided that α and β as above are not too far apart, we prove entrywise lower and upper bounds of the relative error of x to the Perron vector of A.

Uplink-Downlink Duality of Multi-Cell Non-Orthogonal Multiple Access Systems

Towards the sixth generation (6G) wireless communications, multiple access is a potential technology to achieve thousand times of traffic capacity enhancement comparing with 5G. This paper investigates the feasible signal-to-interference-plus-noise-ratio (SINR) region for the multi-cell downlink and uplink non-orthogonal multiple access (NOMA) systems. Within a cell, the signals of different users are multiplexed on a channel with different power levels and successive interference cancellation is applied at the receivers to decode the signals. For the downlink, based on Perron-Frobenius Theory, we first derive a necessary and sufficient condition for an SINR vector to be feasible under a fixed decoding order. Then, a closed-form expression for the feasible SINR region is given by the union of the SINR regions under all possible decoding orders. Following a similar idea, the expression of the feasible SINR region for the uplink is also derived. Furthermore, the duality between the multi-cell downlink and uplink NOMA systems is explored. It is proved that the two systems have same feasible SINR region under dual channels. Finally, we propose an efficient algorithm to approximate the SINR region boundary. Simulation results are provided to validate the efficiency of the algorithm and compare the performance with orthogonal multiple access scheme.

The Value of Cooperation

The well-known Additive Increase-Multiplicative Decrease (AIMD) abstraction for network congestion control was first published by Dah-Ming Chiu and Raj Jain in their seminal work [4] in 1989 and soon played a prominent part in TCP algorithm design for the Internet. The ingenuity of AIMD lies in the abstraction of Internet congestion control, and ever since its inception has also been a staple part of teaching curriculum for performance evaluation and computer networking courses at universities worldwide. In this paper, we describe teaching examples for university students to appreciate the AIMD abstraction from the theoretical aspects such as convex optimization and Perron-Frobenius theory to the data science aspect. The essence of cooperation encompassed by AIMD reverberates even in teaching networks formed by students and educators, giving rise to online classroom flipping teaching tools and data analytics to close the gap between teachers and students.

Perron–Frobenius theory for some classes of nonnegative tensors in the max algebra

An analog of Perron–Frobenius theory is proposed for some classes of nonnegative tensors in the max algebra. In the first part some important characterizations of nonnegative matrices can be extended to nonnegative tensors over max algebra, especially the Perron–Frobenius theorem for weakly irreducible nonnegative tensors and the Collatz–Wielandt minimax theorem for nonnegative tensors. Then, in the second part, an iterative method is proposed for finding the largest max eigenvalue of a nonnegative tensor based on diagonal similar tensors. The iterative method is convergent for weakly irreducible nonnegative tensors. Some numerical results are provided to illustrate the efficiency of the iterative method.

Nonexpansive maps with surjective displacement

We investigate necessary and sufficient conditions for a nonexpansive map $f$ on a Banach space $X$ to have surjective displacement, that is, for $f - \mathrm{id}$ to map onto $X$. In particular, we give a computable necessary and sufficient condition when $X$ is a finite dimensional space with a polyhedral norm. We give a similar computable necessary and sufficient condition for a fixed point of a polyhedral norm nonexpansive map to be unique. We also consider applications to nonlinear Perron-Frobenius theory and suggest some additional computable sufficient conditions for surjective displacement and uniqueness of fixed points.

Cocycles on groupoids arising from -actions

AbstractWe consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space and study generalized Ruelle operators and $ C^{\ast } $ -algebras associated to these groupoids. We provide a new characterization of $ 1 $ -cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $ -tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle–Perron–Frobenius theory for dynamical systems of several commuting operators ( $ k $ -Ruelle triples and commuting Ruelle operators). Results on KMS states on $ C^{\ast } $ -algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence and uniqueness results for KMS states associated to the graphs.

A spectral property for concurrent systems and some probabilistic applications

We study trace theoretic concurrent systems. This setting encompasses safe (1-bounded) Petri nets. We introduce a notion of irreducible concurrent system and we prove the equivalence between irreducibility and a “spectral property”. The spectral property states a strict inequality between radii of convergence of certain growth series associated with the system. The proof that we present relies on Analytic combinatorics techniques. The spectral property is the cornerstone of our theory, in a framework where the Perron-Frobenius theory does not apply directly. This restriction is an inherent difficulty in the study of concurrent systems.We apply the spectral property to the probabilistic theory of concurrent systems. We prove on the one hand the uniqueness of the uniform measure, a question left open in a previous paper. On the other hand, we prove that this uniform measure can be realized as a Markov chain of states-and-cliques on a state space that can be precisely characterized.

Analytic methods for reachability problems

We consider a function-analytic approach to study synchronizing automata, primitive and ergodic matrix families. This gives a new way to establish some criteria for primitivity and for ergodicity of families of nonnegative matrices. We introduce a concept of canonical partition and use it to construct a polynomial-time algorithm for finding a positive matrix product and an ergodic matrix product whenever they exist. This also provides a generalization of some results of the Perron-Frobenius theory from one nonnegative matrix to families of matrices. Then we define the h-synchronizing automata and prove that the existence of a reset tuple is polynomially decidable. The question whether the functional-analytic approach can be extended to the h-primitivity is addressed and several open problems are formulated.

NOMA-Enabled Multi-Beam Satellite Systems: Joint Optimization to Overcome Offered-Requested Data Mismatches

Non-Orthogonal Multiple Access (NOMA) has potentials to improve the performance of multi-beam satellite systems. The performance optimization in satellite-NOMA systems can be different from that in terrestrial-NOMA systems, e.g., considering distinctive channel models, performance metrics, power constraints, and limited flexibility in resource management. In this paper, we adopt a metric, Offered Capacity to requested Traffic Ratio (OCTR), to measure the requested-offered data (or rate) mismatch in multi-beam satellite systems. In the considered system, NOMA is applied to mitigate intra-beam interference while precoding is implemented to reduce inter-beam interference. We jointly optimize power, decoding orders, and terminal-timeslot assignment to improve the max-min fairness of OCTR. The problem is inherently difficult due to the presence of combinatorial and non-convex aspects. We first fix the terminal-timeslot assignment, and develop an optimal fast-convergence algorithmic framework based on Perron-Frobenius theory (PF) for the remaining joint power-allocation and decoding-order optimization problem. Under this framework, we propose a heuristic algorithm for the original problem, which iteratively updates the terminal-timeslot assignment and improves the overall OCTR performance. Numerical results verify that max-min OCTR is a suitable metric to address the mismatch issue, and is able to improve the fairness among terminals. In average, the proposed algorithm improves the max-min OCTR by 40.2% over Orthogonal Multiple Access (OMA).

A dynamical approach to the Perron-Frobenius theory and generalized Krein-Rutman type theorems

We present a dynamical approach to the classical Perron-Frobenius theory by using some elementary knowledge on linear ODEs. It is completely self-contained and significantly different from those in the literature. As a result, we develop a complex version of the Perron-Frobenius theory and prove some generalized Krein-Rutman type theorems.

On the Convergence of Max-Min Fairness Power Allocation in Massive MIMO Systems

Power allocation techniques, among which the max-min fairness power allocation (MMFPA) is one of the most widely used, are essential to guarantee good data throughput for all users in a cell. Recently, an efficient MMFPA algorithm for massive multiple-input multiple-output (MIMO) systems has been proposed. However, this algorithm is susceptible to the initial search interval employed by the underlying bisection search. Even if the optimal point belongs to the initial search interval, this algorithm may fail to converge to such a point. In this letter, we use the Perron-Frobenius theory to explain this issue and provide search intervals that guarantee convergence to the optimal point. Furthermore, we propose the bound test procedure as an efficient way of initializing the search interval. Simulation results corroborate our findings.

Conjecture 𝒪 holds for some horospherical varieties of Picard rank 1

Property 𝒪 for an arbitrary complex, Fano manifold X is a statement about the eigenvalues of the linear operator obtained from the quantum multiplication of the anticanonical class of X. Conjecture 𝒪 is a conjecture that property 𝒪 holds for any Fano variety. Pasquier classified the smooth nonhomogeneous horospherical varieties of Picard rank 1 into five classes. Conjecture 𝒪 has already been shown to hold for the odd symplectic Grassmannians, which is one of these classes. We will show that conjecture 𝒪 holds for two more classes and an example in a third class of Pasquier’s list. Perron–Frobenius theory reduces our proofs to be graph-theoretic in nature.

Understanding and monitoring the evolution of the Covid-19 epidemic from medical emergency calls: the example of the Paris area

We portray the evolution of the Covid-19 epidemic during the crisis of March–April 2020 in the Paris area, by analyzing the medical emergency calls received by the EMS of the four central departments of this area (Centre 15 of SAMU 75, 92, 93 and 94). Our study reveals strong dissimilarities between these departments. We show that the logarithm of each epidemic observable can be approximated by a piecewise linear function of time. This allows us to distinguish the different phases of the epidemic, and to identify the delay between sanitary measures and their influence on the load of EMS. This also leads to an algorithm, allowing one to detect epidemic resurgences. We rely on a transport PDE epidemiological model, and we use methods from Perron–Frobenius theory and tropical geometry.

On the decoupled Markov group conjecture

The Markov group conjecture, a long-standing open problem in the theory of Markov processes with countable state space, asserts that a strongly continuous Markov semigroup $T = (T_t)_{t \in [0,\infty)}$ on $\ell^1$ has bounded generator if the operator $T_1$ is bijective. Attempts to disprove the conjecture have often aimed at glueing together finite dimensional matrix semigroups of growing dimension - i.e., it was tried to show that the Markov group conjecture is false even for Markov processes that decouple into (infinitely many) finite dimensional systems. In this article we show that such attempts must necessarily fail, i.e., we prove the Markov group conjecture for processes that decouple in the way described above. In fact, we even show a more general result that gives a universal norm estimate for bounded generators $Q$ of positive semigroups on any Banach lattice. Our proof is based on a filter product technique, infinite dimensional Perron-Frobenius theory and Gelfand's $T = \operatorname{id}$ theorem.

Perron–Frobenius theory for kernels and Crump–Mode–Jagers processes with macro-individuals

AbstractPerron–Frobenius theory developed for irreducible non-negative kernels deals with so-called R-positive recurrent kernels. If the kernel M is R-positive recurrent, then the main result determines the limit of the scaled kernel iterations $R^nM^n$ as $n\to\infty$ . In Nummelin (1984) this important result is proven using a regeneration method whose major focus is on M having an atom. In the special case when $M=P$ is a stochastic kernel with an atom, the regeneration method has an elegant explanation in terms of an associated split chain. In this paper we give a new probabilistic interpretation of the general regeneration method in terms of multi-type Galton–Watson processes producing clusters of particles. Treating clusters as macro-individuals, we arrive at a single-type Crump–Mode–Jagers process with a naturally embedded renewal structure.