Nonzero Laplacian Eigenvalues Research Articles

Dynamics of a susceptible-infected-recovered model on complex networks with interregional migration.

We present a susceptible-infected-recovered model based on a dynamic flow network that describes the epidemic process on complex metapopulation networks. This model views population regions as interconnected nodes and describes the evolution of each region using a system of differential equations. The next-generation matrix method is used to derive the global basic reproduction number for three cases: a general network with homogeneous infection rates in all regions, a fully connected network, and a star network with heterogeneous infection and recovery rates. For the homogeneous case, we show that this global basic reproduction number is independent of the migration rates between regions. However, the rate of convergence of each region to an equilibrium state exhibits a much larger variance in random (Erdős-Rényi) networks compared to small-scale (Barabási-Albert) networks. For the general heterogeneous case, we report interesting results, namely that the global basic reproduction number decays exponentially with respect to the smallest nonzero Laplacian eigenvalue (algebraic connectivity). Furthermore, we demonstrate both analytically and numerically that as the network's algebraic connectivity increases, either by increasing the average node degree of each region or the global migration rate, the global basic reproduction number decreases and converges to the ratio of the average local infection rate to the average local recovery rate, meaning that the lower bound of the global basic reproduction rate does not equal the mean of local basic reproduction rates.

Property analysis and coherence dynamics for tree-symmetric networks with noise disturbance

Abstract In this paper, we investigate the leaderless and the leader–follower coherence of tree-symmetric networks. Firstly, the analytical expressions for the product and the sum of the reciprocals of all nonzero Laplacian eigenvalues of tree-symmetric networks are computed. Secondly, comparing the leaderless and leader–follower coherence, the existence of leader nodes and network parameters have a great impact on the coherence in the noisy environment. Finally, the Laplacian eigenvalues of tree-symmetric networks were utilized to analyze other properties of the network. These research results have certain theoretical significance for the promotion of distributed system and block chain technology.

Laplacian Spectra of Persistent Structures in Taiwan, Singapore, and US Stock Markets.

An important challenge in the study of complex systems is to identify appropriate effective variables at different times. In this paper, we explain why structures that are persistent with respect to changes in length and time scales are proper effective variables, and illustrate how persistent structures can be identified from the spectra and Fiedler vector of the graph Laplacian at different stages of the topological data analysis (TDA) filtration process for twelve toy models. We then investigated four market crashes, three of which were related to the COVID-19 pandemic. In all four crashes, a persistent gap opens up in the Laplacian spectra when we go from a normal phase to a crash phase. In the crash phase, the persistent structure associated with the gap remains distinguishable up to a characteristic length scale ϵ* where the first non-zero Laplacian eigenvalue changes most rapidly. Before ϵ*, the distribution of components in the Fiedler vector is predominantly bi-modal, and this distribution becomes uni-modal after ϵ*. Our findings hint at the possibility of understanding market crashs in terms of both continuous and discontinuous changes. Beyond the graph Laplacian, we can also employ Hodge Laplacians of higher order for future research.

Scale fragilities in localized consensus dynamics

We consider distributed consensus in networks where the agents have integrator dynamics of order two or higher (n≥2). We assume all feedback to be localized in the sense that each agent has a bounded number of neighbors and consider a scaling of the network through the addition of agents in a modular manner, i.e., without re-tuning controller gains upon addition. We show that standard consensus algorithms, which rely on relative state feedback, are subject to what we term scale fragilities, meaning that stability is lost as the network scales. For high-order agents (n≥3), we prove that no consensus algorithm with fixed gains can achieve consensus in networks of any size. That is, while a given algorithm may allow a small network to converge, it causes instability if the network grows beyond a certain finite size. This holds in families of network graphs whose algebraic connectivity, that is, the smallest non-zero Laplacian eigenvalue, is decreasing towards zero in network size (e.g. all planar graphs). For second-order consensus (n=2) we prove that the same scale fragility applies to directed graphs that have a complex Laplacian eigenvalue approaching the origin (e.g. directed ring graphs). The proofs for both results rely on Routh–Hurwitz criteria for complex-valued polynomials and hold true for general directed network graphs. We survey classes of graphs subject to these scale fragilities, discuss their scaling constants, and finally prove that a sub-linear scaling of nodal neighborhoods can suffice to overcome the issue.

On Exponential Synchronization Rates for High-Dimensional Kuramoto Models With Identical Oscillators and Digraphs

For a high-dimensional Kuramoto model with identical oscillators under a general digraph that has a directed spanning tree, although the exponential synchronization has been proved under some initial state constraints, exponential synchronization rates have not been described exactly until now. In this article, the supremum of exponential synchronization rates is precisely determined as the smallest real part of the nonzero Laplacian eigenvalues of the digraph. Our obtained result extends the existing results from the special case of strongly connected balanced digraphs to the condition of general digraphs owning directed spanning trees, which is the weakest condition for synchronization from the aspect of network structure. Moreover, our adopted method is completely different from and much more elementary than the previous differential geometry method.

Critical groups of strongly regular graphs and their generalizations

We determine the maximum order of an element in the critical group of a strongly regular graph, and show that it achieves the spectral bound due to Lorenzini. We extend the result to all graphs with exactly two non-zero Laplacian eigenvalues, and study the signed graph version of the problem. We also study the monodromy pairing on the critical groups, and suggest an approach to study the structure of these groups using the pairing.

Spectrum of corona products based on splitting graphs

Let G be a simple undirected graph. Three new corona products of graphs based on splitting graph of G are defined. The adjacency spectra of the three new graphs based on splitting graph of G are determined. The number of spanning trees and the Kirchoff index of the new graphs are determined using their nonzero Laplacian eigenvalues.

Spectral analysis of three invariants associated to random walks on rounded networks with 2n-pentagons

A network is defined as an abstract structure that consists of nodes that are connected by links. In this paper, we study two types of rounded networks (resp., ). By using the recursive relation, we obtain all the eigenvalues and their multiplicities with regards to the associated Laplacian matrix. Meanwhile, we utilize the corresponding relationship between the roots and the coefficients of the characteristic polynomial. Based on these relationships, we obtain the analytical expressions for the sum of the reciprocals of all nonzero Laplacian eigenvalues. By the decomposition theorem of Laplacian polynomial, we obtained an explicit closed-form formula of the Kirchhoff index for . As an application of the Laplacian spectra, we reduce the number of spanning trees and the global mean-first passage time for . Furthermore, we show that the Kirchhoff index of is approximately to of its Wiener index. In view of our obtained results, all the corresponding results are considered for .

Concensus-Based ALADIN Method to Faster the Decentralized Estimation of Laplacian Spectrum

With the upcoming fifth Industrial Revolution, humans and collaborative robots will dance together in production. They themselves act as an agent in a connected world, understood as a multi-agent system, in which the Laplacian spectrum plays an important role since it can define the connection of the complex networks as well as depict the robustness. In addition, the Laplacian spectrum can locally check the controllability and observability of a dynamic controlled network, etc. This paper presents a new method, which is based on the Augmented Lagrange based Alternating Direction Inexact Newton (ALADIN) method, to faster the convergence rate of the Laplacian Spectrum Estimation via factorization of the average consensus matrices, that are expressed as Laplacian-based matrices problems. Herein, the non-zero distinct Laplacian eigenvalues are the inverse of the stepsizes {αt,t=1,2,…} of those matrices. Therefore, the problem now is to carry out the agreement on the stepsize values for all agents in the given network while ensuring the factorization of average consensus matrices to be accomplished. Furthermore, in order to obtain the entire Laplacian spectrum, it is necessary to estimate the relevant multiplicities of these distinct eigenvalues. Consequently, a non-convex optimization problem is formed and solved using ALADIN method. The effectiveness of the proposed method is evaluated through the simulation results and the comparison with the Lagrange-based method in advance.

CONVERGENCE RATE AND GLOBAL MEAN WEIGHTED FIRST-PASSAGE TIME IN A 1D CHAIN NETWORK WITH A WEIGHTED ADDING REVERSE EDGE

In this paper, a 1D chain network with a reverse weighted edge is introduced. We focus on studying the relationships including the convergence rate and the length, the convergence rate and weight of adding reverse edge relationships. Laplacian characteristic determinant is calculated and subsequently, the sum of the reciprocals of all nonzero Laplacian eigenvalues is obtained. Hence, the analytic expression of global mean weighted first-passage time (GMWFPT) can be deduced. The obtained results show that there exists a linearly positive relationship between GMWFPT and the weight [Formula: see text].

Eigentime identity of the weighted (m,n)-flower networks

The eigentime identity for random walks on the weighted networks is the expected time for a walker going from a node to another node. Eigentime identity can be studied by the sum of reciprocals of all nonzero Laplacian eigenvalues on the weighted networks. In this paper, we study the weighted [Formula: see text]-flower networks with the weight factor [Formula: see text]. We divide the set of the nonzero Laplacian eigenvalues into three subsets according to the obtained characteristic polynomial. Then we obtain the analytic expression of the eigentime identity [Formula: see text] of the weighted [Formula: see text]-flower networks by using the characteristic polynomial of Laplacian and recurrent structure of Markov spectrum. We take [Formula: see text], [Formula: see text] as example, and show that the leading term of the eigentime identity on the weighted [Formula: see text]-flower networks obey superlinearly, linearly with the network size.

Explicit determination of three invariants associated with random walks on n-prism networks

ABSTRACT In this paper, the Laplacian matrix of an n-prism network and its applications are studied. Firstly, the relation between the Laplacian matrix of and that of is established. Secondly, the analytical expression for the product of all the nonzero Laplacian eigenvalues of is obtained. At the same time, the sum of the reciprocals of all these nonzero Laplacian eigenvalues is deduced. Through these closed analytical results, the expressions for MFPT, the Kirchhoff index and the number of spanning trees of an n-prism network are determined, respectively. Finally, a few numerical examples are given to illustrate the results.

Bounds for Kirchhoff index and Laplacian-energy-like invariant of some derived graphs of a regular graph

The Kirchhoff index and Laplacian-energy-like invariant of a connected graph [Formula: see text], denoted by [Formula: see text] and [Formula: see text], are given by the number of vertex times the sum of the reciprocals of all nonzero Laplacian eigenvalues of [Formula: see text] and the sum of the square roots of all Laplacian eigenvalues of [Formula: see text], respectively. In this paper, we have obtained the Laplacian eigenvalues of some derived graphs, such as double graph, extended double cover and Mycielskian of an [Formula: see text]-regular graph [Formula: see text], in terms of the adjacency eigenvalues of [Formula: see text] and hence, we obtain some upper bounds of Kirchhoff index and Laplacian-energy-like (LEL) invariant of those derived graphs in terms of [Formula: see text], number of vertices and algebraic connectivity of [Formula: see text]. We have shown that the bounds obtained here are better than some existing bounds. We have also obtained the exact formulae for Kirchhoff index and LEL invariant of those derived graph when [Formula: see text] is a complete graph or a complete bipartite graph.

Coherence analysis of a family of weighted star-composed networks

Recently, the study of many kinds of weighted networks has received the attention of researchers in the scientific community. In this paper, first, a class of weighted star-composed networks with a weight factor is introduced. We focus on the network consistency in linear dynamical system for a class of weighted star-composed networks. The network consistency can be characterized as network coherence by using the sum of reciprocals of all nonzero Laplacian eigenvalues, which can be obtained by using the relationship of Laplacian eigenvalues at two successive generations. Remarkably, the Laplacian matrix of the class of weighted star-composed networks can be represented by the Kronecker product, then the properties of the Kronecker product can be used to obtain conveniently the corresponding characteristic roots. In the process of finding the sum of reciprocals of all nonzero Laplacian eigenvalues, the key step is to obtain the relationship of Laplacian eigenvalues at two successive generations. Finally, we obtain the main results of the first- and second-order network coherences. The obtained results show that if the weight factor is 1 then the obtained results in this paper coincide with the previous results on binary networks, otherwise the scalings of the first-order network coherence are related to the node number of attaching copy graph, the weight factor and generation number. Surprisingly, the scalings of the first-order network coherence are independent of the node number of initial graph. Consequently, it will open up new perspectives for future research.

Coherence analysis and Laplacian energy of recursive trees with controlled initial states

We study the consensus of a family of recursive trees with novel features that include the initial states controlled by a parameter. The consensus problem in a linear system with additive noises is characterized as network coherence, which is defined by a Laplacian spectrum. Based on the structures of our recursive treelike model, we obtain the recursive relationships for Laplacian eigenvalues in two successive steps and further derive the exact solutions of first- and second-order coherences, which are calculated by the sum and square sum of the reciprocal of all nonzero Laplacian eigenvalues. For a large network size N, the scalings of the first- and second-order coherences are lnN and N, respectively. The smaller the number of initial nodes, the better the consensus bears. Finally, we numerically investigate the relationship between network coherence and Laplacian energy, showing that the first- and second-order coherences increase with the increase of Laplacian energy at approximately exponential and linear rates, respectively.

First-order network coherence in 5-rose graphs

We consider first-order network coherence in 5-rose graphs with stochastic disturbances. Our analysis for first-order systems is facilitated by connections between first-order network coherence and the sum of reciprocals of all nonzero Laplacian eigenvalues. Our analysis reveals that 5-rose networks with small size have high network coherence and can be considered to be more robust to noise than networks with low coherence. The method used in this paper is practical and easy to handle.

Spectra analysis and network coherence for weighted folded hypercube

Weighted folded hypercube is an charming variance of the famous hypercube and is superior to the weighted hypercube in many criteria. We mainly study the scaling of network coherence for the weighted folded hypercube that is controlled by a weight factor. Network coherence quantifies the steady-state variance of these fluctuations, and it can be regarded as a measure of robustness of the consensus process to the additive noise. If networks with small steady-state variance have better network coherence, it can be regarded as more robust to noise than networks with low coherence. We firstly calculate the spectra of weighted folded hypercube and obtain the leading terms of network coherence that are quantified as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues. Finally, the results show that network coherence depends on iterations and weight factor. Meanwhile, with larger order, the scatings of the first- and second-order network coherence of weighted folded hypercube decrease with the increasing of weight factor.

Consensus dynamics on a family of weighted recursive trees

The consensus dynamics with additive stochastic disturbances are characterized by the network coherence, which is the robustness of consensus algorithms when the nodes are subject to external perturbations. In this paper, the research goal is to obtain the first- and second-order network coherence quantifying as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues. One innovation point of this paper is the structure of a family of the weighted recursive trees with weight factor. We mainly obtain the exact expressions and scalings of network coherence on the family of weighted recursive trees. The scalings of first-order network coherence with network size obey three laws along with the range of the weight factor, while the scalings of second-order network coherence obey four laws along with the range of the weight factor. In addition, the scalings of first- and second-order network coherence on our studied networks are smaller than those performed on other studied networks when [Formula: see text]. The obtained results indicate that the efficiency of network coherence on the weighted network has close relation to the weight distribution, and we can design a better weight distribution to make the coherence of network more efficient.

Laplacian spectrum and coherence analysis of weighted hypercube network

Hypercube network is one of the most important and attractive network topologies so far. In this paper, we consider the scaling for first- and second-order network coherence on the hypercube network controlled by a weight factor. Our objective is to quantify the robustness of algorithms to stochastic disturbances at the nodes by using a quantity called network coherence which can be characterized as Laplacian spectrum. Network coherence can capture how well a network maintains its formation in the face of stochastic external disturbances. Firstly, we deduce the recursive relationships of its eigenvalues at two successive generations of Laplacian matrix. Then, we obtain the Laplacian spectrum of Laplacian matrix. Finally, we calculate the first- and second-order network coherence quantified as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues by using Squeeze Theorem. The obtained results show that the network coherence depends on generation number and weight factor. Meanwhile, the scalings of the first- and second-order network coherence of weighted hypercube decrease with the increasing of weight factor r, when 0 < r < 1.

Coherence analysis of a class of weighted networks.

This paper investigates consensus dynamics in a dynamical system with additive stochastic disturbances that is characterized as network coherence by using the Laplacian spectrum. We introduce a class of weighted networks based on a complete graph and investigate the first- and second-order network coherence quantifying as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues. First, the recursive relationship of its eigenvalues at two successive generations of Laplacian matrix is deduced. Then, we compute the sum and square sum of reciprocal of all nonzero Laplacian eigenvalues. The obtained results show that the scalings of first- and second-order coherence with network size obey four and five laws, respectively, along with the range of the weight factor. Finally, it indicates that the scalings of our studied networks are smaller than other studied networks when 1d<r≤1.