Matrix Tree Theorem Research Articles

Oriented hypergraphic matrix-tree type theorems and bidirected minors via Boolean order ideals

Restrictions of incidence-preserving path maps produce an oriented hypergraphic All Minors Matrix-tree Theorems for Laplacian and adjacency matrices. The images of these maps produce a locally signed graphic, incidence generalization, of cycle covers and basic figures that correspond to incidence-k-forests. When restricted to bidirected graphs the natural partial ordering of maps results in disjoint signed boolean lattices whose minor calculations correspond to principal order ideals. As an application, (1) the determinant formula of a signed graphic Laplacian is reclaimed and shown to be determined by the maximal positive-circle-free elements, and (2) spanning trees are equivalent to single-element order ideals.

Quantized feedback control scheme on coupled systems with time delay and distributed delay: A finite-time inner synchronization analysis

In this paper, finite-time inner synchronization of coupled systems on a network with time delay and distributed delay (CSNTD) is investigated. And here, time delay and distributed delay are both taken into consideration when modelling a realistic network. Different from common feedback control, the controller we design is quantized, which is more realistic and reasonable. By using Lyapunov method and Kirchhoff’s Matrix Tree Theorem, some sufficient criteria are derived to guarantee finite-time inner synchronization of CSNTD. It should be underlined that the method is first applied to studying the issue of finite-time inner synchronization of CSNTD and the synchronization time we obtain has a close relationship with the topological structure of the network. Moreover, to verify our theoretical results, we present an application to coupled oscillators with time delay and distributed delay, and a sufficient criterion is obtained. Ultimately, a numerical example is given to verify the validity and feasibility of theoretical results.

Synchronization analysis of stochastic coupled systems with time delay on networks by periodically intermittent control and graph-theoretic method

In this paper, the issue of inner exponential synchronization (IES) for a class of stochastic coupled systems with time-varying delay on networks (SCSTVDNs) is investigated. A periodically intermittent control strategy is introduced to drive such SCSTVDNs to achieve IES. By exploiting Lyapunov method and Kirchhoffs matrix tree theorem in graph theory, two types of sufficient IES criteria are respectively established in the form of Lyapunov-type theorem and coefficients-type theorem. Furthermore, to show the practicability of the theoretical results, the analysis of IES for the second-order Kuramoto oscillators with stochastic perturbations and time-varying delay is performed. A numerical example is finally provided to demonstrate the validity and feasibility of our analytical results.

Stabilization of stochastic complex-valued coupled delayed systems with Markovian switching via periodically intermittent control

This paper focuses on exponential stabilization of stochastic complex-valued coupled delayed systems with Markovian switching (SCCDM) via periodically intermittent control. The features of complex-valued systems, time-varying delays, stochastic disturbances and Markovian switching are taken into account. The mathematical model of this kind of complex-valued coupled systems is studied for the first time. To study the stabilization problem of SCCDM, two new differential inequalities on delayed Markovian jump systems are established. Then by utilizing Lyapunov method combined with Kirchhoff’s Matrix Tree Theorem, sufficient criteria promising the stability of SCCDM via periodically intermittent control are derived, which have a close relationship with control period, control rate, control gain and the topological structure of the considered coupled network. Then we employ the theoretical results to study the stabilization problem of stochastic complex-valued coupled oscillators with Markovian switching via periodically intermittent control. Finally, numerical examples are provided to demonstrate the effectiveness and feasibility of the proposed results.

Synchronization of coupled stochastic complex-valued dynamical networks with time-varying delays via aperiodically intermittent adaptive control.

This paper deals with the synchronization problem of a class of coupled stochastic complex-valued drive-response networks with time-varying delays via aperiodically intermittent adaptive control. Different from the previous works, the intermittent control is aperiodic and adaptive, and the restrictions on the control width and time delay are removed, which lead to a larger application scope for this control strategy. Then, based on the Lyapunov method and Kirchhoff's Matrix Tree Theorem as well as differential inequality techniques, several novel synchronization conditions are derived for the considered model. Specially, impulsive control is also considered, which can be seen as a special case of the aperiodically intermittent control when the control width tends to zero. And the corresponding synchronization criteria are given as well. As an application of the theoretical results, a class of stochastic complex-valued coupled oscillators with time-varying delays is studied, and the numerical simulations are also given to demonstrate the effectiveness of the control strategies.

A weighted cellular matrix-tree theorem, with applications to complete colorful and cubical complexes

We present a version of the weighted cellular matrix-tree theorem that is suitable for calculating explicit generating functions for spanning trees of highly structured families of simplicial and cell complexes. We apply the result to give weighted generalizations of the tree enumeration formulas of Adin for complete colorful complexes, and of Duval, Klivans and Martin for skeleta of hypercubes. We investigate the latter further via a logarithmic generating function for weighted tree enumeration, and derive another tree-counting formula using the unsigned Euler characteristics of skeleta of a hypercube.

Synchronization of stochastic coupled systems with time-varying coupling structure on networks via discrete-time state feedback control

Abstract This paper is concerned with the exponential synchronization of stochastic coupled systems on networks. Time-varying coupling structure of coupled systems and discrete-time state feedback control are focuses and difficulties of our research. Based on the Lyapunov method and Kirchhoff’s Matrix Tree Theorem, two sufficient criteria are obtained, which guarantee the exponential synchronization of stochastic coupled systems with time-varying coupling structure on networks via discrete-time state feedback control. Finally, we apply theoretical results to stochastic coupled oscillators and propose two numerical examples to illustrate the effectiveness and feasibility of the obtained theoretical results.

Outer synchronization of delayed coupled systems on networks without strong connectedness: A hierarchical method

We consider the outer synchronization between drive-response systems on networks with time-varying delays, where we focus on the case when the underlying networks are not strongly connected. A hierarchical method is proposed to characterize large-scale networks without strong connectedness. The hierarchical algorithm can be implemented by some programs to overcome the difficulty resulting from the scale of networks. This method allows us to obtain two different kinds of sufficient outer synchronization criteria without the assumption of being strongly connected, by combining the theory of asymptotically autonomous systems with Lyapunov method and Kirchhoff's Matrix Tree Theorem in graph theory. The theory improves some existing results obtained by graph theory. As illustrations, the theoretic results are applied to delayed coupled oscillators and a numerical example is also given.

The Stability of Stochastic Coupled Systems With Time-Varying Coupling and General Topology Structure.

We introduce a class of novel stochastic coupled systems in which the coupling structure is time-varying and the topology structure is not strongly connected, and first establish the system on a digraph with a time-varying weight matrix. Motivated by Du and Li (2014), we give a hierarchical method to deal with digraphs without strong connectivity and establish the corresponding hierarchical algorithm to realize this approach. Also, an example is given to illustrate our hierarchical algorithm and its feasibility. In the sequel, based on the theory of asymptotically autonomous systems, Kirchhoff's matrix tree theorem, and Lyapunov method, several moment exponential stability criteria are presented, including a Lyapunov-type theorem and a coefficient-type criterion. Furthermore, theoretical results are applied to stochastic coupled oscillators with time-varying coupling structure (SCTCS), and the stability criterion of SCTCS is obtained. Finally, the effectiveness of theoretical results is illustrated by two numerical examples.

A higher Boltzmann distribution

We characterize the classical Boltzmann distribution as the unique solution to a combinatorial Hodge theory problem in homological degree zero on a finite graph. By substituting for the graph a CW complex X and a choice of degree $$d \le \dim X$$ , we define by direct analogy a higher dimensional Boltzmann distribution $$\rho ^B$$ as a certain real-valued cellular $$(d-1)$$ -cycle. We then give an explicit formula for $$\rho ^B$$ . We explain how these ideas relate to the Higher Kirchhoff Network Theorem of Catanzaro et al. (Homol Homotopy Appl 17:165–189, 2015). We also deduce an improved version of the Higher Matrix-Tree Theorems of Catanzaro et al. (Homol Homotopy Appl 17:165–189, 2015).

Graph-theoretic approach to exponential synchronization of discrete-time stochastic coupled systems with time-varying delay

Abstract In this paper, we investigate the exponential synchronization problem for discrete-time stochastic drive-response coupled systems with time-varying delay. By employing the Lyapunov method combined with Kirchhoff’s matrix tree theorem in graph theory as well as stochastic analysis technique, some novel sufficient criteria are established to guarantee the exponential synchronization of two identical delayed coupled systems with stochastic disturbances. These sufficient criteria have a close relationship with the topological property of the coupled network. Moreover, the theoretical results are applied to a coupled oscillators system to demonstrate the applicability of the proposed synchronization approaches. Finally, a numerical example is provided to illustrate the effectiveness of our theoretical results.

Periodic solutions of discrete time periodic time-varying coupled systems on networks

In this paper, we consider the existence of periodic solutions for discrete time periodic time-varying coupled systems on networks (DPTCSN). Some novel sufficient conditions are obtained to guarantee the existence of periodic solutions for DPTCSN, which have a close relation to the topology property of the corresponding network. Our approach is based on the continuation theorem of coincidence degree theory, generalized Kirchhoff’s matrix tree theorem in graph theory, Lyapunov method and some new analysis techniques. The approach is applied to the existence of periodic solutions for discrete time Cohen–Grossberg Neural Networks (CGNN). Finally, an example and numerical simulations are provided to illustrate the effectiveness of our theoretical results.

Razumikhin method conjoined with graph theory to input-to-state stability of coupled retarded systems on networks

This paper is concerned with input-to-state stability (ISS) of coupled retarded systems on networks (CRSNs). By incorporating Razumikhin method with generalized Kirchhoff’s Matrix Tree Theorem, sufficient conditions are established to guarantee ISS of CRSNs. For the convenience of application, sufficient conditions in the form of coefficients are also obtained. What is more, a sufficient criterion on ISS of the coupled retarded oscillators system on networks is derived. Finally, as an illustration, a numerical example is given to demonstrate the effectiveness of the theoretical results.

Graph-theoretic approach to exponential synchronization of coupled systems on networks with mixed time-varying delays

In this paper, the issue of exponential synchronization for coupled systems on networks with mixed time-varying delays is concerned. An approach combining Kirchhoff’s matrix tree theorem in graph theory with Lyapunov method and periodically intermittent control is taken to investigate the problem. This method is different from the corresponding previous works. Two different kinds of synchronization conditions in the form of Lyapunov-type theorem and coefficients-type criterion are derived. They both reveal synchronization has a close relation with the topology structure of the network. Finally, the feasibility and effectiveness of the proposed method are illustrated by several numerical simulation figures.

Principal minor version of Matrix-Tree theorem for mixed graphs

In Yu, et al. (2017), an analytical expression of the determinant of the Hermitian (quasi-)Laplacian matrix of mixed graphs has been proven. In this paper, we are going to extend those results and derive an analytical expression for the principal minors of the Hermitian (quasi-)Laplacian matrix, which is the principal minor version of the Matrix-Tree theorem.

Asymptotics for the determinant of the combinatorial Laplacian on hypercubic lattices

In this paper, we compute asymptotics for the determinant of the combinatorial Laplacian on a sequence of d-dimensional orthotope square lattices as the number of vertices in each dimension grows at the same rate. It is related to the number of spanning trees by the well-known matrix tree theorem. Asymptotics for 2 and 3 component rooted spanning forests in these graphs are also derived. Moreover, we express the number of spanning trees in a 2-dimensional square lattice in terms of the one in a 2-dimensional discrete torus and also in the quartered Aztec diamond. As a consequence, we find an asymptotic expansion of the number of spanning trees in a subgraph of Z2 with a triangular boundary.

Stability analysis of stochastic coupled systems on networks without strong connectedness via hierarchical approach

In this paper, the issues pertaining with the networks without strong connectedness (NWSC) and a class of stochastic coupled systems on them are considered in totality. Towards NWSC, a hierarchical approach is proposed explicitly based on the topological structure of networks, and we also provide some examples to demonstrate the practical applicability of this approach. In addition, through employing the theory of asymptotically autonomous systems along with the Lyapunov method and Kirchhoff׳s matrix tree theorem in graph theory, several sufficient conditions to guarantee the moment exponential stability of stochastic coupled systems on NWSC are presented in the form of Lyapunov-type theorem and coefficients-type criterion. As a subsequent result, the proposed theory is applied to a class of stochastic coupled oscillators on NWSC with sufficient conditions given to determine their stability. Ultimately, a numerical example is provided that lends insight to certifying the effectiveness and feasibility of our theoretical results.

Threshold dynamics of a delayed multi-group heroin epidemic model in heterogeneous populations

The aim of this paper is to investigate the threshold dynamics of a heroin epidemic in heterogeneous populations. The model is described by a delayed multi-group model, which allows us to model interactions both within-group and inter-group separately. Here we are able to prove the existence of heroin-spread equilibrium and the uniform persistence of the model. The proofs of main results come from suitable applications of graph-theoretic approach to the method of Lyapunov functionals and Krichhoff's matrix tree theorem. Numerical simulations are performed to support the results of the model for the case where $n=2$.

Stability Analysis of SIR Model with Distributed Delay on Complex Networks.

In this paper, by taking full consideration of distributed delay, demographics and contact heterogeneity of the individuals, we present a detailed analytical study of the Susceptible-Infected-Removed (SIR) epidemic model on complex population networks. The basic reproduction number of the model is dominated by the topology of the underlying network, the properties of individuals which include birth rate, death rate, removed rate and infected rate, and continuously distributed time delay. By constructing suitable Lyapunov functional and employing Kirchhoff’s matrix tree theorem, we investigate the globally asymptotical stability of the disease-free and endemic equilibrium points. Specifically, the system shows threshold behaviors: if , then the disease-free equilibrium is globally asymptotically stable, otherwise the endemic equilibrium is globally asymptotically stable. Furthermore, the obtained results show that SIR models with different types of delays have different converge time in the process of contagion: if , then the system with distributed time delay stabilizes fastest; while , the system with distributed time delay converges most slowly. The validness and effectiveness of these results are demonstrated through numerical simulations.

Counting spanning trees on fractal graphs and their asymptotic complexity

Using the method of spectral decimation and a modified version of Kirchhoff's matrix-tree theorem, a closed form solution to the number of spanning trees on approximating graphs to a fully symmetric self-similar structure on a finitely ramified fractal is given in theorem . We show how spectral decimation implies the existence of the asymptotic complexity constant and obtain some bounds for it. Examples calculated include the Sierpiński gasket, a non-post critically finite analog of the Sierpiński gasket, the Diamond fractal, and the hexagasket. For each example, the asymptotic complexity constant is found.