Linear stability analysis of large dynamical systems on random directed graphs

Let$\{D_M\}_{M\geq 0}$be then-vertex random directed graph process, where$D_0$is the empty directed graph onnvertices, and subsequent directed graphs in the sequence are obtained by the addition of a new directed edge uniformly at random. For each$$\varepsilon > 0$$, we show that, almost surely, any directed graph$D_M$with minimum in- and out-degree at least 1 is not only Hamiltonian (as shown by Frieze), but remains Hamiltonian when edges are removed, as long as at most$1/2-\varepsilon$of both the in- and out-edges incident to each vertex are removed. We say such a directed graph is$(1/2-\varepsilon)$-resiliently Hamiltonian. Furthermore, for each$\varepsilon > 0$, we show that, almost surely, each directed graph$D_M$in the sequence is not$(1/2+\varepsilon)$-resiliently Hamiltonian.This improves a result of Ferber, Nenadov, Noever, Peter and Škorić who showed, for each$\varepsilon > 0$, that the binomial random directed graph$D(n,p)$is almost surely$(1/2-\varepsilon)$-resiliently Hamiltonian if$p=\omega(\log^8n/n)$.

We study the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles. For every 08γ81/2 we find a constantc=c(γ) such that the following holds. LetG= (V, E) be a (pseudo)random directed graph onnvertices and with at least a linear number of edges, and letG′ be a subgraph ofGwith (1/2 + γ)|E| edges. ThenG′ contains a directed cycle of length at least (c−o(1))n. Moreover, there is a subgraphG′′ofGwith (1/2 + γ −o(1))|E| edges that does not contain a cycle of length at leastcn. © 2011 Wiley Periodicals, Inc. J Graph Theory 70: 284–296, 2012

On the probability of chaos in large dynamical systems: A Monte Carlo study

Let be the random directed graph on n vertices where each of the possible arcs is present independently with probability p. A celebrated result of Frieze shows that if then typically has a directed Hamilton cycle, and this is best possible. In this paper, we obtain a strengthening of this result, showing that under the same condition, the number of directed Hamilton cycles in is typically . We also prove a hitting‐time version of this statement, showing that in the random directed graph process, as soon as every vertex has in‐/out‐degrees at least 1, there are typically directed Hamilton cycles.

ight about an estimated weighted mean orbit element formation barycenter is investigated. Consensus over random directed graphs in the presence of time delays is reviewed and applied to the problem. On-orbit spacecraft formations are considered random directed graphs with time delays and agreement over formation parameters is shown. Consensus of formation state estimates and formation barycenter using a distributed formation sensor network is proven. A simulation is described, demonstrating the functionality and applicability of the approach. Conclusions and future work are discussed.

Understanding the relationship between complexity and stability in large dynamical systems—such as ecosystems—remains a key open question in complexity theory which has inspired a rich body of work developed over more than fifty years. The vast majority of this theory addresses asymptotic linear stability around equilibrium points, but the idea of ‘stability’ in fact has other uses in the empirical ecological literature. The important notion of ‘temporal stability’ describes the character of fluctuations in population dynamics, driven by intrinsic or extrinsic noise. Here we apply tools from random matrix theory to the problem of temporal stability, deriving analytical predictions for the fluctuation spectra of complex ecological networks. We show that different network structures leave distinct signatures in the spectrum of fluctuations, and demonstrate the application of our theory to the analysis of ecological time-series data of plankton abundances.

The spectral radius of a directed graph is a metric that can only be computed when the structure of the network is completely known. However, in many practical scenarios, it is not possible to exactly retrieve the whole structure of the network; hence, the exact value of the spectral radius is not computable. Even in these scenarios, it is typically possible to extract local structural properties of a network using, for example, graph crawlers. In this paper, we develop a novel measure-theoretic framework to upper and lower bound the spectral radius of a directed graph using local structural information, in particular, using the counts of a collection of small subgraphs or motifs. Our framework is based on recent results relating the multivariate moment problem with semidefinite programming. Using these results, we develop a hierarchy of (small) semidefinite programs whose solutions provide upper and lower bounds on the spectral radius of a directed graph using, solely, subgraph and motif counts. We numerically validate the quality of our bounds using both random and real-world directed graphs.

This study presents a new approach to determine the controllability and observability of a large scale nonlinear dynamic thermal system using graph-theory. The novelty of this method is in adapting graph theory for nonlinear class and establishing a graphic condition that describes the necessary and sufficient terms for a nonlinear class system to be controllable and observable, which equivalents to the analytical method of Lie algebra rank condition. The directed graph (digraph) is utilized to model the system, and the rule of its adaptation in nonlinear class is defined. Subsequently, necessary and sufficient terms to achieve controllability and observability condition are investigated through the structural property of a digraph called connectability. It will be shown that the connectability condition between input and states, as well as output and states of a nonlinear system are equivalent to Lie-algebra rank condition (LARC). This approach has been proven to be easier from a computational point of view and is thus found to be useful when dealing with a large system.

RANDOMIZED CELLULAR AUTOMATA

In spite of intensive research, little progress has been made towards fast and work-efficient parallel algorithms for the single source shortest path problem. Our Δ-stepping algorithm, a generalization of Dial's algorithm and the Bellman-Ford algorithm, improves this situation at least in the following "average-case" sense: For random directed graphs with edge probability d/n and uniformly distributed edge weights a PRAM version works in expected time O(log3 n/ log log n) using linear work. The algorithm also allows for efficient adaptation to distributed memory machines. Implementations show that our approach works on real machines. As a side effect, we get a simple linear time sequential algorithm for a large class of not necessarily random directed graphs with random edge weights.

In this paper, we study the (h/sub 0/, h)-practical stability of large scale dynamic system with impulsive solution. By using comparison principle, scalar and vector Lyapunov functions and impulsive different inequalities, we obtain some sufficient conditions of the (h/sub 0/, h)-practical stability for large scale impulsive dynamic system in terms of two measurements.

Large discrete linear dynamical systems are described by the equation of motion with symmetrical and non-symmetrical matrices. With the solution of the associated right-hand eigenproblem state transition matrix is calculated in an efficient manner, because a complex inversion of a big matrix is substituted by three small, real inversions. Now the system is assumed to be altered. The new parameters cause a change of system matrices. The new eigenvalues and eigenvectors are represented as sums of unperturbed and perturbational terms up to fourth order. Inserting yields algebraic equations arranged in the order of perturbation. They are solved successively by expanding the perturbation eigenvectors in terms of unperturbed ones and employing the appropriately defined left-hand eigenvectors. The perturbation-eigenvalues are obtained directly from simple quotients, the expansion coefficients are solutions of linear systems of equations. Repeated solution of the eigenproblem is unnecessary. For special systems formulas become even more simple. Perturbed modal matrix of state equation and its inverse are represented as simple sums. Hence, also the state transition matrix of the altered system is known.

Principal component analysis is used to decompose a large scale dynamic system into robust subsystems. Decentralized state estimators are then designed to observe the states of the individual subsystems. The estimators are useful in the identification of states which are strongly controllable and/or strongly observable in the inputs and outputs, respectively. The decentralized estimators have few elements and are easy to implement and tune in real time. The decentralized estimators were verified by simulation. Simulation results are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

We discuss the stabilization of large scale linear time invariant dynamical systems via feedback. An overview of efficient schemes based on the discrete Riccati difference equation are presented. In particular, results are given for a Newton-like approach to the problem.

In this paper, we propose an extended block Krylov process to construct two biorthogonal bases for the extended Krylov subspaces \(\mathbb {K}_{m}^e(A,V)\) and \(\mathbb {K}_{m}^e(A^{T},W)\), where \(A \in \mathbb {R}^{n \times n}\) and \(V,~W \in \mathbb {R}^{n \times p}\). After deriving some new theoretical results and algebraic properties, we apply the proposed algorithm with moment matching techniques for model reduction in large scale dynamical systems. Numerical experiments for large and sparse problems are given to show the efficiency of the proposed method.