Eduardo Alberto Canale

Almost Global Synchronization of Symmetric Kuramoto Coupled Oscillators

A few decades ago, Y. Kuramoto introduced a mathematical model of weakly coupled oscillators that gave a formal framework to some of the works of A.T. Winfree on biological clocks [Kuramoto (1975), Kuramoto (1984), Winfree (1980)]. The model proposes the idea that several oscillators can interact in a way such that the individual oscillation properties change in order to achieve a global behavior for the interconnected system. The Kuramoto model serves as a good representation of many systems in several contexts: biology, engineering, physics, mechanics, etc. [Ermentrout (1985), York (1993), Strogatz (1994), Dussopt et al. (1999), Strogatz (2000), Jadbabaie et a. (2003), Rogge et al. (2004), Marshall et al. (2004), Moshtagh et al. (2005)]. Recently, many works on the control community have focused on the analysis of the Kuramoto model, specially the one with sinusoidal coupling. The consensus or collective synchronization of the individuals is particularly important in many applications representing coordination, cooperation, emerging behavior, etc. Local stability properties of the consensus have been initially explored in [Jadbabaie et al. (2004)]. It must be noted that little attention has been devoted to the influence of the underlying interconnection graph on the stability properties of the system. The reason could be the fact that the local stability does not depend on the interconnection [van Hemmen et al. (1993)]. Global or almost global dynamical properties were studied in [Monzon et al. (2005), Monzon (2006), Monzon et al. (2006)]. In these works, the relevance of the interconnection graph of the system was hinted. In the present chapter, we go deeper on the analysis of the relationships between the dynamical properties of the system and the algebraic properties of the interconnection graph, exploiting the strong algebraic structure that every graph has. We step forward into a classification of the interconnection graphs that ensure almost global attraction of the set of synchronized states. In Section 2 we present the Kuramoto model for sinusoidally coupled oscillators, its general properties and the notion of almost global synchronization; in Section 3 we review some basic facts on algebraic graph theory; the symmetric Kuramoto model and the block analysis are presented in Sections 4 and 5; Section 6 gives some examples and applications of the main results; Section 7 presents the problem of classification of almost global synchronizing topologies.

On computing the 2‐diameter ‐constrained K ‐reliability of networks

This article considers a communication network modeled by a graph and a distinguished set of terminal nodes . We assume that the nodes never fail, but the edges fail randomly and independently with known probabilities. The classical K -reliability problem computes the probability that the subnetwork is composed only by the surviving edges in such a way that all terminals communicate with each other. The d -diameter -constrained K -reliability generalization also imposes the constraint that each pair of terminals must be the extremes of a surviving path of approximately d length. It allows modeling communication network situations in which limits exist on the acceptable delay times or on the amount of hops that packets can undergo. Both problems have been shown to be NP -hard, yet the complexity of certain subproblems remains undetermined. In particular, when , it was an open question whether the instances with were solvable in polynomial time. In this paper, we prove that when and is a fixed parameter (i.e. not an input) the problem turns out to be polynomial in the number of nodes of the network (in fact linear). We also introduce an algorithm to compute these cases in such time and also provide two numerical examples.

Gluing Kuramoto coupled oscillators networks

In this work we prove that the problem of almost global synchronization of the Kuramoto model of sinusoidally symmetric coupled oscillators with a given topology could be reduced to the analysis of the blocks of the underlying interconnection graph.

Global Synchronization Properties for Different Classes of Underlying Interconnection Graphs for Kuramoto Coupled Oscillators

This article deals with the general ideas of almost global synchronization of Kuramoto coupled oscillators and synchronizing graphs. We review the main existing results and introduce new results for some classes of graphs that are important in network optimization: complete k -partite graphs and what we have called Monma graphs.

On the Characterization of Families of Synchronizing Graphs for Kuramoto Coupled Oscillators

Abstract Kuramoto model of coupled oscillators represents situations where several individual agents interact and reach a collective behavior. The interaction is naturally described by a interconnection graph. Frequently, the desired performance is the synchronization of all the agents. Almost global synchronization means that the desire objective is reached for every initial conditions, with the possible exception of a zero Lebesgue measure set. This is a useful concept, specially when global synchronization can not be stated, due, for example, to the existence of multiple equilibria. In this survey article, we give an analysis of the influence of the interconnection graph on this dynamical property. We present in a ordered way several known and new results that help on the characterization of what we have called synchronizing topologies.

The wheels: an infinite family of bi-connected planar synchronizing graphs

Almost global synchronization property of Kuramoto coupled oscillations was recently introduced and stands for the case where almost every initial condition of the dynamical system leads to the synchronization of all the agents. When the oscillators are all identical, the property only depends on the the underlying interconnection graph. If the property is present, the interconnection graph is called synchronizing. It is known that a graph is synchronizing if and only if its block are. So, the characterization of synchronizing graphs can be restricted to the class of bi-connected graphs. In this work, we present the first known infinite family of biconnected planar synchronizing graphs, named, the wheels. Besides, we present two graph wich are the first known chordal graph and Halin graphs that not synchronize.

On the Complexity of the Classification of Synchronizing Graphs

This article deals with the general ideas of almost global synchronization of Kuramoto coupled oscillators and synchronizing graphs. It reviews the main existing results and gives some new results about the complexity of the problem. It is proved that any connected graph can be transformed into a synchronized one by making suitable groups of twin vertices. As a corollary it is deduced that any connected graph is the induced subgraph of a synchronizing graph. This implies a big structural complexity of synchronizability. Finally the former is applied to find a two integer parameter family G(a,b) of connected graphs such that if b is the k-th power of 10, the synchronizability of G(a,b) is equivalent to find the k-th digit in the expansion in base 10 of the square root of 2. Thus, the complexity of classify G(a,b) is of the same order than the computation of square root of 2. This is the first result so far about the computational complexity of the synchronizability problem.

Diameter-Constrained Reliability: Complexity, Factorization and Exact computation in Weak Graphs

In this paper we address a problem from the field of network reliability, called diameter-constrained reliability. Specifically, we are given a simple graph G = (V, E) with [V] = n nodes and [E] = m links, a subset K ⊆ V of terminals, a vector p = (p1,...,pm) &epsis; [0, 1]m and a positive integer d, called diameter. We assume nodes are perfect but links fail stochastically and independently, with probabilities qi = 1 --- pi. The diameter-constrained reliability (DCR for short), is the probability that the terminals of the resulting subgraph remain connected by paths composed by d links, or less. This number is denoted by RdK,G(p). The general DCR computation is inside the class of NP-Hard problems, since is subsumes the complexity that a random graph is connected. In this paper the computational complexity of DCR-subproblems is discussed in terms of the number of terminal nodes k = [K] and diameter d. A factorization formula for exact DCR computation is provided, that runs in exponential time in the worst case. Finally, a revision of graph-classes that accept DCR computation in polynomial time is then included. In this class we have graphs with bounded co-rank, graphs with bounded genus, planar graphs, and, in particular, Monma graphs, which are relevant in robust network design. We extend this class adding arborescence graphs. A discussion of trends for future work is offered in the conclusions.

Optimal design of an IP/MPLS over DWDM network

Different approaches for deploying resilient optical networks of low cost constitute a traditional group of NP-Hard problems that have been widely studied. Most of them are based on the construction of low cost networks that fulfill connectivity constraints. However, recent trends to virtualize optical networks over the legacy fiber infrastructure, modified the nature of network design problems and turned inappropriate many of these models and algorithms. In this paper we study a design problem arising from the deployment of an IP/MPLS network over an existing DWDM infrastructure. Besides cost and resiliency, this problem integrates traffic and capacity constraints. We present: an integer programming formulation for the problem, theoretical results, and describe how several metaheuristics were applied in order to find good quality solutions, for a real application case of a telecommunications company.

Diameter-constrained K-reliability evaluation: complexity and heuristics

Consider a communication network with perfect nodes, links with independent failures, special nodes K called terminals, and a diameter d. The corresponding d-diameter constrained K-reliability (d-DCKR) is the probability that the K terminals remain connected by paths composed by d hops or less. This problem has valuable applications in hop-constrained communication. The general d-DCKR evaluation is NP-Hard. However, we prove that the computational complexity of the 2-DCKR is linear in |K| when |K| is fixed, and an analytic expression for the target probability is provided. We introduce two Monte Carlo-based heuristics to tackle the general d-DCKR problem. The first one connects the target problem with. Numerical experiments with Petersen, dodecahedron and a series-parallel graph confirm the effectiveness of both approaches. The article concludes with a discussion of trends for future work.