Fig 2 - uploaded by Dirk Aeyels
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Long term average velocities (horizontal axis) for varying coupling strength K (vertical axis). One of the agents is represented in bold.
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We introduce a model of mutually attracting agents in an arbitrary network, for which the long term behavior results in the emergence of several clusters. The cluster structure is independent of the initial condition and is characterized by a set of inequalities in the parameters of the model. With varying coupling strength, transitions between dif...
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... structures may take place. However, contrary to the all-to-all case, the number of possible cluster configurations with varying K may be larger than N. This follows from the fact that clusters may split with increasing coupling strength, a phenomenon that cannot occur in the all-to-all coupled case described by (1), and which is illustrated in Fig. 2, where the long term average velocities of the agents are shown for varying coupling ...
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Citations
... Due to the complexity and richness of the dynamics of some of these models, analytical results are often hard to come by and restricted to the existence and local stability properties of some of their solutions [10], [20], [2], and exploration of the parameter space is usually done through simulations [11], [13]. In previous papers [5], [1], [6] we introduced a dynamical model for clustering, corresponding to a system of nonidentical attracting agents, which may be connected according to an arbitrary network structure. Each agent has an autonomous component — its natural velocity — and attracts other agents by a saturating interaction function. ...
Clustering is a phenomenon that may emerge in multi-agent systems through self-organization: groups arise consisting of agents with similar dynamic behavior. It is observed in fields ranging from the exact sciences to social and life sciences; consider e.g. swarm behavior of animals or social insects, dynamics of opinion formation, or the synchronization (which corresponds to cluster formation in the phase space) of coupled oscillators modeling brain or heart cells. We consider the formation of clusters in a multi-agent system with each agent belonging to a multi-dimensional space. We mainly focus on cluster transitions with varying coupling strength. While for the one-dimensional case the transition generically involves at most two clusters, this is not necessarily true for higher dimensions. For a system with three agents, all-to-all coupling, and equal weights, we investigate necessary and sufficient conditions guaranteeing the occurrence of only two possible cluster structures: a single cluster containing all three agents, and a configuration with three clusters; for no value of the coupling strength is there an intermediate stage consisting of two clusters. We illustrate the results with a numeric example on opinion formation.
... See, for example, the survey article of Olfati-Saber et al. [2007]. Other works describe different contexts and approaches that show the relevance of the underlying interconnection graph (Jadbabaie et al. [2004], Marshall et al. [2004], Rogge and Aeyels [2004], Verwoerd and Mason [2007], Wang and Ghosh [2007], Smet and Aeyels [2008], Aeyels and Smet [2008], Carareto et al. [2009], Chopra and Spong [2009]). For the Kuramoto model, when all the oscillators are identical, the dynamical properties of the system relay totally on the algebraic and topological properties of this graph. ...
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.
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 xample, 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.
In this work, we give a complete presentation of the idea of synchronizing graphs for the Kuramoto model of coupled oscillators. We present the dynamical model and the main relationships between the system dynamics and the underlying interconnection topology. A synchronizing graph is an interconnection that ensures synchronization of all the oscillators for almost every initial condition. We present the main properties that help in the classification of synchronizing graphs and also some considerations about the structure of this family of graphs and the complexity of the classification task.
The formation of several clusters, arising from attracting forces between nonidentical entities or agents, is a phenomenon observed in diverse fields. Think of people gathered through a mutual interest, swarm behaviour of animals or clustering of oscillators in brain cells.We introduce a dynamic model of mutually attracting agents for which we prove that the long-term behaviour consists of agents organized into several groups or clusters. We have completely characterized the cluster structure (i.e. the number of clusters and their composition) by means of a set of inequalities in the parameters of the model and have identified the intensity of the attraction as a key parameter governing the transition between different cluster structures.The versatility of the model will be illustrated by discussing its relation to the Kuramoto model and by describing how it applies to a system of interconnected water basins.
