A Cucker--Smale Model with Noise and Delay

Erban, Radek; Haškovec, Jan; Sun, Yongzheng

doi:10.1137/15m1030467

Cited by 157 publications

(124 citation statements)

References 30 publications

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“…In the literature, the study on the C-S type models with time-delay is all restricted to the special network topologies. In [10,11,20,34], flocking estimates for the C-S type models with time-delay on a complete network have been discussed using Lyapunov type approach. In a recent work [35], the C-S type model with time-delay in velocity component has been also discussed on a hierarchical network where the asymptotic velocities of particles are given by that of leader.…”

Section: Description Of Main Resultsmentioning

confidence: 99%

Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph

Dong¹,

Ha²,

Kim³

2017

Discrete &Amp; Continuous Dynamical Systems - B

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We study dynamic interplay between time-delay and velocity alignment in the ensemble of Cucker-Smale (C-S) particles(or agents) on time-varying networks which are modeled by digraphs containing spanning trees. Time-delayed dynamical systems often appear in mathematical models from biology and control theory, and they have been extensively investigated in literature. In this paper, we provide sufficient frameworks for the mono-cluster flocking to the continuous and discrete C-S models, which are formulated in terms of system parameters and initial data. In our proposed frameworks, we show that the continuous and discrete C-S models exhibit exponential flocking estimates. For the explicit C-S communication weights which decay algebraically, our results exhibit threshold phenomena depending on the decay rate and depth of digraph. We also provide several numerical examples and compare them with our analytical results.

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Section: Description Of Main Resultsmentioning

confidence: 99%

Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph

Dong¹,

Ha²,

Kim³

2017

Discrete &Amp; Continuous Dynamical Systems - B

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“…The CS model has received lots of attention in literature from different perspectives, i.e., global and local flocking [8,18,19,32,34,36], collision avoiding [1,16], time-delay effect [9,10,21,24,27,44], hierarchical and rooted leadership, general digraph [20,38,39,47], application to flight navigation [43], noisy effects [2,17,23,33], mean-field limit [5,32,34], kinetic and hydrodynamic description [6,7,25,36,45], and variants of C-S model [41,42], etc. (see recent survey papers [14,40] for details).…”

Section: Introductionmentioning

confidence: 99%

Emergent behaviors of continuous and discrete thermomechanical Cucker–Smale models on general digraphs

Dong

Kim

2019

Math. Models Methods Appl. Sci.

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We present emergent dynamics of continuous and discrete thermomechanical Cucker-Smale(TCS) models equipped with temperature as an extra observable on general digraph. In previous literature, the emergent behaviors of the TCS models were mainly studied on a complete graph, or symmetric connected graphs. Under this symmetric setting, the total momentum is a conserved quantity. This determines the asymptotic velocity and temperature a priori using the initial data only. Moreover, this conservation law plays a crucial role in the flocking analysis based on the elementary ℓ 2 energy estimates. In this paper, we consider a more general connection topology which is registered by a general digraph, and the weights between particles are given to be inversely proportional to the metric distance between them. Due to this possible symmetry breaking in communication, the total momentum is not a conserved quantity, and this lack of conservation law makes the asymptotic velocity and temperature depend on the whole history of solutions. To circumvent this lack of conservation laws, we instead employ some tools from matrix theory on the scrambling matrices and some detailed analysis on the state-transition matrices. We present two sufficient frameworks for the emergence of mono-cluster flockings on a digraph for the continuous and discrete models. Our sufficient frameworks are given in terms of system parameters and initial data.Definition 1.1. [31, 35] Let C := (X, V, Θ) be a time-dependent configuration on the extended state space R N d × R N d × R d . Then the configuration C exhibits asymptotic mono-cluster flocking, if and only if the following conditions hold: sup 0≤t<∞ D(X(t)) < ∞, lim t→∞ D(V (t)) = 0, lim t→∞ D(Θ(t)) = 0. December 10, 2018 1:46 WSPC/INSTRUCTION FILE

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“…There are many social, biological, and physical systems in which a number of discrete individuals adjust an internal variable based on mutual interactions, leading the group to converge towards some sort of consensus (see [1] and its references). Examples such as social convention or social consensus [2][3][4], flocking [5,6], swarming [7][8][9][10], and other collective motion [1,11,12] have been extensively studied recently, with some modelling approaches treating the dynamics of opinions [2,3,[13][14][15][16][17][18][19][20][21][22][23][24] or flocks [6,[25][26][27][28][29][30] as examples of selforganized behavior [1,11]. These models depend sensitively on the connectedness of the individuals and many of these models have been considered in the context of networks, for instance in social systems [31][32][33][34][35][36][37] and natural phenomena (see e.g., [38]…”

Section: Introductionmentioning

confidence: 99%

Consensus and Information Cascades in Game-Theoretic Imitation Dynamics with Static and Dynamic Network Topologies

Griffin¹,

Rajtmajer²,

Squicciarini³

et al. 2019

SIAM J. Appl. Dyn. Syst.

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We construct a model of strategic imitation in an arbitrary network of players who interact through an additive game. Assuming a discrete time update, we show a condition under which the resulting difference equations converge to consensus. Two conjectures on general convergence are also discussed. We then consider the case where players not only may choose their strategies, but also affect their local topology. We show that for prisoner's dilemma, the graph structure converges to a set of disconnected cliques and strategic consensus occurs in each clique. Several examples from various matrix games are provided. A variation of the model is then used to create a simple model for the spreading of trends, or information cascades in (e.g., social) networks. We provide theoretical and empirical results on the trend-spreading model.3. Finally, we show how these dynamics can be adapted to model trends in (social) networks and study the qualitative properties of trends theoretically and empirically. This is a significant extension of work [62] in which we present the basic game-theoretic model for weighted imitation and consider its evolution on static graphs, without showing a condition on strategic consensus or studying either topological changes or trends. Consensus in a changing network topology is considered in [12] and is related to this work in so far as we also analyze the case of an agent-controlled changing network topology. A changing topology is also considered in [63], but the authors consider a specific opinion formation game consistent with the DeGroot model rather than a general matrix game.The model considered in this paper is distinct from the unifying model in [1] because:1. the coupling strength in the evolution equation is defined by a game payoff, rather than by an increasing (or non-negative) influence function φ, and 2. the agents' state is the strategy played, rather than a physical location, opinion or velocity.The coupling function we investigate may not be monotonic as a function of any metric on the strategy space over which our dynamics operate. This difference leads to a distinct set of consensus criteria than those given in [1]. As a result, our model cannot be subsumed by the unifying model in [1].

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A Cucker--Smale Model with Noise and Delay

Cited by 157 publications

References 30 publications

Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph

Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph

Emergent behaviors of continuous and discrete thermomechanical Cucker–Smale models on general digraphs

Consensus and Information Cascades in Game-Theoretic Imitation Dynamics with Static and Dynamic Network Topologies

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