On the mathematics of emergence

By Felipe Cucker, Steve Smale. 2007

565 citations recorded by the source corpus.

Abstract.We describe a setting where convergence to consensus in a population of autonomous agents can be formally addressed and prove some general results establishing conditions under which such convergence occurs. Both continuous and discrete time are considered and a number of particular examples, notably the way in which a population of animals move together, are considered as particular instances of our setting.

Subfields: Biomedical Engineering, Computer Networks and Communications, Sociology and Political Science

View Paper: https://www.semanticscholar.org/paper/bf0b218ec413992e38324ddc864349b23c9d2863

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In Biomedical Engineering

In Computer Networks and Communications

In Sociology and Political Science

Citations in the corpus

On Krause's Multi-Agent Consensus Model With State-Dependent Connectivity (2008, background): “The same authors relax the all-to-all assumption in [9], and study communications restricted to arbitrarily changing but connected topologies.”
Differential inclusions in Wasserstein spaces: The Cauchy-Lipschitz framework (2020, background): “Multi-agent systems are ubiquitous in a wide number of applications, ranging from the understanding of patterns formation in several branches of the animal kingdom [14, 19, 33] to the analysis of pedestrian dynamics [2, 32], ensembles of autonomous vehicles [15, 26], and opinion formation on network”
Flocking and asymptotic velocity of the Cucker–Smale model with processing delay (2014, methodology): “The Cucker–Smale model [4] is given by dxi dt = vi, dvi dt = α N ∑”
From particle to kinetic and hydrodynamic descriptions of flocking (2008, background): “3a) A prototype example is the interaction kernel with a polynomial decay of order 2β, [6], r(|x − y|) ≥ A (1 + |x− y|2)β , β ≥ 0.”
Emergent dynamics of the Cucker-Smale flocking model and its variants (2016, background): “ the system is under rooted leadership at each time slice, but the leader agent, denoted by rt, is dependent on time t. Note that for the ﬂocking with symmetric interactions or a ﬁxed leader, such as [27, 31, 32, 41, 46, 48, 50, 60], the asymptotic velocity is a priori known, either the average of t”
Flocking estimates for the Cucker–Smale model with time lag and hierarchical leadership (2017, background): “4) The potential initially considered by Cucker and Smale in [19, 20] is ψ(s) = H (σ+s2)β with H,σ > 0 and β > 0.”
Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system (2009, methodology): “When noise is turned off, the above definition coincides with flocking in the deterministic case (see [2, 3, 8, 10]).”
Fokker–Planck equations in the modeling of socio-economic phenomena (2017, background): “[28, 82, 84] In the biological context, a consistent part of the recent research in population and behavioral biology and ecology has been looking for the emergent behavior of bird flocks, fish schools or bacteria aggregations.[26, 27, 38, 39, 40, 60, 61, 62] Other important examples of emergent beh”
The Pontryagin Maximum Principle in the Wasserstein Space (2017, background): “More generally, the study of other kind of interacting agents, such as opinion dynamics on networks [9, 28], or animal flocks [7, 18], has been conveyed with similar techniques.”
A New Model for Self-organized Dynamics and Its Flocking Behavior (2011, methodology): “The starting point for our discussion is the pioneering work of Cucker-Smale, [8, 9], which led to many subsequent studies [3, 6, 14, 15, 16, 23].”
Sparse stabilization and optimal control of the Cucker-Smale model (2013, unknown): “One of the motivations of this model proposed by Cucker and Smale was in fact to describe the formation and evolution of languages [22, Section 6], although, due to its simplicity, it has been eventually related mainly to the description of the emergence of consensus in a group of moving agents, for”
Cucker-Smale model with normalized communication weights and time delay (2016, methodology): “The Cucker-Smale model was introduced and studied in the seminal papers [12, 13], originally as a model for language evolution.”
Mean-field optimal control as Gamma-limit of finite agent controls (2018, background): “A particular and interesting example for a is given by the following decreasing function a(|q|) = 1/(1+|q|2)γ for some γ ≥ 0, which yields the Cucker-Smale flocking model [31,32].”
Asymptotic Flocking Dynamics for the Kinetic Cucker-Smale Model (2010, background): “482], the results yet obtained are sub-optimal though, in the sense that they do not reproduce at the continuous level the analysis in [16, 17]; no uniform bound of the spatial support of the density is provided, resulting in the sub-optimal estimate for β ≤ 1 4 , instead of β ≤ 12 valid for the dis”
Heterophilious Dynamics Enhances Consensus (2013, methodology): “Many models of self-organized dynamics are driven by examples from biology: these include aggregation of bacteria and amoeba [6, 51, 59, 74, 98], dynamics of insects [14, 33], schools of fish [1, 67, 111], flocking of birds [4, 26, 25, 24, 36, 37, 61, 93, 103, 107], and related models in ecology [58”
Double milling in self-propelled swarms from kinetic theory (2009, background): “A similar program has recently carried over in [19] for a kinetic model of flocking introduced in [20] based on the particle velocity-averaging interaction studied in [11, 12].”