# On the mathematics of emergence

@article{Cucker2007OnTM, title={On the mathematics of emergence}, author={Felipe Cucker and Stephen Smale}, journal={Japanese Journal of Mathematics}, year={2007}, volume={2}, pages={197-227} }

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.

## 409 Citations

The emergence on isolated time scales

- Mathematics2016 21st International Conference on Methods and Models in Automation and Robotics (MMAR)
- 2016

The Cucker-Smale model on isolated time scales models a consensus of emergence in a population of autonomous agents and the results establishing conditions under which such consensus occurs are presented.

Exponentially stable solution of mathematical model of agents dynamics on time scales

- MathematicsAdvances in Difference Equations
- 2019

This work is an attempt at studying leader-following model on the arbitrary time scale. The step size is treated as a function of time. Our purpose is establishing conditions ensuring a…

Towards a unified theory of consensus

- Mathematics
- 2014

This work revisits the classic multi-agent distributed consensus problem and applies its approach to a wide variety of linear, nonlinear consensus and flocking algorithms proposed in the literature and derives new conditions for asymptotic consensus.

Emergence in Random Noisy Environments

- Mathematics
- 2009

We investigate the emergent behavior of four types of generic dynamical systems under random environmental perturbations. Sufficient conditions for nearly-emergence in various scenarios are…

Exponentially stable solution of mathematical model based on graph theory of agents dynamics on time scales

- Mathematics
- 2019

In this paper an emergence of leader-following model based on graph theory on the arbitrary time scales is investigated. It means that the step size is not necessarily constant but it is a function…

Un)conditional consensus emergence under feedback controls

- Mathematics
- 2014

We study the problem of consensus emergence in multi-agent systems via external feedback controllers. We consider a set of agents interacting with dynamics given by a Cucker-Smale type of model, and…

Convergence Results for Two Models of Interaction

- Mathematics
- 2018

I investigate two models interacting agent systems: the first is motivated by the flocking and swarming behaviors in biological systems, while the second models opinion formation in social networks.…

Un)conditional consensus emergence under perturbed and decentralized feedback controls

- Mathematics
- 2015

A characterization of consensus emergence for systems with different feedback structures, such as leader-based configurations, perturbed information feedback, and feedback computed upon spatially confined information, as a parameter-dependent transition regime between self-regulation and centralized feedback stabilization.

Learning and Sparse Control of Multiagent Systems

- Computer Science
- 2016

The question of whether it is possible to externally and parsimoniously influence the dynamics, to promote the formation of certain desired patterns is addressed, and the issue of finding the sparsest control strategy for finite agent models in order to lead the dynamics optimally towards a given outcome is addressed.

## References

SHOWING 1-10 OF 21 REFERENCES

Modeling Language Evolution

- Computer ScienceFound. Comput. Math.
- 2004

This work describes a model for the evolution of the languages used by the agents of a society and proves convergence of these languages to a common one under certain conditions.

Emergent Behavior in Flocks

- MathematicsIEEE Transactions on Automatic Control
- 2007

The main result shows that when beta<1/2 convergence of the flock to a common velocity is guaranteed, while for betages1/ 2 convergence is guaranteed under some condition on the initial positions and velocities of the birds only.

Distributed Asynchronous Deterministic and Stochastic Gradient Optimization Algorithms

- Computer Science1984 American Control Conference
- 1984

A model for asynchronous distributed computation is presented and it is shown that natural asynchronous distributed versions of a large class of deterministic and stochastic gradient-like algorithms retain the desirable convergence properties of their centralized counterparts.

Coordination of groups of mobile autonomous agents using nearest neighbor rules

- MathematicsIEEE Trans. Autom. Control.
- 2003

A theoretical explanation for the observed behavior of the Vicsek model, which proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function.

Stable flocking of mobile agents part I: dynamic topology

- Mathematics42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475)
- 2003

This is the second of a two-part paper, investigating the stability properties of a system of multiple mobile agents with double integrator dynamics. In this second part, we allow the topology of the…

Stable flocking of mobile agents, part I: fixed topology

- Mathematics42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475)
- 2003

This first part generates stable flocking motion for the group using a coordination control scheme which gives rise to smooth control laws for the agents, ensuring collision avoidance and cohesion of the group and an aggregate motion along a common heading direction.

Multi-Vehicle Flocking: Scalability of Cooperative Control Algorithms using Pairwise Potentials

- Computer ScienceProceedings 2007 IEEE International Conference on Robotics and Automation
- 2007

Cooperative control algorithms using pairwise interactions, for the purpose of controlling flocks of unmanned vehicles, show critical thresholds exist between coherent, stable, and scalable flocking and dispersed or collapsing motion of the group.

A Nonlocal Continuum Model for Biological Aggregation

- PhysicsBulletin of mathematical biology
- 2006

A continuum model for biological aggregations in which individuals experience long-range social attraction and short-range dispersal is constructed, and energy arguments are used to understand the nonlinear selection of clump solutions, and to predict the internal density in the large population limit.

Self-organization and selection in the emergence of vocabulary

- BiologyComplex.
- 2002

The combination of self-organization and cultural selection provides a plausible explanation for cultural evolution, which progresses with high transmission rate and is observed that as the vocabulary tends to convergence there is a uniform tendency to exhibit a sharp phase transition.