Spatio-Temporal Patterns for a Generalized Innovation Diffusion Model

We construct a model of innovation diffusion that incorporates a spatial component into a classical imitation-innovation dynamics first introduced by F. Bass. Relevant for situations where the imitation process explicitly depends on the spatial proximity between agents, the resulting nonlinear field dynamics is exactly solvable. As expected for nonlinear collective dynamics, the imitation mechanism generates spatio-temporal patterns, possessing here the remarkable feature that they can be explicitly and analytically discussed. The simplicity of the model, its intimate connection with the original Bass' modeling framework and the exact transient solutions offer a rather unique theoretical stylized framework to describe how innovation jointly develops in space and time.Comment: 20 pages, 4 figure

A Solvable Nonlinear Reaction-Diffusion Model

We construct a coupled set of nonlinear reaction-diffusion equations which are exactly solvable. The model generalizes both the Burger equation and a Boltzman reaction equation recently introduced by Th. W. Ruijgrok and T. T. Wu.Comment: 6 pages, LATe

Cooperative dynamics of loyal customers in queueing networks

We consider queueing networks (QN's) with feedback loops roamed by "intelligent” agents, able to select their routing on the basis of their measured waiting times at the QN nodes. This is an idealized model to discuss the dynamics of customers who stay loyal to a service supplier, provided their service time remains below a critical threshold. For these QN's, we show that the traffic flows may exhibit collective patterns typically encountered in multi-agent systems. In simple network topologies, the emergent cooperative behaviors manifest themselves via stable macroscopic temporal oscillations, synchronization of the queue contents and stabilization by noise phenomena. For a wide range of control parameters, the underlying presence of the law of large numbers enables us to use deterministic evolution laws to analytically characterize the cooperative evolution of our multi-agent systems. In particular, we study the case where the servers are sporadically subject to failures altering their ordinary behavio

Increasing Risk: Dynamic Mean-Preserving Spreads

We extend the celebrated Rothschild and Stiglitz (1970) definition of Mean-Preserving Spreads to a dynamic framework. We adapt the original integral conditions to transition probability densities, and give sufficient conditions for their satisfaction. We then prove that a specific nonlinear scalar diffusion process, super-diffusive ballistic noise, is the unique process that satisfies the integral conditions among a broad class of processes. This process can be generated by a random superposition of linear Markov processes with constant drifts. This exceptionally simple representation enables us to systematically revisit, by means of the properties of Dynamic Mean-Preserving Spreads, four workhorse economic models originally based on White Gaussian Noise

Networks of Self-Adaptive Dynamical Systems

We discuss the adaptive behaviour of a collection of heterogeneous dynamical systems interacting via a weighted network. At each vertex, the network is endowed with a dynamical system with individual (initially different) control parameters governing the local dynamics. We then implement a class of network interactions which generates a self-adaptive behaviour, driving all local dynamics to adopt a set of consensual values for their local parameters. While for ordinary synchronization each individual dynamical system is restored to its original dynamics once network interactions are removed, here the consensual values of control parameters are definitively acquired—even if interactions are removed. For a wide class of dynamical systems, we show analytically how such a plastic and self-adaptive training of control parameters can be realized. We base our study on local dynamics characterized by dissipative ortho-gradient vector fields encompassing a vast class of attractors (in particular limit cycles). The forces generated by the coupling network are derived from a generalized potential. A set of numerical experiments enables us to observe the transient dynamics and corroborate the analytical results obtaine