Large time behavior of averaged kinetic models on networks

Abstract

Communicated by P. DegondWe are interested in flows on general networks and derive a kinetic equation describing general production, social or transportation networks. Corresponding macroscopic transport equations for large time and homogenized behavior are obtained and studied numerically. This work continues a recent discussion [Averaged kinetic models for flows on unstructured networks, Kinetic Related Models 4 (2011) 1081-1096] and provides additionally explicit equilibrium solutions, second-order macroscopic approximations as well as numerical simulations on a large scale homogenized network.Possible applications are production networks. Here, a good is flowing from a raw material supplier through a certain number of layers (nodes in the networks) of intermediate producers to a final consumer. Due to possible machine breakdowns and service distributions travel and waiting times of goods may be described using probability distributions. Production networks of this type have been introduced originally in Refs. 5,8, 9 and 11, and optimized in Refs. 15 and 16. Another application might be air traffic where we consider passengers arriving and leaving airports. The links are the possible flight routes and waiting times reflect delays at airports due to, for example, stochastic weather conditions. A description of effects of possible network structures in application examples for traffic and social networks has been studied in Ref. 7. Therein, the so-called small world network structure is discussed. Those networks are constructed by a process described in Ref. 4 and the mean connection between nodes grows at most logarithmic with the number of nodes. 30 Within this paper we want to explore the connection between a simple dynamics with possible applications in production and air traffic and the structure of the network more closely. To this end we consider a general mathematical model for transport on graphs described as a multi-agent model. We apply asymptotic techniques (borrowed from classical kinetic theory) to derive a simplified model for flows on large networks on large time scales. This reduces the computational complexity of the study of long time phenomena in such flows. We develop a kinetic model for flows on arbitrary graphs, originally proposed in Ref. 20, under some simplifying assumptions, which make the large time scale model practically applicable. The macroscopic model consists of a convection-diffusion equation for the agent density, posed on a continuum in space, representing the graph of the network. In order for this model to be "reasonably smooth", i.e. not to involve measure-valued transport coefficients, it is necessary to locate the nodes of the graph in a certain way, i.e. to "draw the graph in R 2 " in a special way. Reorganizing the network this way results in the solution of an optimization problem for the coordinates of the network nodes. This represents the generalization of an idea, originally proposed in Ref. 7 to higher dimensions. The final macroscopic model is a convection-d...

Deterministic Versus Stochastic Consensus Dynamics on Graphs

“…We hope to characterize this behavior through passing to a continuous description of the graph to obtain in a certain limit a partial differential equation describing the dynamics. One possible avenue to this description would be through a kinetic description of the model combined with a coarsening approach [15,20]. However, as it has been observed [8], propagation of chaos is not always applicable to these dynamics which could hinder this approach.…”

“…If L is symmetric, λ 2 is known as the algebraic connectivity of the graph G and is a measure of how well connected G is. In this context the bound (20) can be interpreted as stating that the consensus model converges faster if the network on which agents interact is better connected.…”

“…We are also interested in examining the dynamics from a control perspective -we include a brief comment discussing how given control over the network structure, the deterministic dynamics we study can be used as an algorithm to induce consensus to any opinion in the convex hull of initial agent opinions [3]. Finally, we aim to study limiting behavior of the model as the number of agents approaches infinity to develop a global description of the dynamics through a partial differential equation derived possibly through a kinetic description of the model [15,20].…”