Mathematical Models and Methods in Applied Sciences

This paper presents the papers published in a special issue devoted to the modeling of behavioral social systems. Some perspective ideas on possible objectives of future research are extracted from the contents of this issue and brought to the reader’s attention. The final ambitious aim is the development of a mathematical theory for complex living systems.

An alternative voting scheme is proposed to fill the democratic gap between a pre-sident elected democratically via universal suffrage (deterministic outcome, the actual majority decides), and a president elected by one person randomly selected from the population (probabilistic outcome depending on respective supports). Indeed, moving from one voting agent to a group of $r$ randomly selected voting agents reduces the probabilistic character of the outcome. Accordingly, building $r$ such groups, each one electing its president (elementary bricks), to constitute a group of the groups with the $r$ local presidents electing a higher-level president, does reduce further the outcome probabilistic aspect. The process is then repeated $n$ times to reach a bottom-up pyramidal structure with $n$ levels, $r^{n-1}$ elementary bricks at the bottom and a president at the top. Agents at the bottom are randomly selected but higher-level presidents are all designated according to the respective local majorities within the groups which elect them. At the top of the hierarchy the president is still elected with a probability but the distance from a deterministic outcome reduces quickly with increasing $n$. At a critical value $n_{c,r}$ the outcome turns deterministic recovering the same result a universal suffrage would yield. This alternative hierarchical scheme introduces several social advantages like the distribution of local power to the competing minority, which thus makes the structure more resilient, yet preserving the presidency allocation to the actual majority. It also produces an area around 50% for which the president is elected with an almost-equiprobability slightly biased in favor of the actual majority. However, our results reveal the existence of a severe geometric vulnerability to lobbying. It is shown that a tiny lobbying group is able to kill the democratic balance by seizing the presidency democratically. It is sufficient to complete a correlated distribution of a few agents at the hierarchy bottom. Moreover, at the present stage, identifying an actual killing distribution is not feasible, which sheds a disturbing light on the devastating effect geometric lobbying can have on democratic hierarchical institutions.

Motivated by the study of social insects, we introduce a stochastic model based on interacting particle systems in order to understand the effect of communication on the division of labor. Members of the colony are located on the vertex set of a graph representing a communication network. They are characterized by one of two possible tasks, which they update at a rate equal to the cost of the task they are performing by either defecting by switching to the other task or cooperating by anti-imitating a random neighbor in order to balance the amount of energy spent in each task. We prove that, at least when the probability of defection is small, the division of labor is poor when there is no communication, better when the communication network consists of a complete graph, but optimal on bipartite graphs with bipartite sets of equal size, even when both tasks have very different costs. This shows a non-monotonic relationship between the number of connections in the communication network and how well individuals organize themselves to accomplish both tasks equally.

This paper introduces and analyzes some models in the framework of mean field games (MFGs) describing interactions between two populations motivated by the studies on urban settlements and residential choice by Thomas Schelling. For static games, a large population limit is proved. For differential games with noise, the existence of solutions is established for the systems of partial differential equations of MFG theory, in the stationary and in the evolutive case. Numerical methods are proposed with several simulations. In the examples and in the numerical results, particular emphasis is put on the phenomenon of segregation between the populations.

We present and discuss various one-dimensional linear Fokker–Planck-type equations that have been recently considered in connection with the study of interacting multi-agent systems. In general, these Fokker–Planck equations describe the evolution in time of some probability density of the population of agents, typically the distribution of the personal wealth or of the personal opinion, and are mostly obtained by linear or bilinear kinetic models of Boltzmann type via some limit procedure. The main feature of these equations is the presence of variable diffusion, drift coefficients and boundaries, which introduce new challenging mathematical problems in the study of their long-time behavior.

We investigate the long-time dynamics of an opinion formation model inspired by a work by Borghesi, Bouchaud and Jensen. First, we derive a Fokker–Planck-type equation under the assumption that interactions between individuals produce little consensus of opinion (grazing collision approximation). Second, we study conditions under which the Fokker–Planck equation has non-trivial equilibria and derive the macroscopic limit (corresponding to the long-time dynamics and spatially localized interactions) for the evolution of the mean opinion. Finally, we compare two different types of interaction rates: the original one given in the work of Borghesi, Bouchaud and Jensen (symmetric binary interactions) and one inspired from works by Motsch and Tadmor (non-symmetric binary interactions). We show that the first case leads to a conservative model for the density of the mean opinion whereas the second case leads to a non-conservative equation. We also show that the speed at which consensus is reached asymptotically for these two rates has fairly different density dependence.

We introduce a novel first-order stochastic swarm intelligence (SI) model in the spirit of consensus formation models, namely a consensus-based optimization (CBO) algorithm, which may be used for the global optimization of a function in multiple dimensions. The CBO algorithm allows for passage to the mean-field limit, which results in a nonstandard, nonlocal, degenerate parabolic partial differential equation (PDE). Exploiting tools from PDE analysis we provide convergence results that help to understand the asymptotic behavior of the SI model. We further present numerical investigations underlining the feasibility of our approach.

In this paper, we develop two efficient numerical methods for a multiscale kinetic equation in the context of crowd dynamics with emotional contagion [A. Bertozzi, J. Rosado, M. Short and L. Wang, Contagion shocks in one dimension, J. Stat. Phys. 158 (2014) 647–664]. In the continuum limit, the mesoscopic kinetic equation produces a natural Eulerian limit with nonlocal interactions. However, such limit ceases to be valid when the underlying microscopic particle characteristics cross, corresponding to the blow up of the solution in the Eulerian system. One method is to couple these two situations — using Eulerian dynamics for regions without characteristic crossing and kinetic evolution for regions with characteristic crossing. For such a hybrid setting, we provide a regime indicator based on the macroscopic density and fear level, and propose an interface condition via continuity to connect these two regimes. The other method is based on a level set formulation for the continuum system. The level set equation shares similar forms as the kinetic equation, and it successfully captures the multi-valued solution in velocity, which implies that the multi-valued solution other than the viscosity solution should be the physically relevant ones for the continuum system. Numerical examples are presented to show the efficiency of these new methods.