# American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

All Issues

## Networks & Heterogeneous Media

September 2020 , Volume 15 , Issue 3

Special issue on mathematical models for collective dynamics

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2020, 15(3): i-i doi: 10.3934/nhm.2020020 +[Abstract](732) +[HTML](267) +[PDF](90.18KB)
Abstract:
2020, 15(3): 307-352 doi: 10.3934/nhm.2020021 +[Abstract](875) +[HTML](94) +[PDF](8461.66KB)
Abstract:

Motivated by experiments on cell segregation, we present a two-species model of interacting particles, aiming at a quantitative description of this phenomenon. Under precise scaling hypothesis, we derive from the microscopic model a macroscopic one and we analyze it. In particular, we determine the range of parameters for which segregation is expected. We compare our analytical results and numerical simulations of the macroscopic model to direct simulations of the particles, and comment on possible links with experiments.

2020, 15(3): 353-368 doi: 10.3934/nhm.2020022 +[Abstract](1114) +[HTML](221) +[PDF](606.45KB)
Abstract:

This paper presents a development of the mathematical theory of swarms towards a systems approach to behavioral dynamics of social and economical systems. The modeling approach accounts for the ability of social entities are to develop a specific strategy which is heterogeneously distributed by interactions which are nonlinearly additive. A detailed application to the modeling of the dynamics of prices in the interaction between buyers and sellers is developed to describe the predictive ability of the model.

2020, 15(3): 369-387 doi: 10.3934/nhm.2020023 +[Abstract](916) +[HTML](98) +[PDF](384.02KB)
Abstract:

We show how to obtain general nonlinear aggregation-diffusion models, including Keller-Segel type models with nonlinear diffusions, as relaxations from nonlocal compressible Euler-type hydrodynamic systems via the relative entropy method. We discuss the assumptions on the confinement and interaction potentials depending on the relative energy of the free energy functional allowing for this relaxation limit to hold. We deal with weak solutions for the nonlocal compressible Euler-type systems and strong solutions for the limiting aggregation-diffusion equations. Finally, we show the existence of weak solutions to the nonlocal compressible Euler-type systems satisfying the needed properties for completeness sake.

2020, 15(3): 389-404 doi: 10.3934/nhm.2020024 +[Abstract](725) +[HTML](90) +[PDF](347.72KB)
Abstract:

The global Cauchy problem for a local alignment model with a relaxational inter-particle interaction operator is considered. More precisely, we consider the global-in-time existence of weak solutions of BGK model with local velocity-alignment term when the initial data have finite mass, momentum, energy, and entropy. The analysis involves weak/strong compactness based on the velocity moments estimates and several velocity-weighted \begin{document}$L^\infty$\end{document} estimates.

2020, 15(3): 405-426 doi: 10.3934/nhm.2020025 +[Abstract](1051) +[HTML](117) +[PDF](2548.7KB)
Abstract:

Experiments with pedestrians revealed that the geometry of the domain, as well as the incentive of pedestrians to reach a target as fast as possible have a strong influence on the overall dynamics. In this paper, we propose and validate different mathematical models at the micro- and macroscopic levels to study the influence of both effects. We calibrate the models with experimental data and compare the results at the micro- as well as macroscopic levels. Our numerical simulations reproduce qualitative experimental features on both levels, and indicate how geometry and motivation level influence the observed pedestrian density. Furthermore, we discuss the dynamics of solutions for different modeling approaches and comment on the analysis of the respective equations.

2020, 15(3): 427-461 doi: 10.3934/nhm.2020026 +[Abstract](835) +[HTML](91) +[PDF](458.19KB)
Abstract:

We study measurable stationary solutions for the kinetic Kuramoto-Sakaguchi (in short K-S) equation with frustration and their stability analysis. In the presence of frustration, the total phase is not a conserved quantity anymore, but it is time-varying. Thus, we can not expect the genuinely stationary solutions for the K-S equation. To overcome this lack of conserved quantity, we introduce new variables whose total phase is conserved. In the transformed K-S equation in new variables, we derive all measurable stationary solution representing the incoherent state, complete and partial phase-locked states. We also provide several frameworks in which the complete phase-locked state is stable, whereas partial phase-locked state is semi-stable in the space of Radon measures. In particular, we show that the incoherent state is nonlinearly stable in a large frustration regime, whereas it can exhibit stable behavior or concentration phenomenon in a small frustration regime.

2020, 15(3): 463-487 doi: 10.3934/nhm.2020027 +[Abstract](901) +[HTML](195) +[PDF](6109.38KB)
Abstract:

We consider mean-field models for data–clustering problems starting from a generalization of the bounded confidence model for opinion dynamics. The microscopic model includes information on the position as well as on additional features of the particles in order to develop specific clustering effects. The corresponding mean–field limit is derived and properties of the model are investigated analytically. In particular, the mean–field formulation allows the use of a random subsets algorithm for efficient computations of the clusters. Applications to shape detection and image segmentation on standard test images are presented and discussed.

2020, 15(3): 489-517 doi: 10.3934/nhm.2020028 +[Abstract](874) +[HTML](118) +[PDF](1984.56KB)
Abstract:

A generic feature of bounded confidence type models is the formation of clusters of agents. We propose and study a variant of bounded confidence dynamics with the goal of inducing unconditional convergence to a consensus. The defining feature of these dynamics which we name the No one left behind dynamics is the introduction of a local control on the agents which preserves the connectivity of the interaction network. We rigorously demonstrate that these dynamics result in unconditional convergence to a consensus. The qualitative nature of our argument prevents us quantifying how fast a consensus emerges, however we present numerical evidence that sharp convergence rates would be challenging to obtain for such dynamics. Finally, we propose a relaxed version of the control. The dynamics that result maintain many of the qualitative features of the bounded confidence dynamics yet ultimately still converge to a consensus as the control still maintains connectivity of the interaction network.

2020, 15(3): 519-542 doi: 10.3934/nhm.2020029 +[Abstract](711) +[HTML](92) +[PDF](699.9KB)
Abstract:

Unlike the classical kinetic theory of rarefied gases, where microscopic interactions among gas molecules are described as binary collisions, the modelling of socio-economic phenomena in a multi-agent system naturally requires to consider, in various situations, multiple interactions among the individuals. In this paper, we collect and discuss some examples related to economic and gambling activities. In particular, we focus on a linearisation strategy of the multiple interactions, which greatly simplifies the kinetic description of such systems while maintaining all their essential aggregate features, including the equilibrium distributions.

2020, 15(3): 543-553 doi: 10.3934/nhm.2020030 +[Abstract](853) +[HTML](102) +[PDF](292.91KB)
Abstract:

We discuss the complete synchronization for a Kuramoto-like model for power grids with frustration. For identical oscillators without frustration, it will converge to complete phase and frequency synchronization exponentially fast if the initial phases are distributed in a half circle. For nonidentical oscillators with frustration, we present a framework leading to complete frequency synchronization where the initial phase configurations are located inside the half of a circle. Our estimates are based on the monotonicity arguments of extremal phase and frequency.

2020 Impact Factor: 1.213
5 Year Impact Factor: 1.384
2020 CiteScore: 1.9