# American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

All Issues

## Networks & Heterogeneous Media

September 2015 , Volume 10 , Issue 3

Special issue on modeling and control in social dynamics

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2015, 10(3): i-ii doi: 10.3934/nhm.2015.10.3i +[Abstract](1957) +[PDF](105.5KB)
Abstract:
This Special Issue is based on research presented at the Workshop Modeling and Control of Social Dynamics", hosted by the Center of Computational and Integrative Biology and the Department of Mathematical Sciences at Rutgers University - Camden. The Workshop is part of the activities of the NSF Research Network in Mathematical Sciences: Kinetic description of emerging challenges in multiscale problems of natural sciences" Grant # 1107444, which is also acknowledged for funding the workshop.

2015, 10(3): 421-441 doi: 10.3934/nhm.2015.10.421 +[Abstract](1542) +[PDF](506.7KB)
Abstract:
This paper proposes a systems theory approach to the modeling of onset and evolution of criminality in a territory. This approach aims at capturing the complexity features of social systems. Complexity is related to the fact that individuals have the ability to develop specific heterogeneously distributed strategies, which depend also on those expressed by the other individuals. The modeling is developed by methods of generalized kinetic theory where interactions and decisional processes are modeled by theoretical tools of stochastic game theory.
2015, 10(3): 443-475 doi: 10.3934/nhm.2015.10.443 +[Abstract](2024) +[PDF](1969.5KB)
Abstract:
We introduce and analyze several variants of a system of differential equations which model the dynamics of social outbursts, such as riots. The systems involve the coupling of an explicit variable representing the intensity of rioting activity and an underlying (implicit) field of social tension. Our models include the effects of exogenous and endogenous factors as well as various propagation mechanisms. From numerical and mathematical analysis of these models we show that the assumptions made on how different locations influence one another and how the tension in the system disperses play a major role on the qualitative behavior of bursts of social unrest. Furthermore, we analyze here various properties of these systems, such as the existence of traveling wave solutions, and formulate some new open mathematical problems which arise from our work.
2015, 10(3): 477-509 doi: 10.3934/nhm.2015.10.477 +[Abstract](1863) +[PDF](2603.2KB)
Abstract:
By a simple extension of the bounded confidence model, it is possible to model the influence of a radical group, or a charismatic leader on the opinion dynamics of normal' agents that update their opinions under both, the influence of their normal peers, and the additional influence of the radical group or a charismatic leader. From a more abstract point of view, we model the influence of a signal, that is constant, may have different intensities, and is heard' only by agents with opinions, that are not too far away. For such a dynamic a Constant Signal Theorem is proven. In the model we get a lot of surprising effects. For instance, the more intensive signal may have less effect; more radicals may lead to less radicalization of normal agents. The model is an extremely simple conceptual model. Under some assumptions the whole parameter space can be analyzed. The model inspires new possible explanations, new perspectives for empirical studies, and new ideas for prevention or intervention policies.
2015, 10(3): 511-526 doi: 10.3934/nhm.2015.10.511 +[Abstract](1299) +[PDF](448.8KB)
Abstract:
Aggregate production planning for highly re--entrant production processes is typically generated by finding optimal release rates based on clearing function models. For production processes with very long cycle times, like in semiconductor production, dispatch policies are used to cover short term fluctuations. We extend the concept of a clearing function to allow control over both, the release rates and priority allocations in re-entrant production. This approach is used to improve the production planning problem using combined release and the allocation dispatch policy. The control parameter for priority allocation, called the push-pull point (PPP), separates the beginning of the factory which employs a push policy from the end of the factory, which uses a pull policy. The extended clearing function model describes the output of the factory as a function of the work in progress (wip) and the position of the PPP. The model's qualitative behavior is analyzed. Numerical optimization results are compared to production planning based only on releases. It is found that controlling the PPP significantly reduces the average wip in the system and hence leads to much shorter cycle times.
2015, 10(3): 527-542 doi: 10.3934/nhm.2015.10.527 +[Abstract](1766) +[PDF](1812.7KB)
Abstract:
A kinetic model for a specific agent based simulation to generate the sales curves of successive generations of high-end computer chips is developed. The resulting continuum market model consists of transport equations in two variables, representing the availability of money and the desire to buy a new chip. In lieu of typical collision terms in the kinetic equations that discontinuously change the attributes of an agent, discontinuous changes are initiated via boundary conditions between sets of partial differential equations. A scaling analysis of the transport equations determines the different time scales that constitute the market forces, characterizing different sales scenarios. It is argued that the resulting model can be adjusted to generic markets of multi-generational technology products where the innovation time scale is an important driver of the market.
2015, 10(3): 543-557 doi: 10.3934/nhm.2015.10.543 +[Abstract](1421) +[PDF](1311.0KB)
Abstract:
We introduce and discuss a new kinetic system for a financial market composed by agents that may belong to two different trader populations, whose behavior determines the price dynamic of a certain stock. Our mesoscopic description is based on the microscopic Lux--Marchesi model [16,17], and share analogies with the recent kinetic model by Maldarella and Pareschi [18], from which it differs in various points. In particular, it takes into account price acceleration, as well as a microscopic binary interaction for the exchange between the two populations of agents. Various numerical simulations show that the model can describe realistic situations, like regimes of boom and crashes, as well as the invariance of the large-time behavior with respect to the number of agents of the market.
2015, 10(3): 559-578 doi: 10.3934/nhm.2015.10.559 +[Abstract](1588) +[PDF](510.4KB)
Abstract:
In this paper we discuss the necessity of insight in the cognitive processes involved in environment navigation into mathematical models for pedestrian motion. We first provide a review of psychological literature on the cognitive processes involved in walking and on the quantitative one coming from applied mathematics, physics, and engineering. Then, we present a critical analysis of the experimental setting for model testing and we show experimental results given by observation. Finally we propose a cognitive model making use of psychological insight as well as optimization models from robotics.
2015, 10(3): 579-608 doi: 10.3934/nhm.2015.10.579 +[Abstract](1617) +[PDF](1193.2KB)
Abstract:
We use the results of a pedestrian tracking experiment to identify a follow-the-leader model for pedestrians walking-in-line. We demonstrate the existence of a time-delay between a subject's response and the predecessor's corresponding behavior. This time-delay induces an instability which can be damped out by a suitable relaxation. By comparisons with the experimental data, we show that the model reproduces well the emergence of large-scale structures such as congestions waves. The resulting model can be used either for modeling pedestrian queuing behavior or can be incorporated into bi-dimensional models of pedestrian traffic.
2015, 10(3): 609-630 doi: 10.3934/nhm.2015.10.609 +[Abstract](1767) +[PDF](2970.8KB)
Abstract:
A number of different interaction modalities have been proposed for human engagement with networked systems. In this paper, we establish formal guarantees for whether or not a given human-swarm interaction (HSI) strategy is appropriate for achieving particular multi-robot tasks, such as guiding a swarm of robots into a particular geometric configuration. In doing so, we define what it means to impose an HSI control structure on a multi-robot system. Control Lyapunov functions are used to establish feasibility for a user to achieve a particular geometric configuration with a multi-robot system under some selected HSI control structure. Several examples of multi-robot systems with unique HSI control structures are provided to illustrated the use of CLFs to establish feasibility. Additionally, we also uses these examples to illustrate how to use optimal control tools to compute three metrics for evaluating an HSI control structure: attention, effort, and scalability.
2015, 10(3): 631-646 doi: 10.3934/nhm.2015.10.631 +[Abstract](2037) +[PDF](2855.1KB)
Abstract:
Recent incidents such as the Asiana Flight 214 crash in San Francisco on July 6, 2013 have brought attention to the need for safer aircraft evacuation plans. In this paper we propose an emergency aircraft evacuation model inspired by Particle Swarm Optimization (PSO). By introducing an attraction-replusion force from swarm modeling we considered realistic behaviors such as feeling push-back from physical obstacles as well as reducing gaps between passengers near emergency exits. We also incorporate a scaled emotion quantity to simulate passengers experiencing fear or panic. In our model elevating emotion increases the velocity of most passengers and decreases the effect of forces exerted by nearby passengers. We also allow a small percentage of passengers to experience a sense of panic that slows their motion. Our first simulations model a Boeing 737-800 with a single class of seats that are distributed uniformly throughout the aircraft. We also simulate the evacuation of a Boeing 777-200ER with multiple service classes. We observed that increasing emotion causes most passengers to move more quickly to the exits, but that passengers experiencing panic can slow down the evacuation. Our simulations also suggest that blocking exits in locations with high seat density significantly delays the evacuation.
2015, 10(3): 647-697 doi: 10.3934/nhm.2015.10.647 +[Abstract](1644) +[PDF](854.7KB)
Abstract:
For high dimensional particle systems, governed by smooth nonlinearities depending on mutual distances between particles, one can construct low-dimensional representations of the dynamical system, which allow the learning of nearly optimal control strategies in high dimension with overwhelming confidence. In this paper we present an instance of this general statement tailored to the sparse control of models of consensus emergence in high dimension, projected to lower dimensions by means of random linear maps. We show that one can steer, nearly optimally and with high probability, a high-dimensional alignment model to consensus by acting at each switching time on one agent of the system only, with a control rule chosen essentially exclusively according to information gathered from a randomly drawn low-dimensional representation of the control system.
2015, 10(3): 699-715 doi: 10.3934/nhm.2015.10.699 +[Abstract](2065) +[PDF](604.4KB)
Abstract:
We present a control approach for large systems of interacting agents based on the Riccati equation. If the agent dynamics enjoys a strong symmetry the arising high dimensional Riccati equation is simplified and the resulting coupled system allows for a formal mean--field limit. The steady--states of the kinetic equation of Boltzmann and Fokker Planck type can be studied analytically. In case of linear dynamics and quadratic objective function the presented approach is optimal and is compared to the model predictive control approach introduced in [2].

2018  Impact Factor: 0.871