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Discrete & Continuous Dynamical Systems - B

July 2014 , Volume 19 , Issue 5

Special Issue on Mathematics of Social Systems

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Preface to special issue on mathematics of social systems
Andrea L. Bertozzi
2014, 19(5): i-v doi: 10.3934/dcdsb.2014.19.5i +[Abstract](944) +[PDF](220.8KB)
Abstract:
This special issue is an outgrowth of a minisyposium titled ``Mathematics of Social Systems" held at the 9th AIMS Conference on Dynamical Systems, Differential Equations and Applications, held in Orlando, FL in July 2012. Presenters from that session were invited to submit papers that were reviewed using the usual procedures of the DCDS journals, along with additional authors from the field. Mathematics has already had a significant impact on basic research involving fundamental problems in physical sciences, biological sciences, computer science and engineering. Examples include understanding of the equations of incompressible fluid dynamics, shock wave theory and compressible gas dynamics, ocean modeling, algorithms for image processing and compressive sensing, and biological problems such as models for invasive species, spread of disease, and more recently systems biology for modeling of complex organisms and complex patterns of disease. This impact has yet to come to fruition in a comprehensive way for complex social behavior. While computational models such as agent-based systems and well-known statistical methods are widely used in the social sciences, applied mathematics has not to date had a core impact in the social sciences at the level that it achieves in the physical and life sciences. However in recent years we have seen a growth of work in this direction and ensuing new mathematics problems that must be tackled to understand such problems. Technical approaches include ideas from statistical physics, nonlinear partial differential equations of all types, statistics and inverse problems, and stochastic processes and social network models. The collection of papers presented in this issue provides a backdrop of the current state of the art results in this developing new research area in applied mathematics. The body of work encompasses many of the challenges in understanding these discrete complex systems and their related continuum approximations.

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Confinement for repulsive-attractive kernels
Daniel Balagué, José A. Carrillo and Yao Yao
2014, 19(5): 1227-1248 doi: 10.3934/dcdsb.2014.19.1227 +[Abstract](631) +[PDF](1235.6KB)
Abstract:
We investigate the confinement properties of solutions of the aggregation equation with repulsive-attractive potentials. We show that solutions remain compactly supported in a large fixed ball depending on the initial data and the potential. The arguments apply to the functional setting of probability measures with mildly singular repulsive-attractive potentials and to the functional setting of smooth solutions with a potential being the sum of the Newtonian repulsion at the origin and a smooth suitably growing at infinity attractive potential.
Phase transition and diffusion among socially interacting self-propelled agents
Alethea B. T. Barbaro and Pierre Degond
2014, 19(5): 1249-1278 doi: 10.3934/dcdsb.2014.19.1249 +[Abstract](741) +[PDF](563.0KB)
Abstract:
We consider a hydrodynamic model of swarming behavior derived from the kinetic description of a particle system combining a noisy Cucker-Smale consensus force and self-propulsion. In the large self-propulsive force limit, we provide evidence of a phase transition from disordered to ordered motion which manifests itself as a change of type of the limit model (from hyperbolic to diffusive) at the crossing of a critical noise intensity. In the hyperbolic regime, the resulting model, referred to as the `Self-Organized Hydrodynamics (SOH)', consists of a system of compressible Euler equations with a speed constraint. We show that the range of SOH models obtained by this limit is restricted. To waive this restriction, we compute the Navier-Stokes diffusive corrections to the hydrodynamic model. Adding these diffusive corrections, the limit of a large propulsive force yields unrestricted SOH models and offers an alternative to the derivation of the SOH using kinetic models with speed constraints.
Inhomogeneous Patlak-Keller-Segel models and aggregation equations with nonlinear diffusion in $\mathbb{R}^d$
Jacob Bedrossian and Nancy Rodríguez
2014, 19(5): 1279-1309 doi: 10.3934/dcdsb.2014.19.1279 +[Abstract](668) +[PDF](654.4KB)
Abstract:
Aggregation equations and parabolic-elliptic Patlak-Keller-Segel (PKS) systems for chemotaxis with nonlinear diffusion are popular models for nonlocal aggregation phenomenon and are a source of many interesting mathematical problems in nonlinear PDEs. The purpose of this work is to give a more complete study of local, subcritical and small-data critical/supercritical theory in $\mathbb{R}^d$, $d \geq 2$. Some existing results can be found in the literature; however, one of the most important cases in biological applications, that is the $\mathbb{R}^2$ case, had not been studied. In this paper, we treat two related systems, which are different generalizations of the classical parabolic-elliptic PKS model. In the first class, nonlocal aggregation is modeled by convolution with a general interaction potential, studied in this generality in our previous work [6]. For this class of models we also present several large data global existence results for critical problems. The second class is a variety of PKS models with spatially inhomogeneous diffusion and decay rate of the chemo-attractant, which is potentially relevant to biological applications and raises interesting mathematical questions.
Mean field games with nonlinear mobilities in pedestrian dynamics
Martin Burger, Marco Di Francesco, Peter A. Markowich and Marie-Therese Wolfram
2014, 19(5): 1311-1333 doi: 10.3934/dcdsb.2014.19.1311 +[Abstract](781) +[PDF](1893.5KB)
Abstract:
In this paper we present an optimal control approach modeling fast exit scenarios in pedestrian crowds. In particular we consider the case of a large human crowd trying to exit a room as fast as possible. The motion of every pedestrian is determined by minimizing a cost functional, which depends on his/her position, velocity, exit time and the overall density of people. This microscopic setup leads in the mean-field limit to a parabolic optimal control problem. We discuss the modeling of the macroscopic optimal control approach and show how the optimal conditions relate to the Hughes model for pedestrian flow. Furthermore we provide results on the existence and uniqueness of minimizers and illustrate the behavior of the model with various numerical results.
Latent self-exciting point process model for spatial-temporal networks
Yoon-Sik Cho, Aram Galstyan, P. Jeffrey Brantingham and George Tita
2014, 19(5): 1335-1354 doi: 10.3934/dcdsb.2014.19.1335 +[Abstract](699) +[PDF](853.3KB)
Abstract:
We propose a latent self-exciting point process model that describes geographically distributed interactions between pairs of entities. In contrast to most existing approaches that assume fully observable interactions, here we consider a scenario where certain interaction events lack information about participants. Instead, this information needs to be inferred from the available observations. We develop an efficient approximate algorithm based on variational expectation-maximization to infer unknown participants in an event given the location and the time of the event. We validate the model on synthetic as well as real-world data, and obtain very promising results on the identity-inference task. We also use our model to predict the timing and participants of future events, and demonstrate that it compares favorably with baseline approaches.
Rethinking centrality: The role of dynamical processes in social network analysis
Rumi Ghosh and Kristina Lerman
2014, 19(5): 1355-1372 doi: 10.3934/dcdsb.2014.19.1355 +[Abstract](930) +[PDF](465.1KB)
Abstract:
Many popular measures used in social network analysis, including centrality, are based on the random walk. The random walk is a model of a stochastic process where a node interacts with one other node at a time. However, the random walk may not be appropriate for modeling social phenomena, including epidemics and information diffusion, in which one node may interact with many others at the same time, for example, by broadcasting the virus or information to its neighbors. To produce meaningful results, social network analysis algorithms have to take into account the nature of interactions between the nodes. In this paper we classify dynamical processes as conservative and non-conservative and relate them to well-known measures of centrality used in network analysis: PageRank and Alpha-Centrality. We demonstrate, by ranking users in online social networks used for broadcasting information, that non-conservative Alpha-Centrality generally leads to a better agreement with an empirical ranking scheme than the conservative PageRank.
The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime
Theodore Kolokolnikov, Michael J. Ward and Juncheng Wei
2014, 19(5): 1373-1410 doi: 10.3934/dcdsb.2014.19.1373 +[Abstract](696) +[PDF](992.8KB)
Abstract:
The existence and stability of localized patterns of criminal activity are studied for the reaction-diffusion model of urban crime that was introduced by Short et. al. [Math. Models. Meth. Appl. Sci., 18, Suppl. (2008), pp. 1249--1267]. Such patterns, characterized by the concentration of criminal activity in localized spatial regions, are referred to as hot-spot patterns and they occur in a parameter regime far from the Turing point associated with the bifurcation of spatially uniform solutions. Singular perturbation techniques are used to construct steady-state hot-spot patterns in one and two-dimensional spatial domains, and new types of nonlocal eigenvalue problems are derived that determine the stability of these hot-spot patterns to ${\mathcal O}(1)$ time-scale instabilities. From an analysis of these nonlocal eigenvalue problems, a critical threshold $K_c$ is determined such that a pattern consisting of $K$ hot-spots is unstable to a competition instability if $K>K_c$. This instability, due to a positive real eigenvalue, triggers the collapse of some of the hot-spots in the pattern. Furthermore, in contrast to the well-known stability results for spike patterns of the Gierer-Meinhardt reaction-diffusion model, it is shown for the crime model that there is only a relatively narrow parameter range where oscillatory instabilities in the hot-spot amplitudes occur. Such an instability, due to a Hopf bifurcation, is studied explicitly for a single hot-spot in the shadow system limit, for which the diffusivity of criminals is asymptotically large. Finally, the parameter regime where localized hot-spots occur is compared with the parameter regime, studied in previous works, where Turing instabilities from a spatially uniform steady-state occur.
Two-species particle aggregation and stability of co-dimension one solutions
Alan Mackey, Theodore Kolokolnikov and Andrea L. Bertozzi
2014, 19(5): 1411-1436 doi: 10.3934/dcdsb.2014.19.1411 +[Abstract](654) +[PDF](2066.3KB)
Abstract:
Systems of pairwise-interacting particles model a cornucopia of physical systems, from insect swarms and bacterial colonies to nanoparticle self-assembly. We study a continuum model with densities supported on co-dimension one curves for two-species particle interaction in $\mathbb{R}^2$, and apply linear stability analysis of concentric ring steady states to characterize the steady state patterns and instabilities which form. Conditions for linear well-posedness are determined and these results are compared to simulations of the discrete particle dynamics, showing predictive power of the linear theory. Some intriguing steady state patterns are shown through numerical examples.
Paladins as predators: Invasive waves in a spatial evolutionary adversarial game
Scott G. McCalla
2014, 19(5): 1437-1457 doi: 10.3934/dcdsb.2014.19.1437 +[Abstract](602) +[PDF](1416.9KB)
Abstract:
Invasive waves are numerically found in a variant of a reaction-diffusion system used to extend an evolutionary adversarial game into space wherein the influence of various strategies is allowed to diffuse. The diffusion of various strategies corresponds to peer-pressure. The invasive waves are driven by a nonlinear instability that enables an otherwise unstable state to travel through an initially uncooperative state leaving a cooperative state behind. The wave speed's dependence on the various diffusion parameters is examined in one- and two-dimensions through numerically solving the reaction-diffusion equations. Various other phenomena, such as pinning near a diffusive inhomogeneity, are also explored.
Gang rivalry dynamics via coupled point process networks
M. B. Short, G. O. Mohler, P. J. Brantingham and G. E. Tita
2014, 19(5): 1459-1477 doi: 10.3934/dcdsb.2014.19.1459 +[Abstract](601) +[PDF](1023.2KB)
Abstract:
We introduce a point process model for inter-gang violence driven by retaliation -- a core feature of gang behavior -- and multi-party inhibition. Here, a coupled system of state-dependent jump stochastic differential equations is used to model the conditional intensities of the directed network of gang rivalries. The system admits an exact simulation strategy based upon Poisson thinning. The model produces a wide variety of transient or stationary weighted network configurations and we investigate under what conditions each type of network forms in the continuum limit. We then fit the model to gang violence data provided by the Hollenbeck district of the Los Angeles Police Department to measure the levels of excitation and inhibition present in gang violence dynamics, as well as the stability of gang rivalries in Hollenbeck.
Cops on the dots in a mathematical model of urban crime and police response
Joseph R. Zipkin, Martin B. Short and Andrea L. Bertozzi
2014, 19(5): 1479-1506 doi: 10.3934/dcdsb.2014.19.1479 +[Abstract](816) +[PDF](696.8KB)
Abstract:
Hotspots of crime localized in space and time are well documented. Previous mathematical models of urban crime have exhibited these hotspots but considered a static or otherwise suboptimal police response to them. We introduce a program of police response to hotspots of crime in which the police adapt dynamically to changing crime patterns. In particular, they choose their deployment to solve an optimal control problem at every time. This gives rise to a free boundary problem for the police deployment's spatial support. We present an efficient algorithm for solving this problem numerically and show that police presence can prompt surprising interactions among adjacent hotspots.

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