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Editor in Chief, Benedetto Piccoli, at nhmaims@camden.rutgers.edu.
NHM offers a strong combination of three features: Interdisciplinary character, specific focus,
and deep mathematical content. Also, the journal aims to create a link between the discrete
and the continuous communities, which distinguishes it from other journals with strong PDE orientation.
NHM publishes original contributions of high quality in networks,
heterogeneous media and related fields.
NHM is thus devoted to research work on complex media arising in mathematical,
physical, engineering, socioeconomical and biomedical problems.
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TOP 10 Most Read Articles in NHM, January 2017
1 
Recognition of crowd behavior from mobile sensors with pattern analysis and graph clustering methods
Volume 6, Number 3, Pages: 521  544, 2011
Daniel Roggen,
Martin Wirz,
Gerhard Tröster
and Dirk Helbing
Abstract
References
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Related Articles
Mobile onbody sensing has distinct advantages for the analysis and understanding of crowd dynamics: sensing is not geographically restricted to a specific instrumented area, mobile phones offer onbody sensing and they are already deployed on a large scale, and the rich sets of sensors they contain allows one to characterize the behavior of users through pattern recognition techniques.
In this paper we present a methodological framework for the machine recognition of crowd behavior from onbody sensors, such as those in mobile phones.
The recognition of crowd behaviors opens the way to the acquisition of largescale datasets for the analysis and understanding of crowd dynamics.
It has also practical safety applications by providing improved crowd situational awareness in cases of emergency.
The framework comprises: behavioral recognition with the user's mobile device, pairwise analyses of the activity relatedness of two users, and graph clustering in order to uncover globally, which users participate in a given crowd behavior.
We illustrate this framework for the identification of groups of persons walking, using empirically collected data.
We discuss the challenges and research avenues for theoretical and applied mathematics arising from the mobile sensing of crowd behaviors.

2 
Numerical network models and entropy principles for isothermal junction flow
Volume 9, Number 1, Pages: 65  95, 2014
Gunhild A. Reigstad
Abstract
References
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We numerically explore network models which are derived for the isothermal Euler equations.
Previously we proved the existence and uniqueness of solutions to the generalized Riemann problem at a junction under the conditions of monotone momentum related coupling constant and equal crosssectional areas for all connected pipe sections.
In the present paper we extend this proof to the case of pipe sections of different crosssectional areas.
We describe a numerical implementation of the network models, where the flow in each pipe section is calculated using a classical highresolution Roe scheme.
We propose a numerical treatment of the boundary conditions at the pipejunction interface, consistent with the coupling conditions. In particular, mass is exactly conserved across the junction.
Numerical results are provided for two different network configurations and for three different network models.
Mechanical energy considerations are applied in order to evaluate the results in terms of physical soundness.
Analytical predictions for junctions connecting three pipe sections are verified for both network configurations.
Long term behaviour of physical and unphysical solutions are presented and compared, and the impact of having pipes with different crosssectional area is shown.

3 
Modeling opinion dynamics: How the network enhances consensus
Volume 10, Number 4, Pages: 877  896, 2015
Marina Dolfin
and Mirosław Lachowicz
Abstract
References
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In this paper we
analyze emergent collective phenomena in the evolution of opinions in a society structured into few interacting nodes of a network. The presented mathematical structure combines two dynamics: a first one on each single node and a second one among the nodes, i.e. in the network. The aim of the model is to analyze the effect of a network structure on a society with respect to opinion dynamics and we show some numerical solutions addressed in this direction, i.e. comparing the emergent behaviors of a consensusdissent dynamic on a single node when the effect of the network is not considered, with respect to the emergent behaviors when the effect of a network structure linking few interacting nodes is considered.
We adopt the framework of the Kinetic Theory for Active Particles (KTAP), deriving a general mathematical structure which allows to deal with nonlinear features of the interactions and representing the conceptual framework toward the derivation of specific models. A specific model is derived from the general mathematical structure by introducing
a consensusdissent dynamics of interactions and a qualitative analysis is given.

4 
Constructing setvalued fundamental diagrams from Jamiton solutions in second order traffic models
Volume 8, Number 3, Pages: 745  772, 2013
Benjamin Seibold,
Morris R. Flynn,
Aslan R. Kasimov
and Rodolfo R. Rosales
Abstract
References
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Fundamental diagrams of vehicular traffic flow are generally multivalued in the congested flow regime. We show that such setvalued fundamental diagrams can be constructed systematically from simple second order macroscopic traffic models, such as the classical PayneWhitham model or the inhomogeneous AwRascleZhang model. These second order models possess nonlinear traveling wave solutions, called jamitons, and the multivalued parts in the fundamental diagram correspond precisely to jamitondominated solutions. This study shows that transitions from functionvalued to setvalued parts in a fundamental diagram arise naturally in wellknown second order models. As a particular consequence, these models intrinsically reproduce traffic phases.

5 
Analysis of a system of nonlocal conservation laws for multicommodity flow on networks
Volume 10, Number 4, Pages: 749  785, 2015
Martin Gugat,
Alexander Keimer,
Günter Leugering
and Zhiqiang Wang
Abstract
References
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We consider a system of scalar nonlocal conservation laws on networks that model a highly reentrant multicommodity manufacturing system as encountered in semiconductor production.
Every single commodity is modeled by a nonlocal conservation law, and the corresponding PDEs are coupled via a collective load, the work in progress.
We illustrate the dynamics for two commodities.
In the applications, directed acyclic networks naturally occur, therefore this type of networks is considered.
On every edge of the network we have a system of coupled conservation laws with nonlocal velocity. At the junctions the right hand side boundary data of the foregoing edges is passed as left hand side boundary data to the following edges and PDEs.
For distributing junctions, where we have more than one outgoing edge, we impose time dependent distribution functions that guarantee conservation of mass.
We provide results of regularity, existence and wellposedness of the multicommodity network model for $L^{p}$, $BV$ and $W^{1,p}$data.
Moreover, we define an $L^{2}$tracking type objective and show the existence of minimizers that solve the corresponding optimal control problem.

6 
A model of riots dynamics: Shocks, diffusion and thresholds
Volume 10, Number 3, Pages: 443  475, 2015
Henri Berestycki,
JeanPierre Nadal
and Nancy Rodíguez
Abstract
References
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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.

7 
Liquidity generated by heterogeneous beliefs and costly estimations
Volume 7, Number 2, Pages: 349  361, 2012
Min Shen
and Gabriel Turinici
Abstract
References
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We study the liquidity, defined as the size of the trading volume, in a situation where an infinite number of agents
with heterogeneous beliefs reach a tradeoff between the cost of a precise estimation (variable depending on the agent) and the expected wealth from trading. The "true" asset price is not known and the market price is set at a level that clears the market. We show that, under some technical assumptions, the model has natural properties such as monotony of supply and demand functions with respect to the price, existence of an equilibrium and monotony with respect to the
marginal cost of information. We also situate our approach within the Mean Field Games (MFG) framework of Lions and Lasry which allows to obtain an interpretation as a limit of Nash equilibrium for an infinite number of agents.

8 
Analyzing humanswarm interactions using control Lyapunov functions and optimal control
Volume 10, Number 3, Pages: 609  630, 2015
JeanPierre de la Croix
and Magnus Egerstedt
Abstract
References
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Related Articles
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 humanswarm interaction (HSI) strategy is appropriate for achieving particular multirobot 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 multirobot system. Control Lyapunov functions are used to establish feasibility for a user to achieve a particular geometric configuration with a multirobot system under some selected HSI control structure. Several examples of multirobot 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.

9 
Opinion dynamics under the influence of radical groups, charismatic leaders, and other constant signals: A simple unifying model
Volume 10, Number 3, Pages: 477  509, 2015
Rainer Hegselmann
and Ulrich Krause
Abstract
References
Full Text
Related Articles
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.

10 
Meanfield control and Riccati equations
Volume 10, Number 3, Pages: 699  715, 2015
Michael Herty,
Lorenzo Pareschi
and Sonja Steffensen
Abstract
References
Full Text
Related Articles
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 meanfield limit. The steadystates
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].

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