Particle Interactions Mediated by Dynamical Networks: Assessment of Macroscopic Descriptions
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Abstract
We provide a numerical study of the macroscopic model of Barré et al. (Multiscale Model Simul, 2017, to appear) derived from an agentbased model for a system of particles interacting through a dynamical network of links. Assuming that the network remodeling process is very fast, the macroscopic model takes the form of a single aggregation–diffusion equation for the density of particles. The theoretical study of the macroscopic model gives precise criteria for the phase transitions of the steady states, and in the onedimensional case, we show numerically that the stationary solutions of the microscopic model undergo the same phase transitions and bifurcation types as the macroscopic model. In the twodimensional case, we show that the numerical simulations of the macroscopic model are in excellent agreement with the predicted theoretical values. This study provides a partial validation of the formal derivation of the macroscopic model from a microscopic formulation and shows that the former is a consistent approximation of an underlying particle dynamics, making it a powerful tool for the modeling of dynamical networks at a large scale.
Keywords
Dynamical networks Crosslinks Microscopic model Kinetic equation Diffusion approximation Meanfield limit Aggregation–diffusion equation Phase transitions Fourier analysis BifurcationsMathematics Subject Classification
82C21 82C22 82C31 65T50 65L07 74G151 Introduction
Complex networks are of significant interest in many fields of life and social sciences. These systems are composed of a large number of agents interacting through local interactions, and selforganizing to reach largescale functional structures. Examples of systems involving highly dynamical networks include neural networks, biological fiber networks such as connective tissues, vascular or neural networks, ant trails, polymers, economic interactions. (Boissard et al. 2013; Mogilner and EdelsteinKeshet 2007; DiDonna and Levine 2006; Broedersz et al. 2010). These networks often offer great plasticity by their ability to break and reform connections, giving to the system the ability to change shape and adapt to different situations (Boissard et al. 2013; Chaudury et al. 2007). For example, the biochemical reactions in a cell involve proteins—DNA, RNA, gene promotors linking/unlinking to create/break large structures–complex of molecules (Kupiec 1997). Because of their paramount importance in biological functions or social organizations, understanding the properties of such complex systems is of great interest. However, they are challenging to model due to the large amount of components and interactions (chemical, biological, social, etc). Due to their simplicity and flexibility, individualbased models are a natural framework to study complex systems. They describe the behavior of each agent and its interaction with the surrounding agents over time, offering a description of the system at the microscopic scale (see, e.g., Barré et al. 2017; Boissard et al. 2013; Degond et al. 2016). However, these models are computationally expensive and are not suited for the study of large systems. To study the systems at a macroscopic scale, meanfield or continuous models are often preferred. These last models describe the evolution in time of averaged quantities such as agent density and mean orientation. As a drawback, these last models lose the information at the individual level. In order to overcome this weakness of the continuous models, a possible route is to derive a macroscopic model from an agentbased formulation and to compare the obtained systems, as was done in, e.g., Barré et al. (2017), Boissard et al. (2013), Degond et al. (2016) for particle interactions mediated by dynamical networks.
A first step in this direction has been made in Barré et al. (2017), following the earlier work (Degond et al. 2016). In this work, the derivation of a macroscopic model for particles interacting through a dynamical network of links is performed. The microscopic model describes the evolution in time of point particles which interact with their close neighbors via local crosslinks modeled by springs that are randomly created and destructed. In the meanfield limit, assuming large number of particles and links as well as propagation of chaos, the corresponding kinetic system consists of two equations: for the individual particle distribution function and for the link densities. The link density distribution provides a statistical description of the network connectivity which turns out to be quite flexible and easily generalizable to other types of complex networks.
In the largescale limit and in the regime where link creation/destruction frequency is very large, it was shown in Barré et al. (2017), following Degond et al. (2016), that the link density distribution becomes a local function of the particle distribution density. The latter evolves on the slow time scale through an aggregation–diffusion equation. Such equations are encountered in many physical systems featuring collective behavior of animals, chemotaxis models, etc. (Topaz et al. 2006; Blanchet et al. 2006; Carrillo et al. 2010; Golestanian 2012; Kolokolnikov et al. 2013) and references therein. The difference between this macroscopic model and the aggregation–diffusion equations studied in the literature Carrillo et al. (2003), Topaz et al. (2006), Bertozzi et al. (2009) lies in the fact that the interaction potential has compact support. As a result, this model has a rich behavior such as metastability in the case of the whole space (Burger et al. 2014; Evers and Kolokolnikov 2016) and exhibits phase transitions in the periodic setting as functions of the diffusion coefficient, the interaction range of the potential, and the links equilibrium length (Barré et al. 2017). By performing the weakly nonlinear stability analysis of the spatially homogeneous steady states, it is possible to characterize the type of bifurcations appearing at the instability onset (Barré et al. 2017). We refer to Barbaro and Degond (2014), Chayes and Panferov (2010), Degond et al. (2015), Barbaro et al. (2016) for related collective dynamics problems showing phase transitions.
If numerous macroscopic models for dynamical networks have been proposed in the literature, most of them are based on phenomenological considerations and very few have been linked to an agentbased dynamics. On the contrary, the macroscopic model proposed in Barré et al. (2017) and its precursor (Degond et al. 2016) have been derived via a formal meanfield limit from an underlying particle dynamics (see also Degond et al. 2014). However, because the derivation performed in Barré et al. (2017) is still formal, its numerical validation as the limit of the microscopic model as well as the persistence of the phase transitions at the micro and macroscopic level as predicted by the weakly nonlinear analysis in Barré et al. (2017) needs to be assessed. This is the goal of the present work.
More precisely, we show that the macroscopic model indeed provides a consistent approximation of the underlying agentbased model for dynamical networks, by confronting numerical simulations of both the micro and macromodels. Moreover, we numerically check that the microscopic system undergoes in onedimensional a phase transition depicted by the values obtained for the limiting macroscopic aggregation–diffusion equation. Furthermore, we numerically validate the weakly nonlinear analysis in Barré et al. (2017) for the type of bifurcation in the twodimensional setting, where simulations for the microscopic model are prohibitively expensive.
In summary, the main contributions of this work are as follows: (i) It provides a numerical validation of the macroscopic model in 1D as its derivation from the microscopic one in Barré et al. (2017) was only formal. It justifies its further use in 2D where the microscopic model is too computationally intensive. (ii) It also provides the experimental validation of the formal bifurcation analysis for the macroscopic model performed in Barré et al. (2017). In particular, it confirms that the two types of bifurcations revealed in Barré et al. (2017)—subcritical and supercritical—do actually occur. (iii) Finally, it shows that this bifurcation structure is indeed relevant for the microscopic model, for which no theoretical analysis exists to date.
The paper is organized as follows. In Sect. 2, we present the microscopic model and sketch the derivation of the kinetic and macroscopic models from the agentbased formulation. In Sect. 3, we focus on the onedimensional case: We first summarize the theoretical results on the stability of homogeneous steady states of the macroscopic model from Barré et al. (2017) and show that both the macroscopic and microscopic simulations are in good agreement with the theoretical predictions made by nonlinear analysis of the macroscopic model. We then compare the profiles of the steady states between the microscopic and macroscopic simulations and show that the two formulations are in very good agreement, also in terms of phase transitions. Finally, in Sect. 4 we provide a numerical study of the twodimensional case for the macroscopic model. The twodimensional numerical simulations on the macroscopic model are able to numerically capture the subcritical and supercritical transitions as predicted theoretically. Because of the computational cost of the microscopic model, the macroscopic model is not only very competitive and efficient in order to detect phase transitions, but also it is almost the only feasible choice showing the main advantage of the limiting kinetic procedure.
2 Derivation of the Macroscopic Model
2.1 Microscopic Model
The twodimensional microscopic model features N particles located at points \(X_i \in \Omega , i\in [1,N]\) linking/unlinking—dynamically in time—to their neighbors which are located in a ball of radius R from their center. The link creation and suppression are supposed to follow Poisson processes in time, of frequencies \(\nu _f^N\) and \(\nu ^N_d\), respectively (see Fig. 1).
2.2 Kinetic Model
In the equation for the limit distribution of particles (4), the first term on the righthand side is a linear diffusion term which is an effect of the random motion of the particles on the microscopic level. The second term is the attractive–repulsive part due to a springlike force between the particles that are linked. Its counterpart appears in the equation for the limit distribution of links (5). This equation has also a diffusion part and the production term (the last two terms on the righthand side) which is due linking processes taking place between the particles that are not yet connected, and unlinking processes breaking the existing links.
2.3 Scaling and Macroscopic Model
In the following, we aim to study theoretically and numerically both the macroscopic model given by Eq. (6) and the corresponding microscopic formulation given by Eq. (3) and rescaled with the scaling introduced in this section. We first focus on the onedimensional case and we show that the numerical solutions behave as theoretically predicted, and that we obtain—numerically—a very good agreement between the micro and macroformulations.
3 Analysis of the Macroscopic Model in the OneDimensional Case
3.1 Theoretical Results
In this section, we apply the results of Barré et al. (2017) to the onedimensional periodic domain \([L,L]\), to study the stability of stationary solutions of the macroscopic model given by Eqs. (6a) and (7) with \(R<L\).
3.1.1 Identification of the Stability Region
3.1.2 Characterization of the Bifurcation Type
Proposition 1

if \(2\hat{V}_{2}  \hat{V}_{1}>0\), the steady state exhibits a supercritical bifurcation;

if \(2\hat{V}_{2}  \hat{V}_{1}<0\), the steady state exhibits a subcritical bifurcation.
Note that the above criterion only involves the potential, but does not involve the parameter D, and it only restricts the values of \(\alpha \) or \(\ell \).
3.2 Numerical Results
3.2.1 Choice of Numerical Parameters
Using (10) and Proposition 1, we check that the bifurcation changes its character for \(\ell =\ell ^*\), where \(\ell ^*=0.75\frac{(\pi 4)\sqrt{2}+2}{\pi (\sqrt{2}1)}\approx 0.4530\). Recall that our criterion did not involve the parameter D; therefore, the bifurcation is supercritical if only \(\ell \in \left( \ell ^*,\ell _c \right) \approx (0.4530, 0.4948)\), and subcritical if \(\ell \in (0,l^*)\approx (0, 0.4530)\). The value of parameter D corresponding to the instability threshold for \(l=l^*\approx 0.4530\) is denoted by \(D^*\), and it is equal to 0.0074. All of these parameters are presented in Fig. 2.
Remark 1
Choosing R comparable to L allows us to observe two types of bifurcation: continuous and discontinuous one. It was observed in Barré et al. (2017) that taking \(R\ll L\) would cause that for most values of \(\ell \) the bifurcation would be subcritical (discontinuous). This effect is captured in Fig. 8, for the twodimensional case.
3.2.2 Macroscopic Model
We now make use of the numerical scheme developed in Carrillo et al. (2015) to analyze the macroscopic equation (6a) with the potential (7) in the unstable regime. The choice of the numerical scheme is due to its free energy decreasing property for equations enjoying a gradient flow structure such as (6a). Keeping this property of gradient flows is of paramount importance in order to compute the right stationary states in the long time asymptotics. In fact, under a suitable CFL condition the scheme is positivity preserving and well balanced, i.e., stationary states are preserved exactly by the scheme.

\(\ell _1= 0.4725\) for different values of the noise D, where we expect a supercritical (continuous) transition for \(D<D_{1} = 0.0040\);

\(\ell _2= 0.3\) for different values of the noise D, where we expect a subcritical (discontinuous) transition for \(D< D_2 = 0.0347\).
Table of parameters \(D_{1,\lambda }\) (supercritical) and \(D_{2,\lambda }\) (subcritical) for the numerical simulations in the macroscopic case with highlighted values corresponding to the phase transition
\(\lambda \)  \(D_{1,\lambda }\)  \(D_{2,\lambda }\)  

1  0.0010  0.0030  0.0338 
2  0.0009  0.0031  0.0339 
3  0.0008  0.0032  0.0340 
4  0.0007  0.0033  0.0340 
5  0.0006  0.0034  0.0341 
6  0.0005  0.0035  0.0342 
7  0.0004  0.0036  0.0343 
8  0.0003  0.0037  0.0344 
9  0.0002  0.0038  0.0345 
10  0.0001  0.0039  0.0346 
11  0  0.0040  0.0347 
12  0.0001  0.0041  0.0348 
13  0.0002  0.0042  0.0349 
14  0.0003  0.0043  0.0350 
As shown in Fig. 3, the quantifier Q indeed undergoes a discontinuous transition around \(D = 0.0347\) for \(\ell = 0.3\) (subcritical case, Fig. 3a) and a continuous transition around \(D = 0.004\) for \(\ell = 0.4725\) (supercritical case, Fig. 3b). These results show that the numerical solutions are in very good agreement with the theoretical predictions.
Theoretical (\(\rho _\mathrm{th}\)) vs numerical (\(\rho _\mathrm{num}\)) values for the size of perturbation
\(\lambda \)  \(\rho _\mathrm{num}\)  A  \(\rho _\mathrm{th}\) 

0.0010  0.3384  0.1094  0.3428 
0.0009  0.3203  0.1058  0.3233 
0.0008  0.3008  0.1017  0.3025 
0.0007  0.2797  0.0968  0.2805 
0.0006  0.2567  0.0912  0.2569 
0.0005  0.2312  0.0847  0.2314 
0.0004  0.2028  0.0770  0.2036 
0.0003  0.1701  0.0678  0.1727 
0.0002  0.1311  0.0562  0.1371 
0.0001  0.0790  0.0403  0.0930 
0  0.0005  0  0 
We now aim at performing the same stability analysis on the microscopic model from Sect. 2.1—the starting point of the derivation of the macroscopic model.
3.2.3 Microscopic Model
Table of parameters (nondimensionalized values)
Parameter  Value  Interpretation 

L  3  Domain half size 
\(\delta \)  0.1  Maximal step 
\(T_\mathrm{f}\)  20  Final simulation time 
\(\xi _\mathrm{init}\)  0.1  Initial fraction \(\frac{K}{N}\) 
\(\nu _d\)  1  Unlinking frequency 
\(\nu _f\)  1  Linking frequency 
R  0.75  Detection radius for creation of links 
\(\ell \)  Adapted  Spring equilibrium length 
\(\kappa \)  2  Spring force between linked fibers 
D  Adapted  Noise intensity 
\(\varepsilon \)  Adapted  Scaling parameter 
We now aim at comparing the profiles of the solutions between the microscopic and macroscopic models, to numerically validate the derivation of the macroscopic model from the microscopic dynamics.
3.2.4 Comparison of the Density Profiles in the Microscopic and Macroscopic Models
Here, we aim at comparing the profiles of the particle densities of the microscopic model with the ones of the macroscopic model as functions of time. As shown in the previous section, for \(\varepsilon \) small enough, we recover the bifurcation and bifurcation types observed from the macroscopic model with the microscopic formulation, with very good quantitative agreement when considering the limiting microscopic model (18) with “\(\varepsilon = 0\).” The simulations of the microscopic model are very timeconsuming for small values of \(\varepsilon \), because we are obliged to consider very small time steps (see “Computational Aspects of the MicroandMacroscopic Models” section of Appendix 1). Here, due to computational time constraints, we therefore compare the results of the macroscopic model (6) with \(\varepsilon =0\) for which the time step can be taken much larger and independent of \(\varepsilon \).
As shown in Fig. 8, we obtain a very good agreement between the solutions of the macroscopic model and of the microscopic one with “\(\varepsilon = 0\).” Close to the transitional D (Fig. 8a, b), the particle density converges in time toward a Gaussianlike distribution for both the microscopic and macroscopic models. Note that the microscopic simulations seem to converge in time toward the steady state faster than the macroscopic model (compare the orange curves on the top panels). This change in speed can be due to the fact that the microscopic model features finite number of particles while the macroscopic model is obtained in the limit of infinite number of particles. Therefore, in the macroscopic setting, each particle interacts with many more particles than in the microscopic model, which could result in a delay in the aggregation process.
When far from the transitional D in the unstable regime (Fig. 8c, d), one can observe the production of several bumps in the steady state of the particle density. The production of several particle clusters in these regimes shows that the noise triggers particle aggregation. For small noise intensity, local particle aggregates are formed which fail to detect neighboring aggregates. As a result, one can observe several clusters in the steady state, for small enough noise intensities. These bumps are observed for both the microscopic and macroscopic models, showing again a good agreement between the two dynamics.
In the next section, we present a numerical study of the macroscopic model in the twodimensional case. As mentioned previously, the microscopic model is in very good agreement with the macroscopic dynamics for small values of \(\varepsilon \) as in the onedimensional case. Its simulations are, however, very timeconsuming, due to the need of very small time steps (see “Computational Aspects of the MicroandMacroscopic Models” section of Appendix 1). As a result, the microscopic model is not suited for the study of very large systems such as the ones considered in the twodimensional case. We therefore provide a numerical twodimensional study using the macroscopic model only.
4 Analysis of the Macroscopic Model in the TwoDimensional Case
4.1 Theoretical Results
In this section, we first recall some theoretical results from Barré et al. (2017) for the twodimensional periodic domain. We will focus on the square periodic domain \([L,L]\times [L,L]\), since the rectangular case can be, in agreement with the analysis in Barré et al. (2017), reduced to the onedimensional case studied above.
Proposition 2

if \(c<d\), the constant steady state exhibits a supercritical bifurcation, and

if \(c>d\), the constant steady state exhibits a subcritical bifurcation.
Note that in the twodimensional case, the bifurcation criterion involves also parameter D. On the other hand, the instability threshold D is given as a function of \(\alpha \) and can be calculated using (20).
4.2 Numerical Results
 1.For \(R/L=1\), \(z_{1,0}=\pi \), the constant steady state is unstable for$$\begin{aligned} \frac{\ell }{R}<\alpha _c:=0.6620\quad \text{ and }\quad D< 0.2334 R^2{\left( \alpha _c\frac{\ell }{R} \right) }. \end{aligned}$$
 2.For \(R/L=1/2\), \(z_{1,0}=\frac{\pi }{2}\), the constant steady state is unstable for$$\begin{aligned} \frac{\ell }{R}<\alpha _c:=0.7333\quad \text{ and }\quad D< 0.1084 R^2{\left( \alpha _c\frac{\ell }{R} \right) }. \end{aligned}$$
 3.For \(R/L=1/4\), \(z_{1,0}=\frac{\pi }{4}\), the constant steady state is unstable for$$\begin{aligned} \frac{\ell }{R}<\alpha _c:=0.7462\quad \text{ and }\quad D< 0.0312 R^2{\left( \alpha _c\frac{\ell }{R} \right) }. \end{aligned}$$
 1.
For \(R/L=1\), the steady state exhibits a supercritical bifurcation for \(\alpha \in (0.1016, 0.5818)\) and a subcritical bifurcation for \(\alpha \in (0,0.1016)\cup (0.5818,0.6620)\).
 2.
For \(R/L=1/2\), the steady state exhibits only a subcritical bifurcation.
 3.
For \(R/L=1/4\), the steady state exhibits only a subcritical bifurcation.
We will study all of the five cases from Fig. 9b corresponding to different values of \(\frac{R}{L}\), but the same value of \(\alpha =\frac{\ell }{R}=0.3\). The theoretical prediction is that the first two cases \(\frac{R}{L}=1\) and \(\frac{R}{L}=0.975\) correspond to a supercritical (continuous) bifurcation while the cases 3–5 correspond to the subcritical (discontinuous) bifurcation.
Moreover, as in the onedimensional case, we can check that the different bifurcation diagrams correspond to different shapes of \(\xi ^*\). In Figs. 12 and 13, we present the graphs of \(\xi ^*(t)\) for all five cases depicted in Fig. 9b. For each of the cases, we present the graph of \(\xi ^*(t)\) for five different values of diffusion parameter D as specified in Table 4.
Table of parameters \(D_{1,\lambda }\) and \(D_{2,\lambda }\) (supercritical), \(D_{3,\lambda }\), \(D_{4,\lambda }\), and \(D_{5,\lambda }\) (subcritical) for the numerical simulations in twodimensional case
\(\lambda \)  \(D_{1,\lambda }\)  \(D_{2,\lambda }\)  \(D_{3,\lambda }\)  \(D_{4,\lambda }\)  \(D_{5,\lambda }\)  

1  0.005  0.7560  0.7254  0.6923  0.6570  0.6201 
2  0.004  0.7569  0.7264  0.6932  0.6579  0.6210 
3  0.003  0.7578  0.7273  0.6941  0.6588  0.6219 
4  0.002  0.7587  0.7282  0.6950  0.6597  0.6229 
5  0.001  0.7596  0.7291  0.6959  0.6607  0.6238 
5 Conclusion
In this paper, we have provided a numerical study of a macroscopic model derived from an agentbased formulation for particles interacting through a dynamical network of links. In the onedimensional case, we were first able to recover numerically the subcritical and supercritical transitions undergone by the steady states of the macroscopic model, in the regime predicted by the theoretical nonlinear analysis of the continuous model. Moreover, the numerical simulations of the rescaled microscopic model revealed the same bifurcations and bifurcation types as obtained with the macroscopic model, with very good precision as \(\varepsilon \) goes to zero in the microscopic setting. Finally, when considering the limiting case “\(\varepsilon =0\)” in the microscopic model, we obtained a very good agreement between the profiles of the solutions of the micro and macromodels. It is noteworthy that both models also feature the same dynamics in time, with a slight delay in the macroscopic simulations compared to the microscopic dynamics. This delay may be due to the fact that the microscopic simulations are performed with a finite number of particles while the macroscopic model is in the limit of infinite number of individuals. However, as for very small values of \(\varepsilon \) the simulations of the microscopic dynamics are very timeconsuming, we were not able to extend the numerical study to a higher number of particles.
For the sake of completeness, we finally presented numerical simulations of the model in the twodimensional case. For computational reasons, we were not able to perform twodimensional simulations of the microscopic model, and we chose to focus on the macroscopic model. In the twodimensional case, we were once again able to numerically recover supercritical and subcritical transitions in the steady states, as function of the noise intensity D, in the same regime as predicted by the theoretical analysis of the macroscopic model. These results validate the theoretical analysis, the numerical method, and the simulations developed for the macroscopic model.
By providing a numerical comparison between the micro and macrodynamics, this study shows that the macroscopic model considered in this paper is indeed a relevant tool to model particles interacting through a dynamical network of links. As a main advantage compared to the microscopic formulation, the macroscopic model enables to explore large systems with low computational cost (such as twodimensional studies) and is therefore believed to be a powerful tool to study network systems on the large scale. Direct perspectives of these works include the derivation of the macroscopic model in a regime of noninstantaneous linking–unlinking of particles. The hope is to understand deeper how the local forces generated by the links are expressed at the macroscopic level. The model could be improved by taking into account other phenomena such as external forces and particle creation/destruction. Finally, rigorously proving the derivation of the macroscopic model from the particle dynamics will be the subject of the future research.
Notes
Acknowledgements
JAC acknowledges support by the Engineering and Physical Sciences Research Council (EPSRC) under Grant No. EP/P031587/1, by the Royal Society and the Wolfson Foundation through a Royal Society Wolfson Research Merit Award, and by the National Science Foundation (NSF) under Grant No. RNMS1107444 (KINet). PD acknowledges support by the Engineering and Physical Sciences Research Council (EPSRC) under Grant Nos. EP/M006883/1, EP/N014529/1 and EP/P013651/1, by the Royal Society and the Wolfson Foundation through a Royal Society Wolfson Research Merit Award No. WM130048, and by the National Science Foundation (NSF) under Grant No. RNMS1107444 (KINet). PD is on leave from CNRS, Institut de Mathématiques de Toulouse, France. DP acknowledges support by the Vienna Science and Technology fund. Vienna Project Number is LS13/029. The work of EZ has been supported by the Polish Ministry of Science and Higher Education Grant “Iuventus Plus” No. 0888/IP3/2016/74.
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