Spontaneous repulsion in the $A+B\to 0$ reaction on coupled networks

We study the transient dynamics of an $A+B\to 0$ process on a pair of randomly coupled networks, where reactants are initially separated. We find that, for sufficiently small fractions $q$ of cross couplings, the concentration of $A$ (or $B$) particles decays linearly in a first stage and crosses over to a second linear decrease at a mixing time $t_x$. By numerical and analytical arguments, we show that for symmetric and homogeneous structures $t_x\propto (\langle k\rangle /q)log(\langle k\rangle /q)$ where $\langle k\rangle$ is the mean degree of both networks. Being this behavior is in marked contrast with a purely diffusive process, where the mixing time would go simply like $\langle k\rangle /q$, we identify the logarithmic slowing down in $t_x$ to be the result of a spontaneous mechanism of repulsion between the reactants $A$ and $B$ due to the interactions taking place at the networks' interface. We show numerically how this spontaneous repulsion effect depends on the topology of the underlying networks.

Figure 1

Illustration of the model. Particles of types $A$ (red) and $B$ (blue) are let to diffuse and react on the top of an interconnected network composed of two layers, each having a given degree distribution and a fraction $q$ of interconnected nodes. Different nodes' size pictorially represent degrees' heterogeneity.

Figure 2

Time evolution of the inverse density in log-log scale for the $A+B\to 0$ process on a pair of randomly coupled networks of equal number of nodes $N={10}^6$ and the initial concentrations ${\rho }_0=0.4$. Different markers in the figures above correspond to different choices of the varying parameters considered. (a) RR networks with $\langle k\rangle =4$ and decreasing values of $q$. (b) ER networks with $\langle k\rangle =40$ and decreasing values of $q$. Simulation's data are represented by dots, while full lines are obtained by integrating numerically Eq. (4). The inset shows the difference between mixed and unmixed initial conditions for $\langle k\rangle =4$ and $q=0.008$. (c) SF networks with $\gamma =3.5$ and decreasing values of $q$. (d) ER networks with fixed fraction $q=0.008$, and increasing values of $\langle k\rangle$. (e) ER networks with $q=0.008,\langle k\rangle =4$, and different initial concentrations. (f) SF networks with $q=0.008,m=2$, and $\gamma =2.5$, with different initial concentrations. Both plots in (e) and (f) have been averaged over 1000 realizations.

Figure 3

Natural log-log plot of the time evolution for the rates of particle's annihilation. Results are obtained for ER networks with $N={10}^6,{\rho }_0=0.4,\langle k\rangle =4$, and decreasing values of $q$ represented by different markers.

Figure 4

Data collapse of the mixing time. (a) Results for ER networks with $N={10}^6$ and different values of $\langle k\rangle$. The inset demonstrates the logarithmic dependency of $t_x$ on $\langle k\rangle /q$ as observed in the same data (circles) and predicted by the theory (full line). (b) Results for SF and SR networks with $N={10}^6,m=2$, and increasing values of the $\gamma$ exponent.

Figure 5

Finite-size effects. (a) Inverse density vs time for ER networks with $\langle k\rangle =4,{\rho }_0=0.4,q=0.008$, and different sizes, as specified by the markers. Besides an earlier extinction of reactants (flat curves, order $1/N$) for smaller networks, the mixing time $t_x$ and the kinetic stages remain unaltered. (b) Plot of $1/t_x$ vs $1/N$ for SR networks with $\gamma =3.5,m=1$ and initial concentration ${\rho }_0=0.4$. Notice that $t_x$ converges to a finite value for large enough networks.