Universality in survivor distributions: Characterizing the winners of competitive dynamics

We investigate the survivor distributions of a spatially extended model of competitive dynamics in different geometries. The model consists of a deterministic dynamical system of individual agents at specified nodes, which might or might not survive the predatory dynamics: all stochasticity is brought in by the initial state. Every such initial state leads to a unique and extended pattern of survivors and nonsurvivors, which is known as an attractor of the dynamics. We show that the number of such attractors grows exponentially with system size, so that their exact characterization is limited to only very small systems. Given this, we construct an analytical approach based on inhomogeneous mean-field theory to calculate survival probabilities for arbitrary networks. This powerful (albeit approximate) approach shows how universality arises in survivor distributions via a key concept—the dynamical fugacity. Remarkably, in the large-mass limit, the survivor probability of a node becomes independent of network geometry and assumes a simple form which depends only on its mass and degree.

Figure 1

Attractor on the chain with $N=4$ in the $a_2\left(0\right)-a_3\left(0\right)$ plane, for fixed $a_1\left(0\right)=a_4\left(0\right)=0.3$.

Figure 2

Attractor on the ring with $N=5$ in the $a_4\left(0\right)-a_5\left(0\right)$ plane, for fixed $a_1\left(0\right)=0.5,a_2\left(0\right)=0.7$, and $a_3\left(0\right)=0.6$. The five attractors meet at point P.

Figure 3

Survival probability $S$ of a typical node, as predicted by our approximate analysis, against mean degree $\langle k\rangle$ for several examples of graphs and networks. For the ER and geometric graphs, the predictions for continuously varying $\langle k\rangle$ are shown as the black (lower) and red (upper) curves. Green filled circles correspond to $K$ regular graphs for integer $K$. The blue square shows the prediction (44) for the BA network, while the vertical bar with arrowheads shows the range of values of $S$ for GPA networks, given by the bounds (49) and (50).

Figure 4

Logarithmic plot of the measured degree-resolved survival probability $S\left[\ell \right]$ against degree $\ell$ for the BA network. Straight line: least-squares fit of all the data points with slope $-0.588$.

Figure 5

Mass-resolved survival probability $S_{\alpha }$ against reduced mass $\alpha$. Full curves: numerical results. Corresponding dashed curves: analytical predictions. Top to bottom near $\alpha =1$: BA network, 1D chain, 2D square lattice.

Figure 6

Density-dependent static complexity $\mathrm{\Sigma }\left(\rho \right)$ against survivor density $\rho$ in the allowed range $\left(1/3\le \rho \le 1/2\right)$.