Limited Imitation Contagion on Random Networks: Chaos, Universality, and Unpredictability

We study a family of binary state, socially inspired contagion models which incorporate imitation limited by an aversion to complete conformity. We uncover rich behavior in our models whether operating with either probabilistic or deterministic individual response functions on both dynamic and fixed random networks. In particular, we find significant variation in the limiting behavior of a population’s infected fraction, ranging from steady state to chaotic. We show that period doubling arises as we increase the average node degree, and that the universality class of this well-known route to chaos depends on the interaction structure of random networks rather than the microscopic behavior of individual nodes. We find that increasing the fixedness of the system tends to stabilize the infected fraction, yet disjoint, multiple equilibria are possible depending solely on the choice of the initially infected node.

Figure 1

Examples of probabilistic and deterministic response functions capturing limited imitation contagion dynamics. At each time $t$, nodes use their given response functions to update their own state based on the perceived fraction of their neighbors in state $S_1$, ${\varphi }_{i,t}$. We construct the tent map $T_2$ (see main text) shown in (a) by averaging over deterministic response functions of the kind shown in (b) by considering a family of the latter with “on” and “off” thresholds uniformly distributed in $[0,\frac{1}{2}]$ and $[\frac{1}{2},1]$, respectively. We then build networked systems whose macroscopic character is tent map-like but differ strongly at the microscopic level.

Figure 2

Comparison of bifurcation diagrams for networked systems with limited imitation contagion based on tent map response functions. (a), (b) For reference, standard bifurcation diagrams for the tent map and the logistic map operating as maps of the unit interval, showing distinct universality classes. The gray scale represents the normalized invariant density $\chi$ (darker indicating higher values). (c)–(f) Orbit diagrams for limited imitation contagion acting on standard random networks for four model classes with probabilistic or deterministic responses and network interactions as indicated. The tunable parameter is average degree $k_{\mathrm{av}}$. The inset in (f) shows the approximate location of the collapse point $k_{\mathrm{av}}^{\mathrm{collapse}}$ as a function of network size, the stars indicating the collapse point for $N={10}^4$. The triangle in (f) indicates $k_{\mathrm{av}}=30$ for the system examined in Fig. 3. Simulation details: network size $N={10}^4$; one random seed per network; granularity of 0.1 in $k_{\mathrm{av}}$; binning size of 0.001 in $\varphi$; for $P\mathrm{\text{−}}R$, $D\mathrm{\text{−}}R$, and $P\mathrm{\text{−}}F$ systems, we ran simulations for ${10}^4$ time steps on ${10}^3$ networks, taking values of ${\varphi }_t$ for $t\ge {10}^3$; and for $D\mathrm{\text{−}}F$ systems, which exhibit long transient behavior (see text), we ran simulations for ${10}^5$ time steps on 200 networks, ignoring the first $5\times {10}^4$ time steps.

Figure 3

Comprehensive survey of the possible limited imitation contagion dynamics for a single fixed random network ($k_{\mathrm{av}}=30$) with its $N={10}^4$ nodes having fixed deterministic response functions. We ran $N$ simulations, activating each node as a single seed and allowed the fully deterministic dynamics to run until collapse. (a)–(c) Partial time series for three different initial seeds leading to the narrowest and widest postcollapse dynamics (a),(b) and the longest collapse time $t_c$ (c). The insets in (a)–(c) magnify two cycles of the underlying periodic behavior. (d) Complementary cumulative distribution of collapse times (black squares) compared with an example distribution from another network (gray triangles). (e) Invariant densities, ordered by variance. (f) Histogram of postcollapse microperiod $\tau$. (g) Histogram of postcollapse macroperiod $T$.