Self-reinforcing cascades: A spreading model for beliefs or products of varying intensity or quality

Abstract

Models of how things spread often assume that transmission mechanisms are fixed over time. However, social contagions--the spread of ideas, beliefs, innovations--can lose or gain in momentum as they spread: ideas can get reinforced, beliefs strengthened, products refined. We study the impacts of such self-reinforcement mechanisms in cascade dynamics. We use different mathematical modeling techniques to capture the recursive, yet changing nature of the process. We find a critical regime with a range of power-law cascade size distributions with non-universal scaling exponents. This regime clashes with classic models, where criticality requires fine tuning at a precise critical point. Self-reinforced cascades produce critical-like behavior over a wide range of parameters, which may help explain the ubiquity of power-law distributions in empirical social data.

Figures (5)

• Figure 1: Schematic of a self-reinforcing cascade. Start with a seed of intensity one. At each generation $n$, the process gains a unit of intensity when reaching active neighbors (orange); or, loses a unit of intensity when reaching inactive neighbors (blue). Paths of the cascade end when they reach a node with no children (dead-end) or when the intensity falls to zero (absorbing boundary). The final cascade consists of all nodes where the process had positive intensity.
• Figure 2: Phase transitions of SRC and directed percolation on Poisson trees of average branching number $\ell=3$. For the SRC, we compare our recursive exact solution based on Eq. (\ref{['eq:recursion']}) to simulations. The critical point marking the emergence of a supercritical cascade is at $p=1/\ell=1/3$ for percolation and at $p = (1-2\sqrt{2}/3)/2 \approx 0.0286$, as computed from Eq. (\ref{['eq:pc']}), for the SRC.
• Figure 3: Critical threshold $p_c$ of SRC on Poisson trees of different average branching number $\ell$. Results are obtained by solving the exact recursion in Eq. (\ref{['eq:recursion']}), the explicit solution in Eq. (\ref{['eq:pc']}), and the critical condition of the traveling wave in Eq. (\ref{['eq:tv_imax']}). The results match up to the numerical precision at which we solve the recursion. The inset validates the explicit solution in Eq. (\ref{['eq:m_k']}) for the expected cascade size $m_k$, comparing it with $10^4$ simulations per value of initial intensity, performed at $p=0.01$ and $\ell = 3$.
• Figure 4: Extended critical behavior around the critical point $p_c$ for a Poisson tree of $\ell = 3$ ($p_c\approx 0.0286$). (a) Cascade size distributions for $p$ above and below $p_c$. Above $p_c$, we find a scaling relationship with exponential cutoff $s^{-\tau(p)} \times e^{-s/\bar{s}(n_c(p))}$ based on the critical generation $n_c(p)$ given in Eq. (\ref{['eq:n_c']}) if $p$ is close to $p_c$. Below $p_c$, we find arbitrarily steep power-law decays as a function of $p$; for instance, $\tau\approx 3.5$ for $p=0.01$. Results from $10^8$ simulations are reported for two $p$ values with logarithmic binning. The recursion is exact. (b) Scaling exponents versus $p$ obtained with our approximate solution, Eq. (\ref{['eq:taup']}), and by fitting power-law tails to our exact solution from recursion. Goodness-of-fit is evaluated with the Kolmogorov–Smirnov (KS) statistic clauset2009power; markers show its minima and lines show an acceptable range (KS $< 0.05$).
• Figure 5: Expected maximal intensity over generations $n$ produced by the logarithmically-corrected $I_{\textrm{max}}(n,p)$ for different values $p > p_c$. We compare the solution with the average maximal intensity observed in at least $10^6$ given the process is not extinct at generation $n-1$. By definition, surviving cascades in simulations are always at intensity greater than zero. Nonetheless, the traveling wave solution captures the general long-time behavior of $I_{\textrm{max}}(n,p)$.