Seed size strongly affects cascades on random networks

The average avalanche size in the Watts model of threshold dynamics on random networks of arbitrary degree distribution is determined analytically. Existence criteria for global cascades are shown to depend sensitively on the size of the initial seed disturbance. The dependence of cascade size upon the mean degree $z$ of the network is known to exhibit several transitions—these are typically continuous at low $z$ and discontinuous at high $z$; here it is demonstrated that the low-$z$ transition may in fact be discontinuous in certain parameter regimes. Connections between these results and the zero-temperature random-field Ising model on random graphs are discussed.

Figure 1

(Color online) Average density $\rho$ of active nodes in a Poisson random graph of mean degree $z$ and uniform threshold value $R$. (a) Color-coded values of $\rho$ from Eq. (1) on the $R,z$ plane with seed fraction ${\rho }_0={10}^{-2}$. Lines show approximations to the global cascade boundaries: the solid line encloses the region where (5) is satisfied; the dashed line represents the extended cascade boundary from Eq. (6). (b) Values of $\rho$ at $R=0.18$ from Eq. (1) (lines) and numerical simulations (symbols), averaged over 100 realizations with $N={10}^5$. Seed fractions are ${\rho }_0={10}^{-3}$ (solid), $5\times {10}^{-3}$ (dashed), and ${10}^{-2}$ (dot-dashed). The arrow marks the cascade boundary given by Eq. (5) in the limit of zero seed fraction.

Figure 2

(Color online) As Fig. 1, but threshold distribution is Gaussian with mean $R$ and standard deviation 0.2, and seed fraction ${\rho }_0=0$. (a) Cascade boundaries as in Fig. 1; the arrow marks the critical point $(R_c,z_c)$ described in the text; (b) theory and numerical simulation results ($N={10}^5$, average over 100 realizations) at $R=0.2$ (solid), 0.362 (dashed), and 0.38 (dash-dotted).

Figure 3

(Color online) Bifurcation diagrams as described in text for dependence of $q_{\infty }$ on $z$ for $\sigma =0.2$ and ${\rho }_0=0$ at $R=\left(\mathrm{a}\right)$ 0.35, (b) 0.371, and (c) 0.375.