Figure 1

(a) Average number of new infections $n(t)$ of the SI model with uncorrelated identical power law $P(\tau )$ with exponent $\alpha =2.1$ ($\square$), $2.5$ ($ˆ$), $2.9$ ($▵$), and with an exponential $P(\tau )$ ($▿$), respectively. All $P(\tau )$ have the same mean waiting time $\langle \tau \rangle =2$. It decays with the exponent $\approx \alpha -1$ for power law $P(\tau )$, whereas it decays exponentially for exponential $P(\tau )$. (b) Generation time distribution $g(\Delta )$ for power law $P(\tau )$. It decays as a power law with the same exponent as $n(t)$. Simulations were performed on a scale-free network with $\gamma =3$ and size $N={10}^4$, and averaged over ${10}^8$ independent runs. Initial infected node is selected to be the hub. In both panels, dotted lines have slopes $-1.1$, $-1.5$, and $-1.9$, drawn as a guide. Data are binned logarithmically.Reuse & Permissions

Figure 2

Schematic illustration of the SI model on the PQN. Filled (open) nodes denote infected (susceptible). The arrows from an infected to a susceptible node indicate potential subsequent infection channels.Reuse & Permissions

Figure 3

(a) Average number of new infection $n(t)$ of the SI model on the PQN ($ˆ$) and on the random execution queue network ($▿$). After the initial increase, $n(t)$ after the peak decays with a power-law tail with exponent $\beta \approx 2$ for the PQN ($ˆ$) and exponentially for the random execution case ($▿$). (b) The waiting time distribution of the whole network, $P_{\mathrm{PQN}}(\tau )$, and the waiting time distributions of links connecting nodes with degree $k$ and $k^{\prime }$, $P_{{\mathit{kk}}^{\prime }}(\tau )$, of the same process for the PQN. Both $P_{\mathrm{PQN}}(\tau )$ and $P_{{\mathit{kk}}^{\prime }}(\tau )$ decay as a power law, yet with different exponents dependent on the local topological position, ${\alpha }_{\mathrm{PQN}}\approx 2$, ${\alpha }_{196,1}\approx 1$, ${\alpha }_{2,1}\approx 3$, and ${\alpha }_{4,4}\approx 2$. (c) The generation time distribution $g_{\mathrm{PQN}}(\Delta )$ of the same process for the PQN. It decays as a power law with the same exponent as $n_{\mathrm{PQN}}(t)$. In all panels, dotted lines have slope $-2$, drawn as a guide. Simulations were performed on a scale-free network with $\gamma =3$ and size $N={10}^4$, and averaged over ${10}^3$ different initial conditions. Data are binned logarithmically.Reuse & Permissions

Figure 4

$n(t)$ (left scale) and $g(\Delta )$ (right scale) of the SI model on the Erdős-Rényi (ER) network with mean degree $\langle k\rangle =4$ and size $N={10}^4$ and Cayley tree with branching number $k=4$ with six generations ($N=1456$). In each case, $n(t)$ and $g(\Delta )$ show power-law tails with the same exponent. Dotted line has slope $-2$, drawn as a guide. Data are binned logarithmically.Reuse & Permissions