Critical behavior of a two-step contagion model with multiple seeds

A two-step contagion model with a single seed serves as a cornerstone for understanding the critical behaviors and underlying mechanism of discontinuous percolation transitions induced by cascade dynamics. When the contagion spreads from a single seed, a cluster of infected and recovered nodes grows without any cluster merging process. However, when the contagion starts from multiple seeds of $O\left(N\right)$ where $N$ is the system size, a node weakened by a seed can be infected more easily when it is in contact with another node infected by a different pathogen seed. This contagion process can be viewed as a cluster merging process in a percolation model. Here we show analytically and numerically that when the density of infectious seeds is relatively small but $O\left(1\right)$, the epidemic transition is hybrid, exhibiting both continuous and discontinuous behavior, whereas when it is sufficiently large and reaches a critical point, the transition becomes continuous. We determine the full set of critical exponents describing the hybrid and the continuous transitions. Their critical behaviors differ from those in the single-seed case.

Figure 1

Schematic illustration of epidemic spreading processes in the SWIR model with multiple seeds. (a) Epidemic spreading begins from each infectious node. (b) These nodes can infect susceptible neighbor nodes and change their state to either $I$ or $W$. (c) A node in state $I$ contacts a node in state $W$ from a different root. (d) Then the node in state $I$ infects the node in state $W$ and changes its state to $I$. The two clusters merge.

Figure 2

Schematic plot of $F(m,{\rho }_0)$ versus $m$ for $0<{\rho }_0<{\rho }_c$. Curves represent $F(m,{\rho }_0)$ for different $\kappa$. $m_d, m_m$, and $m_u$ are the solutions of $F(m,{\rho }_0)=0$, where $m_d<m_m<m_u$. $m_0^{\pm }$ are the solutions of $F(m,{\rho }_0)=dF(m,{\rho }_0)/dm=0$ with $m_0^{-}<m_0^{+}$.

Figure 3

Schematic plot of $F(m,{\rho }_c)$ versus $m$ for ${\rho }_0={\rho }_c$. There exists a solution $m_0$ at which the self-consistency equations $F(m_0,{\rho }_c)=0$ and $d^{2}F(m,{\rho }_c)/d^2{m|}_{m=m_0}=0$ hold.

Figure 4

(a) Schematic plot of the order parameter $m\left(\kappa \right)$ versus $\kappa$ for ${\rho }_0<{\rho }_c$. Thick solid (dashed) curves represent stable (unstable) solutions of the self-consistency equation. Dashed-dotted lines represent the pandemic probability $P_{\infty }\left(\kappa \right)$. (b) Plot of $m\left(\kappa \right)$ versus $\kappa$ for ER networks with mean degree $z=8$ and ${\rho }_0=2\times {10}^{-3}$. Solid curve represents analytic solution of the self-consistency equation. Red dots (blue squares) represent averaged values of $m_u$ ($m_d$) obtained by numerical simulations on ER networks of system size $N=4.096\times {10}^7$. (c) Plot of $P_{\infty ,N}$ versus $\kappa$ for various system sizes $N$. Data are obtained for ER networks with mean degree $z=8$ and ${\rho }_0=2\times {10}^{-3}$. At $\kappa ={\kappa }_c^{-}\approx 0.11495, d{P}_{\infty ,N}/d\kappa \sim N^{1/2}$, which implies that $P_{\infty }\left(\kappa \right)$ behaves like a step function in the limit $N\to \infty$, as depicted schematically in (a) with dashed-dotted lines.

Figure 5

Plot of $m\left(\kappa \right)$ versus $\kappa$ for ER networks with mean degree $z=8$ and ${\rho }_0={\rho }_c\approx 0.00747762$. Solid curve represents analytic solution of the self-consistency equation. Red dots (blue squares) represent average values of $m_u$ ($m_d$) obtained by numerical simulations on ER networks of the system size $N=1.024\times {10}^7$.

Figure 6

Plot of $m_0^{-}-m_d$ versus $\mathrm{\Delta }\kappa ={\kappa }_c^{-}-\kappa$ in a double logarithmic scale. Data are obtained for ER networks with mean degree $z=8$ with ${\rho }_0=2\times {10}^{-3}$. The black dashed curve both in the main panel and in the inset represents the analytical solution. For $N=1.6384\times {10}^8$ ($▵$), crossover behavior is likely to occur at $\mathrm{\Delta }\kappa \approx {10}^{-4.2}$, which is roughly close to the point from which the analytical solution (black dashed curve in the inset) of $m_0^{-}-m_d$ deviates from the straight red line with a slope of 0.5.

Figure 7

Plot of $m_0^{-}-\langle m_0^{-}\left(N\right)\rangle$ versus $N$ at $\kappa ={\kappa }_c^{-}$ in a double logarithmic scale. Data are obtained for ER networks with mean degree $z=8$. ${\rho }_0=2\times {10}^{-3}$ is used. The slope of the data point corresponds to $\beta /\overline{\nu }$. As the system size is increased, crossover behavior appears in the slope from $-0.2$ to $-0.24$, which indicates that $\beta /\overline{\nu }\approx 0.24$ in the limit $N\to \infty$.

Figure 8

Plot of the susceptibility $\chi$ versus $N$ at $\kappa ={\kappa }_c^{-}$ in a double logarithmic scale. Data are obtained for ER networks with mean degree $z=8$. ${\rho }_0=2\times {10}^{-3}$. Here the slope of the data points corresponds to $\gamma /\overline{\nu }$, which is estimated to be $\approx 0.5\pm 0.01$.

Figure 9

Plot of the histogram of the order parameter $p\left(m\right)$ at ${\kappa }_c\approx 0.108021$. Data are obtained for ER networks of $N=2.048\times {10}^7$ with mean degree $z=8$ and ${\rho }_0={\rho }_c\approx 0.00747762$. Even though simulations were performed at ${\kappa }_c$ and ${\rho }_c$, the distribution of the order parameter exhibits two peaks, a prototypical pattern of a discontinuous transition due to the finite size effect. As $N$ is increased, we expect that the two peaks converge and become a single peak.

Figure 10

Plot of $m_0-\langle m_0^{-}\left(N\right)\rangle$ versus $N$ at ${\kappa }_c\approx 0.108021$. Data are obtained for ER networks with mean degree $z=8$. ${\rho }_0={\rho }_c$ is taken as $\approx 0.00747762$. $\beta /\overline{\nu }$ is estimated to be $\approx 0.153\pm 0.003$.

Figure 11

Plot of $\langle m_0^{+}\left(N\right)\rangle -m_0$ versus $N$ at ${\kappa }_c\approx 0.108021$. Data are obtained for ER networks with mean degree $z=8$. ${\rho }_0={\rho }_c$ is taken as $\approx 0.00747762$. ${\beta }^{\prime }/{\overline{\nu }}^{\prime }$ is estimated to be $\approx 0.164\pm 0.005$.

Figure 12

Plot of the susceptibility $\chi$, the fluctuation of the order parameter $m_d$, as a function of the system size $N$ at ${\kappa }_c\approx 0.108021$. $\gamma /\overline{\nu }$ is estimated to be $\approx 0.69\pm 0.005$. Data are obtained for ER networks with $z=8$. ${\rho }_0$ is taken as ${\rho }_c\approx 0.00747762$.

Figure 13

Plot of the susceptibility ${\chi }^{\prime }$, the fluctuation of the order parameter $m_u$, as a function of the system size $N$ at ${\kappa }_c\approx 0.108021$. ${\gamma }^{\prime }/{\overline{\nu }}^{\prime }$ is estimated to be $\approx 0.6\pm 0.005$. Data are obtained for ER networks with mean degree $z=8$. ${\rho }_0={\rho }_c$ is taken as $\approx 0.00747762$.

Figure 14

Scaling plot of the susceptibility $\chi$ for $\kappa <{\kappa }_c$ in the form $\chi N^{-\gamma /\overline{\nu }}$ versus $(\kappa -{\kappa }_c)N^{1/\overline{\nu }}$, where $\overline{\nu }\approx 2.13$ and $\gamma \approx 1.47$ are used. Data are obtained for ER networks with mean degree $z=8$. ${\rho }_0={\rho }_c$ is taken as $\approx 0.00747762$.

Figure 15

Scaling plot of the susceptibility ${\chi }^{\prime }$ for $\kappa >{\kappa }_c$ in the form ${\chi }^{\prime }{N}^{-{\gamma }^{\prime }/{\overline{\nu }}^{\prime }}$ versus $(\kappa -{\kappa }_c)N^{1/{\overline{\nu }}^{\prime }}$, where ${\overline{\nu }}^{\prime }\approx 2.13$ and ${\gamma }^{\prime }\approx 1.28$ are used. Data are obtained for ER networks with mean degree $z=8$. ${\rho }_0={\rho }_c$ is taken as $\approx 0.00747762$.