Mixed-order phase transition in a two-step contagion model with a single infectious seed

Percolation is known as one of the most robust continuous transitions, because its occupation rule is intrinsically local. As one of the ways to break the robustness, occupation is allowed to more than one species of particles and they occupy cooperatively. This generalized percolation model undergoes a discontinuous transition. Here we investigate an epidemic model with two contagion steps and characterize its phase transition analytically and numerically. We find that even though the order parameter jumps at a transition point $r_c$, then increases continuously, it does not exhibit any critical behavior: the fluctuations of the order parameter do not diverge at $r_c$. However, critical behavior appears in mean outbreak size, which diverges at the transition point in a manner that the ordinary percolation shows. Such a type of phase transition is regarded as a mixed-order phase transition. We also obtain scaling relations of cascade outbreak statistics when the order parameter jumps at $r_c$.

Figure 1

For $\langle k\rangle >4$, schematic plot of $G\left(m\right)$ versus $m$ for fixed reaction probabilities $r=r_{*}$ (bottom, green), $r_{*}<r<r_a$ (middle, blue), and $r=r_a$ (top, red). $G\left(m\right)$ becomes zero at $m=0,m_d$, and $m_u$, which are determined using Eqs. (17) and (18).

Figure 2

Schematic plot of $m\left(r\right)$ for $\langle k\rangle >4$. Stable solutions of $m\left(r\right)$ are represented by blue (solid or dashed) curve and line, whereas unstable solutions are done by green (solid or dashed) curve and line. Physically accessible states are indicated by solid lines, whereas inaccessible states are indicated by dashed lines. The probability $P_{\infty }\left(r\right)$ is indicated by dashed-dotted curve. The order parameter $m\left(r\right)$ jumps from $m=0$ to $m_u\left(r\right)$ at $r_a$ with the probability $P_{\infty }\left(r\right)$.

Figure 3

Scaling plot of the outbreak size distribution $s^{{\tau }_a}{p}_s\left(r\right)$ versus $s/s_c$ for several values of $r<r_a$, in which ${\tau }_a=1.5$ and $s_c\sim {(r_a-r)}^{-1/{\sigma }_a}$ with ${\sigma }_a=0.5$ are used. Data are obtained from systems of $N=5.12\times {10}^6$ and $\langle k\rangle =8$.

Figure 4

Scaling plot of the susceptibility ${\chi }_a$ versus the reaction probability $r<r_a$ in the form ${\chi }_a{N}^{-{\gamma }_a/{\overline{\nu }}_a}$ and $(r_a-r)N^{1/{\overline{\nu }}_a}$, respectively. Data are obtained from systems of different system sizes $N$ but with the same mean degree $\langle k\rangle =8$. With the choice of ${\gamma }_a=1$ and ${\overline{\nu }}_a=3$, data from different system sizes are well collapsed onto a single curve.

Figure 5

Scaling plots of the outbreak size distribution versus $s$ for several values of $r>r_a$, in which ${\tau }_a=1.5$ and $s_c\sim {(r-r_a)}^{-1/{\sigma }_a}$ with ${\sigma }_a=0.5$ are used. Data of macroscopic-scale outbreak sizes appear away from the curves of finite outbreaks. Data are obtained from systems of $N=5.12\times {10}^6$ with $\langle k\rangle =8$.

Figure 6

Scaling plot of the susceptibility ${\chi }_a$ versus the reaction probability $r$ for $r>r_a$ and $\langle k\rangle =8$. Data are obtained from systems of different system sizes $N$. With the choice of ${\gamma }_a=1$ and ${\overline{\nu }}_a=3$, data from different system sizes are well collapsed on a single curve.

Figure 7

(a) Plot of the fraction $p\left(m\right)$ of the samples having $m$ versus $m$. The distribution is separated into two curves composed of finite and infinite outbreak samples. (b) Plot of numerical data of $m\left(r\right)$ versus $r$ on the theoretical curve shown in Fig. 2. Data are obtained in two different ways, averaged over all samples (green $•)$, over respective finite-outbreak and infinite-outbreak samples (orange $\square )$. Data for both (a) and (b) are obtained from systems of $N=1.28\times {10}^6$ and $\langle k\rangle =8.$

Figure 8

Scaling plot of rescaled outbreak probability $P_{\infty ,N}\left(r\right)N^{{\beta }_p/{\overline{\nu }}_p}$ that an infinite outbreak occurs in a certain sample versus rescaled reaction probability $\mathrm{\Delta }r{N}^{1/{\overline{\nu }}_p}$ for $\langle k\rangle =8$. With the choice of known values ${\beta }_p=1$ and ${\overline{\nu }}_p=3$, the data are well collapsed onto a single curve.

Figure 9

Plot of the mean outbreak time of finite outbreaks $\langle t_{\mathrm{finite}}\rangle$ as a function of (a) $\mathrm{\Delta }r=r-r_a$ and (b) $N$ on semilogarithmic scales. Systems of $\langle k\rangle =8$ are used.

Figure 10

Plot of temporal evolution of the order parameter as a function of time $t$ for system size $N/{10}^6=2^8,2^9$, and $2^{10}$ from left to right for infinite outbreaks. Data are obtained from single trial of the process for each system size with $\langle k\rangle =8$ at $r=r_a$. Inset: plot of the mean maximum slope versus $N$. The slopes show independent behavior of $N$, indicating that the increase rate of the infinite outbreak size is independent of the system size.

Figure 11

Plot of the mean outbreak time of infinite outbreaks $\langle t_c\left(N\right)\rangle$ versus $N$ on double logarithmic scale. Data are obtained from systems of different system sizes $N$ and $\langle k\rangle =8$ at $r=r_a$. The guideline has a slope of 0.35.

Figure 12

Schematic plot of the order parameter $m\left(r\right)$ as a function of the reaction probability $r$ for the case $\langle k\rangle =4$. Stable solutions of $m\left(r\right)$ are represented by blue (solid) line and curve, whereas unstable solutions are represented by green (solid) line. Physically accessible states are represented by solid lines, whereas inaccessible states are done by dashed lines. The probability $P_{\infty }\left(r\right)$ is indicated by a dashed-dotted curve.

Figure 13

Plot of the susceptibility ${\chi }_m$ versus the reaction probability for $r>r_a$ and $\langle k\rangle =4$. Data are obtained from systems of different sizes $N$. Here, the guideline has a slope of $-1.5$, which implies that ${\gamma }_m\approx 1.5$. We remark that data statistics in the plateau region are uncertain because finite and infinite outbreaks are indistinguishable.

Figure 14

Data collapse of the order parameter averaged over all configurations in the form of $m_t(r,N)N^{({\beta }_m+{\beta }_p)/{\overline{\nu }}_m}$ versus $(r-r_a)N^{1/{\overline{\nu }}_m}$ for $\langle k\rangle =4$. ${\beta }_m=0.5,{\beta }_p=1$, and ${\overline{\nu }}_m=2.5$ are used.

Figure 15

Plot of the susceptibility ${\chi }_a$ versus the reaction probability $\mathrm{\Delta }r=r_a-r$ for the case $\langle k\rangle =4$. Data are obtained from systems of different sizes $N$. Here, the guideline has a slope of $-1$, which implies ${\gamma }_a^{\prime }\approx 1$ for $r<r_a$. We remark that the data statistics in the plateau region are uncertain because finite and infinite outbreaks are indistinguishable.

Figure 16

Plot of the susceptibility ${\chi }_a$ versus the reaction probability $\mathrm{\Delta }=r-r_a$ for the case $\langle k\rangle =4$. Data are obtained from systems of different sizes $N$. Here, the guideline has a slope of $-1$, which implies ${\gamma }_a\approx 1$. We remark that the data statistics in the plateau region are uncertain because finite and infinite outbreaks are indistinguishable.

Figure 17

Schematic plot of the order parameter $m\left(r\right)$ as a function of the reaction probability $r$ for the case $\langle k\rangle <4$. Stable solutions of $m\left(r\right)$ are represented by blue (solid or dashed) line and curve, while unstable solutions are done by green (solid or dashed) curve and line. Physically accessible state is represented as solid curve, while inaccessible state is represented as dashed curve. The probability $P_{\infty }\left(r\right)$ is represented by dashed-dotted curve.

Figure 18

Plot of the susceptibility ${\chi }_a$ versus the reaction probability $\mathrm{\Delta }r=r_a-r$ for $r<r_a$. Data are obtained from systems of different sizes $N$ and $\langle k\rangle =3$. The guideline has a slope of $-1$, implying that the susceptibility exponent ${\gamma }_a^{\prime }\approx 1.0$. We remark that the data statistics in the plateau region are uncertain because finite and infinite outbreaks are indistinguishable.

Figure 19

Plot of the susceptibility ${\chi }_a$ versus the reaction probability $\mathrm{\Delta }r=r-r_a$ for $r>r_a$. Data are obtained from systems of different sizes $N$ and $\langle k\rangle =3$. The guideline has a slope of $-1$, implying ${\gamma }_a\approx 1.0$. We remark that the data statistics in the plateau region are uncertain because finite and infinite outbreaks are indistinguishable.