Figure 1

(Color online) For $N={10}^2$, ${10}^4$, and ${10}^6$, respectively, in (a), (b), and (c), $1-f_c$ versus $\lambda$. The solid line represents the results of Monte Carlo simulations; the dashed line is the prediction of Eq. (4). (d) The difference $\Delta$ [see Eq. (7)] between the prediction of Eq. (4) and Monte Carlo simulations for (from top to bottom) $N={10}^2,{10}^3,{10}^4,{10}^6$. Note that if we had used a larger value of the upper cutoff $K$, then $\Delta$ would decrease monotonically from $\lambda =3$ to 1 instead of having a minimum near $\lambda =2$.Reuse & Permissions

Figure 2

(Color online) $P\left(k\right)$ versus $k$ for $N={10}^2$, ${10}^4$, and ${10}^6$ in (a), (b), and (c), respectively. The solid line represents $P\left(k\right)$ after network construction using the Molloy-Reed method; the dashed line is the distribution after the removal of self-loops and multiple edges.Reuse & Permissions

Figure 3

(Color online) Correlation $r$ as a function of $\lambda$ for (from top to bottom at left) $N={10}^2$, ${10}^3$, ${10}^4$, and ${10}^6$ for distributions after removal of self-loops and multiple edges. Note that the correlation increases with $N$ for $\lambda \lesssim 3$ and decreases with $N$ for $\lambda \gtrsim 3$.Reuse & Permissions

Figure 4

(Color online) Example illustrating case in which $⟨(i\mid \kappa =2⟩\phantom{)}\ne (i\mid ⟨\kappa ⟩=2)$ for star network of one hub of degree 99 and 99 nodes of degree 1. Thin lines are $\kappa$ vs $i$, where $i$ denotes the number of the step at which a node is deleted, for cases in which the hub is deleted at step (from left to right) 1, 10, 20, 30, 40, 50, 60, 70, 80, 90, and 100. The thick line is the average of the thin lines. Note that the value of $i$ at which the average crosses the horizontal line $\kappa =2$ is much higher than 50, the average of the values of $i$ at which the thin lines cross the horizontal line $\kappa =2$.Reuse & Permissions

Figure 5

(Color online) For $⟨k⟩=2$ and for (from left to right) $N={10}^2$, ${10}^3$, ${10}^4$, ${10}^5$, and ${10}^6$ (a) $1-f_c$ vs number of hubs $q$. The solid lines represent Monte Carlo simulation results. Dashed lines (short) are the approximation $f_c^{\text{low}}$; dashed lines (long) are the approximation $f_c^{\text{high}}$. (b) Number of hubs $q$ versus $N$. Squares represent characteristic values $q^{*}$ at which high and low $q$ approximations intersect. Triangles represent values of $q$ at which the standard deviation in $1-f_c$ is minimal.Reuse & Permissions

Figure 6

(Color online) Number of hubs $q^{*}$ at which approximations for low and high $q$ intersect vs $N$. Squares, triangles, and circles represent networks with $⟨k⟩=2$, 3, and 4, respectively.Reuse & Permissions

Figure 7

(Color online) (a) $P(1-f_c)$, the probability distribution of $1-f_c$ for $N={10}^3$ and $q=1$ (dashed line) and (from left to right in order of increasing position of peaks) $q=5,10$, and 20. (b) Standard deviation $\sigma$ versus $q$ for $N={10}^2$, ${10}^3$, ${10}^4$, ${10}^5$, and ${10}^6$ (from left to right in order of increasing length of the tails of the distributions). Note that the second peak in this plot which is most pronounced for smaller $N$ is an artifact of finite size.Reuse & Permissions

Figure 8

(Color online) For random scale-free networks with $4⩽k⩽N$, standard deviation ${\sigma }_{fc}$ versus $\lambda$ for $N={10}^2$, ${10}^3$, ${10}^4$, ${10}^5$, and ${10}^6$ (from top to bottom).Reuse & Permissions