Figure 1

(Color online) For the ring backbone network of $N_0=1000$ nodes and average degree $\langle k\rangle =100$, (a) evolution of the eigenratio as a function of time for three values of the infection probability: $P_s=0.5$ (circle), 0.8 (triangle), and 1.0 (square), (b) $R$ versus the size of the infected network, which is apparently independent of the value of $P_s$, and (c) eigenvalues ${\lambda }_2$ and ${\lambda }_N$ versus $N$. The solid curves are analytic predictions. The existence of the local maximum in $R$ for large $N$ can be explained theoretically by noting that a crossover from the lattice to the ring behaviors necessarily occurs for dynamic networks of large sizes (see Sec. 3B). Here, all simulation results are obtained by averaging over 100 independent runs of the infection dynamics. For regular networks, since the network structure is fixed for given size and average degree, ensemble average using different network realizations is not necessary.Reuse & Permissions

Figure 2

(Color online) For the ring backbone network of $N_0=1000$ nodes and $m=10$, (a) $\lambda$ versus spatial Fourier frequency $f$ from Eq. (4), and (b) the left-hand side (LHS, horizontal line) and right-hand side (RHS, dots) of Eq. (6). The vertical lines from left to right indicate the values of $f$ for which the equality $\left(m+1\right)\pi f/\left(2N\right)=\left(j+1/2\right)\pi$ holds for $j=0,1,\dots ,m/2$.Reuse & Permissions

Figure 3

(Color online) For a regular backbone network with a ring topology, (a) schematic illustration of a dynamic network in the transient phase where the infected nodes (solid circle) constitute an open latticelike network, (b) illustration of a large infected network with a ring-type topology in the steady-state phase, and (c) crossover of numerically obtained values (circles) of ${\lambda }_2$ from prediction based on the latticelike topology to that with the ring topology on network of $N_0=1000$ and $\langle k\rangle =100$.Reuse & Permissions

Figure 4

(Color online) For an ensemble of random networks of fixed average degree $\langle k\rangle =20$ and $N_0=1000$ nodes, (a) evolution of the eigenratio $R$ in time for three values of the spreading probability: $P_s=0.3$ (circle), 0.5 (triangle), and 0.8 (square), (b) $R$, and (c) numerical eigenvalues ${\lambda }_2$ and ${\lambda }_N$ (symbols) versus the size of the dynamic network and theoretical predictions (curves). Circles: numerically obtained ${\lambda }_N$; triangles: $k_{\text{max}}+1$; squares: numerically obtained ${\lambda }_2$; and diamonds: minimum degree. The results are obtained by averaging over 300 network realizations.Reuse & Permissions

Figure 5

(Color online) For scale-free networks of fixed average degree $\langle k\rangle =20$ and $N_0=1000$ nodes, (a) evolution of the eigenratio $R$ in time for three values of the spreading probability: $P_s=0.5$ (square), 0.8 (triangle), and 1 (circle), [(b) and (c)] $R$, ${\lambda }_2$, and ${\lambda }_N$ versus the size of the dynamic network and theoretical predictions (curves). Circles: numerically obtained ${\lambda }_N$; triangles: $k_{\text{max}}+1$; squares: numerically obtained ${\lambda }_2$; and diamonds: minimum degree. The results are obtained by averaging over 300 network realizations.Reuse & Permissions