5 0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0 1 2 3 4 5 6 0.05 0.1 0.15 0.2 0.25 0 1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 p$_{k}$ −theory P ( k,k’ )−theory Numerical simulation  on original network (a) Facebook Oklahoma (b) AS Internet (c) PGP Network (d) Power Grid t I ( t ) FIG. 4: (Color online) SIS dynamics, which we display as plots of infected fraction I ( t ) versus time t for the networks from Fig. 1. The parameters in Eq. (17) of Ref. [6] are the recovery rate µ and the spreading rate λ . We use the value µ = 1 in all figure panels; we use I (0) = 10 − 3 in panels (a)–(c) and I (0) = 0 . 002 in panel (d); and we use λ = 0 . 02 in panel (a); λ = 0 . 2 in (b) and (c); and λ = 0 . 8 in panel (d). A. Watts-Strogatz Networks D. Susceptible-Infected-Susceptible Model III. MEASURE OF PREDICTION QUALITY In Fig. 4, we show a comparison between theory and numerical simulation results for the time evolution of a susceptible-infected-susceptible (SIS) epidemic model on various networks. Unlike the other processes that we have discussed, the theory for this case—as given, for ex- ample, by Eq. (17) of Ref. [6]—is expected to apply accu- rately only to the early-time development of the infection. In view of this restriction, the results of Fig. 4 are consis- tent with those of Figs. 1–3. That is, the P ( k, k $^{′}$)-theory once again provides accurate results for certain networks for a variety of processes of interest but is rather inaccu- rate for other networks. Using the small-world networks introduced by Watts and Strogatz [29], one can conduct a systematic study of the effects of clustering C and the mean intervertex distance ℓ . We start with a ring of N = 10000 nodes and connect each node to z = 10 nearest neighbors. We then randomly rewire a fraction f of the links in the net- work [53]. When f = 0, the values of C and ℓ are both high. When f = 1, the rewired network is connected completely at random, which gives it low C and ℓ val- ues. For each value of f between 0 and 1, we numerically calculate the clustering coefficient C$_{f}$ , the mean interver- tex distance ℓ$_{f}$ , and the GCC size S$_{f}$ ( p ) for all values of the bond occupation probability p between 0 and 1. The difference between S$_{f}$ ( p ) and the P ( k, k $^{′}$)-theory curve, which we denote by S$_{th}$ ( p ), gives a quantitative measure for the inaccuracy of the theory for this particular value of the rewiring parameter f . We define the error measure We now aim to characterize the types of networks for which P ( k, k $^{′}$)-theory can be expected to give good re- sults. Because Figs. 1–4 demonstrate that this charac- terization holds for several processes, we will concentrate hereafter primarily on the bond percolation case. E$_{f}$ = 1 M M ∑ i =1 | S$_{th}$ ( p$_{i}$ ) − S$_{f}$ ( p$_{i}$ ) | , (1)