Network Name N N$_{E}$ < k > < k 2 > β rand c β k$_{S}$$_{max}$ Contact Network of Inpatients 8622 151649 35.1 1633 1.7% 4% 66 Actor Network 47719 1028537 46.0 17483 0.21% 1% 199 Email Contacts 12701 20417 3.2 351.1 0.73% 8% 23 Live Journal 3453394 21378154 12.38 892.45 1.1% 1.5% 100 Cond-mat Collaboration Network 17628 52884 7.0 109.4 5.1% 10% 22 RL Internet 493312 808844 3.3 71.9 4.6% 6% 36 AS Internet 20556 62920 6.1 2111.2 0.23% n/a 41 Product Space 765 40164 104.8 16931 0.50% n/a 100 TABLE I: Properties of the real-world networks studied in this work. Here N is the number of nodes, N$_{E}$ is the number of edges, < k > is the average degree in the network, < k 2 > is the average squared degree in the network, β rand c is the epidemic threshold for a corresponding random network ( β rand c ≈ λ < k > / < k 2 > ), λ = 0 . 8 in SIS simulations, β is the value we used in SIR simulations and k$_{S}$$_{max}$ is the highest k -shell index of the network. We consider only the largest connected cluster of the network if the original network is disconnected. nodes of the network represent the authors, who are connected if they have co-authored at least one paper. The cond-mat collaboration dataset consists of 17628 authors with average degree 6.0 f) The Internet at the router level (RL) . The nodes of the RL Internet network are the Internet routers. Two routers are connected if there exists a physical connection between them. Data have been gathered from the DIMES project [13]. The largest connected component of the analyzed dataset contains 493312 routers with an average degree of 3.3. g) The Internet at the autonomous system level (AS) . The nodes are autonomous systems which are connected if there exists a physical connection between them. An autonomous system is a collection of connected IP routing prefixes under the control of one or more network operators that presents a common, clearly defined routing policy to the Internet. Data have been gathered by the DIMES project [13]. The largest connected component of the AS Internet consists of 20556 autonomous systems with average degree 6.1. h) Product space of economic goods . This is the network of proximity between products according to Ref. [22]. We use a proximity threshold 0.3, and we recover similar results for 2