Table 1. Properties of the real-world networks studied in this work.

Network	N	E		kmax	Hk	r	C	kSmax	λc	λ
Email	1133	5451	9.6	71	1.942	0.078	0.220	11	0.06	0.08
CA-Hep	8638	24806	5.7	65	2.261	0.239	0.482	31	0.08	0.12
Hamster	2000	16097	16.1	273	2.719	0.023	0.540	24	0.02	0.04
Blog	3982	6803	3.4	189	4.038	−0.133	0.284	7	0.08	0.27
PGP	10680	24340	4.6	206	4.153	0.240	0.266	31	0.06	0.19
Astro	14845	119652	16.1	360	2.820	0.228	0.670	56	0.02	0.05
Router	5022	6258	2.5	106	5.503	−0.138	0.012	7	0.08	0.27
Emailcontact	12625	20362	3.2	576	34.249	−0.387	0.109	23	0.01	0.10
AS	22963	48436	4.2	2390	61.978	−0.198	0.230	25	0.004	0.13

Structural properties include number of nodes (N), number of edges (E), average degree , maximum degree (k_max), degree heterogeneity (), degree assortativity (r), clustering coefficient (C), maximum k_s index (k_Smax), epidemic threshold (λ_c), infection probability used in the SIR spreading in the main text (λ) (see Method for details). For the first six networks, there exist core-like groups. While for the last three networks, there is no core-like group in the network, which we will discuss in the last part.