Table 1 Main properties of the data sets used in the present study. N: number of nodes, L: number of edges, \(\langle k \rangle\): average degree, \(k_{\max }\): maximum degree, \(\langle ks \rangle\): average value of the \(k\)-shell index, D: maximum value of the \(k\)-shell index, \(N_\text {c}^{{\texttt {Lvn}{}}}\), \(Q^{\texttt {Lvn}{}}\): number of communities determined by the Louvain method and the corresponding modularity, respectively, \(N_\text {c}^{{\texttt {SBM}{}}}\), \(Q^{\texttt {SBM}{}}\): number of communities determined by the SBM and the corresponding modularity, respectively.

Data set	N	L	\(\langle k \rangle\)	\(k_{\max }\)	\(\langle k_\text {s}\rangle\)	D	\(N_\text {c}^{{\texttt {Lvn}{}}}\)	\(Q^{\texttt {Lvn}{}}\)	\(N_\text {c}^{{\texttt {SBM}{}}}\)	\(Q^{\texttt {SBM}{}}\)	References
Facebook 1	4039	88234	43.691	1045	26.880	115	16	0.835	62	0.551	^55,74
Facebook 2	6386	217662	68.168	930	35.712	56	19	0.419	198	0.158	^56,57,58,75
Facebook 3	2235	90954	81.391	467	44.508	63	8	0.436	87	0.139	^56,57,58,76
Facebook 4	11247	351358	62.480	415	32.413	63	10	0.438	274	0.193	^56,57,58,77
Facebook 5	27737	1034802	74.615	2555	38.681	81	18	0.470	547	0.172	^56,57,58,78
Twitter	81306	1342296	33.018	3383	17.762	96	73	0.808	510	0.511	^55,79
Web-blogs	1490	16715	22.436	351	12.154	36	275	0.426	17	0.076	^60,80
Emails	1005	16064	31.968	345	17.063	34	26	0.410	33	0.232	^81,82,83
Cond. Matter	23133	93439	8.078	279	4.900	25	619	0.730	203	0.633	^83,84
Comp. Science	317080	1049866	6.622	343	4.215	113	209	0.822	676	0.726	^59,85,86
Global airline	3376	19179	11.362	248	6.123	31	26	0.665	40	0.311	⁶¹
Words	146005	656999	9.000	1008	5.289	31	378	0.759	548	0.583	^59,87,88
Cookpad Greece	32235	745178	46.234	8196	23.709	158	40	0.166	76	0.020	–
Cookpad Spain	122158	1749751	28.647	12637	14.547	162	262	0.270	90	0.035	–
Cookpad UK	13758	47525	6.909	1880	3.558	33	199	0.350	8	0.114	–

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