Table 2 Comparison between the LS and Louvain algorithms on networks with ground-truth community labels

	N	E	N_c	Louvain			LS			Δt (ms)
				F₁	N_c	t (ms)	F₁	N_c	t (ms)
Karate	34	78	2	0.63	4	8	0.83	2	6	2
Football⁵⁹	115	613	10	0.87	10	18	0.35	6	20	–2
Polbooks	105	441	3	0.70	5	13	0.80	2	8	5
Polblogs	1490	19,090	2	0.85	9	328	0.69	3	212	116
Cora	2708	5429	7	0.32	28	380	0.33	7	139	241
Citeseers	3264	9072	6	0.27	35	384	0.45	7	131	253
PubMed	19,717	44,327	3	0.20	43	8745	0.46	8	2298	6447
DBLP	317,080	1,049,866	–	–	220	256,000	–	8; 1859	45,000	211,000

N_c denotes the number of ground-truth communities in the network or identified by different methods, and F₁-score is a common performance measure in machine learning between predictions and ground-truth labels (see more details in Supplementary Note 2), and t (ms) is the running time of the algorithm when implemented in Python. As there is no ground truth labels but only meta data for DBLP⁴⁷ (see Supplementary Note 2 for more discussions), we are unable to report F₁-score. As LS is able to detect multiscale structure, we report the number of communities detected with notable gaps: 8 large communities, 1859 smaller communities. Both the Louvain and LS algorithm are of linear complexity in time, and our LS method is faster. In addition, the LS method performs better in most cases. The algorithm with a better performance is highlighted in bold. Comparisons with a broader range of classical community detection algorithms are shown in Table 3.

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