Table 1. Empirical statistics and analysis results of real data sets.

No.
1	206779	18217	1.323	0.756	0.725	0.738
2	20516	5671	0.969	1	0.858	0.859
3	109854	13906	1.063	0.941	0.845	0.817
4	449205	20220	1.464	0.683	0.667	0.679
5	68458	9191	1.095	0.913	0.823	0.810
6	81037	13254	1.025	0.976	0.859	0.832
7	63742	16622	1.057	0.946	0.840	0.852
8	138985	15550	1.188	0.842	0.787	0.765
9	101940	12667	1.117	0.895	0.818	0.799
10	504610	116800	0.893	1	0.936	0.863
11	53214	34194	0.540	1	0.983	0.946
12	310853	69185	0.939	1	0.913	0.871
13	30852	17562	0.595	1	0.972	0.939
14	2761	2328	0.397	1	0.964	0.978
15	58300	22599	0.786	1	0.941	0.914
16	20660	8155	0.790	1	0.921	0.890
17	226090	69251	0.692	1	0.977	0.894
18	176291	62567	0.572	1	0.989	0.920
19	44735	19933	0.685	1	0.961	0.915
20	1924	1323	0.463	1	0.946	0.939
21	5093	2985	0.593	1	0.941	0.920
22	3490	2442	0.500	1	0.952	0.950
23	1403	787	0.524	1	0.926	0.931
24	7469	4142	0.654	1	0.936	0.925
25	7710	3857	0.658	1	0.935	0.930
26	3232	2658	0.416	1	0.964	0.976
27	13165	7743	0.612	1	0.959	0.936
28	3749	2353	0.568	1	0.943	0.940
29	30092	11002	0.815	1	0.924	0.891
30	21894	8666	0.776	1	0.930	0.900
31	7627	3841	0.685	1	0.933	0.930
32	4185	2242	0.675	1	0.921	0.929
33	23822	10753	0.648	1	0.959	0.917
34	8829	40	3.0	0.33	0.34	0.35
35	237982	56961	0.462	1	0.993	0.929

is the total number of elements, is the total number of distinct elements, is the Zipf's exponent obtained by the maximum likelihood estimation [3], [43], is the asymptotic solution of the Heaps' exponent as shown in Eq. 7, is the numerical value of the Heaps' exponent given and as shown in Fig. 3, and is the empirical result of the Heaps' exponent obtained by the least square method. The effective number of the 34th data set is only two digits since the size of this data set is very small. Except the 4th data set, in all other 34 real data sets, the numerical results based on Eq. 6 outperform the asymptotic solution shown in Eq. 7. Detailed description of these data sets can be found in Materials and Methods .