Ego-zones: non-symmetric dependencies reveal network groups with large and dense overlaps

Applied Network Science

Table 15 Properties of zone overlaps in real-world networks

Network	Overlaps	Overlap size		Zones in overlaps		Zones in O size
		Max	Avg	Total	Percent	Max	Avg
artist	750	26	3.667	292	38.9	22	6.007
as-22july06	58095	583	2.763	3628	6.2	13	4.089
astro-ph	500677	311	3.705	33119	6.6	203	23.743
Brightkite	27925	114	4.779	4942	17.7	55	10.169
com-amazon	65057	128	4.601	29576	45.5	24	5.498
com-dblp	925712	177	2.859	116482	12.6	163	12.946
cond-mat	27516	47	2.771	4519	16.4	47	8.483
cond-mat-2005	375065	184	2.345	30731	8.2	78	10.124
email-Enron	302590	169	1.817	15609	5.2	41	6.379
facebook	26322	42	1.759	973	3.7	34	12.807
ChCh-Miner	670	73	10.213	238	35.5	43	16.008
new_sites	3819	41	5.933	1637	42.9	28	9.296
power	125	12	3.272	66	52.8	12	4.712
PP-Decagon	86548	382	19.909	4453	5.1	382	128.530
PP-Pathways	17880	122	11.439	2510	14.0	122	43.175
Yeast	227	17	3.802	112	49.3	8	4.688
LFR _{20 500 2000}	5420	81	9.782	2290	42.3	35	8.932
LFR _{7 60 4000}	14577	27	2.060	2331	16.0	17	5.961

Except for the two biological networks, the average size of the overlaps for most networks is small. However, it can be seen in all the networks that a non-trivial number of larger zones exists in the overlaps; some of the networks have zones (nested in overlaps) with a high number of nodes. In the last column, there are average zone sizes detected in overlaps of two other zones. These sizes are greater than the average overlap sizes in ‘avg overlap size’ column. Thus, the zones exist more likely in larger overlaps. The size of a nested zone can correspond with the size of the entire overlap; e.g., in two biological networks (PP-Decagon and PP-Pathways), the maximum overlap of two zones is also a zone