Introduction
Related Work
Pattern Mining
Local Pattern Mining on Attributed Networks
Community Detection on Attributed Graphs
Background
Mining Closed Patterns to Enumerate Core Subgraphs
-
The intersection operator int(X) returns the most specific pattern occurring in the vertex subset X.
-
The core operator p(X) returns the core, according to some core definition, of the subgraph GX of G induced by the vertex subset X. p is an interior operator (see below).
Pruning Local Patterns in Graphs Using Optimistic Estimates
Local Pattern Exploration, Abstraction, and Selection
The MinerLSD Algorithm
Datasets
Nom |
|
V
|
|
|
E
|
|
|
L
|
|
\(\overline {deg(v)}\)
|
\(\overline {|l(v)|}\)
|
---|---|---|---|---|---|
S50 | 50 | 74 | 14 | 2.96 | 7 |
Lawyers | 71 | 556 | 42 | 15.66 | 20 |
CoExp | 151 | 1849 | 36 | 24.49 | 18 |
LastFM | 1892 | 12717 | 17625 | 13.44 | 40.07 |
Delicious | 1867 | 7664 | 52800 | 8.21 | 123.47 |
DBLP.C | 3140 | 10689 | 4588 | 6.81 | 15.02 |
DBLP.P | 45131 | 228173 | 32 | 10.11 | 2.15 |
DBLP.S | 108032 | 276658 | 23254 | 5.12 | 13.93 |
DBLP.XL | 929937 | 3461697 | 92164 | 7.44 | 10.16 |
-
S50 is a standard attributed graph dataset used in a previous work about graph abstractions (Soldano and Santini 2014). 1 It represents 148 friendship relations between 50 pupils of a school in the West of Scotland; the labels concern the students’ substance use (tobacco, cannabis and alcohol) and sporting activity. The values of the corresponding variables are ordered (see (Soldano and Santini 2014) for details).
-
The Lawyers dataset concerns a network study of corporate law partnership that was carried out in a Northeastern US corporate law firm from 1988 to 1991 in New England (Lazega 2001). It concerns 71 attorneys (partners and associates) of this firm who are the vertices of four networks. In the resulting data, each attorney is described using various attributes. 2 We consider the advice network which is originally a directed graph in a undirected version, so that two lawyers are connected if at least one asks for advice to the other one.
-
The CoExp dataset models a representative regulatory network for yeast obtained from Microarray expression data processed by the CoRegNet(Nicolle et al. 2015) program. In the CoExp dataset the vertices are co-regulators and they are linked if they share a common set of target genes. The vertices are labeled with their influence profile along a metabolic transition of the organism. Each influence value represents the regulation activity of the considered co-regulator at some instant of the metabolic transition.
-
LastFM, DBLP.C and DBLP.XL were used in Galbrun et al. (2014). LastFM models the social network of last.fm where individuals are described by the artists or groups they have listened to. DBLP.C contains a co-authorship graph built from a set of publication references extract from DBLP of researchers that have published in the ICDM conference. The authors are labeled by keywords extracted from the papers’ titles. DBLP.XL is the complete labeled DBLP co-authorship network used in Galbrun et al. (2014).
-
DBLP.P was used in Bechara-Prado et al. (Bechara Prado et al. 2013). It represents a co-authorship graph built from a set of publication references extract from DBLP, published between January 1990 and February 2011 in the major conferences or journals of the Data Mining and Database communities. Three labels have been added to the original dataset based on the scope of the conferences and journals, respectively: DB (databases), DM (data mining) and AI (artificial intelligence).
-
Delicious consists of the social (friendship) network of the resource sharing system delicious where individuals are described by their bookmarks’ tags. The dataset is publicly available and was obtained from the HetRec workshop (Cantador et al. 2011) at Recsys 2011.3
-
DBLP.S was used in Silva et al. (2012). It also represents a co-authorship network from a set of publication references extracted from DBLP.
Experiments and Results
Baseline Methods
MinerLC
COMODO
Similarities and Differences in Pattern Selection
-
In COMODO the vertex subset W is obtained as the extremities of the set of edges in which a pattern occurs and a pattern occurs in an edge whenever it occurs, in the original dataset, in both connected vertices. That is, for each edge we assign the set of common items of both nodes, such that a pattern always covers two nodes connected by an edge. As a consequence, W ignores isolated nodes in which p occurs. To obtain the same vertex subset in MinerLC (and MinerLSD) it is necessary to remove isolated nodes, which is enabled by applying a 1-core graph abstraction.
-
Since COMODO does not enumerate closed patterns, the same subgroup may be associated to several patterns. For that case, a post-processing is needed to eliminate the duplicates from the list of subgroups which may then be compared to the subgroups in the MinerLC pairs. This post-processing is one of the standard post-processing options of COMODO.
-
MinerLC is run with a core definition while COMODO uses various parameters to limit the enumeration, as for instance the top-k parameter.
Results and Discussion
Parameters and Datasets
-
#c the number of pairs (c,e).
-
#lme: the number of pairs (c,e) such that oe(MODL)(e)≥lm.
-
#nec: the number of (necessary) pairs (c,e) a top-down search has to consider to ensure that no pair with oe(MODL)(e)≥lm is lost. See “Pruning: Efficiency of the Local Modularity Estimate” section for details and results on #nec.
-
#lm the number of pairs (c,e) such that MODL(e)≥lm
-
#lmeSD: the number of pairs (c,e) such that oe(MODL)(e)≥lm as generated by COMODO.
Impact of Closed Patterns in Reducing the Search Space
Data / #c | 0.005 | 0.01 | 0.02 | 0.03 | 0.05 | 0.15 |
---|---|---|---|---|---|---|
S50 | 83 | |||||
#lmeSD | 493 | 493 | 357 | 326 | 259 | 83 |
#lme | 83 | 83 | 77 | 72 | 67 | 36 |
CoExp | 196 | |||||
#lmeSD | 1232895 | 991231 | 806911 | 468991 | 285183 | 77823 |
#lme | 178 | 166 | 150 | 133 | 114 | 64 |
DBLP.P | 2396 | |||||
#lmeSD | 148 | 32 | 18 | 9 | 5 | 3 |
#lme | 34 | 22 | 15 | 9 | 5 | 3 |
Lawyers | 3221 | |||||
#lmeSD | 3021675 | 1535949 | 677089 | 420699 | 168689 | 10339 |
#lme | 2929 | 2512 | 1970 | 1640 | 1146 | 295 |
DBLP.C | 14820 | |||||
#lmeSD | 179 | 66 | 24 | 16 | 7 | 1 |
#lme | 179 | 66 | 24 | 16 | 7 | 1 |
k-core sizes of the various networks
Modularity Distributions
Pruning: Efficiency of the Local Modularity Estimate
Small Datasets
Data / #c | / 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
---|---|---|---|---|---|---|---|
S50 | 83 | ||||||
#lme | 83 | 83 | 77 | 72 | 67 | 67 | 36 |
#nec | 83 | 79 | 72 | 66 | 62 | 48 | 0 |
#lm | 81 | 77 | 68 | 63 | 55 | 46 | 0 |
CoExp | 196 | ||||||
#lme | 178 | 166 | 150 | 133 | 125 | 114 | 64 |
#nec | 146 | 137 | 104 | 64 | 34 | 10 | 0 |
#lm | 83 | 65 | 35 | 16 | 8 | 1 | 0 |
DBLP.P | 2396 | ||||||
#lme | 34 | 22 | 15 | 9 | 7 | 5 | 3 |
#nec | 29 | 21 | 8 | 5 | 4 | 4 | 0 |
#lm | 28 | 20 | 7 | 4 | 3 | 3 | 0 |
Lawyers | 3221 | ||||||
#lme | 2929 | 2512 | 1970 | 1640 | 1365 | 1146 | 295 |
#nec | 1792 | 1131 | 495 | 201 | 99 | 38 | 0 |
#lm | 1238 | 738 | 308 | 87 | 39 | 5 | 0 |
DBLP.C | 14820 | ||||||
#lme | 179 | 66 | 24 | 16 | 9 | 7 | 1 |
#nec | 145 | 43 | 15 | 4 | 3 | 2 | 0 |
#lm | 144 | 42 | 14 | 3 | 2 | 1 | 0 |
Lawyers | 1-core | #c = 3221 | time = 1 | ||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 2929 | 2512 | 1970 | 1640 | 1365 | 1146 | 295 |
#lm | 1238 | 738 | 308 | 87 | 39 | 5 | 0 |
time (s) | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
2-core | #c = 2080 | time<1 | |||||
#lme | 2080 | 1938 | 1670 | 1454 | 1265 | 1089 | 291 |
#lm | 1262 | 775 | 322 | 104 | 41 | 7 | 0 |
time (s) | 0 | 0 | 1 | 1 | 0 | 0 | 1 |
3-core | #c = 1302 | time<1 | |||||
#lme | 1302 | 1302 | 1215 | 1118 | 1024 | 920 | 282 |
#lm | 1030 | 746 | 348 | 108 | 43 | 7 | 0 |
time (s) | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
5-core | #c = 463 | time<1 | |||||
#lme | 463 | 463 | 463 | 459 | 449 | 432 | 202 |
#lm | 413 | 366 | 253 | 119 | 36 | 9 | 0 |
time (s) | 0 | 0 | 1 | 1 | 0 | 0 | 0 |
7-core | #c = 155 | time<1 | |||||
#lme | 155 | 155 | 155 | 155 | 155 | 155 | 115 |
#lm | 147 | 133 | 97 | 62 | 36 | 13 | 0 |
time (s) | 1 | 1 | 0 | 0 | 0 | 1 | 0 |
CoExp | 1-core | #c = 196 | time<1 | ||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 178 | 166 | 150 | 133 | 125 | 114 | 64 |
#lm | 83 | 65 | 35 | 16 | 8 | 1 | 0 |
time (s) | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
2-core | #c = 172 | time<1 | |||||
#lme | 162 | 153 | 141 | 125 | 118 | 108 | 64 |
#lm | 89 | 78 | 51 | 26 | 12 | 3 | 0 |
time (s) | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
3-core | #c = 138 | time<1 | |||||
#lme | 134 | 129 | 118 | 109 | 102 | 95 | 56 |
#lm | 75 | 64 | 42 | 23 | 12 | 0 | 0 |
time (s) | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
5-core | #c = 62 | time<1 | |||||
#lme | 62 | 60 | 57 | 51 | 48 | 47 | 31 |
#lm | 31 | 23 | 12 | 4 | 2 | 1 | 0 |
time (s) | 1 | 0 | 0 | 0 | 1 | 0 | 0 |
7-core | #c = 37 | time<1 | |||||
#lme | 37 | 37 | 36 | 34 | 33 | 32 | 19 |
#lm | 27 | 22 | 17 | 5 | 3 | 2 | 0 |
time (s) | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
DBLP.C | 1-core | #C = 14820 | time = 31 | ||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 179 | 66 | 24 | 16 | 9 | 7 | 1 |
#lm | 144 | 42 | 14 | 3 | 2 | 1 | 0 |
#time (s) | 41 | 36 | 31 | 30 | 25 | 25 | 17 |
2-core | #c = 1991 | time = 20 | |||||
#lme | 101 | 35 | 19 | 10 | 6 | 5 | 1 |
#lm | 78 | 29 | 11 | 4 | 2 | 1 | 0 |
#time (s) | 23 | 22 | 21 | 20 | 18 | 19 | 15 |
3-core | #c = 319 | time = 11 | |||||
#lme | 46 | 23 | 11 | 5 | 4 | 2 | 1 |
#lm | 39 | 15 | 5 | 3 | 2 | 1 | 0 |
#time (s) | 12 | 11 | 11 | 11 | 10 | 10 | 9 |
5-core | #c = 20 | time = 2 | |||||
#lme | 8 | 3 | 2 | 2 | 2 | 1 | 1 |
#lm | 7 | 3 | 2 | 2 | 1 | 1 | 0 |
#time (s) | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
7-core | #c = 2 | time = 1 | |||||
#lme | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
#lm | 1 | 1 | 1 | 1 | 1 | 1 | 0 |
#time (s) | 1 | 1 | 0 | 1 | 1 | 1 | 0 |
DBLP.P | 1-core | #c = 2396 | time = 9 | ||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 34 | 22 | 15 | 9 | 7 | 5 | 3 |
#lm | 28 | 20 | 7 | 4 | 3 | 3 | 0 |
time (s) | 42 | 42 | 42 | 40 | 38 | 37 | 33 |
2-core | #c = 661 | time = 7 | |||||
#lme | 31 | 21 | 12 | 9 | 7 | 5 | 3 |
#lm | 25 | 19 | 7 | 4 | 3 | 3 | 0 |
time (s) | 38 | 39 | 39 | 38 | 37 | 37 | 33 |
3-core | #c = 261 | time = 7 | |||||
#lme | 27 | 20 | 10 | 7 | 6 | 5 | 3 |
#lm | 21 | 12 | 5 | 4 | 3 | 3 | 0 |
time (s) | 32 | 33 | 34 | 34 | 32 | 33 | 30 |
5-core | #c = 84 | time = 6 | |||||
#lme | 12 | 9 | 7 | 7 | 5 | 4 | 5 |
#lm | 12 | 9 | 6 | 4 | 4 | 3 | 0 |
time (s) | 20 | 21 | 20 | 20 | 19 | 19 | 17 |
7-core | #c = 42 | time = 5 | |||||
#lme | 10 | 8 | 7 | 4 | 4 | 4 | 2 |
#lm | 10 | 7 | 5 | 4 | 4 | 3 | 0 |
time (s) | 12 | 12 | 12 | 12 | 11 | 11 | 10 |
Medium Size Datasets
LastFM | 1-core | #c=1555292 | time=2874 | ||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 59528 | 16163 | 6817 | 3475 | 1920 | 52 | |
#lm | 17627 | 3633 | 1238 | 575 | 276 | 0 | |
time (s) | 5816 | 3400 | 2252 | 1605 | 1187 | 196 | |
2-core | #c = 471546 | time = 2320 | |||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 50507 | 14752 | 6464 | 3349 | 1856 | 52 | |
#lm | 16751 | 3646 | 1252 | 583 | 282 | 0 | |
time (s) | 4668 | 2915 | 1995 | 1452 | 1073 | 178 | |
3-core | #c = 161764 | time = 1878 | |||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 87211 | 39127 | 12694 | 5753 | 3039 | 1720 | 50 |
#lm | 46400 | 14637 | 3377 | 1219 | 572 | 276 | 0 |
time (s) | 4149 | 3422 | 2262 | 1596 | 1174 | 885 | 147 |
5-core | #c = 26312 | time = 1069 | |||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 24807 | 18103 | 8224 | 4272 | 2352 | 1412 | 46 |
#lm | 20562 | 9507 | 2680 | 1035 | 496 | 239 | 0 |
time (s) | 2148 | 2013 | 1580 | 1206 | 857 | 706 | 117 |
7-core | #c = 5859 | time = 531 | |||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 5854 | 5620 | 4031 | 2533 | 1517 | 994 | 39 |
#lm | 5814 | 4482 | 1737 | 775 | 402 | 189 | 0 |
time (s) | 902 | 953 | 877 | 738 | 594 | 486 | 87 |
Delicious | 1-core | #c=11833577 | time=121934 | ||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 5655 | 776 | 255 | 121 | 71 | 4 | |
#lm | 2214 | 165 | 31 | 6 | 1 | 0 | |
time (s) | 5296 | 2018 | 1173 | 825 | 643 | 179 | |
2-core | #c = 130458 | time = 1845 | |||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 7251 | 1421 | 288 | 116 | 65 | 37 | 3 |
#lm | 5440 | 879 | 138 | 39 | 11 | 6 | 0 |
time (s) | 1499 | 920 | 569 | 426 | 358 | 298 | 129 |
3-core | #c = 11076 | time = 269 | |||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 1729 | 430 | 114 | 51 | 25 | 17 | 1 |
#lm | 1419 | 311 | 71 | 25 | 9 | 6 | 0 |
time (s) | 331 | 259 | 208 | 182 | 158 | 149 | 87 |
5-core | #c = 576 | time = 68 | |||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 296 | 89 | 25 | 14 | 7 | 6 | 1 |
#lm | 241 | 70 | 19 | 10 | 5 | 4 | 0 |
time (s) | 77 | 71 | 66 | 64 | 62 | 61 | 55 |
7-core | #c = 77 | time = 21 | |||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 67 | 41 | 13 | 7 | 4 | 1 | 1 |
#lm | 66 | 34 | 10 | 5 | 2 | 1 | 1 |
time (s) | 23 | 23 | 20 | 20 | 19 | 18 | 18 |
Large Datasets
DBLP.S | 1-core | #c ≥ 3457143 | time = STOPPED AFTER 36h | ||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 1150 | 351 | 103 | 50 | 26 | 18 | 1 |
#lm | 778 | 230 | 68 | 25 | 12 | 6 | 0 |
time (s) | 59989 | 37645 | 24906 | 20634 | 17299 | 16167 | 8332 |
2-core | #c ≥ 3584834 | time = STOPPED AFTER 36h | |||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 958 | 303 | 94 | 44 | 24 | 16 | 1 |
#lm | 722 | 218 | 64 | 24 | 12 | 6 | 0 |
time (s) | 36302 | 25949 | 19065 | 16068 | 13869 | 12907 | 7073 |
3-core | #c = 1576164 | time = 45720 | |||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 621 | 208 | 72 | 28 | 17 | 9 | 1 |
#lm | 533 | 165 | 49 | 20 | 9 | 6 | 0 |
time (s) | 19799 | 15531 | 12329 | 10221 | 9149 | 8276 | 5143 |
5-core | #c = 44345 | time = 3791 | |||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 200 | 71 | 26 | 10 | 6 | 4 | 1 |
#lm | 180 | 59 | 21 | 7 | 3 | 2 | 0 |
time (s) | 4410 | 3760 | 3173 | 2877 | 2709 | 2533 | 2044 |
7-core | #c = 5659 | time = 881 | |||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 62 | 24 | 10 | 4 | 2 | 1 | 1 |
#lm | 62 | 23 | 10c | 3 | 1 | 1 | 0 |
time (s) | 1005 | 908 | 812 | 784 | 756 | 687 | 689 |
DBLP.XL | 7-core | #c = 9206 | time = 93906 | ||||
l | 0.005 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.15 |
#lme | 10 | 5 | 4 | 3 | 2 | 1 | 1 |
#lm | 9 | 5 | 3 | 1 | 1 | 1 | 0 |
time (s) | 113790 | 111079 | 110142 | 107967 | 103363 | 97326 | 96811 |
Structural Pattern Set Analysis
k | No lm | lm=0.005 | lm=0.01 | lm=0.02 | lm=0.04 |
---|---|---|---|---|---|
Lawyers: n=71m=556 | |||||
1 | 12.7 (9.6) | 17.2 (9.5) | 20.11 (9.6) | 24.4 (9.5) | 29.8 (4.59) |
2 | 15.3 (10.0) | 16.8 (9.4) | 19.35 (9.3) | 23.4 (9.2) | 27.7 (5.6) |
3 | 18.0 (10.6) | 18.0 (9.5) | 19.49 (9.2) | 22.9 (9.0) | 26.8 (6.3) |
5 | 23.2 (11.9) | 21.7 (10.1) | 21.50 (9.0) | 23.7 (8.3) | 27.3 (5.6) |
CoExp: n=151, m=1849 | |||||
1 | 59.5 (44.0) | 51.9 (35.8) | 51.5 (32.8) | 54.6 (31.6) | 53.9 (21.9) |
2 | 56.9 (41.5) | 49.2 (31.9) | 51.9 (29.4) | 57.2 (25.9) | 63.0 (18.5) |
3 | 54.7 (37.9) | 47.2 (31.6) | 47.4 (28.8) | 52.2 (24.7) | 60.2 (18.3) |
5 | 50.0 (36.2) | 48.3 (34.2) | 51.6 (33.1) | 54.9 (28.9) | 85.5 (9.2) |
DBLP.C: n=3140, m=10689 | |||||
1 | 8.3 (30.8) | 124.0 (106.1) | 231.3 (144.0) | 373.9 (163.6) | 631.0 (63.6) |
2 | 12.5 (68.3) | 124.6 (325.9) | 159.5 (103.9) | 248.0 (119.0) | 434.5 (64.4) |
3 | 21.7 (126.4) | 117.3 (350.3) | 250.7 (549.6) | 604.6 (906.4) | 1256.5 (1366.8) |
5 | 62.6 (199.6) | 164.9 (327.7) | 351.3 (480.6) | 511.0 (555.8) | 904.0 (0.0) |
k | No lm | lm=0.005 | lm=0.01 | lm=0.02 | lm=0.04 |
---|---|---|---|---|---|
Lawyers: n=71m=556 | |||||
1 | 38.2 (58.0) | 61.4 (61.4) | 78.4 (64.0) | 103.2 (67.1) | 25.6 (63.9) |
2 | 53.8 (66.3) | 62.7 (61.9) | 78.3 (64.4) | 104.1 (67.3) | 126.1 (68.1) |
3 | 73.2 (75.4) | 72.9 (65.5) | 82.2 (64.4) | 104.5 (65.0) | 124.8 (67.9) |
5 | 121.7 (95.5) | 109.9 (77.0) | 108.2 (67.7) | 123.7 (62.6) | 135.7 (59.8) |
CoExp: n=151, m=1849 | |||||
1 | 365.4 (473.9) | 323.1 (512.8) | 327.1 (497.0) | 392.0 (579.7) | 448.6 (653.1) |
2 | 404.7 (488.8) | 317.2 (492.9) | 325.0 (483.7) | 352.2 (492.9) | 377.4 (529.4) |
3 | 445.1 (496.5) | 385.1 (535.0) | 375.7 (525.1) | 394.6 (529.5) | 481.5 (598.9) |
5 | 525.4 (499.6) | 551.8 (563.1) | 625.0 (548.9) | 729.3 (544.9) | 1495.0 (132.9) |
DBLP.C: n=3140, m=10689 | |||||
1 | 7.4 (91.4) | 164.9 (192.5) | 347.3 (282.5) | 627.8 (340.3) | 1278.0 (356.4) |
2 | 22.8 (240.5) | 316.81 (1183.3) | 349.2 (271.3) | 577.7 (328.6) | 1126.0 (357.8) |
3 | 66.7 (531.01) | 419.7 (1487.5) | 945.3 (2350.3) | 2393.0 (3924.2) | 5229.0 (5897.3) |
5 | 347.3 (1246.9) | 950.4 (2066.5) | 2101.3 (3055.2) | 3085.0 (3586.5) | 5621.0 (0.0) |
k | No lm | lm=0.005 | lm=0.01 | lm=0.02 | lm=0.04 |
---|---|---|---|---|---|
Lawyers: n=71m=556 | |||||
1 | 4.59 (2.63) | 6.34 (2.51) | 7.26 (2.46) | 8.13 (2.61) | 8.98 (3.27) |
2 | 5.92 (2.56) | 6.63 (2.47) | 7.50 (2.38) | 8.50 (2.42) | 9.43 (2.82) |
3 | 7.23 (2.47) | 7.42 (2.32) | 7.92 (2.27) | 8.80 (2.27) | 9.45 (2.67) |
5 | 9.89 (2.28) | 9.74 (2.10) | 9.82 (1.96) | 10.38 (1.80) | 11.38 (1.95) |
7 | 12.23 (2.08) | 12.15 (2.05) | 11.99 (1.86) | 12.20 (1.40) | 13.14 (1.20) |
CoExp: n=151, m=1849 | |||||
1 | 9.58 (8.34) | 9.38 (9.54) | 10.14 (9.83) | 11.17 (11.12) | 12.19 (13.90) |
2 | 11.15 (8.37) | 9.78 (9.09) | 10.14 (9.18) | 10.41 (9.85) | 10.19 (11.46) |
3 | 13.07 (8.60) | 12.02 (9.66) | 12.15 (9.83) | 12.22 (10.44) | 13.22 (13.20) |
5 | 17.92 (8.25) | 18.40 (9.00) | 20.68 (8.50) | 25.22 (5.72) | 35.42 (0.71) |
DBLP.C: n=3140, m=10689 | |||||
1 | 1.73 (0.57) | 2.60 (0.82) | 2.93 (0.75) | 3.38 (0.65) | 4.02 (0.73) |
2 | 3.15 (0.68) | 4.15 (0.98) | 4.33 (0.72) | 4.66 (0.68) | 5.13 (0.89) |
3 | 4.73 (0.87) | 5.90 (0.78) | 6.35 (0.79) | 6.74 (1.14) | 7.89 (0.80) |
5 | 7.30 (1.57) | 8.55 (1.94) | 10.16 (2.02) | 10.92 (2.17) | 12.45 (0) |
k | No lm | lm=0.005 | lm=0.01 | lm=0.02 | lm=0.04 |
---|---|---|---|---|---|
Lawyers: n=71m=556 | |||||
1 | 10.99 (9.07) | 4.90 (2.55) | 3.68 (1.72) | 2.68 (1.19) | 1.80 (0.65) |
2 | 6.58 (4.40) | 4.81 (2.53) | 3.67 (1.72) | 2.64 (1.16) | 1.81 (0.65) |
3 | 4.62 (2.81) | 4.14 (2.18) | 3.46 (1.58) | 2.58 (1.10) | 1.84 (0.66) |
5 | 2.74 (1.62) | 2.82 (1.53) | 2.69 (1.32) | 2.19 (0.89) | 1.56 (0.52) |
CoExp: n=151, m=1849 | |||||
1 | 4.95 (8.72) | 2.46 (1.59) | 2.13 (1.36) | 1.62 (1.11) | 0.69 (0.34) |
2 | 3.88 (4.76) | 2.69 (1.79) | 2.48 (1.52) | 2.24 (1.35) | 1.69 (1.10) |
3 | 3.58 (4.48) | 2.83 (2.33) | 2.58 (1.77) | 2.41 (1.49) | 1.82 (1.14) |
5 | 3.75 (4.12) | 2.89 (2.75) | 2.24 (2.04) | 1.57 (1.07) | 0.16 (0.10) |
DBLP.C: n=3140, m=10689 | |||||
1 | 14.23 (13.72) | 6.85 (1.94) | 5.28 (1.19) | 3.94 (0.80) | 2.39(0.41) |
2 | 11.29 (10.37) | 5.99 (2.09) | 4.88 (1.17) | 3.78 (0.77) | 2.44 (0.50) |
3 | 11.44 (8.04) | 5.96 (1.90) | 4.57 (1.64) | 2.96 (1.82) | 1.27 (1.68) |
5 | 10.99 (8.22) | 5.99 (3.07) | 3.13 (2.53) | 1.99 (2.22) | 0.42 (0) |