Table 1 Community detection on real networks with Louvain algorithm

From: Machine learning meets complex networks via coalescent embedding in the hyperbolic space

Method

Karate

Opsahl 8

Opsahl 9

Opsahl 10

Opsahl 11

Polbooks

Football

Polblogs

Mean

% Impr.

 

N = 34

N = 43

N = 44

N = 77

N = 77

N = 105

N = 115

N = 1222

  
 

E = 78

E = 193

E = 348

E = 518

E = 1088

E = 441

E = 613

E = 16,714

  
 

m = 2.29

m = 4.49

m = 7.91

m = 6.73

m = 14.13

m = 4.20

m = 5.33

m = 13.68

  
 

T = 0.43

T = 0.43

T = 0.32

T = 0.35

T = 0.28

T = 0.51

T = 0.60

T = 0.68

  
 

γ = 2.12

γ = 8.20

γ = 5.92

γ = 5.06

γ = 4.87

γ = 2.62

γ = 9.09

γ = 2.38

  
 

N c = 2

N c = 7

N c = 7

N c = 4

N c = 4

N c = 3

N c = 12

N c = 2

  

EBC-ncISO-EA

1.00

0.57

0.47

1.00

0.93

0.59

0.90

0.68

0.77

+13.2

RA-MCE-EA

0.83

0.51

0.47

1.00

0.96

0.57

0.82

0.67

0.73

+7.4

RA-ncMCE-EA

0.73

0.55

0.47

1.00

1.00

0.57

0.83

0.67

0.73

+7.4

EBC-MCE-EA

0.83

0.47

0.41

1.00

0.96

0.57

0.90

0.62

0.72

+5.9

EBC-ncMCE-EA

0.88

0.46

0.41

1.00

0.96

0.57

0.85

0.62

0.72

+5.9

EBC-ISO-EA

0.83

0.42

0.47

1.00

0.89

0.59

0.88

0.66

0.72

+5.9

LPCS

0.83

0.49

0.41

1.00

0.96

0.55

0.87

0.67

0.72

+5.9

ncMCE-EA

0.73

0.47

0.47

1.00

0.96

0.57

0.89

0.62

0.71

+4.4

RA-LE-EA

0.67

0.48

0.53

1.00

0.92

0.56

0.82

0.70

0.71

+4.4

RA-ncISO-EA

0.67

0.54

0.42

1.00

0.92

0.56

0.86

0.67

0.70

+2.9

ncISO-EA

0.73

0.50

0.41

1.00

0.88

0.54

0.87

0.66

0.70

+2.9

EBC-LE-EA

0.85

0.42

0.41

0.96

0.92

0.56

0.85

0.62

0.70

+2.9

MCE-EA

0.64

0.47

0.47

0.96

0.92

0.55

0.86

0.62

0.69

+1.5

unweighted

0.46

0.55

0.41

1.00

0.96

0.50

0.93

0.64

0.68

0.0

LE-EA

0.63

0.55

0.41

1.00

0.78

0.55

0.82

0.67

0.68

0.0

RA-ISO-EA

0.57

0.43

0.44

1.00

0.88

0.54

0.86

0.67

0.67

−1.5

ISO-EA

0.34

0.50

0.41

0.96

0.93

0.56

0.82

0.67

0.65

−4.4

HyperMap

0.56

0.60

0.28

0.92

0.85

0.50

0.83

0.69

0.65

−4.4

HyperMap-CN

0.55

0.47

0.41

0.93

0.79

0.54

0.79

0.70

0.65

−4.4

  1. Normalized mutual information (NMI) computed between the ground truth communities and the ones detected by the Louvain algorithm for eight real networks. NMI = 1 indicates a perfect match between the two partitions of the nodes. For each method, the network has been embedded in the hyperbolic space and the embedding coordinates are used to weight the input matrix for the Louvain algorithm: observed links are weighted using the hyperbolic distances between the nodes and non-observed links using the hyperbolic shortest paths (see “Methods” section for details). As a reference, the Louvain algorithm has been run giving in input also the unweighted adjacency matrix, the related row is highlighted in italic. The table contains also some statistics for each network: number of nodes N, number of edges E, temperature T (inversely related to the clustering coefficient), power-law degree distribution exponent γ, half of average degree m, and number of ground truth communities N c. Due to the higher performance, only the EA methods are here reported, whereas the complete table is shown as Supplementary Table 1. The NMI values highlighted in bold for the Karate and Opsahl_11 networks are the ones whose embedding is shown in Fig. 7. The rightmost column reports the percentage of improvement with respect to the unweighted variant, the best result is highlighted in bold. The results obtained only weighting the observed links are shown in Supplementary Table 5
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