Table 2.
Community detection on real networks with Infomap algorithm
| 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 | 0.68 | 0.75 | 0.47 | 1.00 | 1.00 | 0.54 | 0.92 | 0.53 | 0.74 | +4.2 |
| ncMCE-EA | 0.68 | 0.74 | 0.47 | 1.00 | 0.93 | 0.50 | 0.92 | 0.52 | 0.72 | +1.4 |
| unweighted | 0.55 | 0.69 | 0.47 | 1.00 | 1.00 | 0.52 | 0.92 | 0.52 | 0.71 | 0.0 |
| EBC-MCE-EA | 0.68 | 0.55 | 0.53 | 1.00 | 0.96 | 0.52 | 0.93 | 0.52 | 0.71 | 0.0 |
| EBC-ncMCE-EA | 0.58 | 0.55 | 0.53 | 1.00 | 1.00 | 0.52 | 0.93 | 0.52 | 0.70 | −1.4 |
| ISO-EA | 0.68 | 0.53 | 0.47 | 1.00 | 0.96 | 0.52 | 0.92 | 0.53 | 0.70 | −1.4 |
| LE-EA | 0.68 | 0.54 | 0.47 | 1.00 | 0.96 | 0.53 | 0.92 | 0.51 | 0.70 | −1.4 |
| EBC-LE-EA | 0.68 | 0.55 | 0.47 | 0.95 | 0.96 | 0.52 | 0.93 | 0.53 | 0.70 | −1.4 |
| ncISO-EA | 0.68 | 0.53 | 0.47 | 1.00 | 0.96 | 0.47 | 0.92 | 0.53 | 0.69 | −2.8 |
| EBC-ISO-EA | 0.55 | 0.55 | 0.47 | 1.00 | 1.00 | 0.52 | 0.92 | 0.53 | 0.69 | −2.8 |
| MCE-EA | 0.68 | 0.54 | 0.47 | 0.95 | 0.93 | 0.51 | 0.92 | 0.52 | 0.69 | −2.8 |
| RA-ncISO-EA | 0.55 | 0.55 | 0.47 | 1.00 | 1.00 | 0.52 | 0.92 | 0.52 | 0.69 | −2.8 |
| RA-ISO-EA | 0.58 | 0.55 | 0.47 | 1.00 | 0.96 | 0.52 | 0.92 | 0.52 | 0.69 | −2.8 |
| RA-ncMCE-EA | 0.47 | 0.55 | 0.53 | 1.00 | 1.00 | 0.52 | 0.92 | 0.50 | 0.69 | −2.8 |
| LPCS | 0.55 | 0.55 | 0.53 | 1.00 | 0.96 | 0.52 | 0.93 | 0.51 | 0.69 | −2.8 |
| RA-LE-EA | 0.55 | 0.55 | 0.47 | 1.00 | 0.93 | 0.52 | 0.92 | 0.52 | 0.68 | −4.2 |
| RA-MCE-EA | 0.47 | 0.55 | 0.53 | 1.00 | 0.92 | 0.52 | 0.92 | 0.51 | 0.68 | −4.2 |
| HyperMap-CN | 0.52 | 0.55 | 0.41 | 1.00 | 0.86 | 0.57 | 0.89 | 0.46 | 0.66 | −7.0 |
| HyperMap | 0.52 | 0.60 | 0.32 | 1.00 | 0.92 | 0.49 | 0.90 | 0.46 | 0.65 | −8.5 |
Normalized mutual information (NMI) computed between the ground truth communities and the ones detected by the Infomap 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 hyperbolic distances between the nodes are used to weight the observed links in the input matrix for the Infomap algorithm (see “Methods” section for details). As a reference, the Infomap 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 2. The rightmost column reports the percentage of improvement with respect to the unweighted variant