CDLIB: a python library to extract, compare and evaluate communities from complex networks

Rossetti, Giulio; Milli, Letizia; Cazabet, Rémy

doi:10.1007/s41109-019-0165-9

Applied Network Science

Table 4 Clustering comparison functions implemented in CDLIB

From: CDLIB: a python library to extract, compare and evaluate communities from complex networks

Name	Description	Formula	Reference
ami	Adjusted Mutual Information is an adjustment of the Mutual Information score to account for chance.	\(\frac {(X, Y) - E(MI(X, Y))} {max(H(X), H(Y)) - E(MI(X, Y))}\)	(Vinh et al. 2010)
ari	The Rand Index computes a similarity measure between two clusterings by considering all pairs of samples and counting pairs that are assigned in the same or different clusters in the predicted and true clusterings.	\(\frac {RI - Expected_{RI}}{max(RI - Expected_{RI})}\)	(Hubert and Arabie 1985)
closeness	Closeness of community size distributions.	\(\frac {1}{2} \sum _{i=1}^{r} \sum _{j=1}^{s} \min \left \{\frac {x_{a}\left (n^{i}_{a}\right)}{N^{a}}, \frac {x_{b}\left (x^{j}_{b}\right)}{N_{b}}\right \} \delta _{1}\left (n^{i}_{a}, n^{j}_{b}\right)\)	(Dao et al. 2018)
f1	Average F1 score (harmonic mean of Precision and Recall) of the optimal matches among the partitions in input. clustering.	\(2 \times \frac {precision \times recall} {precision + recall}\)	(Rossetti et al. 2016)
nf1	Normalized version of F1 that corrects the resemblance score taking into account degree of node overlap and clutering coverage.	\( \frac {F1\times Coverage}{Redundancy}\)	(Rossetti 2017),(Rossetti et al. 2016)
nmi	Normalized Mutual Information (NMI) is an normalization of the Mutual Information (MI) score to scale the results between 0 (no mutual information) and 1 (perfect correlation)	\(\frac {H(X) + H(Y) - H(X, Y)}{(H(X) + H(Y))/2}\)	(Lancichinetti et al. 2009)
onmi-LFK	Original extension of the Normalized Mutual Information (NMI) score to cope with overlapping partitions.	\( 1 - \frac {1}{2}\bigg (\frac {H(X\|Y)}{H(X)}+\frac {H(Y\|X)}{H(Y)}\bigg)\)	(Lancichinetti et al. 2009)
onmi-MGH	Extension of the Normalized Mutual Information (NMI) score to cope with overlapping partitions, based on max normalization.	\( \frac {I(X:Y)}{max(H(X),H(Y))}\)	(McDaid et al. 2011)
omega	Resemblance index defined for overlapping, complete coverage, clusterings.	\( \frac {Obs(s1, s2) - Exp(s1, s2)} {1 - Exp(s1, s2)}\)	(Murray et al. 2012)
vi	Variation of Information among two nodes partitions.	H(X)+H(Y)−2MI(X,Y)	(Meilă 2007)

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