Table 2 Description of the symbols used in the formulas for fitness and comparison functions
From: CDLIB: a python library to extract, compare and evaluate communities from complex networks
Symbol | Description |
|---|---|
C | partition of the graph |
Ci | community ∈ partition C |
bC | number of edges on the community boundary |
dm | internal degree median value |
eC | number of community internal edges in C |
k | the total degree |
\(k^{int}_{iC}\) | the degree of node i within C |
\(k^{out}_{iC}\) | the degree of node i outside C |
lC | number of edges from nodes in C to nodes outside C |
Mint | total possible internal edges: \(\sum _{C} \binom {n_{C}}{2}\) |
Nc | Total number of communities of all sizes detected by a given algorithm, \(\sum _{i} n_{c} x_{c}^{i}\) |
nC | number of community nodes in C |
Γ(i) | degree of the node i |
p | density of the graph |
pC | the density of community C. \(p_{C}=m_{C}/ \binom {n_{C}}{2} \) |
〈q〉 | the expected fraction of internal edges |
xc | number of communities having the same number of nodes of c |
Coverage | percentage of communities in Y that are matched by at least an object in X \(\frac {|Y_{id}|}{|Y|}\) where Yid is the subset of communities in Y matched by community in X |
D(x||y) | KL divergence |
Dc | the sum of the degrees of the vertices in community C |
δ(ci,cj) | indicator function: it assumes value 1 iff i and j belong to the same community, 0 otherwise |
\(\delta _{1}(n_{a}^{i}, n_{b}^{j})\) | indicator function: it assumes value 1 iff two communities a and b have the same number of nodes, \(n_{a}^{i} = n_{b}^{j}\), 0 otherwise |
Exp(s1,s2) | the expected agreement between solutions s1 and s2. \(Exp(s1, s2) = \sum _{j=0}^{min (J,K)}N_{j1}N_{j2}/N^{2}\) |
H(X) | partition entropy of X |
H(X) | the entropy of the random variable X associated to an algorithm community |
H(Y) | the entropy of the random variable Y associated to a ground truth community |
H(X,Y) | the joint entropy |
I(X:Y) | mutual information \(\frac {1}{2} \left [ H(X)-H(X|Y)+ H(Y)-H(Y|X)\right ]\) |
MI(X,Y) | mutual information of X and Y |
Obs(s1,s2) | the observed agreement between solutions s1 and s2. \(Obs(s1, s2) = \sum _{j=0}^{min (J,K)} A_{j}/N\) |
Redundancy | percentage of communities in Y that are matched by at least an object in X \(\frac {|X|}{|Y_{id}|}\) where Yid is the subset of communities in Y matched by community in X |