Skip to main content

Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

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

SymbolDescription
Cpartition of the graph
Cicommunity partition C
bCnumber of edges on the community boundary
dminternal degree median value
eCnumber of community internal edges in C
kthe total degree
\(k^{int}_{iC}\)the degree of node i within C
\(k^{out}_{iC}\)the degree of node i outside C
lCnumber of edges from nodes in C to nodes outside C
Minttotal possible internal edges: \(\sum _{C} \binom {n_{C}}{2}\)
NcTotal number of communities of all sizes detected by a given algorithm, \(\sum _{i} n_{c} x_{c}^{i}\)
nCnumber of community nodes in C
Γ(i)degree of the node i
pdensity of the graph
pCthe density of community C. \(p_{C}=m_{C}/ \binom {n_{C}}{2} \)
〈q〉the expected fraction of internal edges
xcnumber of communities having the same number of nodes of c
Coveragepercentage 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
Dcthe 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. \(Obs(s1, s2) = \sum _{j=0}^{min (J,K)} A_{j}/N\)
Redundancypercentage 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