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

Symbol | Description |
---|---|

C | partition of the graph |

C_{i} | community ∈ partition C |

b_{C} | number of edges on the community boundary |

d_{m} | internal degree median value |

e_{C} | 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 |

l_{C} | number of edges from nodes in C to nodes outside C |

M_{int} | total possible internal edges: \(\sum _{C} \binom {n_{C}}{2}\) |

N_{c} | Total number of communities of all sizes detected by a given algorithm, \(\sum _{i} n_{c} x_{c}^{i}\) |

n_{C} | number of community nodes in C |

Γ(i) | degree of the node i |

p | density of the graph |

p_{C} | the density of community C. \(p_{C}=m_{C}/ \binom {n_{C}}{2} \) |

〈q〉 | the expected fraction of internal edges |

x_{c} | 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 Y_{id} is the subset of communities in Y matched by community in X |

D(x||y) | KL divergence |

D_{c} | the sum of the degrees of the vertices in community C |

δ(c_{i},c_{j}) | 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 Y_{id} is the subset of communities in Y matched by community in X |