Table 4 Symbols used for calculating permanence in Eq. 1

Perm(v)

Permanence of vertex v

I(v)

Internal neighbors of vertex v

\(E_{max}(v)\)

Maximum connections to a single external community

\(C_{in}\)

Internal clustering coefficient

\(C_{in}=\frac{x}{C(I(v),2)}\)

\(x=\) existing connections among the internal neighbors of v and C(I(v), 2) means the total number of possible connections among the internal neighbors of v

\(Perm(v)=1\)

v strongly connected to assigned community

\(Perm(v)=0\)

v equally pulled by all neighbors (singleton community)

\(Perm(v)=-1\)

v weakly connected to assigned community (wrong community)