Mathematical description
Notations
- $G(V,E)$ graph with vertex set $V$ and edge set $E$, $E \subset V\times V$
- $\mathcal{N}(x)$ is the neighborhood of $v$, i.e., the set of nodes adjacent to vertex $x$, not including $x$
- $|S|$ is the cardnal of set $S$
- $A$ the adjacency matrix of $G(V,E)$
Vertex degree
\[k(v) = | \mathcal{N}(v) |, \quad \forall v \in V \]
Local cluster coefficient
\[\textrm{lcc}(v) = \sum_{ (v,u) \in E} \frac{ \textrm{cn}(v,u) }{ k(x)(k(x)-1)} = \frac{| \mathcal{N}(u) \cap \mathcal{N}(v)|}{ k(x)(k(x)-1)} \]
see \cite{Watss-xxx}
(Perron) eigenvector centrality
\[\textrm{eig-c}(v) = \textrm{to fill-in}\]
see \cite{newman-girvan-2004}
hits
\[\textrm{hits}(v) = \]
see \cite{Kleinberg-1999}
Edge degree
\[k( \ell ) = k(u) + k(v) -2, \quad \ell = (u,v) \in E\]
Common neighbors
\[\textrm{cn}(\ell) = | \mathcal{N}(u) \cap \mathcal{N}(v)|, \quad \ell = (u,v) \in E \]
It is equal to the number of triangles containing edge $\ell$.
See \cite{newman-girvan-2004}
Adamic-Adar index
\[\textrm{aa}(x,y) = \sum_{u \in \mathcal{N}(x) \cap \mathcal{N}(y) } \frac{1}{ \log( k(u) ) } \]
see \cite{adamic-adar-2023}