# All-subgraphs Centrality

#### Definition

The main principle of this approach is very simple: the more relevant subgraphs around a vertex, the more central it is in the network. The notion of “relevant subgraphs” formalized by choosing a family of subgraphs that, give a graph $G$ and a vertex $v$ in $G$, which assigns a subset of connected subgraphs of $G$ that contains $v$. Any of such families defines a measure of centrality by counting the number of subgraphs assigned to the vertex. This centrality measure takes every subgraph into account.

The all-subgraphs centrality of $v$ in $G$ is defined as:

$$C_A(v,G):=log (|A(v,G)|)$$

Considering a graph $G$ and a vertex $v \in V(G)$. $A(v,G)$ denotes by the set of all connected subgraphs of $G$ that contains $v$, formally, $A(v) = {G'\subseteq G|G' is connected$\wedge v\in V(G')}$. The function$C_A$naturally induces a ranking between nodes: the higher the centrality$C_A(v,G)$, the more important is$v$in$G\$.

#### References

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