Clustering coefficient

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Example clustering coefficient on an undirected graph for the shaded node i. Black edges are nodes connecting neighbors of i, and dotted red edges are for unused possible edges.

Duncan J. Watts and Steven Strogatz (1998) introduced the clustering coefficient¹ graph measure to determine whether or not a graph is a small-world network.

First, let us define a graph in terms of a set of $n$ vertices $V={v_1,v_2,...v_n}$ and a set of edges $E$, where $e_{ij}$ denotes an edge between vertices $v_i$ and $v_j$. Below we assume $v_i$, $v_j$ and $v_k$ are members of V.

We define the neighbourhood N for a vertex $v_i$ as its immediately connected neighbours as follows:

$N_i = \{v_j\} : e_{ij} \in E.$

The degree $k_i$ of a vertex is the number of vertices, $|N_i|$, in its neighbourhood $N_i$.

The clustering coefficient $C_i$ for a vertex $v_i$ is the proportion of links between the vertices within its neighbourhood divided by the number of links that could possibly exist between them. For a directed graph, $e_{ij}$ is distinct from $e_{ji}$, and therefore for each neighbourhood $N_i$ there are $k_i(k_i-1)$ links that could exist among the vertices within the neighbourhood. Thus, the clustering coefficient is given as:

$C_i = \frac{|\{e_{jk}\}|}{k_i(k_i-1)} : v_j,v_k \in N_i, e_{jk} \in E.$

An undirected graph has the property that $e_{ij}$ and $e_{ji}$ are considered identical. Therefore, if a vertex $v_i$ has $k_i$ neighbours, $\frac{k_i(k_i-1)}{2}$ edges could exist among the vertices within the neighbourhood. Thus, the clustering coefficient for undirected graphs can be defined as:

$C_i = \frac{2|\{e_{jk}\}|}{k_i(k_i-1)} : v_j,v_k \in N_i, e_{ij} \in E.$

Let $\lambda_G(v)$ be the number of triangles on $v \in V(G)$ for undirected graph $G$. That is, $\lambda_G(v)$ is the number of subgraphs of $G$ with 3 edges and 3 vertices, one of which is $v$. Let $\tau_G(v)$ be the number of triples on $v \in G$. That is, $\tau_G(v)$ is the number of subgraphs (not necessarily induced) with 2 edges and 3 vertices, one of which is $v$ and such that $v$ is incident to both edges. Then we can also define the clustering coefficient as:

$C_i = \frac{\lambda_G(v)}{\tau_G(v)}$

It is simple to show that the two preceding definitions are the same, since $\tau_G(v) = C({k_i},2) = \frac{1}{2}k_i(k_i-1)$.

These measures are 1 if every neighbour connected to $v_i$ is also connected to every other vertex within the neighbourhood, and 0 if no vertex that is connected to $v_i$ connects to any other vertex that is connected to $v_i$.

The clustering coefficient for the whole system is given by Watts and Strogatz as the average of the clustering coefficient for each vertex:

$\overline{C} = \frac{1}{n}\sum_{i=1}^{n} C_i.$

References

¹ Watts, D. J. and Strogatz, S. H. "Collective dynamics of 'small-world' networks." [1]

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