Clustering coefficient

In graph theory, a clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. Evidence suggests that in most real-world networks, and in particular social networks, nodes tend to create tightly knit groups characterised by a relatively high density of ties; this likelihood tends to be greater than the average probability of a tie randomly established between two nodes (Holland and Leinhardt, 1971; Watts and Strogatz, 1998).
Two versions of this measure exist: the global and the local. The global version was designed to give an overall indication of the clustering in the network, whereas the local gives an indication of the embeddedness of single nodes.
Local clustering coefficient.
 The local clustering coefficient of a vertex (node) in a graph quantifies how close its neighbours are to being a clique (complete graph). Duncan J. Watts and Steven Strogatz introduced the measure in 1998 to determine whether a graph is a small-world network.
A graph formula_1 formally consists of a set of vertices formula_2 and a set of edges formula_3 between them. An edge formula_4 connects vertex formula_5 with vertex formula_6.
The neighbourhood formula_7 for a vertex formula_5 is defined as its immediately connected neighbours as follows:
We define formula_10 as the number of vertices, formula_11, in the neighbourhood, formula_12, of a vertex.
The local clustering coefficient formula_13 for a vertex formula_5 is then given by a proportion of the number of links between the vertices within its neighbourhood divided by the number of links that could possibly exist between them. For a directed graph, formula_4 is distinct from formula_16, and therefore for each neighbourhood formula_12 there are formula_18 links that could exist among the vertices within the neighbourhood (formula_10 is the number of neighbours of a vertex). Thus, the local clustering coefficient for directed graphs is given as 
An undirected graph has the property that formula_4 and formula_16 are considered identical. Therefore, if a vertex formula_5 has formula_10 neighbours, formula_25 edges could exist among the vertices within the neighbourhood. Thus, the local clustering coefficient for undirected graphs can be defined as
Let formula_27 be the number of triangles on formula_28 for undirected graph formula_29. That is, formula_27 is the number of subgraphs of formula_29 with 3 edges and 3 vertices, one of which is formula_32. Let formula_33 be the number of triples on formula_34. That is, formula_33 is the number of subgraphs (not necessarily induced) with 2 edges and 3 vertices, one of which is formula_32 and such that formula_32 is incident to both edges. Then we can also define the clustering coefficient as
It is simple to show that the two preceding definitions are the same, since
These measures are 1 if every neighbour connected to formula_5 is also connected to every other vertex within the neighbourhood, and 0 if no vertex that is connected to formula_5 connects to any other vertex that is connected to formula_5.
Since any graph is fully specified by its adjacency matrix "A", the local clustering coefficient for a simple undirected graph can be expressed in terms of "A" as:
where:
and "Ci"=0 when "ki" is zero or one. In the above expression, the numerator counts twice the number of complete triangles that vertex "i" is involved in. In the denominator, "ki2" counts the number of edge pairs that vertex "i" is involved in plus the number of single edges traversed twice. "ki" is the number of edges connected to vertex i, and subtracting "ki" then removes the latter, leaving only a set of edge pairs that could conceivably be connected into triangles. For every such edge pair, there will be another edge pair which could form the same triangle, so the denominator counts twice the number of conceivable triangles that vertex "i" could be involved in.
Global clustering coefficient.
The global clustering coefficient is based on triplets of nodes. A triplet is three nodes that are connected by either two (open triplet) or three (closed triplet) undirected ties. A triangle graph therefore includes three closed triplets, one centered on each of the nodes (n.b. this means the three triplets in a triangle come from overlapping selections of nodes). The global clustering coefficient is the number of closed triplets (or 3 x triangles) over the total number of triplets (both open and closed). The first attempt to measure it was made by Luce and Perry (1949). This measure gives an indication of the clustering in the whole network (global), and can be applied to both undirected and directed networks (often called transitivity, see Wasserman and Faust, 1994, page 243).
The global clustering coefficient is defined as:
The number of closed triplets has also been referred to as 3 × triangles in the literature, so:
A generalisation to weighted networks was proposed by Opsahl and Panzarasa (2009), and a redefinition to two-mode networks (both binary and weighted) by Opsahl (2009).
Since any graph is fully specified by its adjacency matrix "A", the global clustering coefficient for an undirected graph can be expressed in terms of "A" as:
where:
and "C"=0 when the denominator is zero.
Network average clustering coefficient.
As an alternative to the global clustering coefficient, the overall level of clustering in a network is measured by Watts and Strogatz as the average of the local clustering coefficients of all the vertices formula_49 :
It is worth noting that this metric places more weight on the low degree nodes, while the transitivity ratio places more weight on the high degree nodes.
A graph is considered small-world, if the graph has a small mean-shortest path length that scales with the natural log of the number of nodes in the network,
formula_51. For example, a random graph is small-world, while a lattice is not, and scale-free graphs are often considered ultra-small world, as their mean-shortest path length scales with formula_52.
A generalisation to weighted networks was proposed by Barrat et al. (2004), and a redefinition to bipartite graphs (also called two-mode networks) by Latapy et al. (2008) and Opsahl (2009).
Alternative generalisations to weighted and directed graphs have been provided by Fagiolo (2007) and Clemente and Grassi (2018).
This formula is not, by default, defined for graphs with isolated vertices; see Kaiser (2008) and Barmpoutis et al. The networks with the largest possible average clustering coefficient are found to have a modular structure, and at the same time, they have the smallest possible average distance among the different nodes.
Percolation of clustered networks.
For studying the robustness of clustered networks a percolation approach is developed. The robustness with respect to localized attacks has been studied using localized percolation.
Wave localization in clustered complex networks has also been studied.