Local Clustering Coefficient
Essential knowledge:

The Local Clustering Coefficient tells us how connected the network is around a particular node. The clustering coefficient is a fraction, representing the number of connections that exist as a proportion of the number that could exist.

So for example, if I have three friends (Derek, Ed, and Freya), then those friends have three potential connections (Derek  Ed, Ed  Freya, Freya  Derek). If only one of those potential connections actually exists (for example if Derek is friends with Ed) then one out of three connections exist, giving me a local clustering coefficient of 1/3.
You may be interested that:
The local clustering coefficient is a measure introduced by Watts and Strogatz in 1998 in their work to identify small world networks. It is calculated for each node in the network to examine the existing connections between its neighbouring nodes. In other words, it checks the existing connections between the neighbours of a given node to see whether they form a clique around that node.
Let’s look at the following example to illustrate the clustering coefficient.
Lets calculate the local clustering coefficient for the node C:

C’s neighbours are A, B, D and E.

There is only one existing connection between C’s neighbours, and that is AB.

There are six possible connections between its neighbours: AD, AE, BD, BE, DE and the one that is already there which is AB.
So the clustering coefficient for C is 1/6
The clustering coefficent is essentially a measure of how densely connected the network is around a particular node. So for example in a social network a person with a high clustering coefficient is one whose friends tend to be friends with one another, forming a clique.
Thinking about your own social networks, are there ones in which your friends tend not to know one another, or ones in which they do (in other words where you have a particularly high or low clustering coefficient)?
If you’re really curious:
The definition of the local clustering coefficient is different for directed and undirected networks. In an undirected network, a relationship between node A and node B is the same as the relationship between node B and node A (for example, a friendship relationship in a social network). In a directed network, the relationship between node A and node B can be different than the relationship between node B and node A (for example, a follower relationship in a social network).
In an undirected network: The local clustering coefficient for a node i is defined as:
Where n_{i} is the number of neighbours for the node i. So in our example above when calculating the local clustering coefficient for node C we get:
In a directed network: The local clustering coefficient for a node i is defined as:
And to calculate the average clustering coefficient in a network we can just calculate the clustering coefficient for each node and then take an average, for example with n nodes:
Knowing the average clustering coefficients allows us to compare different networks in terms of how densely their nodes are connected.
Optional further reading
 Watts, DJ and Strogatz, SH (1998) ‘Collective dynamics of `smallworld’ networks’, Nature, vol. 393, no. 6684, pp. 440–442, Jun
© University of Southampton 2016