Average clustering coefficient of a network (igraph)

Question

I want to calculate the average clustering coefficient of a graph (from igraph package). However, I am not sure which approach I should follow.

library(igraph)
graph <- erdos.renyi.game(10000, 10000, type = "gnm")

# Global clustering coefficient
transitivity(graph)
# Average clustering coefficient
transitivity(graph, type = "average")
# The same as above
mean(transitivity(graph, type = "local"), na.rm = TRUE)

I would be grateful for some guidance.

Solution 1

Using transitivity(graph) computes a global clustering coefficient (transitivity):

This is simply the ratio of the triangles and the connected triples in the graph. For directed graph the direction of the edges is ignored.

Meanwhile, transitivity(graph, type = "average") being an average of transitivity(graph, type = "local") first computes the local clustering coefficients and then averages them:

The local transitivity of an undirected graph, this is calculated for each vertex given in the vids argument. The local transitivity of a vertex is the ratio of the triangles connected to the vertex and the triples centered on the vertex. For directed graph the direction of the edges is ignored.

See, e.g., ?transitivity and Clustering coefficient.

So firstly both of them are valid measures and the choice should depend on your purposes. The difference between them is quite clear (see the wikipedia page):

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. In fact, a weighted average where each local clustering score is weighted by k_i(k_i-1) is identical to the global clustering coefficient

where k_i is the number of vertex i neighbours. Hence, perhaps using both of them would be quite fair too.

Solution 2

@Julius Vainora answered it. Additional note for those that are trying to look for some answers for what does type = "average" do since igraph documentation doesn't say anything about it:

transitivity(graph, type = "average")

is the same as

transitivity(graph, type = "localaverage")

is the same as

transitivity(graph, type = "localaverageundirected")

is the same as

mean(transitivity(graph, type = "local"), na.rm = TRUE)

Solution 3

transitivity(g, type="local") 

How about the order of the output vector? Is it same as the order of:

degree(g, mode="all")

The degree vector has indexs but clustering coefficient not