Transitivity of a graph

Transitivity measures the probability that the adjacent vertices of a vertex are connected. This is sometimes also called the clustering coefficient.

transitivity(
  graph,
  type = c("undirected", "global", "globalundirected", "localundirected", "local",
    "average", "localaverage", "localaverageundirected", "barrat", "weighted"),
  vids = NULL,
  weights = NULL,
  isolates = c("NaN", "zero")
)

Arguments

graph

The graph to analyze.

type

The type of the transitivity to calculate. Possible values:

"global"

The global transitivity of an undirected graph. This is simply the ratio of the count of triangles and connected triples in the graph. In directed graphs, edge directions are ignored.

"local"

The local transitivity of an undirected graph. It is calculated for each vertex given in the vids argument. The local transitivity of a vertex is the ratio of the count of triangles connected to the vertex and the triples centered on the vertex. In directed graphs, edge directions are ignored.

"undirected"

This is the same as global.

"globalundirected"

This is the same as global.

"localundirected"

This is the same as local.

"barrat"

The weighted transitivity as defined by A. Barrat. See details below.

"weighted"

The same as barrat.

vids

The vertex ids for the local transitivity will be calculated. This will be ignored for global transitivity types. The default value is NULL, in this case all vertices are considered. It is slightly faster to supply NULL here than V(graph).

weights

Optional weights for weighted transitivity. It is ignored for other transitivity measures. If it is NULL (the default) and the graph has a weight edge attribute, then it is used automatically.

isolates

Character scalar, for local versions of transitivity, it defines how to treat vertices with degree zero and one. If it is ‘NaN’ then their local transitivity is reported as NaN and they are not included in the averaging, for the transitivity types that calculate an average. If there are no vertices with degree two or higher, then the averaging will still result NaN. If it is ‘zero’, then we report 0 transitivity for them, and they are included in the averaging, if an average is calculated. For the global transitivity, it controls how to handle graphs with no connected triplets: NaN or zero will be returned according to the respective setting.

Value

For ‘global’ a single number, or NaN if there are no connected triples in the graph.

For ‘local’ a vector of transitivity scores, one for each vertex in ‘vids’.

Details

Note that there are essentially two classes of transitivity measures, one is a vertex-level, the other a graph level property.

There are several generalizations of transitivity to weighted graphs, here we use the definition by A. Barrat, this is a local vertex-level quantity, its formula is

$$C_i^w=\frac{1}{s_i(k_i-1)}\sum_{j,h}\frac{w_{ij}+w_{ih}}{2}a_{ij}a_{ih}a_{jh}$$

\(s_i\) is the strength of vertex \(i\), see strength(), \(a_{ij}\) are elements of the adjacency matrix, \(k_i\) is the vertex degree, \(w_{ij}\) are the weights.

This formula gives back the normal not-weighted local transitivity if all the edge weights are the same.

The barrat type of transitivity does not work for graphs with multiple and/or loop edges. If you want to calculate it for a directed graph, call as_undirected() with the collapse mode first.

References

Wasserman, S., and Faust, K. (1994). Social Network Analysis: Methods and Applications. Cambridge: Cambridge University Press.

Alain Barrat, Marc Barthelemy, Romualdo Pastor-Satorras, Alessandro Vespignani: The architecture of complex weighted networks, Proc. Natl. Acad. Sci. USA 101, 3747 (2004)

See also

Other structural.properties: bfs(), component_distribution(), connect(), constraint(), coreness(), degree(), dfs(), distance_table(), edge_density(), feedback_arc_set(), girth(), is_acyclic(), is_dag(), is_matching(), k_shortest_paths(), knn(), reciprocity(), subcomponent(), subgraph(), topo_sort(), unfold_tree(), which_multiple(), which_mutual()

Author

Gabor Csardi csardi.gabor@gmail.com

Examples

g <- make_ring(10)
transitivity(g)
#> [1] 0
g2 <- sample_gnp(1000, 10 / 1000)
transitivity(g2) # this is about 10/1000
#> [1] 0.01046946

# Weighted version, the figure from the Barrat paper
gw <- graph_from_literal(A - B:C:D:E, B - C:D, C - D)
E(gw)$weight <- 1
E(gw)[V(gw)[name == "A"] %--% V(gw)[name == "E"]]$weight <- 5
transitivity(gw, vids = "A", type = "local")
#>   A 
#> 0.5 
transitivity(gw, vids = "A", type = "weighted")
#>    A 
#> 0.25 

# Weighted reduces to "local" if weights are the same
gw2 <- sample_gnp(1000, 10 / 1000)
E(gw2)$weight <- 1
t1 <- transitivity(gw2, type = "local")
t2 <- transitivity(gw2, type = "weighted")
all(is.na(t1) == is.na(t2))
#> [1] TRUE
all(na.omit(t1 == t2))
#> [1] FALSE