transitivity: Transitivity of a graph

Description

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

Usage

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: [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

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 tha

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, defines how to treat vertices with degree zero and one. If it is NaN then they local transitivity is reported as NaN and they are not included in the averaging, for the transitivity types that c

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

$C_{i}^{w} = \frac{1}{s_{i} (k_{i} - 1)} \sum_{j, h} \frac{w_{i j} + w_{i h}}{2} a_{i j} a_{i h} a_{j h}$

$s_i$ 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)

Examples

Run this code

g <- make_ring(10)
transitivity(g)
g2 <- sample_gnp(1000, 10/1000)
transitivity(g2)   # this is about 10/1000

# 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")
transitivity(gw, vids="A", type="weighted")

# 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))
all(na.omit(t1 == t2))

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