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")
)
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
’.
The graph to analyze.
The type of the transitivity to calculate. Possible values:
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.
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.
This is the same as global
.
This is the same as global
.
This is the same as local
.
The weighted transitivity as defined by A. Barrat. See details below.
The same as barrat
.
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)
.
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.
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 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.
Gabor Csardi csardi.gabor@gmail.com
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
strength
,
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.
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)
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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