transitivity {igraph}  R Documentation 
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"))
graph 
The graph to analyze. 
type 
The type of the transitivity to calculate. Possible values:

vids 
The vertex ids for the local transitivity will be
calculated. This will be ignored for global transitivity types.
The default value is 
weights 
Optional weights for weighted transitivity. It is
ignored for other transitivity measures. If it is 
isolates 
Character scalar, defines how to treat vertices with
degree zero and one. If it is ‘ 
Note that there are essentially two classes of transitivity measures, one is a vertexlevel, 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 vertexlevel quantity, its formula is
weighted C_i = 1/s_i 1/(k_i1) sum( (w_ij+w_ih)/2 a_ij a_ih a_jh, j, h)
s_i is the strength of vertex i, see
graph.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 notweighted 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.
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
’.
Gabor Csardi csardi.gabor@gmail.com
Wasserman, S., and Faust, K. (1994). Social Network Analysis: Methods and Applications. Cambridge: Cambridge University Press.
Alain Barrat, Marc Barthelemy, Romualdo PastorSatorras, Alessandro Vespignani: The architecture of complex weighted networks, Proc. Natl. Acad. Sci. USA 101, 3747 (2004)
g < graph.ring(10) transitivity(g) g2 < erdos.renyi.game(1000, 10/1000) transitivity(g2) # this is about 10/1000 # Weighted version, the figure from the Barrat paper gw < graph.formula(AB:C:D:E, BC:D, CD) 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 < erdos.renyi.game(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))