1% Generated by roxygen2: do not edit by hand 2% Please edit documentation in R/structural.properties.R 3\name{transitivity} 4\alias{transitivity} 5\title{Transitivity of a graph} 6\usage{ 7transitivity( 8 graph, 9 type = c("undirected", "global", "globalundirected", "localundirected", "local", 10 "average", "localaverage", "localaverageundirected", "barrat", "weighted"), 11 vids = NULL, 12 weights = NULL, 13 isolates = c("NaN", "zero") 14) 15} 16\arguments{ 17\item{graph}{The graph to analyze.} 18 19\item{type}{The type of the transitivity to calculate. Possible values: 20\describe{ \item{"global"}{The global transitivity of an undirected 21graph (directed graphs are considered as undirected ones as well). This is 22simply the ratio of the triangles and the connected triples in the graph. 23For directed graphs the direction of the edges is ignored.} 24\item{"local"}{The local transitivity of an undirected graph, this is 25calculated for each vertex given in the \code{vids} argument. The local 26transitivity of a vertex is the ratio of the triangles connected to the 27vertex and the triples centered on the vertex. For directed graph the 28direction of the edges is ignored. Note that calculations do not work 29reliably in non-simple graphs, and since the presence of mutual edges in 30a directed graph becomes non-simple when treated as undirected, therefore 31it is advised to avoid using this function on non-simple and/or directed 32graphs. igraph prints a warning in these cases; the warning will be turned 33into an error in igraph 1.3.0.} \item{"undirected"}{This is the 34same as \code{global}.} \item{"globalundirected"}{This is the same as 35\code{global}.} \item{"localundirected"}{This is the same as 36\code{local}.} \item{"barrat"}{The weighted transitivity as defined A. 37Barrat. See details below.} \item{"weighted"}{The same as 38\code{barrat}.} }} 39 40\item{vids}{The vertex ids for the local transitivity will be calculated. 41This will be ignored for global transitivity types. The default value is 42\code{NULL}, in this case all vertices are considered. It is slightly faster 43to supply \code{NULL} here than \code{V(graph)}.} 44 45\item{weights}{Optional weights for weighted transitivity. It is ignored for 46other transitivity measures. If it is \code{NULL} (the default) and the 47graph has a \code{weight} edge attribute, then it is used automatically.} 48 49\item{isolates}{Character scalar, defines how to treat vertices with degree 50zero and one. If it is \sQuote{\code{NaN}} then they local transitivity is 51reported as \code{NaN} and they are not included in the averaging, for the 52transitivity types that calculate an average. If there are no vertices with 53degree two or higher, then the averaging will still result \code{NaN}. If it 54is \sQuote{\code{zero}}, then we report 0 transitivity for them, and they 55are included in the averaging, if an average is calculated.} 56} 57\value{ 58For \sQuote{\code{global}} a single number, or \code{NaN} if there 59are no connected triples in the graph. 60 61For \sQuote{\code{local}} a vector of transitivity scores, one for each 62vertex in \sQuote{\code{vids}}. 63} 64\description{ 65Transitivity measures the probability that the adjacent vertices of a vertex 66are connected. This is sometimes also called the clustering coefficient. 67} 68\details{ 69Note that there are essentially two classes of transitivity measures, one is 70a vertex-level, the other a graph level property. 71 72There are several generalizations of transitivity to weighted graphs, here 73we use the definition by A. Barrat, this is a local vertex-level quantity, 74its formula is 75 76\deqn{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}}{ 77weighted C_i = 1/s_i 1/(k_i-1) sum( (w_ij+w_ih)/2 a_ij a_ih a_jh, j, h)} 78 79\eqn{s_i}{s_i} is the strength of vertex \eqn{i}{i}, see 80\code{\link{strength}}, \eqn{a_{ij}}{a_ij} are elements of the 81adjacency matrix, \eqn{k_i}{k_i} is the vertex degree, \eqn{w_{ij}}{w_ij} 82are the weights. 83 84This formula gives back the normal not-weighted local transitivity if all 85the edge weights are the same. 86 87The \code{barrat} type of transitivity does not work for graphs with 88multiple and/or loop edges. If you want to calculate it for a directed 89graph, call \code{\link{as.undirected}} with the \code{collapse} mode first. 90} 91\examples{ 92 93g <- make_ring(10) 94transitivity(g) 95g2 <- sample_gnp(1000, 10/1000) 96transitivity(g2) # this is about 10/1000 97 98# Weighted version, the figure from the Barrat paper 99gw <- graph_from_literal(A-B:C:D:E, B-C:D, C-D) 100E(gw)$weight <- 1 101E(gw)[ V(gw)[name == "A"] \%--\% V(gw)[name == "E" ] ]$weight <- 5 102transitivity(gw, vids="A", type="local") 103transitivity(gw, vids="A", type="weighted") 104 105# Weighted reduces to "local" if weights are the same 106gw2 <- sample_gnp(1000, 10/1000) 107E(gw2)$weight <- 1 108t1 <- transitivity(gw2, type="local") 109t2 <- transitivity(gw2, type="weighted") 110all(is.na(t1) == is.na(t2)) 111all(na.omit(t1 == t2)) 112 113} 114\references{ 115Wasserman, S., and Faust, K. (1994). \emph{Social Network 116Analysis: Methods and Applications.} Cambridge: Cambridge University Press. 117 118Alain Barrat, Marc Barthelemy, Romualdo Pastor-Satorras, Alessandro 119Vespignani: The architecture of complex weighted networks, Proc. Natl. Acad. 120Sci. USA 101, 3747 (2004) 121} 122\author{ 123Gabor Csardi \email{csardi.gabor@gmail.com} 124} 125\keyword{graphs} 126