| Title: | Introduction to Social Network Analysis with R |
|---|---|
| Description: | Functions and datasets accompanying the workshop "Introduction to Social Network Analysis with R" on annual INSNA Sunbelt conferences. |
| Authors: | Michal Bojanowski [aut, cre] |
| Maintainer: | Michal Bojanowski <[email protected]> |
| License: | GPL-2 |
| Version: | 1.0-0 |
| Built: | 2025-02-03 04:09:19 UTC |
| Source: | https://github.com/mbojan/isnar |
Assortativity coefficient is a measure of segregation for social networks due to Mark Newman (2002).
assort(object, ...) ## S3 method for class 'table' assort(object, ...) ## S3 method for class 'igraph' assort(object, vattr, ...) ## Default S3 method: assort(object, ...)assort(object, ...) ## S3 method for class 'table' assort(object, ...) ## S3 method for class 'igraph' assort(object, vattr, ...) ## Default S3 method: assort(object, ...)
object |
R object, see available methods |
... |
other arguments to/from other methods |
vattr |
character, name of the vertex attribute for which the measure is to be calculated |
The measure evaluates the relative prevalence of within-group ties. It is based on the contact layer of the mixing matrix.
Assortativity coefficient is 1 if all ties are within-group. The minimum can be negative, but not less than -1, and depends on the relative number of ties of nodes in different groups. If the network conforms to "proportionate mixing", the coefficient is 0.
If object is a table it is treated as a mixing matrix.
Two-dimensional table is interpreted as a contact layer. Three-dimensional
table is interpreted as a full mixing matrix
cross-classyfying all dyads, in which 'g' and 'h' correspond to group
membership of ego and alter respectively. Layers y=1 and y=2 are assumed to
be non-contact and contact layers respectively. In the 3-d case only
g[,,2] is used.
If g is an object of class "igraph" the measure is
calculated for the vertex attribute specified with vattr.
For any other classes, object g are coerced to a table and the
table method is called.
Numeric value of the index.
Newman, M. J. and Girvan, M. (2002) "Mixing patterns and community structure in networks", arXiv:cond-mat/0210146v1
Newman, M. J. (2003) "Mixing patterns in networks" arXiv:cond-mat/0209450v2
Mixing matrices: mixingm
Other segregation measures: coleman,
ei, freeman,
gamix, orwg,
smi, ssi
assort(Wnet, "gender") assort(EFnet, "type") if( require(igraph, quietly = TRUE) ) { # value of 'assort' for full networks of different sizes f <- function(n) { gfull <- graph.full(n, directed=FALSE) V(gfull)$type <- rep(1:2, length=vcount(gfull)) assort(gfull, "type") } set.seed(1) x <- sort(sample(5:100, 25) * 2) y <- sapply(x, f) plot(x, y, type="o", xlab="Network size", ylab="Assortativity coefficient", main="Assortativity coef. for full networks of different sizes") }assort(Wnet, "gender") assort(EFnet, "type") if( require(igraph, quietly = TRUE) ) { # value of 'assort' for full networks of different sizes f <- function(n) { gfull <- graph.full(n, directed=FALSE) V(gfull)$type <- rep(1:2, length=vcount(gfull)) assort(gfull, "type") } set.seed(1) x <- sort(sample(5:100, 25) * 2) y <- sapply(x, f) plot(x, y, type="o", xlab="Network size", ylab="Assortativity coefficient", main="Assortativity coef. for full networks of different sizes") }
Contact layer of the mixing matrix of men and women in US based on "Aids in Multi-Ethnic Neighborhoods" (AMEN). Based on Newman (2003).
Four-by-four numeric matrix with dimnames.
woman
man black hispanic white other
black 0.258 0.016 0.035 0.013
hispanic 0.012 0.157 0.058 0.019
white 0.013 0.023 0.306 0.035
other 0.005 0.007 0.024 0.016
Newman, M. (2003) "Mixing patterns in networks" Arxiv:cond-mat/0209450 v2
Catania et al. (1992) "The population-based AMEN (AIDS in Multi-Ethnic Neighborhoods) study" American Journal of Public Health 82, 284-287
Morris, M. (1995) "Data driven network models for the spread of infectious disease". In D. Mollison (ed.) "Epidemic Models: Their Structure and Relation to Data", pp. 302-322, Cambridge University Press, Cambridge
Newman, M. (2003) "Mixing patterns in networks" Arxiv:cond-mat/0209450 v2
data(Catania) # assortativity ep <- sum(Catania %*% Catania) ( sum(diag(Catania)) - ep ) / ( 1 - ep )data(Catania) # assortativity ep <- sum(Catania %*% Catania) ( sum(diag(Catania)) - ep ) / ( 1 - ep )
Coauthorship network of scientists from University of Warsaw and their externa co-workers. Relation indicates pairs of people who has authored at least one publication together.
Object of class igraph of size 10114, undirected.
Node attributes include authors' direct affiliation and also their affiliation to faculty.
Own calculation
Colemans's homphily index for directed networks.
coleman(object, ...) ## S3 method for class 'table' coleman(object, gsizes = NULL, loops = FALSE, ...) ## S3 method for class 'igraph' coleman(object, vattr, ...) ## Default S3 method: coleman(object, ...)coleman(object, ...) ## S3 method for class 'table' coleman(object, gsizes = NULL, loops = FALSE, ...) ## S3 method for class 'igraph' coleman(object, vattr, ...) ## Default S3 method: coleman(object, ...)
object |
R object, see Details for available methods |
... |
other arguments passed to/from methods |
gsizes |
numeric vector of group sizes |
loops |
logical, whether loops are allowed |
vattr |
character, vertex attribute |
Coleman's homophily index computes homophily scores for each group defined by a vertex attribute.
If object is a table it is interpreted as a mixing matrix. If it is
only the contact layer (2-dimensional), then vector of group sizes need to
be supplied via gsizes.
object can be of class "igraph"
Default method tries to coerce object to table and use other methods.
Vector of numeric values of the index for each group
Coleman, J. (1958) "Relational analysis: The study of social organizations with survey methods", Human Organization 17:28–36.
Other segregation measures: assort,
ei, freeman,
gamix, orwg,
smi, ssi
if( require(igraph, quietly = TRUE)) { coleman(as.directed(Wnet, "mutual"), "gender") coleman(as.directed(EFnet, "mutual"), "type") }if( require(igraph, quietly = TRUE)) { coleman(as.directed(Wnet, "mutual"), "gender") coleman(as.directed(EFnet, "mutual"), "type") }
Subset of Polish General Social Survey data with data on incomes
SPSS system file
Polish Social Science Data Archive (ADS)
PGSS reference
Artificial example data from Echenique & Fryer (2006) representing a city with black and white neighbourhoods.
Object of class "igraph". An undirected network with vertex attributes
type (values 1 or 2).
name
This data is taken from Echenique & Fryer (2006, figure III). The data represent a fictional city composed of 30 neighborhoods that are either black or white.
Echenique, Federico and Roland G. Fryer, Jr. (2006) "A Measure of Segregation Based On Social Interactions" Quarterly Journal of Economics CXXII(2):441-485
if( require(igraph, quietly = TRUE) ) { set.seed(2992) plot(EFnet, layout=layout.fruchterman.reingold, vertex.color=V(EFnet)$type+1, vertex.label.family="", sub="Source: Echenique & Fryer (2006)", main="Neighborhood racial segregation\n in a fictional city" ) }if( require(igraph, quietly = TRUE) ) { set.seed(2992) plot(EFnet, layout=layout.fruchterman.reingold, vertex.color=V(EFnet)$type+1, vertex.label.family="", sub="Source: Echenique & Fryer (2006)", main="Neighborhood racial segregation\n in a fictional city" ) }
An index proposed by Krackhard and Stern (1988) to capture relative prevalence of between- and within-group ties. From that perspective it can be interpreted as a measure of network segregation.
ei(object, ...) ## S3 method for class 'table' ei(object, ...) ## S3 method for class 'igraph' ei(object, vattr, directed = is.directed(object), loops = any(is.loop(object)), ...) ## Default S3 method: ei(object, ...)ei(object, ...) ## S3 method for class 'table' ei(object, ...) ## S3 method for class 'igraph' ei(object, vattr, directed = is.directed(object), loops = any(is.loop(object)), ...) ## Default S3 method: ei(object, ...)
object |
R object, see Details for available methods |
... |
other arguments passed to/from other methods |
vattr |
character scalar or vector of length equal to the size of
|
directed |
logical whether the network is directed |
loops |
logical, whether loops are allowed |
If object is a table, it is assumed to be a mixing matrix.
Method for igraphs
Default method tries to coerce object to table.
Numerical value of the E-I index.
Other segregation measures: assort,
coleman, freeman,
gamix, orwg,
smi, ssi
ei(Wnet, "gender")ei(Wnet, "gender")
Fold a square matrix by collapsing lower triangle on upper triangle, or vice versa, through addition.
fold(x, direction = c("upper", "lower"))fold(x, direction = c("upper", "lower"))
x |
square numeric matrix |
direction |
character, one of |
By default, for direction=="upper", the function takes the values in
the lower triangle of x and adds them symetrically to the values in
the upper triangle. The values on the diagonal remain unchanged. The lower
triangle is filled with 0s. If direction=="lower" the upper triangle
is collapsed on to the lower triangle.
Square matrix of the same dim as x with the lower
(upper) triangle folded onto the upper (lower) triangle.
(m <- matrix(1:4, 2, 2)) (f1 <- fold(m)) (f2 <- fold(m, "lower")) stopifnot( all.equal(diag(m), diag(f1)) ) stopifnot( all.equal(diag(m), diag(f2)) ) stopifnot( all.equal(f1[1,2], m[2,1] + m[1,2]) ) stopifnot( all.equal(f2[2,1], m[2,1] + m[1,2]) )(m <- matrix(1:4, 2, 2)) (f1 <- fold(m)) (f2 <- fold(m, "lower")) stopifnot( all.equal(diag(m), diag(f1)) ) stopifnot( all.equal(diag(m), diag(f2)) ) stopifnot( all.equal(f1[1,2], m[2,1] + m[1,2]) ) stopifnot( all.equal(f2[2,1], m[2,1] + m[1,2]) )
Calculate Freeman's segregation index for undirected netoworks with arbitrary number of groups.
freeman(object, ...) ## S3 method for class 'table' freeman(object, gsizes = NULL, more = FALSE, loops = FALSE, ...) ## S3 method for class 'igraph' freeman(object, vattr, gsizes = NULL, loops = any(is.loop(object)), ...) ## Default S3 method: freeman(object, ...)freeman(object, ...) ## S3 method for class 'table' freeman(object, gsizes = NULL, more = FALSE, loops = FALSE, ...) ## S3 method for class 'igraph' freeman(object, vattr, gsizes = NULL, loops = any(is.loop(object)), ...) ## Default S3 method: freeman(object, ...)
object |
R object, see Details for available methods |
... |
other arguments passed to/from other methods |
gsizes |
numeric, optional true distribution of types, see Details |
more |
logical, should some more output be returned |
loops |
logical, whether loops are allowed |
vattr |
character scalar or any vector of length equal to
|
Freeman's segregation index (Freeman, 1978) is designed to capture the extent to which the defined groups of vertices tend to have more edges with vertices from the same group than with other groups. Formally, the index compares the observed number of between-group ties with the number of between-group ties that would be expected if ties would be created randomly.
The index has some discontinuity as there are network and group configurations that are characterized by the higher number of between-group ties that is expected under a random graph. The index is truncates these situations and takes a value of 0.
The original Freeman's formulation was for only two types of vertices. Here it is extended to the arbitrary number of types. The modification only affects the way in which the expected number of inter-type edges under pure random graph is calculated.
The function internally calculates the frequency of types of vertices in the
supplied attributer vattr. However, it is possible to override this
by specifying “true” type distribution with the dis argument. It is
assumed to be a table (as returned by table) or a numeric vector with
frequencies of types of vertices. This may be especially usefull when
dealing with large graphs with larger number of isolates.
Method for mixing matrices
Method for "igraph"s
The value of the Freeman's index.
If more is TRUE, some intermediate results are returned in a
list.
Freeman, Linton C. (1978) Segregation in Social Networks, Sociological Methods & Research 6(4):411–429
Other segregation measures: assort,
coleman, ei,
gamix, orwg,
smi, ssi
## White's data from Freeman's article # segregation level freeman(Wnet, "gender") # using 'more' argument freeman(Wnet, "gender", more=TRUE)## White's data from Freeman's article # segregation level freeman(Wnet, "gender") # using 'more' argument freeman(Wnet, "gender", more=TRUE)
Given contact layer of the mixing matrix compute a full 3-dimensional mixing matrix by rebuilding the non-contact layer from supplied group sizes and other arguments.
full_mm(cl, gsizes, directed = TRUE, loops = FALSE)full_mm(cl, gsizes, directed = TRUE, loops = FALSE)
cl |
numeric matrix with contact layer of the mixing matrix |
gsizes |
numeric vector or matrix with group sizes, see Details. |
directed |
logical, whether the network is directed |
loops |
logical, whether loops (self-ties) are allowed Contact layer of the mixing matrix is a cross-classification of ties
according to the attributes of tie sender (ego) and tie receiver (alter).
Classically, the same attribute is used for ego and alter resulting in a
square mixing matrix. In such cases In more general case we can use different node attributes for ego and
alter. Then |
full_mm returns a full three-dimenstional mixing matrix as an array
with dim attribute equal to c( nrow(cl), ncol(cl), 2 ).
### Square example # Contact layer of the mixing matrix mm1 <- matrix( c( 20, 10, 5, 12, 30, 10, 3, 11, 25 ), byrow=TRUE, ncol=3, nrow=3) dimnames(mm1) <- list(ego=letters[1:3], alter=letters[1:3]) mm1 # Assuming some group sizes gs1 <- c(a=9, b=12, c=10) # Full mixing matrix full_mm( mm1, gs1) ### Non-square example # Mixing matrix # Now using different attributes for ego and alter mm2 <- cbind(mm1, c(20, 10, 5)) colnames(mm2) <- LETTERS[1:4] names(dimnames(mm2)) <- c("ego", "alter") mm2 # Create artificial distribution of attributes set.seed(123) a1 <- sample(letters[1:3], sum(gs1), replace=TRUE, prob=gs1/sum(gs1)) table(a1) a2 <- sample(LETTERS[1:4], sum(gs1), replace=TRUE) table(a2) (x <- table(a1, a2)) # Cross-tablulation # Full mixing matrix full_mm( mm2, gsizes=x)### Square example # Contact layer of the mixing matrix mm1 <- matrix( c( 20, 10, 5, 12, 30, 10, 3, 11, 25 ), byrow=TRUE, ncol=3, nrow=3) dimnames(mm1) <- list(ego=letters[1:3], alter=letters[1:3]) mm1 # Assuming some group sizes gs1 <- c(a=9, b=12, c=10) # Full mixing matrix full_mm( mm1, gs1) ### Non-square example # Mixing matrix # Now using different attributes for ego and alter mm2 <- cbind(mm1, c(20, 10, 5)) colnames(mm2) <- LETTERS[1:4] names(dimnames(mm2)) <- c("ego", "alter") mm2 # Create artificial distribution of attributes set.seed(123) a1 <- sample(letters[1:3], sum(gs1), replace=TRUE, prob=gs1/sum(gs1)) table(a1) a2 <- sample(LETTERS[1:4], sum(gs1), replace=TRUE) table(a2) (x <- table(a1, a2)) # Cross-tablulation # Full mixing matrix full_mm( mm2, gsizes=x)
Data from Columbia University Drug Study on diffusion of medical innovation in the form of a new drug (gammanym). Its a network of 31 medical doctors connected with friendship or discussion links. Vertex attribute "adoption" specifies time of first prescription of the new drug.
Directed network (class 'igraph') with edge attributes:
logical, discussion link
logical, friendship nomination
and vertex attributes:
numeric, time of adoption
Pajek datasets http://vlado.fmf.uni-lj.si/pub/networks/data/esna/Galesburg2.htm
Coleman, J.S., E. Katz, H. Menzel (1966) "Medical Innovation. A Diffusion Study", Indianapolis: Bobbs-Merrill
if(require(igraph, quietly = TRUE)) { plot(Galesburg2, edge.arrow.size=.3) }if(require(igraph, quietly = TRUE)) { plot(Galesburg2, edge.arrow.size=.3) }
Measure of within-group mixing in networks proposed in Gupta, Anderson and May (1989).
gamix(object, ...) ## S3 method for class 'table' gamix(object, debug = FALSE, ...) ## S3 method for class 'igraph' gamix(object, vattr, ...) ## Default S3 method: gamix(object, ...)gamix(object, ...) ## S3 method for class 'table' gamix(object, debug = FALSE, ...) ## S3 method for class 'igraph' gamix(object, vattr, ...) ## Default S3 method: gamix(object, ...)
object |
R object, see Details for available methods |
... |
other objects passed to/from other methods |
debug |
logical, return some intermediate results as attributes to the returned value |
vattr |
character, name of vertex attribute |
The measure varies between -1/vcount(g) for dissassortative mixing
and 1 for perfect within-group mixing. It takes a value of 0 for
proportionate mixing.
Method for mixing matrices
Method for igraphs
Numerical value of the measure.
Gupta, S., Anderson, R., May, R. (1989) "Networks of sexual contacts: implications for the pattern of spread of HIV", AIDS 3:807–817
Other segregation measures: assort,
coleman, ei,
freeman, orwg,
smi, ssi
gamix(Wnet, "gender") gamix(EFnet, "type")gamix(Wnet, "gender") gamix(EFnet, "type")
Subset of Polish General Social Survey data with response to questions on gender roles
Plain text file with header, space-sparated values, and the following columns
Polish Social Science Data Archive (ADS)
PGSS reference
group_sizes recomputes group sizes from a full mixing matrix. This is only
limited to square (single-attribute) mixing matrices.Computing group sizes from square mixing matrices
group_sizes recomputes group sizes from a full mixing matrix. This is only
limited to square (single-attribute) mixing matrices.
group_sizes(mm, directed = TRUE, loops = FALSE)group_sizes(mm, directed = TRUE, loops = FALSE)
mm |
numeric array with |
directed |
logical, whether network is directed |
loops |
logical, whether loops are allowed |
A numeric vector of group sizes
# For White's data mm <- mixingm( Wnet, "gender", full=TRUE )# For White's data mm <- mixingm( Wnet, "gender", full=TRUE )
Classroom network (directed multigraph) based on a study by (Polish) Educational Research Institute. Relations come from sociometric questions (loose translation from Polish):
With whom would you like to play with?
With whom would you share a secret?
Imagine you are to work in groups during class. With whom would you like to work in a group?
With whom would you rather not play?
Node attributes include gender, IQ score (Raven's test), socioeconomic position of parents.
Object of class igraph with a directed multigraph (possible multiple edges within the same pair of vertices) of size 26. Edge attributes:
gender
Educational Research Institute
Educational Research Institute
Based on data from TODO
22-by-22 numeric matrix with dimnames.
TODO
TODO: Bojanowski () "Industrial structure and interfirm collaboration".
Functions and data accompanying my workshop "Introduction to Social Network Analysis with R".
This package is about analyzing social networks. Social Network Analysis (SNA) uses mathematical graph theory to represent its ideas. Nevertheless, for, among other things, historical reasons SNA developed its own terminology of networks that functions somewhat in parallel to graph-theoretical terms. To clarify, these are the synonyms for strict mathematical terms corresponding to graphs:
a collection of vertices and lines connecting them. Sometimes called "network"
sometimes called "node"
sometimes called a (directed/undirected) "tie". Edge starting point can be called a "tie sender" or "ego". Edge endpoint can be called "tie receiver" or "alter"
Network of judges from one of the Polish regional courts. Relation indicates
which judges have ruled in at least one case together. This network is
a projection from bipartite network judge_net_bp.
Object of class igraph of size 40, undirected, with predefined layout.
Node attributes include gender and code of division.
Own calculation based on SAOS
Two-mode network with classes representing judges and judgments from one of the Polish regional courts. Relation indicates which judges were involved in each case.
Object of class igraph of size 1189, undirected, bipartite.
Node attributes include judges' gender and code of division. Attribute 'type'
indicates classes of nodes, in accordance with igraph representation of
bipartite networks, TRUE for judges and FALSE for judgments.
Own calculation based on SAOS
Creating network mixing matrices.
mixingm(object, ...) ## S3 method for class 'igraph' mixingm(object, rattr, cattr = rattr, full = FALSE, directed = is.directed(object), loops = any(is.loop(object)), ...)mixingm(object, ...) ## S3 method for class 'igraph' mixingm(object, rattr, cattr = rattr, full = FALSE, directed = is.directed(object), loops = any(is.loop(object)), ...)
object |
R object, see Details for available methods |
... |
other arguments passed to/from other methods |
rattr |
name of the vertex attribute or an attribute itself as a
vector. If |
cattr |
name of the vertex attribute or an attribute itself as a vector. If supplied, used for columns in the mixing matrix. |
full |
logical, whether two- or three-dimensional mixing matrix should be returned. |
directed |
logical, whether the network is directed. By default,
directedness of the network is determined with
|
loops |
logical, whether loops are allowed. By default it is TRUE
whenever there is at least one loop in |
Network mixing matrix is, traditionally, a two-dimensional cross-classification of edges depending on the values of a specified vertex attribute for tie sender and tie receiver. It is an important tool for assessing network homophily or segregation.
Let be the number of distinct values of the vertex attribute in
question. We may say that we have mutually exclusive groups in the
network. The mixing matrix is a matrix such that
is the number of ties send by vertices in group
to vertices in group . The diagonal of that matrix is of special
interest as, say, is the number of ties within
group .
A full mixing matrix is a three-dimensional array that cross-classifies all network dyads depending on:
the value of the vertex attribute for tie sender
the value of the vertex attribute for tie receiver
the status of the dyad, i.e. whether it is connected or not
The two-dimensional version is a so-called "contact layer" of the three-dimensional version.
If object is of class "igraph," mixing matrix is created for the
network in object based on vertex attributes supplied in arguments
rattr and optionally cattr.
If only rattr is specified (or, equivalently, rattr and
cattr are identical), the result will be a mixing matrix if full is FALSE or
if full is TRUE. Where is the number of categories of
vertex attribute specified by rattr.
If rattr and cattr can be used to specify different vertex
attributes for tie sender and tie receiver.
Depending on full argument a two- or three-dimensional array
crossclassifying connected or all dyads in object.
For undirected network and if foldit is TRUE (default), the matrix is
folded onto the upper triangle (entries in lower triangle are 0).
if(require(igraph, quietly = TRUE)) { # some directed network net <- graph(c(1,2, 1,3, 2,3, 4,5, 1,4, 1,5, 4,2, 5,3)) V(net)$type <- c(1,1,1, 2,2) mixingm(net, "type") mixingm(net, "type", full=TRUE) # as undirected mixingm( as.undirected(net), "type") mixingm(net, "type") mixingm(net, "type", full=TRUE) }if(require(igraph, quietly = TRUE)) { # some directed network net <- graph(c(1,2, 1,3, 2,3, 4,5, 1,4, 1,5, 4,2, 5,3)) V(net)$type <- c(1,1,1, 2,2) mixingm(net, "type") mixingm(net, "type", full=TRUE) # as undirected mixingm( as.undirected(net), "type") mixingm(net, "type") mixingm(net, "type", full=TRUE) }
Odds ratio for connected, as opposed to disconnected, dyads depending whether it is between- or within-group, i.e. how much more likely the dyad will be connected if it is within-group.
orwg(object, ...) ## S3 method for class 'table' orwg(object, ...) ## S3 method for class 'igraph' orwg(object, vattr, ...) ## Default S3 method: orwg(object, ...)orwg(object, ...) ## S3 method for class 'table' orwg(object, ...) ## S3 method for class 'igraph' orwg(object, vattr, ...) ## Default S3 method: orwg(object, ...)
object |
R object, see Details for available methods |
... |
other arguments passed to/from other methods |
vattr |
character scalar or any vector, name of the vertex attribute or the attribute itself (as a vector) |
The measure takes values, like all odds ratios, from (0; Inf).
Method for mixing matrices.
Method for igraphs
Numeric value of the measure.
Moody, Jim (2001) "Race, school integration, and friendship segregation in America", American Journal of Sociology, 107(3):679–377
Other segregation measures: assort,
coleman, ei,
freeman, gamix,
smi, ssi
orwg(Wnet, "gender")orwg(Wnet, "gender")
Measuring relative redundancy of ties.
pairwise_redundancy(g, v = igraph::V(g)) pr_invdistance(g, v = igraph::V(g)) pr_sharedp(g, v = igraph::V(g))pairwise_redundancy(g, v = igraph::V(g)) pr_invdistance(g, v = igraph::V(g)) pr_sharedp(g, v = igraph::V(g))
g |
graph |
v |
nodes to be treated as egos |
pairwise_redundancy computes a combination of the two measures:
Every function returns a data frame with columns ego, v1,
v2 with codes for, respectively, ego, alter1, and alter2. Additional column
depends on function and it is:
inverse distance returned by pr_invdistance
number of shared partners returned by pr_sharedp
the two above combined returned by pairwise_redundancy
Hage & Harary (1983) use the Gahuku-Gama system of the Eastern Central Highlands of New Guinea, described by Read (1954), to illustrate a clusterable signed graph. Read's ethnography portrayed an alliance structure among three tribal groups containing balance as a special case; among Gahuku-Gama the enemy of an enemy can be either a friend or an enemy.
read_highland_tribesread_highland_tribes
Igraph object with undirected network with 16 vertices.
Vertices have original names of the tribes (vertex attribute name).
Edge attributes
logical, whether a tie is positive or negative
UCINET IV datasets retrieved from Pajek data collection http://vlado.fmf.uni-lj.si/pub/networks/data/ucinet/ucidata.htm#gama
Hage P. and Harary F. (1983). Structural models in anthropology. Cambridge: Cambridge University Press. (See p 56-60).
Read K. (1954). Cultures of the central highlands, New Guinea. Southwestern Journal of Anthropology, 10, 1-43.
if( require(igraph, quietly=TRUE) ) { plot(read_highland_tribes, vertex.color=ifelse(E(read_highland_tribes)$positive, "black", "red"), layout=layout.fruchterman.reingold(delete.edges(read_highland_tribes, E(read_highland_tribes)[!positive])), edge.curved=0.1) }if( require(igraph, quietly=TRUE) ) { plot(read_highland_tribes, vertex.color=ifelse(E(read_highland_tribes)$positive, "black", "red"), layout=layout.fruchterman.reingold(delete.edges(read_highland_tribes, E(read_highland_tribes)[!positive])), edge.curved=0.1) }
Segregation Matrix Index due to Freshtman (1997). A measure of network segregation. Currently (and originally) supports only two groups.
smi(object, ...) ## S3 method for class 'table' smi(object, normalize = TRUE, ...) ## S3 method for class 'igraph' smi(object, vattr, ...) ## Default S3 method: smi(object, ...)smi(object, ...) ## S3 method for class 'table' smi(object, normalize = TRUE, ...) ## S3 method for class 'igraph' smi(object, vattr, ...) ## Default S3 method: smi(object, ...)
object |
R object, see Details for available methods |
... |
other arguments passed to/from other methods |
normalize |
logical, whether normalized values should be returned,
defaults to |
vattr |
character, name of the node attribute designating groups |
The Segregation Matrix Index (SMI) is calculated for every group separately. It compares the density within group to the density of between group ties of nodes belonging to that group.
Non-normalized version is the ratio of the within-group density to the between-group density, so vary between 0 and infinity. The normalized version varies between 0 and 1.
Method for mixing matrices.
Method for igraphs
Numeric vector of length equal to the number of groups in g according
to vattr with the values of SMI for the groups.
Freshtman, M. (1997) "Cohesive Group Segregation Detection in a Social Network by the Segregation Matrix Index", Social Networks, 19:193–207
Other segregation measures: assort,
coleman, ei,
freeman, gamix,
orwg, ssi
if(require(igraph, quietly = TRUE)) { data(Wnet) smi( as.directed(Wnet, "mutual"), "gender") }if(require(igraph, quietly = TRUE)) { data(Wnet) smi( as.directed(Wnet, "mutual"), "gender") }
These functions implement Spectral Segregation Index as proposed by Echenique & Fryer (2006). This index is a node-level measure of segregation in a given network.
ssi(g, vattr)ssi(g, vattr)
g |
object of class "igraph" representing a network |
vattr |
character, name of the vertex attribute |
For a full description and axiomatization see Echenique & Fryer (2006).
The network g is converted to adjacency matrix and normalized so that
all rows sum-up to 1.
The procedure essentially consists of creating a submatrix, say, of
the adjacency matrix, say . This submatrix contains only
vertices of the given type. It may be viewed as a type-homogeneous
subnetwork of A. This subnetwork is further decomposed into connected
components. Then, for every component, an eigenvalue decomposition is
applied. The value of the index for the component is simply the largest
eigenvalue, and the individual-level indices are obtained by distributing it
according to the corresponding eigenvector.
Named vector of individual level values of SSI. Names correspond to vertex
ids in g.
Echenique, Federico and Roland G. Fryer, Jr. (2006) A Measure of Segregation Based On Social Interactions
Other segregation measures: assort,
coleman, ei,
freeman, gamix,
orwg, smi
if( require(igraph, quietly = TRUE)) { ### artificial EF data x <- ssi(EFnet, "type") x # show it on picture a <- V(EFnet)$type # rescale SSI values to use as shades of gray k <- 1 - scale(x, center=min(x), scale=max(x) - min(x)) plot( EFnet, layout=layout.fruchterman.reingold, vertex.color= gray(k), vertex.label.family="", vertex.shape=c("circle", "square")[a], vertex.label.color=gray( (1-k) > .4 ) ) ### For White's kinship data x <- ssi(Wnet, "gender") x # plot it a <- V(Wnet)$gender k <- 1 - scale(x, center=min(x), scale=max(x) - min(x)) set.seed(1234) plot( Wnet, layout=layout.fruchterman.reingold, main="Node segregation in White's kinship data", vertex.label.family="", vertex.label=V(Wnet)$name, vertex.color= gray(k), vertex.shape=c("circle", "csquare")[a], vertex.label.color="black") legend( "topleft", legend=c("Men", "Women"), pch=c(0,1), col=1) }if( require(igraph, quietly = TRUE)) { ### artificial EF data x <- ssi(EFnet, "type") x # show it on picture a <- V(EFnet)$type # rescale SSI values to use as shades of gray k <- 1 - scale(x, center=min(x), scale=max(x) - min(x)) plot( EFnet, layout=layout.fruchterman.reingold, vertex.color= gray(k), vertex.label.family="", vertex.shape=c("circle", "square")[a], vertex.label.color=gray( (1-k) > .4 ) ) ### For White's kinship data x <- ssi(Wnet, "gender") x # plot it a <- V(Wnet)$gender k <- 1 - scale(x, center=min(x), scale=max(x) - min(x)) set.seed(1234) plot( Wnet, layout=layout.fruchterman.reingold, main="Node segregation in White's kinship data", vertex.label.family="", vertex.label=V(Wnet)$name, vertex.color= gray(k), vertex.shape=c("circle", "csquare")[a], vertex.label.color="black") legend( "topleft", legend=c("Men", "Women"), pch=c(0,1), col=1) }
(De)symmetrize square binary matrix in various ways.
symmetrize(mat, rule = c("upper", "lower", "div", "intdiv"))symmetrize(mat, rule = c("upper", "lower", "div", "intdiv"))
mat |
square numeric matrix |
rule |
character, direction of copying, see Details |
Argument mat is to be a square numeric matrix. The way it is made
symmetric, or asymetric, depends on the value of the rule argument.
If rule is "upper" or "lower" then mat is made symmetric by
copying, respectively, upper triangle onto lower, or lower onto upper. The
value of rule specifies values of which triangle will stay in the
returned value.
If rule is "intdiv" then the off-diagonal values are distributed
approximately equally between the lower/upper triangles. If r is the
computed result, then r[i,j] will be equal to (x[i,j] + x[j,i])
%/% 2 if r[i,j] is in the lower triangle. It will be equal to
(x[i,j] + x[j,i]) %/% 2 + 1 if in the upper triangle.
If rule is "div" then the off-diagonal values are distributed equally
between the lower/upper triangles: as with "intdiv" but using normal
/ division.
A matrix: symmetrized version of mat.
m <- matrix(1:16, 4, 4) # copy upper triangle onto lower symmetrically symmetrize(m, "upper") # copy lower triangle onto upper symmetrically symmetrize(m, "lower") # distribute off-diagonal values exactly # r[i,j] = (m[i,j] + m[j,i]) / 2 r1 <- symmetrize(m, "div") r1 all.equal(sum(m), sum(r1)) # distribute off-diagonal values using integer division r2 <- symmetrize(m, "intdiv") r2 all.equal(sum(m), sum(r2))m <- matrix(1:16, 4, 4) # copy upper triangle onto lower symmetrically symmetrize(m, "upper") # copy lower triangle onto upper symmetrically symmetrize(m, "lower") # distribute off-diagonal values exactly # r[i,j] = (m[i,j] + m[j,i]) / 2 r1 <- symmetrize(m, "div") r1 all.equal(sum(m), sum(r1)) # distribute off-diagonal values using integer division r2 <- symmetrize(m, "intdiv") r2 all.equal(sum(m), sum(r2))
This data is taken from Freeman (1978) who uses data from White (1975) to illustrate the segregation measure.
Object of class "igraph" with an undirected network of size 10. Vertex
attribute gender, takes values 1=woman, 2=man.
Based on Freeman (1978):
White dealt with the problem of segregation among social positions rather than among individual persons. He specified a set of standard kinship positions that he called the “effective kinship network”.
Traditional analysis (e.g. Murdock, 1971) have argued that societies sometimes proscribe interaction among some kinship positions as an extension of icest taboos. Thus, given this reasoning, kinship positions should be segregated according to the gender of their occupants. White's data provide possibility to test of this hypothesis.
White collected data on the rules governing various kinds of interaction among occupants of his ten standard kinship positions for a sample of 219 societies. For every pair of positions White specified whether or not interaction between their occupants was ever restricted in any society in the sample.
Freeman, Linton C. (1978) "Segregation in Social Networks" Sociological Methods and Research 6(4):411–429
Freeman, Linton C. (1978) "Segregation in Social Networks" Sociological Methods and Research 6(4):411–429
Murdock, G. P. (1971) "Cross-Sex Patterns of Kin Behavior" Ethnology 1: 359–368
White, D. R. (1975) "Communicative Avoidance in Social Networks". University of California, Irvine. (mimeo)
if( require(igraph, quietly = TRUE) ) { data(Wnet) set.seed(2992) plot(Wnet, layout=layout.fruchterman.reingold, vertex.color=V(Wnet)$gender + 1, vertex.label=V(Wnet)$name, vertex.label.family="", main="White's (1975) data on kinship networks") legend("topleft", col=2:3, legend=c("Woman", "Man"), pch=19) }if( require(igraph, quietly = TRUE) ) { data(Wnet) set.seed(2992) plot(Wnet, layout=layout.fruchterman.reingold, vertex.color=V(Wnet)$gender + 1, vertex.label=V(Wnet)$name, vertex.label.family="", main="White's (1975) data on kinship networks") legend("topleft", col=2:3, legend=c("Woman", "Man"), pch=19) }