sna (version 2.8)

rguman: Draw Dyad Census-Conditioned Random Graphs

Description

rguman generates random draws from a dyad census-conditioned uniform random graph distribution.

Usage

rguman(n, nv, mut = 0.25, asym = 0.5, null = 0.25, 
    method = c("probability", "exact"), return.as.edgelist = FALSE)

Value

A matrix or array containing the drawn adjacency matrices

Arguments

n

the number of graphs to generate.

nv

the size of the vertex set (\(|V(G)|\)) for the random graphs.

mut

if method=="probability", the probability of obtaining a mutual dyad; otherwise, the number of mutual dyads.

asym

if method=="probability", the probability of obtaining an asymmetric dyad; otherwise, the number of asymmetric dyads.

null

if method=="probability", the probability of obtaining a null dyad; otherwise, the number of null dyads.

method

the generation method to use. "probability" results in a multinomial dyad distribution (conditional on the underlying rates), while "exact" results in a uniform draw conditional on the exact dyad distribution.

return.as.edgelist

logical; should the resulting graphs be returned in edgelist form?

Author

Carter T. Butts

Details

A simple generalization of the Erdos-Renyi family, the U|MAN distributions are uniform on the set of graphs, conditional on order (size) and the dyad census. As with the E-R case, there are two U|MAN variants. The first (corresponding to method=="probability") takes dyad states as independent multinomials with parameters \(m\) (for mutuals), \(a\) (for asymmetrics), and \(n\) (for nulls). The resulting pmf is then $$ p(G=g|m,a,n) = \frac{(M+A+N)!}{M!A!N!} m^M a^A n^N, $$ where \(M\), \(A\), and \(N\) are realized counts of mutual, asymmetric, and null dyads, respectively. (See dyad.census for an explication of dyad types.)

The second U|MAN variant is selected by method=="exact", and places equal mass on all graphs having the specified (exact) dyad census. The corresponding pmf is $$ p(G=g|M,A,N) = \frac{M!A!N!}{(M+A+N)!}. $$

U|MAN graphs provide a natural baseline model for networks which are constrained by size, density, and reciprocity. In this way, they provide a bridge between edgewise models (e.g., the E-R family) and models with higher order dependence (e.g., the Markov graphs).

References

Holland, P.W. and Leinhardt, S. (1976). “Local Structure in Social Networks.” In D. Heise (Ed.), Sociological Methodology, pp 1-45. San Francisco: Jossey-Bass.

Wasserman, S. and Faust, K. (1994). Social Network Analysis: Methods and Applications. Cambridge: Cambridge University Press.

See Also

rgraph, rgnm, dyad.census

Examples

#Show some examples of extreme U|MAN graphs
gplot(rguman(1,10,mut=45,asym=0,null=0,method="exact")) #Clique
gplot(rguman(1,10,mut=0,asym=45,null=0,method="exact")) #Tournament
gplot(rguman(1,10,mut=0,asym=0,null=45,method="exact")) #Empty

#Draw a sample of multinomial U|MAN graphs
g<-rguman(5,10,mut=0.15,asym=0.05,null=0.8)

#Examine the dyad census
dyad.census(g)