rgraph_ba function

Scale-free and Homophilic Random Networks

Generates a scale-free random graph based on Bollabas et al. (2001), also know as Linearized Chord Diagram (LCD) which has nice mathematical propoerties. And also scale-free homophilic networks when an vertex attribute eta is passed.

rgraph_ba(m0 = 1L, m = 1L, t = 10L, graph = NULL, self = TRUE, eta = NULL)

Arguments

• m0: Integer scalar. Number of initial vertices in the graph.
• m: Integer scalar. Number of new edges per vertex added.
• t: Integer scalar. Number of time periods (steps).
• graph: Any class of accepted graph format (see netdiffuseR-graphs).
• self: Logical scalar. When TRUE autolinks (loops, self edges) are allowed (see details).
• eta: Numeric vector of length t+m0. When specified, it generates a scale-free homophilic network (see details).

Returns

If graph is not provided, a static graph, otherwise an expanded graph (t aditional vertices) of the same class as graph.

The resulting graph will have graph$meta$undirected = FALSE if it is of class diffnet and attr(graph, "undirected")=FALSE otherwise.

Details

Based on Ballobás et al. (2001) creates a directed random graph of size t + m0. A big difference with B-A model is that this allows for loops (self/auto edges) and further multiple links, nevertheless, as $t$ increases, the number of such cases reduces.

By default, the degree of the first m0 vertices is set to be 2 (loops). When m>1, as described in the paper, each new link from the new vertex is added one at a time counting ‘outward half’ of the edge being added as already contributingto the degrees .

When self=FALSE, the generated graph is created without autolinks. This means that at the beginning, if the number of links equals zero, all vertices have the same probability of receiving a new link.

When eta is passed, it implements the model specified in De Almeida et al. (2013), a scale-free homophilic network. To do so eta is rescaled to be between 0 and 1 and the probability that the node $i$ links to node $j$

is as follows:

Where $A(ij) = |eta(i) - eta(j)|$ and $k(j)$ is the degree of the $j$-th vertex.

Examples

# Using another graph as a base graph ---------------------------------------
graph <- rgraph_ba()
graph

graph <- rgraph_ba(graph=graph)

# Generating a scale-free homophilic graph (no loops) -----------------------
set.seed(112)
eta <- rep(c(1,1,1,1,2,2,2,2), 20)
ans <- rgraph_ba(t=length(eta) - 1, m=3, self=FALSE, eta=eta)

# Converting it to igraph (so we can plot it)
ig  <- igraph::graph_from_adjacency_matrix(ans)

# Neat plot showing the output
oldpar <- par(no.readonly = TRUE)
par(mfrow=c(1,2))
plot(ig, vertex.color=c("red","blue")[factor(eta)], vertex.label=NA,
    vertex.size=5, main="Scale-free homophilic graph")
suppressWarnings(plot(dgr(ans), main="Degree distribution"))
par(oldpar)

References

Bollobás, B´., Riordan, O., Spencer, J., & Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures & Algorithms, 18(3), 279–290. tools:::Rd_expr_doi("10.1002/rsa.1009")

Albert-László Barabási, & Albert, R. (1999). Emergence of Scaling in Random Networks. Science, 286(5439), 509–512. tools:::Rd_expr_doi("10.1126/science.286.5439.509")

Albert-László Barabási. (2016). Network Science: (1st ed.). Cambridge University Press. Retrieved from https://barabasi.com/book/network-science

De Almeida, M. L., Mendes, G. A., Madras Viswanathan, G., & Da Silva, L. R. (2013). Scale-free homophilic network. European Physical Journal B, 86(2). tools:::Rd_expr_doi("10.1140/epjb/e2012-30802-x")

See Also

Other simulation functions: permute_graph(), rdiffnet(), rewire_graph(), rgraph_er(), rgraph_ws(), ring_lattice()

Author(s)

George G. Vega Yon