How to generate a random directed cyclic graph with a defined "average" number of edges for each node? (in R language)

I want to generate random directed cyclic graphs while defining the number of nodes and the average number of edges for each node. By "average number of edges" i mean that the degree of each node is usually the one i define with lower chances of being lower or higher.

For example i want a DCG with 3 edges per node i will have 80% nodes with degree 3, 15% nodes with degree 2, 5% nodes with degree 1 (note: all nodes should have edges, no disconnected nodes)

I would also like to define the number of cycles in the generated DCG.

I don't know if there is something that is already implemented that does this.

I tried with igraph with the erdos.renyi.game

library(igraph)  
nodes = 20  
edges = 40  
g <- erdos.renyi.game(nodes, edges, type = "gnm",  
          directed = TRUE, loops = FALSE)  
is.dag(g) ##FALSE --> it's a Directed Cyclic Graph  
x = degree(g)  
mean(x)  
table(x)  
names(sort(-table(x)))[1]  
hist(x)  
plot(g)   

with 20 nodes and 40 edges, directed edges and no self loops, i obtain a DCG but i have 2 problems:

First, i cannot define the number of edges per node, here i tried to have a number of edges that is two times the number of nodes in hope to get an average of 2 edges per node but i end up with random degrees ranging up to nodes with degree of 8

Second, i cannot define the number of cycles or the number of nodes that define the cycle, so if i have two nodes pointing at eachother it counts as DCG. It would be better if i could define cycles that involve more than 2 nodes for the tests.

I think this function should do what you need. Essentially, it creates a random edgelist which contains every node at least once. The number of edges in the edgelist is determined by multiplying the number of nodes and the desired average number of edges.

In the course of sampling, some self-loops may happen by chance. These are replaced through resampling until no self-loops remain.

generate_cyclic <- function(nodes, ave_edges) {
  ave_edges <- ave_edges/2
  from <- c(nodes, sample(nodes, floor(length(nodes) * (ave_edges - 1)), TRUE))
  to <- c(sample(nodes, floor(length(nodes) * (ave_edges - 1)), TRUE), nodes)
  while(any(from == to)) {
    i <- which(from == to)
    from[i] <- sample(nodes, length(i), TRUE)
  }
  igraph::graph_from_edgelist(cbind(from, to))
}

Testing, we get

library(igraph)

g <- generate_cyclic(LETTERS[1:20], ave_edges = 3)

is.dag(g)
#> [1] FALSE

x = degree(g)  
mean(x)  
#> [1] 3

hist(x)

plot(g)