library(igraph)

# original data as a list
input_data  = list(c(1,2,3),c(1,4,5),c(1,6),c(3),c(4,6))

## function that makes all pairs
pairing <- function(x) {
  n = length(x)
  if (n<2) {
    res <- NULL
  } else {
  q <- combn(n,2)
  x2 <- x[q]
  #dim(x2) <- dim(q)
  res <- x2
  }
  return(res)
}

## for each paper create all author pairs
pairing_bypaper = lapply(input_data,pairing)

## remove papers that contribute no edges
pair_noedge = sapply(pairing_bypaper,is.null)
pair2_bypaper <- pairing_bypaper[!pair_noedge]

## combine all 'subgraphs'
pair_all <- do.call('c',pair2_bypaper)

## how many authors are there?
n.authors <- length(unique(pair_all))

## make a graph
my_graph = graph(pair_all, directed = FALSE)

## plot
plot(my_graph)

plot(my_graph, vertex.label.cex = 0.8, edge.width = E(my_graph)$weight)

degree(my_graph)
[1] 5 2 2 3 2 2

## size
V(my_graph)$vertex_degree <-  degree(my_graph)

plot(my_graph,
     vertex.label.cex = 0.8,
     edge.width = E(my_graph)$weight,
     vertex.size = V(my_graph)$vertex_degree #add this
     )

scale_factor <- 4
plot(my_graph,
     vertex.label.cex = 0.8,
     edge.width = E(my_graph)$weight,
     vertex.size = V(my_graph)$vertex_degree * scale_factor
     )