How Igraph handle weights?

Question

Good day to everyone.

I have a really simple question that I wasn't able to find an answer, because of lack of terminology I am afraid. In r's package igraph how are considered weights? Are they considered a cost, therefore reducing the capacity of an edge or are they considered indeed as the capacity of the edge?

Thank you very much

Answer 1

In igraph, weights are edge-attributes that represent the friction or cost of traveling that edge in a parth, not the capacity or bandwidth of the edge. Low weight makes for a low weight-sum of a path and get.shortest.paths() returns the path with the lowest weight-sum when run without disabling weights on a weighted graph.

This code example shows a graph with different shortest paths in weighted and unweighted mode, and explains why the path is differently calculated.

library(igraph)

# Colours for the weighted edges
N <- 22
set.seed(7890123)

# Make a random graph with randomly weighted edges coloured in gray
g <- erdos.renyi.game(N, .2, type="gnp", directed=F, loops=F, weighted=T)
E(g)$weight <- sample(1:40, length(E(g)), replace=T)
#E(g)$weight <- E(g)$weight/10
E(g)$color <- "gray"
V(g)$size <- 4
V(g)$size[c(1,N)] <- 12

# Look how the shortest path is calculated differently when taken the graph weihgt into acocunt
(weighted.path <- unlist(get.shortest.paths(g, 1, N)$vpath) )
(unweighted.path <- unlist(get.shortest.paths(g, 1, N, weights=NA)$vpath) )

# Set weights and colours of shortest paths to visualise them
E(g, path=weighted.path)$color <- "red"
E(g, path=unweighted.path)$color <- "green"

# plot the graph with red shortest weighted path, and green shortest path
same.sahpe <- layout_with_fr(g)
plot(g, vertex.color="white", vertex.label=NA, edge.weight=2, edge.width=(E(g)$weight/5), layout=same.sahpe)

# The two paths look like this. Even though path-length might be longer, a weighted
# shortest path is determined using the sum of path-weights. As with path-lengths, the
# lowest value is the path most easily travelled by. In this case, the weighted alternative
# has a longer path but with lower friction.
data.frame(path.length=c(length(weighted.path),
                        length(unweighted.path)
                        ),
           weight.sum=c(sum(E(g, path=unlist(weighted.path))$weight),
                        sum(E(g, path=unlist(unweighted.path))$weight)
           )
)

See the shortest unweighted path between the two larger vertices has the length of 4, but travels over rather thickly weighted green edges. The shortest weighted path has the lowest weight-sum and travels more steps (5) but over wedges with lower weight resulting in a lower weight-sum, or lower cost of travelling the parth, if you like.