## Friday, April 24, 2020

### Generating Random Digraphs

In a recent post, OR consultant and blogger Erwin Kalvelagen discussed generating a random sparse network in GAMS. More specifically, he starts with a fixed set of nodes and a desired number of arcs, and randomly generates approximately that number of arcs. His best results, in terms of execution time, came from exporting the dimensions to R, running a script there, writing out the arcs and importing them back into GAMS.

There are three possible issues with the script. Erwin acknowledged the first, which applies to all his approaches: after removing duplicates, he wound up with fewer than the targeted number of arcs. In many applications this would not be a problems, since you would be looking for "about 1% density" rather than "exactly 1% density". Still, there might be times when you need a specific number of arcs, period. You could supplement Erwin's method with a loop that would test the number of arcs and, if short, would generate more arcs, remove duplicates, add the survivors to the network and repeat.

The second possible issue is the occurrence of self-loops (arcs with head and tail the same, such as (5, 5)). Again, this may or may not be a problem in practice, depending on the application. I rarely encounter network applications where self-loops are expected, or even tolerated. Again, you could modify Erwin's code easily to remove self-loops, and it would not increase execution time much.

The third possible issue is that some nodes may be "orphans" (no arcs in or out), and others may be accessible only one way (either inward degree 0 or outward degree 0). Once again, the application will dictate whether this is a matter of concern.

I decided to test a somewhat different approach to generating the network (using R). It has the advantage of guaranteeing the targeted number of arcs, with no self-loops. (It does not address the issue of orphan nodes.) It has the disadvantage of being slower than Erwin's algorithm (but by what I would call a tolerable amount). My approach is based on assigning an integer index to every possible arc. Assume there are $N$ nodes, indexed $0, \dots, N-1$. (Erwin uses 1-based indexing, but it is trivial to adjust from 0-based to 1-based after the network has been generated.) There are $N^2$ arcs, including self-loops, indexed from $0,\dots,N^2-1$. The arc with index $k$ is given by $$f(k) = (k \div n, k \mod n),$$where $\div$ denotes the integer quotient (so that $7 \div 3 = 2$). As self-loop is an arc whose index $k$ satisfies $k\div n = k \mod n$; those are precisely the arcs with indices $k=m(n + 1)$ for $m=0,\dots,n-1$. So my version of the algorithm is to start with the index set $\lbrace 0,\dots,n^2-1\rbrace$, remove the indices $0, n+1, 2n+2,\dots, n^2-1$, take a random subset of the survivors and apply $f()$ to them.

I have an R Notebook that compares the two algorithms, using the same dimensions Erwin had: $n=5000$ nodes, with a target density of 1% (250,000 arcs). Timing is somewhat random, even though I set a fixed random number seed for each algorithm. The notebook includes both code and output. As expected, my code gets the targeted number of arcs, with no self-loops. Erwin's code, as expected, comes up a bit short on the number of arcs, and contains a few (easily removed) self-loops. Somewhat interestingly, in my test runs every node had both inward and outward degree at least 1 for both algorithms. I think that is a combination of a fairly large arc count and a bit of luck (the required amount of luck decreasing as the network density increases). If orphans, or nodes with either no outbound or no inbound arcs, turn out to be problems, there is a fairly easy fix for both methods. First, randomly generate either one or two arcs incident on each node (depending on whether you need both inward and outward arcs everywhere). Then generate the necessary number of additional arcs by adjusting the sample size. As before, you might come up a few arcs short with Erwin's algorithm (unless you include a loop to add arcs until the target is reached). In my algorithm, you can easily calculate the indices of the initial set of arcs (the index of arc $(i,j)$ is $n\times i + j$) and then just remove those indices at the same time that you remove the indices of the self-loops, before generating the remaining arcs.