How to generate random graphs?

Question

I want to be able to generate random, undirected, and connected graphs in Java. In addition, I want to be able to control the maximum number of vertices in the graph. I am not sure what would be the best way to approach this problem, but here are a few I can think of:

(1) Generate a number between 0 and n and let that be the number of vertices. Then, somehow randomly link vertices together (maybe generate a random number per vertex and let that be the number of edges coming out of said vertex). Traverse the graph starting from an arbitrary vertex (say with Breadth-First-Search) and let our random graph G be all the visited nodes (this way, we make sure that G is connected).

(2) Generate a random square matrix (of 0's and 1's) with side length between 0 and n (somehow). This would be the adjacency matrix for our graph (the diagonal of the matrix should then either be all 1's or all 0's). Make a data structure from the graph and traverse the graph from any node to get a connected list of nodes and call that the graph G.

Any other way to generate a sufficiently random graph is welcomed. Note: I do not need a purely random graph, i.e., the graph you generate doesn't have to have any special mathematical properties (like uniformity of some sort). I simply need lots and lots of graphs for testing purposes of something else.

Here is the Java Node class I am using:

public class Node<T> {
    T data;
    ArrayList<Node> children= new ArrayList<Node>();
    ...}

Here is the Graph class I am using (you can tell why I am only interested in connected graphs at the moment):

public class Graph {
    Node mainNode;
    ArrayList<Node> V= new ArrayList<Node>();

    public Graph(Node node){
        mainNode= node;
    }
    ...}

As an example, this is how I make graphs for testing purposes right now:

//The following makes a "kite" graph G (with "a" as the main node).

/*     a-b
        |/|
        c-d
*/
Node<String> a= new Node("a");
Node<String> b= new Node("b");
Node<String> c= new Node("c");
Node<String> d= new Node("d");
a.addChild(b);
a.addChild(c);
b.addChild(a);
b.addChild(c);
b.addChild(d);
c.addChild(a);
c.addChild(b);
c.addChild(d);
d.addChild(c);
d.addChild(b);
Graph G1= new Graph(a);

Answer 1

Whatever you want to do with your graph, I guess its density is also an important parameter. Otherwise, you'd just generate a set of small cliques (complete graphs) using random sizes, and then connect them randomly.

If I'm correct, I'd advise you to use the Erdős-Rényi model: it's simple, not far from what you originally proposed, and allows you to control the graph density (so, basically: the number of links).

Here's a short description of this model:

1. Define a probability value p (the higher p and the denser the graph: 0=no link, 1=fully connected graph);
2. Create your n nodes (as objects, as an adjacency matrix, or anything that suits you);
3. Each pair of nodes is connected with a (independent) probability p. So, you have to decide of the existence of a link between them using this probability p. For example, I guess you could ranbdomly draw a value q between 0 and 1 and create the link iff q < p. Then do the same thing for each possible pair of nodes in the graph.

With this model, if your p is large enough, then it's highly probable your graph is connected (cf. the Wikipedia reference for details). In any case, if you have several components, you can also force its connectedness by creating links between nodes of distinct components. First, you have to identify each component by performing breadth-first searches (one for each component). Then, you select pairs of nodes in two distinct components, create a link between them and consider both components as merged. You repeat this process until you've got a single component remaining.

Answer 2

The only tricky part is ensuring that the final graph is connected. To do that, you can use a disjoint set data structure. Keep track of the number of components, initially n. Repeatedly pick pairs of random vertices u and v, adding the edge (u, v) to the graph and to the disjoint set structure, and decrementing the component count when the that structure tells you u and v belonged to different components. Stop when the component count reaches 1. (Note that using an adjacency matrix simplifies managing the case where the edge (u, v) is already present in the graph: in this case, adj[u][v] will be set to 1 a second time, which as desired has no effect.)

If you find this creates graphs that are too dense (or too sparse), then you can use another random number to add edges only k% of the time when the endpoints are already part of the same component (or when they are part of different components), for some k.

Answer 3

The following paper proposes an algorithm that uniformly samples connected random graphs with prescribed degree sequence, with an efficient implementation. It is available in several libraries, like Networkit or igraph.

Fast generation of random connected graphs with prescribed degrees. Fabien Viger, Matthieu Latapy

Be careful when you make simulations on random graphs: if they are not sampled uniformly, then they may have hidden properties that impact simulations; alternatively, uniformly sampled graphs may be very different from the ones your code will meet in practice...