Giant components and the Lambert W-function

The Lambert W-function is the function w(z) implicitly defined by w exp(w) = z. When I first saw this, I thought that I’d seen something like this come up many times and that it would be really handy to know that this function has a name and has many software implementations [1]. Since then, however, I’ve hardly ever run into the W-function in application. But here’s an application I ran into recently.

A “giant component” in a random network is a component whose size grows in proportion to the size of the network. For a Poisson random network, let c = p(n − 1) be the expected number of vertices connected to any given vertex. If c > 1 then as n goes to infinity, there will be a giant component with probability 1. S, the proportion of nodes inside the a giant component, satisfies the equation

S = 1 − exp(-cS).

S plays a role similar to that of w in the definition of the W-function, which suggests the W-function might come in handy. And in fact, this book gives this solution for S:

S = 1 + w( −c exp(−c) )/c.

There are a couple issues. First, it’s not obvious that solution is correct. Second, we need to be more careful about how we define w(z) when z is negative.

Given some value of c, let S = 1 + w( −c exp(−c) )/c. Then

w( −c exp(−c) ) = −c(1 − S).

Apply the function f(x) = x exp(x) to both sides of the equation above. By the definition of the W-function, we have

−c exp(−c) = −c(1 − S) exp( −c(1 − S) ).

From there it easily follows that

S = 1 − exp(−cS).

Now let’s look more carefully at the definition of the W-function. For positive real z, w exp(w) = z has a unique real solution defining w. But in the calculations above, we have evaluated w at −c exp(−c), which is negative since c > 1. For negative arguments, we have to pick a branch of w. Which branch guarantees that S = 1 − exp(-cS)? Any branch [2]. No matter which solution to w exp(w) = z we pick, the resulting S satisfies the giant component equation. However, the wrong branch might result in a meaningless solution. Since S is a proportion, S should be between 0 and 1.

Let’s go back and say that for our purposes, we will take w(z) to mean the real number no less than −1 satisfying w exp(w) = z. Then for c > 1, the solution S = 1 + w( −c exp(−c) )/c is between 0 and 1. You can show that S = 1 − exp(−cS) has a unique solution for 0 < S < 1, so any other branch would not give a solution in this interval.

Related post: Measuring graph connectivity with eigenvalues

[1] For example, in Python the Lambert W-function is scipy.special.lambertw. The function takes two arguments: z and an integer denoting the branch. By default, the branch argument is set to 0, giving the branch discussed in this post.

[2] For −1/e < z < 0, there are two real branches of w(z), but there are also infinitely many complex branches.