Taylor’s power law states that the variance function decays as a power law. It is observed for population densities of species in ecology. For random networks another power law, that is, the power law degree distribution is widely studied. In this paper the original Taylor’s power law is considered for random networks. A precise mathematical proof is presented that Taylor’s power law is asymptotically true for the

Taylor’s power law is a well-known empirical pattern in ecology. Its general form is

However, there is another power law for networks. There are large networks satisfying

There are lot of modifications of the preferential attachment model, here we can list only a few of them. The following general graph evolution model was introduced by Cooper and Frieze in [

In several cases the connection of two edges in a network can be interpreted as co-operation (collaboration). For example in the movie actor network two actors are connected by an edge if they have appeared in a film together. In the collaboration graph of scientists an edge connects two people if they have been co-authors of a paper (see, e.g. [

In [

We are interested in the following general question. Is the original Taylor’s power law true for random networks? First we considered data sets of real life networks. We analysed them and the statistical analysis showed that there are cases when Taylor’s law is true and there are cases when it is not true (our empirical results will be published elsewhere). So we encountered the following more specific problem: Find network structures where Taylor’s power law is true. To this end we analysed the above

In this paper we prove an asymptotic Taylor’s power law for the

First we give a short mathematical description of our random graph model from [

Let

We first explain the model on a high level, before giving a formal definition in the next paragraphs. The general rules of the evolution of our graph are the following. At each time step,

Now we describe the details of the evolution steps of our graph. We have two options in every step of the evolution.

For every vertex we shall use its central weight and its peripheral weight. The central weight of a vertex is

Throughout the paper

In [

Using

We see that

Now we turn to the new results of this paper. First we consider the marginals of the asymptotic joint distribution

Now we turn to the conditional expectations of the asymptotic distribution. Let

Now we turn to the conditional second moments of the asymptotic distribution. Let

Now Propositions

How can we observe the above Taylor’s law in practice? As

If

Now we consider the case when we interchange the roles of

For the in-degree

For the joint limiting distribution we have the following result.

Throughout the proof we shall use the following facts on the

Now we turn to the proofs of the new results. First we deal with the marginal distribution.

To calculate the marginal distribution

For

Now we check if the sum of the values of

Now we consider the expectation.

We calculate

Let us start with

If

Now we turn to the second moment.

To find the second moment

We start with

Now turn to

When

Propositions

Here we present some numerical evidence supporting our result. The scheme of our computer experiment is the following. We fixed the size

In the following five experiments we used various parameter sets. The step size was always

Here

Finally, we show a numerical result when the conditions of Theorem

Let

A case when Taylor’s power law is not satisfied.

The authors are grateful to the referees and to the editor for the careful reading of the paper and for the valuable suggestions.