We consider an inhomogeneous growing network with two types of vertices. The degree sequences of two different types of vertices are investigated, respectively. We not only prove that the asymptotical degree distribution of type
Recently there has been much interest in studying inhomogeneous large-scale networks and attempting to model their topological properties. The classical random graph models are generally homogeneous, in the sense that all the vertices come in the same type (for static random graphs, see [
In this paper, our main purpose is to define an inhomogeneous model which can combine the effects of preferential attachment and difference by birth and then to provide quantitative descriptions of its properties of degree sequence. Our model inherits certain features from homogeneous growing model such that it is capable of producing asymptotic degree distributions such as power law distributions and exponential distributions by choosing proper parameters.
At first, we introduce a type space
Now we can define our model based on the information of
The attachment procedures of the new edge proceeded as follows. With probability With probability
here and thereafter we let
In our model, (
Our model is different from the inhomogeneous model [
What we are interested in is the limit distribution of the degree sequences of different types in the resulting graph
If
(i) If
(ii) For
We want to say that the proof of (
For all
In our model, one has
Note the following fact:
For all
We are also concerned about the strong law of large numbers of degree sequences for different types, respectively, as follows.
For
At last, it is also interesting to find out an expression for the joint-probability distribution for degrees of adjacent vertices. For
The rest of this paper is organized as follows. In Section
The following two lemmas are useful to prove our main results. The readers who are interested in their details can refer to the associated materials.
Suppose that a sequence
Let
In the following, in graph
Now we conclude this section by stating the following two lemmas whose proofs are proposed in the appendix.
For
Let (i) for (ii) one has the following:
For
Now let us come to solve (
Note that, for any
Combining Lemma
For
For
To solve the above two equations ((
For
We suppose that
By Stirling's formula, we have that
In our model, Azuma's inequality no longer works because there is no uniform bound on the change in the number of
At first we note a basic fact as follows:
Now let us come to estimate (
Let
Similarly, for
We also notice a fact that the total number of pairs of adjacent vertices is equal to the number of edges at time
To solve the above equations ((
At first we have
For
Moreover, by (
For
(i) Applying Lemma
(ii) We introduce for
The author declares that there is no conflict of interests regarding the publication of this paper.
The author also thanks the anonymous reviewers for valuable comments and suggestions that improved the presentation of this paper. This work is supported by the National Natural Science Foundation of China no. 11201344.