In the literature, there are many growth models for complex networks (CN) that reproduce some topological properties of real systems1. However, in most growth models it is assumed that all nodes are born with the same amount of outgoing links (i.e., their out-degree is a constant), as in the model proposed by Barabási-Albert2. In other models, such as the one proposed by Dorogovtsev et.al3 and the one proposed by Krapivsky and Redner4, the out-degree distribution decays as an exponential or a poisson distribution, respectively. However, these results differ from the out-degree behavior of several real CN. For example, in metabolic networks5, the Internet6 and WWW7 the out-degree decays as a power-law. Therefore Dorogovtsev et.al.8 and Bollobás et.al.9 developed two models that are able to produce out-degree distributions that decay as a power-law with exponent and respectively, that is, in both models the exponent is greater than 2. It is important to mention that for the average out-degree to be finite in the infinite system size limit the exponent must be larger than 2. Since any exponent smaller or equal to 2 results in a distribution with diverging first moment, i.e. where the average out-degree of nodes is infinite when N → ∞.

In the present work, we propose a simple growth model for directed CN which is able to generate out-degree distributions that decay as a power-law with exponent 0 < γout < 1. In the proposed model, the growth of the network is done by adding nodes one at a time. At the beginning, only the node n0 exists in the network and its out-degree is 0. Then we consider that the out-degree of any new node nnew added to the network is determined as follows:

  • with probability p where 0 < p < 1, nnew copies the out-degree of a randomly selected node from the network. It is important to note that as the quantity Qs of nodes with out-degree s increases, the probability that node nnew has out-degree s also increases to , where N is the total number of nodes in the network.

  • with complementary probability 1 − p, nnew randomly selects an out-degree uniformly distributed from 0 to N. That is, node nnew has out-degree 0, 1, 2, … N. It is important to note that this rule produces unrealistic out-degree of the new node almost all the times it is applied. That is, new nodes may have out-degree of the order N.

By applying the previous considerations and using the continuum method10, we can write the following differential equation:

that describes the variation of the quantity Qs of nodes with out-degree s with respect to the total number N of nodes in the network. The term g1 accounts for the situation that a new node copies the out-degree of a randomly selected node in the network. The term g2 describes the random selection of out-degree for a new node.

Eq. 1 can be written in the standard form for a linear differential equation as follows:

multiplying by the integrating factor , we obtain

Since to the integral of Eq. 3 is not elementary, the solution retrieved is in terms of the Hypergeometrical Function 2F1 as follows:

where k is a constant. To obtain the out-degree distribution Qs(N), we solve Eq. 4 for s = 1, s = 2 and so on as follows:

•  for Q1(N), we need to consider the initial condition

This initial condition is due to the fact that at the beginning, the network is formed only by node n0 with no outgoing links, that is N = 1. For this case the quantity Q1(1) of nodes with out-degree s = 1 is zero (Q1(1) = 0). When the node n1 is added (N = 2), the probability for node n1 to have out-degree s = 1 is . Solving Eq. 4 for the initial condition , we obtain:

•  for Q2(N), we need to consider the initial condition

This initial condition is due to the fact that, before adding node n2, only nodes n0 and n1 are in the network (N = 2) and any of them has s ≥ 2, therefore Q2(2) = 0. When node n2 is added (N = 3), the probability that node n2 has out-degree s = 2 is . Solving Eq. 4 for the initial condition , we obtain:

From the previous results in Eqs. 5 and 6, we can deduce that:

Normalizing Eq. 7 we obtain

Eq. 8, shows that the exponent γout of the out-degree distribution obtained with the proposed model is only determined by the probability p. That is, the out-degree distribution obtained decays as a power-law

with exponent γout = p.

On the other hand, we can deduce that as a consequence of the random out-degree selection by new nodes with probability 1 − p (second rule of the proposed model), the average out-degree of the nodes grows with the network size. To validate this hypothesis, we analytically calculate the average out-degree using the following differential equation:

that describes the increment of the average out-degree with respect to the total number N of nodes in the network. On the right-hand side of Eq. 10, the term describes the mean of the random out-degree uniformly selected from 0 to N by a new node. Thus, the term describes the increment of .

Eq. 10 can be written in the standard form for a linear differential equation as follows:

Solving Eq. 11 we obtain

As the total number of nodes in the network increases (N ≫ 1), we can approximate Eq. 12 as follows:

From Eq. 13 we can see that effectively grows proportionally to the network size, that is, in the proposed model the average out-degree of nodes is infinite when N → ∞.

In order to validate the analytical solutions for the out-degree distribution (Eq. 8) and average out-degree (Eq. 13) of the proposed model, we performed four numerical simulations using p = 0.1, p = 0.3, p = 0.6 and p = 0.9. In each simulation, we considered the growth of a directed network from 1 to 104 nodes. Figure 1a shows that the results of the numerical simulations and the analytical prediction (Eq. 8) for the out-degree distribution fit appropriately. On the other hand, we measure the average out-degree in each simulation for different network sizes. Figure 1b shows that the average out-degree retrieved from the simulations and the analytical prediction (Eq. 13) fit also appropriately. That is, in the proposed model the average out-degree grows linearly with N for any value of 0 < p < 1 as stated by Eq. 13 and consequently the average out-degree of nodes is infinite when N → ∞. This contrasts with some large networks that are sparse where the number of edges is much smaller than the maximum possible and the average out-degree increases slowly as the network grows11.

Figure 1
figure 1

(a) Comparison of the out-degree distribution (symbols) retrieved from the simulations and the analytical predictions (lines). (b) Comparison of the Average out-degree retrieved from the simulations for different network sizes and the analytical predictions (lines).

Full size image

The topological properties of real CN seems to be the result of a set of local processes. We consider that the proposed model in this work can contribute to develop new growth models for directed CN which consider local processes that shape the out-degree of the nodes and, therefore, produce better predictions of the behavior of real CN and thus increases the understanding of these systems.