Abstract
Many growth models have been published to model the behavior of real complex networks. These models are able to reproduce several of the topological properties of such networks. However, in most of these growth models, the number of outgoing links (i.e., outdegree) of nodes added to the network is constant, that is all nodes in the network are born with the same number of outgoing links. In other models, the resultant outdegree distribution decays as a poisson or an exponential distribution. However, it has been found that in real complex networks, the outdegree distribution decays as a powerlaw. In order to obtain outdegree distribution with powerlaw behavior some models have been proposed. This work introduces a new model that allows to obtain outdegree distributions that decay as a powerlaw with an exponent in the range from 0 to 1.
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In the literature, there are many growth models for complex networks (CN) that reproduce some topological properties of real systems^{1}. However, in most growth models it is assumed that all nodes are born with the same amount of outgoing links (i.e., their outdegree is a constant), as in the model proposed by BarabásiAlbert^{2}. In other models, such as the one proposed by Dorogovtsev et.al^{3} and the one proposed by Krapivsky and Redner^{4}, the outdegree distribution decays as an exponential or a poisson distribution, respectively. However, these results differ from the outdegree behavior of several real CN. For example, in metabolic networks^{5}, the Internet^{6} and WWW^{7} the outdegree decays as a powerlaw. Therefore Dorogovtsev et.al.^{8} and Bollobás et.al.^{9} developed two models that are able to produce outdegree distributions that decay as a powerlaw with exponent and respectively, that is, in both models the exponent is greater than 2. It is important to mention that for the average outdegree 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 outdegree of nodes is infinite when N → ∞.
In the present work, we propose a simple growth model for directed CN which is able to generate outdegree distributions that decay as a powerlaw with exponent 0 < γ_{out} < 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 n_{0} exists in the network and its outdegree is 0. Then we consider that the outdegree of any new node n_{new} added to the network is determined as follows:

with probability p where 0 < p < 1, n_{new} copies the outdegree of a randomly selected node from the network. It is important to note that as the quantity Q_{s} of nodes with outdegree s increases, the probability that node n_{new} has outdegree s also increases to , where N is the total number of nodes in the network.

with complementary probability 1 − p, n_{new} randomly selects an outdegree uniformly distributed from 0 to N. That is, node n_{new} has outdegree 0, 1, 2, … N. It is important to note that this rule produces unrealistic outdegree of the new node almost all the times it is applied. That is, new nodes may have outdegree of the order N.
By applying the previous considerations and using the continuum method^{10}, we can write the following differential equation:
that describes the variation of the quantity Q_{s} of nodes with outdegree s with respect to the total number N of nodes in the network. The term g_{1} accounts for the situation that a new node copies the outdegree of a randomly selected node in the network. The term g_{2} describes the random selection of outdegree 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 _{2}F_{1} as follows:
where k is a constant. To obtain the outdegree distribution Q_{s}(N), we solve Eq. 4 for s = 1, s = 2 and so on as follows:
• for Q_{1}(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 n_{0} with no outgoing links, that is N = 1. For this case the quantity Q_{1}(1) of nodes with outdegree s = 1 is zero (Q_{1}(1) = 0). When the node n_{1} is added (N = 2), the probability for node n_{1} to have outdegree s = 1 is . Solving Eq. 4 for the initial condition , we obtain:
• for Q_{2}(N), we need to consider the initial condition
This initial condition is due to the fact that, before adding node n_{2}, only nodes n_{0} and n_{1} are in the network (N = 2) and any of them has s ≥ 2, therefore Q_{2}(2) = 0. When node n_{2} is added (N = 3), the probability that node n_{2} has outdegree s = 2 is . Solving Eq. 4 for the initial condition , we obtain:
Normalizing Eq. 7 we obtain
Eq. 8, shows that the exponent γ_{out} of the outdegree distribution obtained with the proposed model is only determined by the probability p. That is, the outdegree distribution obtained decays as a powerlaw
with exponent γ_{out} = p.
On the other hand, we can deduce that as a consequence of the random outdegree selection by new nodes with probability 1 − p (second rule of the proposed model), the average outdegree of the nodes grows with the network size. To validate this hypothesis, we analytically calculate the average outdegree using the following differential equation:
that describes the increment of the average outdegree with respect to the total number N of nodes in the network. On the righthand side of Eq. 10, the term describes the mean of the random outdegree 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 outdegree of nodes is infinite when N → ∞.
In order to validate the analytical solutions for the outdegree distribution (Eq. 8) and average outdegree (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 10^{4} nodes. Figure 1a shows that the results of the numerical simulations and the analytical prediction (Eq. 8) for the outdegree distribution fit appropriately. On the other hand, we measure the average outdegree in each simulation for different network sizes. Figure 1b shows that the average outdegree retrieved from the simulations and the analytical prediction (Eq. 13) fit also appropriately. That is, in the proposed model the average outdegree grows linearly with N for any value of 0 < p < 1 as stated by Eq. 13 and consequently the average outdegree 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 outdegree increases slowly as the network grows^{11}.
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 outdegree of the nodes and, therefore, produce better predictions of the behavior of real CN and thus increases the understanding of these systems.
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J.E.G., E.S.N., U.P.N. and J.A.E. designed, performed the research and wrote the manuscript.
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EsquivelGómez, J., StevensNavarro, E., PinedaRico, U. et al. A growth model for directed complex networks with powerlaw shape in the outdegree distribution. Sci Rep 5, 7670 (2015). https://doi.org/10.1038/srep07670
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DOI: https://doi.org/10.1038/srep07670
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