Abstract
The outdegree distribution is one of the most reported topological properties to characterize real complex networks. This property describes the probability that a node in the network has a particular number of outgoing links. It has been found that in many real complex networks the outdegree has a behavior similar to a powerlaw distribution, therefore some network growth models have been proposed to approximate this behavior. This paper introduces a new growth model that allows to produce outdegree distributions that decay as a powerlaw with an exponent in the range from 1 to ∞.
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Introduction
Among the topological properties of real complex networks (CN), one of the most studied is the outdegree distribution. This property describes the probability that a node in the network has a particular number of outgoing links. It has been found that in many real CN the outdegree behaves as a powerlaw distribution (P(k) ∝ k^{−γ})^{1,2,3,4,5,6,7}. In order to approximate this type of outdegree distribution, some growth models for CN have been proposed. For example, Dorogovtsev et.al.^{8} and Bollobás et.al.^{9} have each developed a model capable of producing outdegree distributions that decay as a powerlaw with exponent and , respectively. Hence in both models the γ exponent is greater than 2. Esquivel et.al.^{10} proposed a model that produces outdegree distributions that decay as a powerlaw where the γ exponent value is in the range between 0 and 1.
The previous models are not able to produce outdegree distributions with γ exponents in the range between 1 and 2. However, there are real CN where the γ exponent value is within this interval. For example, the social network of Flickr users^{6}, the Any Beat network^{7}, the online social network Epinions^{6} and the network of flights between airports of the world (OpenFlights)^{6} where the γ exponent for the outdegree distribution of these CN is close to 1.74, 1.71, 1.69 and 1.74 respectively.
This paper introduces a new model for growth of directed CN that allows to obtain outdegree distributions that decay as a powerlaw with exponents in the range 1 < γ < ∞. That is, the proposed model is able to generate all exponent values found in documented real CN^{1,2,3,4,5,6,7}.
It has been demonstrated that the growth and evolution of CN is influenced by local processes that shape its topological and dynamical properties^{11}. The model proposed in here incorporates two local processes for adding new nodes to the network: a random outdegree selection and a copy of an already present outdegree value. In many large networks the maximum degree of a node (the degree of a node is the sum of its incoming and outgoing links) is much smaller than the number of nodes^{6}. Thus, the proposed model assumes that the probability that a new node n_{new} selects a random outdegree decreases as the network grows. This probability is expressed as N^{−α} where N is the total number of nodes in the network (including n_{new}) and α is a constant greater than 0. In other words, the probability that new nodes have an outdegree close to N tends to zero as N ≫ 1.
Proposed model
In this model, the growth of the network is performed by adding nodes one at a time. At the beginning, only node n_{0} is present in the network and its outdegree is 0. Then, the outdegree of any new node n_{new} added to this network is determined as follows:

With probability N^{−α}, n_{new} randomly selects an outdegree uniformly distributed from 0 to N − 1. That is, n_{new} may have outdegree 0, 1, 2, …, N − 1. It is important to notice that it is possible that n_{new} has an outdegree of the order of N − 1. In this situation n_{new} would connect to all the other nodes in the network. This scenario may not be realistic for large values of N, because in many real networks, the maximum degree for a node is much smaller than the total number of nodes N^{6}. However, the probability N^{−α} decreases when N increases for α > 0.

With complementary probability 1 − N^{−α}, n_{new} copies the outdegree of a randomly selected node from the network. It is important to notice that as the number Q_{s} of nodes with outdegree s increases, the probability that n_{new} has outdegree s also increases to .
It is possible to employ the continuum method^{12} to obtain the analytical solution for the proposed model. This method is implemented using the following differential equation:
The previous equation describes the variation of the number Q_{s} of nodes with outdegree s with respect to the total number N of nodes in the network. The term g_{1} describes the situation that a new node randomly selects an outdegree value and the term g_{2} the situation that a new node copies this value from a randomly selected node in the network.
Eq. 1 may be written in the standard form for a linear differential equation:
From Eq. 2, it is possible to deduce the integrating factor . Solving for I(N) produces non elementary functions, which complicate the solution of Eq. 2. In order to obtain an integrating factor in terms of elementary functions, it is best to simplify Eq. 2 as follows:
This simplification has little implications for large values of N, because N − 1 ≈ N, as N ≫ 1. This allows to employ the following integrating factor: . Multiplying Eq. 3 by I_{2}(N) produces:
Solving for Q_{s}(N)
where k is a constant and Γ(·) is the incomplete Gamma function. In order to obtain the outdegree distribution Q_{s}(N), it is necessary to solve Eq. 6 for s = 1, s = 2 and so on as follows:

for Q_{1}(N), consider the initial condition
this initial condition is due to the fact that, at the beginning the network only has one node, n_{0}, with no outgoing links (N = 1). When the next node, n_{1}, is added (N = 2), the probability that node n_{1} has outdegree s = 1 is .

Then, solving Eq. 6 for the initial condition produces:

for Q_{2}(N), consider the initial condition
this initial condition is due to the fact that, before adding node n_{2} only n_{0} and n_{1} exist in the network (N = 2) and both have s < 2, therefore Q_{2}(2) = 0. When n_{2} is added (N = 3), the probability that node n_{2} has outdegree s = 2 is .

Then, solving Eq. 6 with the initial condition , one obtains:
From the results in Eqs. 7 and 8, it is possible to deduce that:
Normalizing Eq. 9, yields:
Eq. 10 describes the outdegree distribution P_{s}(N) obtained with the proposed model for 1 < s < N. It can also be noted that, as s → N, Eq. 10 predicts that . That is P_{s}(N) decays to 0 rapidly as s → N and N ≫ 1, therefore the powerlaw behavior exhibits a cutoff (Figure 1a).
In order to obtain the scaling exponent of the outdegree distribution, terms Γ(·) into Eq. 10 are simplified using:
where γ(a, x) and Γ(a, x) are the lower and upper incomplete Gamma functions, respectively. By the following asymptotic property:
it is possible to write:
Using Eq. 11 it is possible rewrite the Γ(·) terms of Eq. 10 as follows:
Substituting Eqs. 12 and 13 into Eq. 10 and considering that s + 1 ≈ s as s ≫ 1, Eq. 10 can be expressed as:
Using the two first terms of the series expansion of in Eq. 15 and simplifying
for s ≫ 1, , thus it is possible to rewrite Eq. 16 as:
Furthermore, in the limit when N → ∞, Eq. 17 takes the form
Eq. 18 shows that the outdegree distribution obtained with the proposed model decays as a powerlaw P_{s} ~ s^{−γ} for 1 < s < N with scaling exponent γ = α + 1.
To validate the analytical solution of the model as described by Eq. 10, four experiments were executed using α = 0.5, 1, 1.5 and 2. Each of these experiments simulated the growth of a directed network from N = 1 to 10^{4} nodes. Figure 1b shows that the outdegree distribution produced by these experiments and the analytical predictions by Eq. 10 fit appropriately.
Comparison with real networks
To verify that the proposed model is able to reproduce the outdegree distribution of real CN, the social network of Flickr users^{6} was selected.
In this network, the users correspond to the nodes and their friendship connections to the links. This network has 2, 302, 925 nodes and 33, 140, 017 links. Figure 2a shows that the outdegree distribution of the nodes in the Flickr network decay as a powerlaw distribution with γ ≈ 1.74. Figure 2b shows that the model proposed by Eq. 10 with α = 0.74 and N = 2, 302, 925 reproduces appropriately the outdegree distribution of the Flickr network for s > 1.
Discussion
The model proposed in this article has been able to reproduce the outdegree distribution of the Flickr social network for values of s > 1. Although this model produces a good fit with the outdegree distribution of a real network, we cannot guarantee that the local processes incorporated in this model are the only ones involved in the behavior of the outdegree distribution of the nodes in this network. Unknown processes may help to explain why for s = 1, this model does not fit. However, the proposed model provides a simplification of these processes and therefore, reproduces the outdegree distribution of the network.
Conclusions
Local processes participate in the growth and evolution of real CN which, in turn, shape the outdegree of its nodes. The model proposed here incorporates two local processes: a random outdegree selection and a copy of an outdegree for the nodes added to the network. This model is able to produce outdegree distributions that decay as a powerlaw with the γ exponent in the range from 1 to ∞. That is, the proposed model reproduces all exponent values found in distributions of documented real complex networks.
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J.E.G., P.D.A.V., E.S.N., U.P.R., R.E.B.N. and J.A.E. contributed to this research and helped to edit this manuscript.
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EsquivelGómez, J., ArjonaVillicaña, P., StevensNavarro, E. et al. On a growth model for complex networks capable of producing powerlaw outdegree distributions with wide range exponents. Sci Rep 5, 9067 (2015). https://doi.org/10.1038/srep09067
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DOI: https://doi.org/10.1038/srep09067
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