Analytical results for the in-degree and out-degree distributions of directed random networks that grow by node duplication

Abstract

We present exact analytical results for the degree distribution in a directed network model that grows by node duplication (ND). Such models are useful in the study of the structure and growth dynamics of gene regulatory networks and scientific citation networks. Starting from an initial seed network, at each time step a random node, referred to as a mother node, is selected for duplication. Its daughter node is added to the network and duplicates each outgoing link of the mother node with probability p . In addition, the daughter node forms a directed link to the mother node itself. Thus, the model is referred to as the corded directed-node-duplication (DND) model. The corresponding undirected ND model was studied before and was found to exhibit a power-law degree distribution. We obtain analytical results for the in-degree distribution P t (K in = k), and for the out-degree distribution P t (K out = k), of the corded DND network at time t. It is found that the in-degrees follow a shifted powerlaw distribution, so the network is asymptotically scale free. In contrast, the out-degree distribution is a narrow distribution, that converges to a Poisson distribution in the limit of p 1 and to a Gaussian distribution in the limit of p 1. Such distinction between a broad in-degree distribution and a narrow out-degree distribution is common in empirical networks such as scientific citation networks. Using these distributions we calculate the mean degree K in t = K out t , which converges to 1/(1 − p) in the large network limit, for the whole range of 0 < p < 1.

Statistical analysis of edges and bredges in configuration model networks

“…However, apart from a few studies [26,[61][62][63][64][65][66] the DSPL has not attracted nearly as much attention as the degree distribution. Recently, an analytical approach was developed for calculating the DSPL in the (ER) network [67], followed by more general formulations that apply to configuration model networks [43,68,69], to modular networks [70] and to networks that form by kinetic growth processes [71][72][73].…”

“…8(b) we present the probability P (e ∈ B) that a random edge in a configuration model network with an exponential degree distribution is a bredge (solid line), as a function of c, obtained from Eq. (73). We also present the probability P (e ∈ B, GC) (dashed line) that a randomly selected edge in the network is a bredge that resides in the giant component and the probability P (e ∈ B, FC) (dotted line) that a randomly selected edge in the network is a bredge that resides in one of the finite components.…”

Cumulative structure and path length in networks of knowledge

“…In this model, nodes are more likely to connect to nodes that already have a large number of knowledge connections, referred to by Price as the 'cumulative advantage effect' 2 also known as, in the context of un-directed links, 'preferential attachment' [7]. In many applications of the Price model, the focus is on degree distributions, which describe how outgoing or incoming links are distributed over nodes [7,41,35]. While these distributions to an important extent determine network structures, they are mainly revealing for the variation in the relative importance of nodes, and perhaps less useful to study to what extent there is knowledge flow in such networks.…”

Analytical results for the distribution of shortest path lengths in directed random networks that grow by node duplication

“…In a more recent paper we introduced a directed version of the corded node duplication network model and studied its in-degree and out-degree distributions [66]. This model is referred to as the corded directed node duplication (DND) model.…”

“…Therefore, the degree distribution consists of two separate distributions, namely the distribution P t (K in = k) of in-degrees and the distribution P t (K out = k) of out-degrees, at time t. These two distributions are related to each other by the constraint that their means must be equal, namely K in t = K out t . We obtained exact analytical results for the in-degree distribution and the out-degree distribution of the corded DND network [66]. It was found that the in-degrees follow a shifted power-law distribution while the out-degrees follow a narrow distribution that converges to a Poisson distribution in the sparse network limit and to a Gaussian distribution in the dense network limit.…”