https://doi.org/10.1140/epjb/e2019-100139-5

Regular Article

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

Racah Institute of Physics, The Hebrew University,
Jerusalem
91904, Israel

^{a} e-mail: biham@phys.huji.ac.il

Received:
9
March
2019

Received in final form:
10
April
2019

Published online: 19 June 2019

We present exact analytical results for the distribution of shortest path lengths (DSPL) in a directed network model that grows by node duplication. 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. In this network not all pairs of nodes are connected by directed paths, in spite of the fact that the corresponding undirected network consists of a single connected component. More specifically, in the large network limit only a diminishing fraction of pairs of nodes are connected by directed paths. To calculate the DSPL between those pairs of nodes that are connected by directed paths we derive a master equation for the time evolution of the probability *P*_{t}(*L* = *ℓ*), *ℓ* = 1, 2, … , where *ℓ* is the length of the shortest directed path. Solving the master equation, we obtain a closed form expression for *P*_{t}(*L* = *ℓ*). It is found that the DSPL at time *t* consists of a convolution of the initial DSPL *P*_{0}(*L* = *ℓ*), with a Poisson distribution and a sum of Poisson distributions. The mean distance 𝔼_{t}[*L*|*L* < ∞] between pairs of nodes which are connected by directed paths is found to depend logarithmically on the network size *N*_{t}. However, since in the large network limit the fraction of pairs of nodes that are connected by directed paths is diminishingly small, the corded DND network is not a small-world network, unlike the corresponding undirected network.

Key words: Statistical and Nonlinear Physics

*© EDP Sciences / Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2019*