Dynamics of network growth and evolution: integrating non-constant edge growth, mixed attachment and reciprocity

Advancements in network science research have enriched our understanding of the mechanisms shaping the topological properties of networks over the past two decades. However, the existing models still grapple with limitations in fully capturing the diverse structures and behaviours observed in real-world networks. This paper addresses these limitations by examining network growth in networks characterised by a distinct in-degree distribution, exhibiting expected new edges in the head and an extended exponential or power-law behaviour in the tail. To overcome these complexities, we propose a comprehensive model encompassing non-constant edge establishment driven by mixed attachment and reciprocal mechanisms. This extension offers a more accurate representation of real-world networks. Utilising various discrete probability distributions, including Poisson, binomial, zeta and log-series, our model accommodates variations in the number of new edges, providing a realistic depiction of network evolution. Analytical expressions for the limit in- and out-degree distributions and evolving dynamics of cumulative complementary in- and out-degree distributions are derived. These findings enable a detailed assessment of each mechanism’s contribution to the head and tail of the in- and out-degree distributions. Furthermore, we validate the practical relevance of our model by fitting it to real-world networks, emphasising the impact of the number of new edges and reciprocity.