A dynamic network in a dynamic population: asymptotic properties

TL;DR

This paper analyzes a stochastic dynamic network model with evolving nodes and edges, deriving conditions for giant component emergence and explicit degree correlation expressions, revealing how social indices influence network connectivity.

Contribution

It introduces a novel dynamic network model with social indices and derives asymptotic properties, including giant component criteria and degree correlations, expanding understanding of evolving social networks.

Findings

Giant component exists under specific growth conditions.

Degree correlation $ ho$ is always positive in one submodel.

Degree correlation $ ho$ can be positive or negative depending on parameters.

Abstract

We derive asymptotic properties for a stochastic dynamic network model in a stochastic dynamic population. In the model, nodes give birth to new nodes until they die, each node being equipped with a social index given at birth. During the life of a node it creates edges to other nodes, nodes with high social index at higher rate, and edges disappear randomly in time. For this model we derive criterion for when a giant connected component exists after the process has evolved for a long period of time, assuming the node population grows to infinity. We also obtain an explicit expression for the degree correlation $\rho$ (of neighbouring nodes) which shows that $\rho$ is always positive irrespective of parameter values in one of the two treated submodels, and may be either positive or negative in the other model, depending on the parameters.