In a random graph, counts for the number of vertices with given degrees will typically be depende... more In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly inhomogeneous distribution, only when the degrees are large can we reasonably approximate the joint counts as independent. The proofs are based on Stein's method and the Stein-Chen method with a new size-biased coupling for such inhomogeneous random graphs, and, hence, bounds on the distributional distance are obtained. Finally, we illustrate that apparent (pseudo-)power-law-type behaviour can arise in such inhomogeneous networks despite not actually following a power-law degree distribution.
Both small-world models of random networks with occasional long-range connections and gossip proc... more Both small-world models of random networks with occasional long-range connections and gossip processes with occasional long-range transmission of information have similar characteristic behaviour. The long-range elements appreciably reduce the effective distances, measured in space or in time, between pairs of typical points. In this paper we show that their common behaviour can be interpreted as a product of the locally branching nature of the models. In particular, it is shown that both typical distances between points and the proportion of space that can be reached within a given distance or time can be approximated by formulae involving the limit random variable of the branching process.
In Gorham et al. [2020] asymptotic consistency for stochastic Stein discrepancies is shown for St... more In Gorham et al. [2020] asymptotic consistency for stochastic Stein discrepancies is shown for Stein operators which yield continuous functions. Due to this continuity assumption the techniques applied in Gorham et al. [2020] differ considerably to then ones we shall employ here. Moreover, while the results in Gorham et al. [2020] are of asymptotic nature, the results in this section give computable bounds.
In a random graph, counts for the number of vertices with given degrees will typically be depende... more In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly inhomogeneous distribution, only when the degrees are large can we reasonably approximate the joint counts as independent. The proofs are based on Stein's method and the Stein-Chen method with a new size-biased coupling for such inhomogeneous random graphs, and, hence, bounds on the distributional distance are obtained. Finally, we illustrate that apparent (pseudo-)power-law-type behaviour can arise in such inhomogeneous networks despite not actually following a power-law degree distribution.
Both small-world models of random networks with occasional long-range connections and gossip proc... more Both small-world models of random networks with occasional long-range connections and gossip processes with occasional long-range transmission of information have similar characteristic behaviour. The long-range elements appreciably reduce the effective distances, measured in space or in time, between pairs of typical points. In this paper we show that their common behaviour can be interpreted as a product of the locally branching nature of the models. In particular, it is shown that both typical distances between points and the proportion of space that can be reached within a given distance or time can be approximated by formulae involving the limit random variable of the branching process.
In Gorham et al. [2020] asymptotic consistency for stochastic Stein discrepancies is shown for St... more In Gorham et al. [2020] asymptotic consistency for stochastic Stein discrepancies is shown for Stein operators which yield continuous functions. Due to this continuity assumption the techniques applied in Gorham et al. [2020] differ considerably to then ones we shall employ here. Moreover, while the results in Gorham et al. [2020] are of asymptotic nature, the results in this section give computable bounds.
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