Connectivity of Large Scale Networks: Emergence of Unique Unbounded Component

TL;DR

This paper proves that large-scale networks with probabilistic connections form a single unbounded component as density increases, extending classical geometric graph results to more realistic connection models.

Contribution

It establishes the emergence of a unique unbounded component in large random networks under a general connection function, broadening the scope of connectivity results beyond the unit disk model.

Findings

Asymptotically almost surely, only isolated nodes and one unbounded component exist at high density.

No finite-size components of fixed order appear as density tends to infinity.

Connectivity is achieved when there are no isolated nodes in the network.

Abstract

This paper studies networks where all nodes are distributed on a unit square $A\triangleq[(-1/2,1/2)^{2}$ following a Poisson distribution with known density $\rho$ and a pair of nodes separated by an Euclidean distance $x$ are directly connected with probability $g(\frac{x}{r_{\rho}})$, independent of the event that any other pair of nodes are directly connected. Here $g:[0,\infty)\rightarrow[0,1]$ satisfies the conditions of rotational invariance, non-increasing monotonicity, integral boundedness and $g(x)=o(\frac{1}{x^{2}\log^{2}x})$; further, $r_{\rho}=\sqrt{\frac{\log\rho+b}{C\rho}}$ where $C=\int_{\Re^{2}}g(\Vert \boldsymbol{x}\Vert)d\boldsymbol{x}$ and $b$ is a constant. Denote the above network by\textmd{}$\mathcal{G}(\mathcal{X}_{\rho},g_{r_{\rho}},A)$. We show that as $\rho\rightarrow\infty$, asymptotically almost surely a) there is no component in $\mathcal{G}(\mathcal{X}_{\rho},g_{r_{\rho}},A)$ of fixed and finite order $k>1$; b) the number of components with an unbounded order is one. Therefore as $\rho\rightarrow\infty$, the network asymptotically almost surely contains a unique unbounded component and isolated nodes only; a sufficient condition for $\mathcal{G}(\mathcal{X}_{\rho},g_{r_{\rho}},A)$ to be asymptotically almost surely connected is that there is no isolated node in the network.{\normalsize{}}The contribution of these results, together with results in a companion paper on the asymptotic distribution of isolated nodes in \textmd{\normalsize $\mathcal{G}(\mathcal{X}_{\rho},g_{r_{\rho}},A)$}, is to expand recent results obtained for connectivity of random geometric graphs from the unit disk model to the more generic and more practical random connection model.