Minimum spanning trees of random geometric graphs with location dependent weights

Abstract

Consider n nodes ${\{{\mathit{X}}_{\mathit{i}}\}}_{1\le \mathit{i}\le \mathit{n}}$ independently distributed in the unit square S, each according to a distribution f. Nodes ${\mathit{X}}_{\mathit{i}}$ and ${\mathit{X}}_{\mathit{j}}$ are joined by an edge if the Euclidean distance $\mathit{d}({\mathit{X}}_{\mathit{i}},{\mathit{X}}_{\mathit{j}})$ is less than ${\mathit{r}}_{\mathit{n}}$, the adjacency distance and the resulting random graph ${\mathit{G}}_{\mathit{n}}$ is called a random geometric graph (RGG). We now assign a location dependent weight to each edge of ${\mathit{G}}_{\mathit{n}}$ and define ${\mathit{MST}}_{\mathit{n}}$ to be the sum of the weights of the minimum spanning trees of all components of ${\mathit{G}}_{\mathit{n}}$. For values of ${\mathit{r}}_{\mathit{n}}$ above the connectivity regime, we obtain upper and lower bound deviation estimates for ${\mathit{MST}}_{\mathit{n}}$ and ${\mathit{L}}^2$-convergence of ${\mathit{MST}}_{\mathit{n}}$ appropriately scaled and centred.