Minimum spanning trees across dense cities

TL;DR

This paper analyzes the length of minimum spanning trees in dense city models and the entire unit square, providing variance estimates and convergence results under certain connectivity and density conditions.

Contribution

It introduces new approximation techniques for variance estimation and convergence analysis of MST lengths in dense urban and uniform settings.

Findings

Variance of MST length grows at most as a power of log(n).

MST length converges to zero after centering and scaling in dense city models.

Almost sure convergence of MST length in the uniform case.

Abstract

Consider~\(n\) nodes distributed independently across~\(N\) cities contained with the unit square~\(S\) according to a distribution~\(f.\) Each city is modelled as an~\(r_n \times r_n\) square contained within~\(S\) and~\(MSTC_n\) denotes the length of the minimum spanning tree containing all the~\(n\) nodes. We use approximation methods to obtain variance estimates for~\(MSTC_n\) and prove that if the cities are well-connected and densely populated in a certain sense, then~\(MSTC_n\) appropriately centred and scaled converges to zero in probability. Using the proof techniques, we alternately derive corresponding results for the length~\(MST_n\) of the minimum spanning tree for the usual case when the nodes are independently distributed throughout the unit square~\(S.\) In particular, we obtain that the variance of~\(MST_n\) grows at most as a power of the logarithm of~\(n\) and use a subsequence argument to get almost sure convergence of~\(MST_n\) appropriately centred and scaled.