Large Deviation Properties of Minimum Spanning Trees for Random Graphs

TL;DR

This work characterizes the entire distribution of the minimum spanning tree weight $W$ on two random-graph ensembles: complete graphs with iid edge weights in $[0,1]$ and connected Erdős-Rényi graphs with $p=c/N$. It employs a Markov-chain Monte Carlo large-deviation sampling with a Boltzmann bias $e^{-W/ heta}$ to extract $P(W)$ over extreme tails, verifying the large-deviation principle. For complete graphs, the mean MST weight converges to $igl\langle W igr angle_N o rac{oldsymbol{ ext{zeta}}(3)}{D}$ with $D=F'(0)=1$, and the right tail displays a stretched-exponential form with exponent $eta o 1.22(2)$; the full rate function $oldsymbol{igPhi}(w)$ collapses with $N$, and edge-weight-based bounds correlate strongly with $W$. For connected ER graphs, $W$ scales linearly with $N-1$ and the central part of the distribution is Gaussian for small $c$, while the tails become increasingly asymmetric as $c$ grows, with a percolation threshold at $c=1$ marking a qualitative change in tail structure. The results illuminate how graph structure and edge-weight randomness drive MST fluctuations and motivate future analytical works on rate functions and other graph ensembles.

Abstract

We study the large-deviation properties of minimum spanning trees for two ensembles of random graphs with $N$ nodes. First, we consider complete graphs. Second, we study Erdős-Rényi (ER) random graphs with edge probability $p=c/N$ conditioned to be connected. By using large-deviation Markov chain sampling, we are able to obtain the distribution $P(W)$ of the spanning-tree weight $W$ down to probability densities as small as $10^{-300}$. For the complete graph, we confirm analytical predictions with respect to the expectation value. For both ensembles, the large deviation principle is fulfilled. For the connected ER graphs, we observe a remarkable change of the distributions at the value of $c=1$, which is the percolation threshold for the original ER ensemble.

Figures (12)

• Figure 1: Equilibration of Markov chain: time series $W(t)$ for ER graphs with $N=512$ and $c=5$ for two different initial configurations with full graphs (all $w_{ij}=1$) and a line graph (all $w_{ij}$ near 0.)
• Figure 2: Average spanning tree weight $W$ as a function of the number $N$ of nodes for the complete graph. The horizontal line indicates the limiting value obtained analytically, see Eq. (\ref{['eq:expectation']}), while the upper solid curve prepresents Eq. (\ref{['eq:power-lawB']}). The lower solid curve indicates a fit to Eq. (\ref{['eq:power-law']}).
• Figure 3: Distribution of spanning tree weight $W$ for the complete graph with $N=256$ nodes.
• Figure 4: Rate functions $\phi(W)$ for the complete graph for different values of the number $N$ of nodes. The line shows the result of a fit of the data for $N=256$ to a power law Eq. (\ref{['eq:power:law:rate']}).
• Figure 5: Average of $W_{G}^{\text{min}}$, $W_{G}^{N-1}$, and (inset) $W_{G}^{\text{mean}}$, conditioned to the value $W$ of the MST of $G$, respectively. The line indicates the diagonal $W_G=W/(N-1)$. The vertical line near $W/(N-1)=0.002$ indicates the typical edge weight of an MST, corresponding to $P(W)$ exhibiting a peak near $W=1$ in Fig. \ref{['fig:dist_complete']} and $N=512$.