Random Weighted Uniform Spanning Trees

• Random Weighted Uniform Spanning Trees are probabilistic models that generate spanning trees in graphs by assigning random weights to edges, generalizing classical USTs.
• They exhibit distinctive phase transitions and scaling behaviors, where changes in disorder strength shift the tree structure from CRT-like to MST-like regimes.
• Advanced sampling algorithms, from Wilson's technique to quantum speedup methods, enable efficient generation and analysis in complex network applications.

Random Weighted Uniform Spanning Trees are probabilistic models and algorithms for generating spanning trees in finite graphs where edge weights are assigned randomly and the likelihood of a tree is determined by the product of its constituent edge weights. The study of such models encompasses both combinatorial properties and computational algorithms, including connections to statistical mechanics (through Gibbs measures), effective resistance metric, percolation theory, and spectral graph theory. These models generalize the classical uniform spanning tree (UST) model by incorporating randomness in the edge weights, yielding rich phase transitions and universality phenomena.

1. Model Definition and Probability Law

Let $G = (V, E)$ be a connected undirected graph of $n$ vertices. Each edge $e$ may be assigned a (random) positive weight $w_e$, often drawn i.i.d. from some continuous distribution (such as $\mathrm{Unif}(0,1)$). The distribution over spanning trees $T\subset E$ is a Gibbs measure

$P_w(T) = \frac{1}{Z(w)} \prod_{e \in T} w_e,$

where $Z(w) = \sum_{T'} \prod_{e \in T'} w_e$ is the partition function summing over all spanning trees $T'$ of $G$ (Makowiec et al., 22 Oct 2024, Makowiec et al., 2023). In special cases such as $w_e \equiv 1$, one recovers the classical uniform spanning tree law.

A prominent variant is the random spanning tree in random environment (RSTRE) on the complete graph $K_n$, with edge weights defined by $w_e = \exp(-\beta \omega_e)$ for i.i.d. $\omega_e \sim \mathrm{Unif}(0,1)$ and disorder strength (inverse temperature) $\beta \geq 0$. As $\beta$ varies, the model interpolates continuously between the UST ($\beta=0$) and the minimum spanning tree ($\beta\to\infty$) (Makowiec et al., 22 Oct 2024).

2. Diameter Scaling and Phase Transitions

The graph distance diameter of a random spanning tree under random weighting exhibits universal scaling laws and sharp transitions determined by the disorder strength $\beta$:

• Low Disorder ($\beta \leq C n/\log n$): The diameter is $\Theta(n^{1/2})$, matching the UST regime (Makowiec et al., 22 Oct 2024, Makowiec et al., 2023).
• High Disorder ($\beta \geq n^{4/3}\log n$): The diameter is $\Theta(n^{1/3})$, as in the MST regime (Makowiec et al., 22 Oct 2024).

An intermediate regime ($\beta = n^\alpha$ with $\alpha\in(1,4/3)$) is conjectured to induce diameter scaling $n^{\gamma(\alpha)+o(1)}$ for a continuous exponent $\gamma(\alpha)\in (1/3,1/2)$, establishing a one-parameter family interpolating between CRT-like and MST-like behavior (Makowiec et al., 22 Oct 2024).

For bounded-degree expander graphs and high-dimensional tori, the diameter of the weighted UST under i.i.d. edge weights remains of order $n^{1/2+o(1)}$ with high probability, provided the base graph maintains robust expansion (Makowiec et al., 2023).

3. Effective Resistance and Marginal Edge Probabilities

Marginal probabilities for edges in random weighted spanning trees are exactly characterized by Kirchhoff's Matrix-Tree Theorem: $P_w(e \in T) = w_e R_w(u, v),$ where $R_w(u, v)$ is the effective resistance between $u$ and $v$ in the weighted network defined by conductances $w_e$ (Makowiec et al., 22 Oct 2024, Madry et al., 2015). Covariances of edge indicators and node degree moments can be written in closed form using Laplacian minors and their inverses (Sanmartín et al., 20 Sep 2024), leveraging

$$P_w((u, v) \in T) = w_{uv}[M_{uu} + M_{vv} - 2M_{uv}],$$

where $M = (L(w)^{[r]})^{-1}$ is the inverse Laplacian minor after deleting root $r$.

Negative correlations between edge inclusion events yield strong concentration bounds, and play a central role both in statistical inference and algorithmic sampling (Dolev et al., 2016).

4. Algorithms for Weighted Random Spanning Trees

A variety of algorithmic paradigms exist for sampling random weighted spanning trees:

• Wilson's Algorithm / Loop-Erased Random Walk: Generalizes to the weighted setting by biasing transitions proportional to edge weights, applicable on planar and general graphs (Madry et al., 2015, Cannon et al., 15 Aug 2025).
• Aldous–Broder Algorithm: Weighted random walks for tree generation; cover time is governed by the maximal effective resistance, yielding $O(m n \log n)$ total time in the worst case (Dolev et al., 2016).
• Recursive Effective Resistance Partitioning: A framework achieving $\tilde{O}(m^{4/3})$ time via recursive partitioning in the effective resistance metric, shortcut random walks, and Laplacian solvers (Madry et al., 2015).
• Laplacian Solver Shortcuts and Schur Complements: Recent advances provide almost-linear-time exact and $\epsilon$-approximate samplers, with weight-independent complexity: $O(m^{1+o(1)}\beta^{o(1)})$ (exact), $O(m^{1+o(1)}\epsilon^{-o(1)})$ (approximate) (Schild, 2017).
• Quantum Speedup: A quantum algorithm samples random weighted spanning trees in $\tilde{O}(\sqrt{mn})$ time, leveraging quantum resistance oracles and sampling-without-replacement techniques, and is provably optimal up to polylogarithmic factors (Apers et al., 22 Apr 2025).

On planar grid-like graphs, divide-and-conquer algorithms using partial duality and separators achieve $O(n \log n)$ exact sampling for both weighted and unweighted cases (Cannon et al., 15 Aug 2025).

5. Spectral Sparsification and Applications

The union of $O(\log n/\epsilon^2)$ independently-sampled random spanning trees yields, after suitable reweighting, a spectral sparsifier of the original graph: for any $x \in \mathbb{R}^n$,

$$(1-\epsilon)x^T L_G x \leq x^T L_H x \leq (1+\epsilon)x^T L_G x,$$

where $L_H$ is the Laplacian of the union and $L_G$ the original Laplacian. This exploits matrix concentration for sums of rank-one projection matrices associated with trees (Dolev et al., 2016). Tree-based sparsification directly impacts algorithms for virtual network security and monitoring in software-defined networks.

6. Universality and Scaling Limits

On high-dimensional graphs and under wide classes of edge weight distributions, the metric space of a weighted uniform spanning tree (after rescaling distances by $n^{-1/2}$) converges in law to the Brownian continuum random tree (CRT) (Addario-Berry et al., 2020, Makowiec et al., 2023). For more general degree sequences, tree-weighted random graphs also yield the CRT limit under mild variance conditions, via an additive coalescent construction (Addario-Berry et al., 2020).

Heavy-tailed weight distributions or unbounded-degree graphs may force the diameter to collapse, with the UST concentrating on the MST-like regime ($n^{1/3+o(1)}$ diameter), showing the necessity of bounded-degree and good expansion for universality (Makowiec et al., 2023, Makowiec et al., 22 Oct 2024).

7. Local Observables and Statistical Properties

Explicit formulas for expectations, variance, and covariance of node degrees in random weighted spanning trees involve Laplacian minors and can be extended to directed graphs (arborescences) (Sanmartín et al., 20 Sep 2024). The distributional structure integrates both edge probability weights and arbitrary degree weights and can be extracted via trace and determinant identities over scaled Laplacians. The full distribution is encoded in the coefficients of determinant expansions, connecting combinatorial structures to spectral graph theory.

---

In sum, the theory and algorithms of random weighted uniform spanning trees fuse combinatorics, spectral theory, probabilistic metric geometry, and algorithmic innovation—yielding a rich framework for both foundational study and diverse applications in sampling, network science, and randomized matrix algorithms.