Yichun Yang

Yichun Yang, Longlong Lin, Rong-Hua Li et al.

The computation of resistance distance is pivotal in a wide range of graph analysis applications, including graph clustering, link prediction, and graph neural networks. Despite its foundational importance, efficient algorithms for computing resistance distances on large graphs are still lacking. Existing state-of-the-art (SOTA) methods, including power iteration-based algorithms and random walk-based local approaches, often struggle with slow convergence rates, particularly when the condition number of the graph Laplacian matrix, denoted by $κ$, is large. To tackle this challenge, we propose two novel and efficient algorithms inspired by the classic Lanczos method: Lanczos Iteration and Lanczos Push, both designed to reduce dependence on $κ$. Among them, Lanczos Iteration is a near-linear time global algorithm, whereas Lanczos Push is a local algorithm with a time complexity independent of the size of the graph. More specifically, we prove that the time complexity of Lanczos Iteration is $\tilde{O}(\sqrtκ m)$ ($m$ is the number of edges of the graph and $\tilde{O}$ means the complexity omitting the $\log$ terms) which achieves a speedup of $\sqrtκ$ compared to previous power iteration-based global methods. For Lanczos Push, we demonstrate that its time complexity is $\tilde{O}(κ^{2.75})$ under certain mild and frequently established assumptions, which represents a significant improvement of $κ^{0.25}$ over the SOTA random walk-based local algorithms. We validate our algorithms through extensive experiments on eight real-world datasets of varying sizes and statistical properties, demonstrating that Lanczos Iteration and Lanczos Push significantly outperform SOTA methods in terms of both efficiency and accuracy.

Weiming Feng, Heng Guo, Yichun Yang

We introduce two new models equivalent to ferromagnetic two-spin systems: a weighted subgraph model and a random cluster type model. Using these new connections, we obtain an efficient sampling algorithm and a new randomised algorithm that efficiently approximates the partition function of ferromagnetic two-spin systems in certain parameter regimes. No efficient sampling algorithms are known before in this regime, and our new estimation algorithm runs in near-quadratic time for bounded degree graphs and in polynomial time for general graphs, improving upon the previous algorithm of Guo, Liu, and Lu (2020).

Pan Peng, Yuyang Wang, Joy Qiping Yang et al.

We study lower bounds for estimating the spectral density of the normalized adjacency matrix of a graph. Previously, Cohen-Steiner et al. [KDD 2018] proposed an algorithm for $\varepsilon$-approximate spectral density estimation in the Wasserstein-1 distance, using $2^{O(1/\varepsilon)}$ random walks initiated from uniformly random nodes in the graph. Later, Jin et al. [COLT 2023] established a nearly matching exponential lower bound for \emph{weighted} graphs, assuming the algorithm has access to samples from random walks started at random nodes. It was left open whether this lower bound could be extended to \emph{unweighted} graphs. In this paper, we answer this question in the affirmative by proving an exponential lower bound for unweighted graphs. Specifically, we show that no algorithm can compute an $\varepsilon$-approximation to the spectrum of a normalized graph adjacency matrix with constant success probability, even when given the full transcripts of $2^{Ω(1/\varepsilon^{1/6})}$ random walks, each of length $2^{Ω(1/\varepsilon^{1/6})}$, started from uniformly random nodes.