Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2405.11688

Performance Analysis of Monte Carlo Algorithms in Dense Subgraph Identification

Wanru Guo
View original source ↗Read source document ↗Export metadata ↓

Abstract

The exploration of network structures through the lens of graph theory has become a cornerstone in understanding complex systems across diverse fields. Identifying densely connected subgraphs within larger networks is crucial for uncovering functional modules in biological systems, cohesive groups within social networks, and critical paths in technological infrastructures. The most representative approach, the SM algorithm, cannot locate subgraphs with large sizes, therefore cannot identify dense subgraphs; while the SA algorithm previously used by researchers combines simulated annealing and efficient moves for the Markov chain. However, the global optima cannot be guaranteed to be located by the simulated annealing methods including SA unless a logarithmic cooling schedule is used. To this end, our study introduces and evaluates the performance of the Simulated Annealing Algorithm (SAA), which combines simulated annealing with the stochastic approximation Monte Carlo algorithm. The performance of SAA against two other numerical algorithms-SM and SA, is examined in the context of identifying these critical subgraph structures using simulated graphs with embeded cliques. We have found that SAA outperforms both SA and SM by 1) the number of iterations to find the densest subgraph and 2) the percentage of time the algorithm is able to find a clique after 10,000 iterations, and 3) computation time. The promising result of the SAA algorithm could offer a robust tool for dissecting complex systems and potentially transforming our approach to solving problems in interdisciplinary fields.

Explore related subjects

stat.COstat.ME

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Wanru Guo. 2024-05-19. Performance Analysis of Monte Carlo Algorithms in Dense Subgraph Identification. https://arxiv.org/abs/2405.11688

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Estimating Hierarchically Rank Structured Covariance Matrices

We consider the problem of estimating a high-dimensional covariance matrix from a very limited number of samples. This problem is ubiquitous in computational fluid dynamics, where a small number of fluid snapshots must be used to construct a Gramian matrix determining a reduced-order model, as well as in computational geoscience, where a small ensemble of Earth system forecasts must be used to estimate the covariance matrix associated with the forecast uncertainty. It is common practice to regularize the small-sample covariance by imposing a "localization" structure that enforces a physically realistic correlation length scale, imposing a sparsity constraint, "shrinking" towards a prescribed target, or attenuating small correlations. We propose an alternate technique that regularizes the small-sample covariance by imposing hierarchical rank structure. Compared to regularization methods that assume sparsity such as spatial localization, hierarchical rank structure accommodates a wider range of covariance matrices, roughly corresponding to situations where long-range correlations vary more smoothly than short-range ones. It also results in a data-sparse matrix format that permits highly efficient matrix-vector products. We present theory and algorithms which show how to efficiently estimate a high-dimensional, hierarchically rank structured covariance matrix from limited samples. Through an error analysis and numerical experiments with a variety of model problems, we demonstrate that these techniques are effective at reducing sampling errors, and that in many cases they achieve smaller estimation error than conventional techniques.

stat.CO↗
arXiv2026

Optimal Slice-Adaptive Tuning of Hybrid Slice Sampling

Slice sampling is a Markov chain Monte Carlo algorithm that draws its next state uniformly from a "slice"---a super-level set of the target density function---at each iteration, thereby providing automatic local adaptivity to the scale of the target. In practice the exact slice is not known, so general-purpose implementations use an approximate slice that is grown from a starting interval of length $w>0$, with a computational cost that depends on $w$. This work presents an analysis of the average per-iteration number of target density evaluations, as a function of $w$, of hybrid slice sampling with various slice-finding schemes for targets with contiguous slices. The paper uses the results of the analysis to develop automated, slice-adaptive tuning schemes along with suboptimality bounds and asymptotic convergence guarantees. Simulations demonstrate that the tuning schemes reliably yield near-optimal slice-adaptive tuning with essentially no dependence on the initial setting of $w$.

stat.CO↗