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.

翻译：通过图论视角探索网络结构已成为理解不同领域复杂系统的基石。在大型网络中识别密集连接的子图，对于揭示生物系统中的功能模块、社交网络中的凝聚群体以及技术基础设施中的关键路径至关重要。最典型的方法SM算法无法定位大尺寸子图，因而无法识别密集子图；而研究者先前使用的SA算法将模拟退火与马尔可夫链的高效移动相结合。然而，除非采用对数冷却调度，包括SA在内的模拟退火方法无法保证定位全局最优解。为此，本研究引入并评估了模拟退火算法（SAA）的性能，该算法将模拟退火与随机近似蒙特卡洛算法相结合。通过在嵌入团结构的模拟图上识别这些关键子图结构，我们检验了SAA与另外两种数值算法——SM和SA的性能对比。研究发现，SAA在以下三个指标上均优于SA和SM：1）找到最密集子图所需的迭代次数；2）经过10,000次迭代后能成功找到团的时间占比；3）计算时间。SAA算法展现出的优异结果为解析复杂系统提供了有力工具，并有可能变革我们解决跨学科问题的方法。