In this study, we explore the impact of network topology on the approximation capabilities of artificial neural networks (ANNs), with a particular focus on complex topologies. We propose a novel methodology for constructing complex ANNs based on various topologies, including Barab\'asi-Albert, Erd\H{o}s-R\'enyi, Watts-Strogatz, and multilayer perceptrons (MLPs). The constructed networks are evaluated on synthetic datasets generated from manifold learning generators, with varying levels of task difficulty and noise. Our findings reveal that complex topologies lead to superior performance in high-difficulty regimes compared to traditional MLPs. This performance advantage is attributed to the ability of complex networks to exploit the compositionality of the underlying target function. However, this benefit comes at the cost of increased forward-pass computation time and reduced robustness to graph damage. Additionally, we investigate the relationship between various topological attributes and model performance. Our analysis shows that no single attribute can account for the observed performance differences, suggesting that the influence of network topology on approximation capabilities may be more intricate than a simple correlation with individual topological attributes. Our study sheds light on the potential of complex topologies for enhancing the performance of ANNs and provides a foundation for future research exploring the interplay between multiple topological attributes and their impact on model performance.

翻译：本研究探讨网络拓扑对人工神经网络（ANN）逼近能力的影响，重点关注复杂拓扑结构。我们提出了一种基于多种拓扑结构构建复杂ANN的新方法，包括Barabási-Albert、Erdős-Rényi、Watts-Strogatz以及多层感知器（MLP）。利用流形学习生成器生成的合成数据集对所建网络进行评估，数据集包含不同任务难度和噪声水平。研究结果表明，与传统MLP相比，复杂拓扑结构在高难度任务条件下具备更优性能。这一性能优势归因于复杂网络利用底层目标函数组合特性的能力。然而，该优势以增加前向传播计算时间和降低对图损伤的鲁棒性为代价。此外，我们探究了多种拓扑属性与模型性能之间的关系。分析表明，单一属性无法解释所观察到的性能差异，这表明网络拓扑对逼近能力的影响可能比与单个拓扑属性的简单相关性更为复杂。本研究揭示了复杂拓扑结构在提升ANN性能方面的潜力，为未来探索多拓扑属性协同作用及其对模型性能影响的研究奠定了基础。