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, and on real-world datasets from the UCI suite. 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：在本研究中，我们探讨了网络拓扑结构对人工神经网络（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人工神经网络的新方法，包括Barabási-Albert、Erdős–Rényi、Watts-Strogatz以及多层感知机（MLP）。这些构建的网络在基于流形学习生成器生成的合成数据集（包含不同难度水平和噪声）以及UCI套件中的真实数据集上进行了评估。我们的研究结果表明，在高难度场景下，复杂拓扑结构相较于传统MLP展现出更优的性能。这一性能优势归因于复杂网络能够利用目标函数的组合结构特性。然而，这种优势是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的。此外，我们还探究了多种拓扑属性与模型性能之间的关系。分析表明，没有任何单一属性能够完全解释观察到的性能差异，这意味着网络拓扑结构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人工神经网络性能方面的潜力，并为未来探索多重拓扑属性间相互作用及其对模型性能影响的研究奠定了基础。