We study the problem of network regression, where one is interested in how the topology of a network changes as a function of Euclidean covariates. We build upon recent developments in generalized regression models on metric spaces based on Fr\'echet means and propose a network regression method using the Wasserstein metric. We show that when representing graphs as multivariate Gaussian distributions, the network regression problem requires the computation of a Riemannian center of mass (i.e., Fr\'echet means). Fr\'echet means with non-negative weights translates into a barycenter problem and can be efficiently computed using fixed point iterations. Although the convergence guarantees of fixed-point iterations for the computation of Wasserstein affine averages remain an open problem, we provide evidence of convergence in a large number of synthetic and real-data scenarios. Extensive numerical results show that the proposed approach improves existing procedures by accurately accounting for graph size, topology, and sparsity in synthetic experiments. Additionally, real-world experiments using the proposed approach result in higher Coefficient of Determination ($R^{2}$) values and lower mean squared prediction error (MSPE), cementing improved prediction capabilities in practice.

翻译：本文研究网络回归问题，即探究网络拓扑结构如何随欧几里得协变量变化。基于近期基于弗雷歇均值的度量空间广义回归模型进展，我们提出了一种利用Wasserstein度量的网络回归方法。研究表明，当将图表示为多元高斯分布时，网络回归问题需要计算黎曼中心质量（即弗雷歇均值）。具有非负权重的弗雷歇均值可转化为重心问题，并能通过定点迭代高效计算。尽管Wasserstein仿射平均计算中定点迭代的收敛性保证仍是开放问题，我们在大量合成与真实数据场景中提供了收敛证据。大量数值结果表明，所提方法在合成实验中通过精确考虑图规模、拓扑结构与稀疏性，改进了现有流程。此外，在真实世界实验中采用该方法获得了更高的决定系数（$R^{2}$）值与更低的均方预测误差（MSPE），在实践中巩固了其提升的预测能力。