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.

我们研究了网络回归问题，通过基于弗雷歇平均和使用 Wasserstein 度量的广义回归模型，提出了一种网络回归方法。通过将图形表示为多变量高斯分布，我们展示了网络回归问题需要计算一个 Riemannian 中心（即 Frechet 平均）。通过固定点迭代可以有效计算具有非负权重的 Frechet 平均，该方法在合成和实际数据情景中的大量数值结果表明改进了现有程序，准确考虑了图形的大小、拓扑和稀疏性。此外，使用该方法的实际实验结果也显示出更高的决定系数（$R^{2}$）值和更低的均方预测误差（MSPE），从而巩固了在实践中改进的预测能力。