In this paper we focus on jointly estimating the edge probabilities for multi-layer networks. We define a novel multi-layer graphon, a ternary function in contrast to the bivariate graphon function in the literature by introducing an additional latent layer position parameter, which is model-free and covers a wide range of multi-layer networks. We develop a computationally efficient two-step neighborhood smoothing algorithm to estimate the edge probabilities of multi-layer networks, which requires little tuning and fully utilize the similarity across both network layers and nodes. Numerical experiments demonstrate the advantages of our method over the existing state-of-the-art ones. A real Worldwide Food Import/Export Network dataset example is analyzed to illustrate the better performance of the proposed method over benchmark methods in terms of link prediction.

翻译：本文聚焦于多层网络边概率的联合估计问题。通过引入额外的潜在层位置参数，我们定义了一种新颖的多层图函数——该函数为三元函数，区别于文献中常见的二元图函数，其具有模型无关性且能涵盖广泛的多层网络类型。我们提出了一种计算高效的两步邻域平滑算法来估计多层网络的边概率，该算法需要极少的参数调优，并能充分利用网络层间与节点间的相似性。数值实验证明了本方法相较于现有最优方法的优势。通过对真实全球食品进出口网络数据集的分析，我们展示了所提方法在链接预测任务中相较于基准方法的更优性能。