We study the problem of change point localization in dynamic networks models. We assume that we observe a sequence of independent adjacency matrices of the same size, each corresponding to a realization of an unknown inhomogeneous Bernoulli model. The underlying distribution of the adjacency matrices are piecewise constant, and may change over a subset of the time points, called change points. We are concerned with recovering the unknown number and positions of the change points. In our model setting, we allow for all the model parameters to change with the total number of time points, including the network size, the minimal spacing between consecutive change points, the magnitude of the smallest change and the degree of sparsity of the networks. We first identify a region of impossibility in the space of the model parameters such that no change point estimator is provably consistent if the data are generated according to parameters falling in that region. We propose a computationally-simple algorithm for network change point localization, called network binary segmentation, that relies on weighted averages of the adjacency matrices. We show that network binary segmentation is consistent over a range of the model parameters that nearly cover the complement of the impossibility region, thus demonstrating the existence of a phase transition for the problem at hand. Next, we devise a more sophisticated algorithm based on singular value thresholding, called local refinement, that delivers more accurate estimates of the change point locations. Under appropriate conditions, local refinement guarantees a minimax optimal rate for network change point localization while remaining computationally feasible.

中文翻译：

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稀疏动态网络中的最佳变化点检测与定位

我们研究动态网络模型中的更改点本地化问题。我们假设我们观察到一系列大小相同的独立邻接矩阵，每个矩阵对应于一个未知的不均匀伯努利模型的实现。邻接矩阵的基础分布是分段恒定的，并且可能会在时间点的一个子集（称为变化点）上发生变化。我们关心的是找回变更点的未知数量和位置。在我们的模型设置中，我们允许所有模型参数随时间点的总数而变化，包括网络大小，连续变化点之间的最小间距，最小变化的幅度和网络的稀疏程度。我们首先在模型参数的空间中确定一个不可能的区域，以便如果根据落入该区域的参数生成数据，则没有任何变化点估计值可证明是一致的。我们提出了一种计算简单的网络更改点定位算法，称为网络二进制分段，该算法依赖于邻接矩阵的加权平均值。我们表明，网络二进制分割在几乎覆盖不可能区域补余的模型参数范围内是一致的，从而说明了手头问题的相变存在。接下来，我们设计基于奇异值阈值的更复杂的算法，称为局部优化，该算法可提供更准确的变化点位置估计。在适当的条件下，