随机块模型图的谱嵌入性能及网络结构的研究
On spectral embedding performance and elucidating network structure in stochastic block model graphs
摘要：
图上的统计推断通常通过涉及矩阵值图表示的低维嵌入的谱方法进行，例如图拉普拉斯或邻接矩阵。在随机块模型图中，利用Laplacian谱嵌入和邻接谱嵌入的信息论性质，分析了块分配恢复算法的渐近信息论相对性能.我们研究了谱嵌入性能与底层网络结构（例如~同质性，亲和性，核心-外围，（不）平衡性）通过综合处理的两块随机块模型和类的$K$ -块模型表现出同质平衡的亲和性结构。我们的研究结果支持这样的说法，即对于稀疏性的特定概念，松散地说，“拉普拉斯谱嵌入有利于相对稀疏的图，而邻接谱嵌入有利于不太稀疏的图。我们还提供了证据来支持“邻接谱嵌入有利于核心-外围网络结构”的说法。"
英文摘要：
Statistical inference on graphs often proceeds via spectral methods involving low-dimensional embeddings of matrix-valued graph representations, such as the graph Laplacian or adjacency matrix. In this paper, we analyze the asymptotic information-theoretic relative performance of Laplacian spectral embedding and adjacency spectral embedding for block assignment recovery in stochastic block model graphs by way of Chernoff information. We investigate the relationship between spectral embedding performance and underlying network structure (e.g.~homogeneity, affinity, core-periphery, (un)balancedness) via a comprehensive treatment of the two-block stochastic block model and the class of $K$-block models exhibiting homogeneous balanced affinity structure. Our findings support the claim that, for a particular notion of sparsity, loosely speaking, "Laplacian spectral embedding favors relatively sparse graphs, whereas adjacency spectral embedding favors not-too-sparse graphs." We also provide evidence in support of the claim that "adjacency spectral embedding favors core-periphery network structure."
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