标题： 渗流与局域化：非回溯矩阵的次主导特征值 显示英文标题

标题： Percolation and localisation: Sub-leading eigenvalues of the nonbacktracking matrix

摘要： 网络的非回溯矩阵的谱已知包含有关网络渗流性质的基本信息。 确实，其主特征值的倒数通常用作渗流阈值的估计值。 然而，对于许多非回溯中心性集中在少数节点上的网络，例如具有核心-边缘结构的网络，这种谱方法会严重低估阈值。 在本工作中，我们通过超越主特征值并深入研究非回溯矩阵的谱，来研究表现出这种局域化效应的网络。 我们发现，当存在局域化时，阈值通常更接近于其中一个次主实特征值的倒数：对应特征向量为“非局域化”的最大实特征值。 我们研究了一个核心-边缘网络模型，并且理论上和实验上确定了参数范围，在此范围内，我们的方法能很好地逼近阈值，而使用主特征值得出的估计则不能。 我们进一步展示了在大规模现实世界网络上的实验结果，这些结果展示了我们方法的实用性。 显示英文摘要

摘要： The spectrum of the nonbacktracking matrix associated to a network is known to contain fundamental information regarding percolation properties of the network. Indeed, the inverse of its leading eigenvalue is often used as an estimate for the percolation threshold. However, for many networks with nonbacktracking centrality localised on a few nodes, such as networks with a core-periphery structure, this spectral approach badly underestimates the threshold. In this work, we study networks that exhibit this localisation effect by looking beyond the leading eigenvalue and searching deeper into the spectrum of the nonbacktracking matrix. We identify that, when localisation is present, the threshold often more closely aligns with the inverse of one of the sub-leading real eigenvalues: the largest real eigenvalue with a "delocalised" corresponding eigenvector. We investigate a core-periphery network model and determine, both theoretically and experimentally, a regime of parameters for which our approach closely approximates the threshold, while the estimate derived using the leading eigenvalue does not. We further present experimental results on large scale real-world networks that showcase the usefulness of our approach.