Departamento de Física da Universidade de Aveiro and I3N, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal.
Phys Rev E. 2023 Jan;107(1-1):014301. doi: 10.1103/PhysRevE.107.014301.
The nonbacktracking matrix and the related nonbacktracking centrality (NBC) play a crucial role in models of percolation-type processes on networks, such as nonrecurrent epidemics. Here we study the localization of NBC in infinite sparse networks that contain an arbitrary finite subgraph. Assuming the local tree likeness of the enclosing network, and that branches emanating from the finite subgraph do not intersect at finite distances, we show that the largest eigenvalue of the nonbacktracking matrix of the composite network is equal to the highest of the two largest eigenvalues: that of the finite subgraph and of the enclosing network. In the localized state, when the largest eigenvalue of the subgraph is the highest of the two, we derive explicit expressions for the NBCs of nodes in the subgraph and other nodes in the network. In this state, nonbacktracking centrality is concentrated on the subgraph and its immediate neighborhood in the enclosing network. We obtain simple, exact formulas in the case where the enclosing network is uncorrelated. We find that the mean NBC decays exponentially around the finite subgraph, at a rate which is independent of the structure of the enclosing network, contrary to what was found for the localization of the principal eigenvector of the adjacency matrix. Numerical simulations confirm that our results provide good approximations even in moderately sized, loopy, real-world networks.
无向跟踪矩阵及其相关的非跟踪中心性（NBC）在网络上的渗流型过程模型中起着至关重要的作用，例如不可重复的传染病。在这里，我们研究了无限稀疏网络中 NBC 的局域化，这些网络包含任意有限的子图。假设包围网络的局部树状相似性，并且从有限子图发出的分支不在有限距离处相交，我们表明复合网络的无向跟踪矩阵的最大特征值等于两个最大特征值中的最高值：有限子图和包围网络的最高值。在局域化状态下，当子图的最大特征值是两个中最高时，我们推导出子图中和网络中其他节点的 NBC 的显式表达式。在这种状态下，非跟踪中心性集中在子图及其在包围网络中的直接邻域上。在不相关的包围网络的情况下，我们得到了简单、精确的公式。我们发现，平均 NBC 在有限子图周围呈指数衰减，其衰减率与包围网络的结构无关，这与邻接矩阵主特征向量的局域化所发现的情况相反。数值模拟证实，即使在中等大小的、有环的、真实网络中，我们的结果也提供了很好的近似。
