SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 3, Page 1229-1247, January 2021.
We develop a complete theory for the combinatorics of walk-counting on a directed graph in the case where each backtracking step is downweighted by a given factor. By deriving expressions for the associated generating functions, we also obtain linear systems for computing centrality measures in this setting. In particular, we show that backtrack-downweighted Katz-style network centrality can be computed at the same cost as standard Katz. Studying the limit of this centrality measure at its radius of convergence also leads to a new expression for backtrack- downweighted eigenvector centrality that generalizes previous work to the case where directed edges are present. The new theory allows us to combine advantages of standard and nonbacktracking cases, avoiding localization while accounting for tree-like structures. We illustrate the behavior of the backtrack-downweighted centrality measure on both synthetic and real networks.

中文翻译：

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Backtrack-Downweighted Walks 的理论

SIAM Journal on Matrix Analysis and Applications，第 42 卷，第 3 期，第 1229-1247 页，2021 年 1 月。
我们为有向图上的步行计数组合开发了一个完整的理论，在这种情况下，每个回溯步骤都被给定的因素降低了权重。通过推导相关生成函数的表达式，我们还获得了在此设置中计算中心性度量的线性系统。特别是，我们表明可以以与标准 Katz 相同的成本计算回溯降低权重的 Katz 式网络中心性。研究这种中心性度量在其收敛半径上的限制也导致了一种新的回溯降低特征向量中心性表达式，它将以前的工作推广到存在有向边的情况​​。新理论使我们能够结合标准和非回溯案例的优势，在考虑树状结构的同时避免定位。