We present a novel method to estimate the dominant eigenvalue and eigenvector pair of any non-negative real matrix via graph infection. The key idea in our technique lies in approximating the solution to the first-order matrix ordinary differential equation (ODE) with the Euler method. Graphs, which can be weighted, directed, and with loops, are first converted to its adjacency matrix A. Then by a naive infection model for graphs, we establish the corresponding first-order matrix ODE, through which A's dominant eigenvalue is revealed by the fastest growing term. When there are multiple dominant eigenvalues of the same magnitude, the classical power iteration method can fail. In contrast, our method can converge to the dominant eigenvalue even when same-magnitude counterparts exist, be it complex or opposite in sign. We conduct several experiments comparing the convergence between our method and power iteration. Our results show clear advantages over power iteration for tree graphs, bipartite graphs, directed graphs with periods, and Markov chains with spider-traps. To our knowledge, this is the first work that estimates dominant eigenvalue and eigenvector pair from the perspective of a dynamical system and matrix ODE. We believe our method can be adopted as an alternative to power iteration, especially for graphs.

翻译：我们提出了一种新颖的方法，通过图感染来估计任何非负实矩阵的主特征值和特征向量对。该方法的核心思想在于利用欧拉法近似求解一阶矩阵常微分方程（ODE）。对于带权、有向且含自环的图，首先将其转换为邻接矩阵A，然后通过一种简单的图感染模型建立相应的一阶矩阵ODE，从中通过最快增长项揭示A的主特征值。当存在多个模长相等的主特征值时，经典的幂迭代法可能失效。相比之下，我们的方法即使存在同模长特征值（无论是复数还是符号相反），也能收敛到主特征值。我们进行了多项实验，比较该方法与幂迭代法的收敛性能。结果表明，在树形图、二分图、周期有向图以及带有陷阱状态的马尔可夫链中，本方法相较于幂迭代法具有明显优势。据我们所知，这是首个从动力系统和矩阵ODE角度估计主特征值-特征值对的工作。我们相信该方法可作为幂迭代法的替代方案，尤其适用于图结构场景。