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意非负实矩阵的主特征值和特征向量对。该技术的核心思想在于利用欧拉方法近似求解一阶矩阵常微分方程（ODE）。首先，将可能带有权重、方向及自环的图转化为其邻接矩阵A，然后通过一种朴素的图感染模型建立相应的一阶矩阵ODE，进而通过增长最快的项揭示A的主特征值。当存在多个模长相同的主特征值时，经典的幂迭代法可能失效；相比之下，即使存在模长相同（无论符号相反或为复数）的特征值，我们的方法也能收敛到主特征值。我们通过多项实验对比了本方法与幂迭代法的收敛性能。结果表明，对于树形图、二分图、含周期有向图以及带有蜘蛛陷阱的马尔可夫链，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与矩阵ODE视角估计主特征值-特征向量对的工作。我们相信该方法可作为幂迭代法的替代方案，尤其适用于图结构数据。