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.Comment: 13 pages, 8 figures, 3 table