  Time dependence is a universal phenomenon in nature, and a variety of
mathematical models in terms of dynamical systems have been developed to
understand the time-dependent behavior of real-world problems. Originally
constructed to analyze the topological persistence over spatial scales,
persistent homology has rarely been devised for time evolution. We propose the
use of a new filtration function for persistent homology which takes as input
the adjacent oscillator trajectories of a dynamical system. We also regulate
the dynamical system by a weighted graph Laplacian matrix derived from the
network of interest, which embeds the topological connectivity of the network
into the dynamical system. The resulting topological signatures, which we call
evolutionary homology (EH) barcodes, reveal the topology-function relationship
of the network and thus give rise to the quantitative analysis of nodal
properties. The proposed EH is applied to protein residue networks for protein
thermal fluctuation analysis, rendering the most accurate B-factor prediction
of a set of 364 proteins. This work extends the utility of dynamical systems to
the quantitative modeling and analysis of realistic physical systems.
