  In this work, we develop a novel framework to measure the similarity between
dynamic financial networks, i.e., time-varying financial networks.
Particularly, we explore whether the proposed similarity measure can be
employed to understand the structural evolution of the financial networks with
time. For a set of time-varying financial networks with each vertex
representing the individual time series of a different stock and each edge
between a pair of time series representing the absolute value of their Pearson
correlation, our start point is to compute the commute time matrix associated
with the weighted adjacency matrix of the network structures, where each
element of the matrix can be seen as the enhanced correlation value between
pairwise stocks. For each network, we show how the commute time matrix allows
us to identify a reliable set of dominant correlated time series as well as an
associated dominant probability distribution of the stock belonging to this
set. Furthermore, we represent each original network as a discrete dominant
Shannon entropy time series computed from the dominant probability
distribution. With the dominant entropy time series for each pair of financial
networks to hand, we develop a similarity measure based on the classical
dynamic time warping framework, for analyzing the financial time-varying
networks. We show that the proposed similarity measure is positive definite and
thus corresponds to a kernel measure on graphs. The proposed kernel bridges the
gap between graph kernels and the classical dynamic time warping framework for
multiple financial time series analysis. Experiments on time-varying networks
extracted through New York Stock Exchange (NYSE) database demonstrate the
effectiveness of the proposed approach.
