  Despite the great advancement of experimental tools and theoretical models, a
quantitative characterization of the microscopic structures of ion aggregates
and its associated water hydrogen-bonding networks still remains a challenging
problem. In this paper, a newly-invented mathematical method called persistent
homology is introduced, for the first time, to quantitatively analyze the
intrinsic topological properties of ion aggregation systems and
hydrogen-bonding networks. Two most distinguishable properties of persistent
homology analysis of assembly systems are as follows. First, it does not
require a predefined bond length to construct the ion or hydrogen network.
Persistent homology results are determined by the morphological structure of
the data only. Second, it can directly measure the size of circles or holes in
ion aggregates and hydrogen-bonding networks. To validate our model, we
consider two well-studied systems, i.e., NaCl and KSCN solutions, generated
from molecular dynamics simulations. They are believed to represent two
morphological types of aggregation, i.e., local clusters and extended ion
network. It has been found that the two aggregation types have distinguishable
topological features and can be characterized by our topological model very
well. For hydrogen-bonding networks, KSCN systems demonstrate much more
dramatic variations in their local circle structures with the concentration
increase. A consistent increase of large-sized local circle structures is
observed and the sizes of these circles become more and more diverse. In
contrast, NaCl systems show no obvious increase of large-sized circles. Instead
a consistent decline of the average size of circle structures is observed and
the sizes of these circles become more and more uniformed with the
concentration increase.
