Structural characterization of water networks in different chemico-physical conditions by means of a graph theory approach

Water, besides being fundamental from several points of view, exhibits thermodynamic and dynamic anomalies compared to other substances. Over the years, scientists have proposed different scenarios to explain these peculiarities. The most accredited one foresees the presence in the supercooled region of a first order phase transition between two liquid phases at different densities: the low-(LDL) and high-(HDL) density liquid. Although several studies support this hypothesis, the experimental measurements and characterizations in the region where this transition should occur are very challenging due to fast crystallization.In this thesis we apply concepts from graph theory, together with molecular dynamics simulations, to advance our understanding of the structure of liquid water under different conditions. We consider liquid water as an undirected and unweighted graph, in which the nodes correspond to water oxygens, and the edges represent the interactions between molecules. In the first part, we analyse simulated liquid water at high pressure, where the coexistence line between the two liquid phases should be crossed. We investigate the performance of several centrality measures when used as order parameters to identify whether water is in the LDL or HDL phase. Among these the total communicability (TC), besides being found useful in this context, turns out to be efficient from a computational point of view. The new order parameter based on TC is also able to show that HDL forms are not homogeneous, but composed by patches at different local density. We explore the structural properties of the two phases using well-established metrics from Network Analysis.In the second part, we study simulated liquid water at ambient pressure in both stable and metastable supercooled conditions. We prove that our new order parameter can identify the two liquid states also in these situations, and that it captures both the structural and dynamic differences between the two states. We investigate the high connectivity regions we identified at high pressure, and we show that these extended patches exhibit a short lifetime, a high mobility and an increased local density. We observe small highly connected patches also at low temperatures, where the prevailing state is LDL-like.In the third part, we describe some attempts to improve the description of liquid water using different graph models: directed, weighted, and dynamic networks, and using non-backtracking walks.