Hydrogen bond network topology in liquid water and methanol: a graph theory approach

Networks are increasingly recognized as important building blocks of various systems in nature and society. Water is known to possess an extended hydrogen bond network, in which the individual bonds are broken in the sub-picosecond range and still the network structure remains intact. We investigated and compared the topological properties of liquid water and methanol at various temperatures using concepts derived within the framework of graph and network theory (neighbour number and cycle size distribution, the distribution of local cyclic and local bonding coefficients, Laplacian spectra of the network, inverse participation ratio distribution of the eigenvalues and average localization distribution of a node) and compared them to small world and Erdős-Rényi random networks. Various characteristic properties (e.g. the local cyclic and bonding coefficients) of the network in liquid water could be reproduced by small world and/or Erdős-Rényi networks, but the ring size distribution of water is unique and none of the studied graph models could describe it. Using the inverse participation ratio of the Laplacian eigenvectors we characterized the network inhomogeneities found in water and showed that similar phenomena can be observed in Erdős-Rényi and small world graphs. We demonstrated that the topological properties of the hydrogen bond network found in liquid water systematically change with the temperature and that increasing temperature leads to a broader ring size distribution. We applied the studied topological indices to the network of water molecules with four hydrogen bonds, and showed that at low temperature (250 K) these molecules form a percolated or nearly-percolated network, while at ambient or high temperatures only small clusters of four-hydrogen bonded water molecules exist.