Path Laplacian operators and superdiffusive processes on graphs. II. Two-dimensional lattice

Many physical systems are best represented by graphs G = (V,E), where the set of nodes (vertices) V represents the entities of the system and the set of edges E describes the interactions between these entities [13]. Among those systems we can mention atomic and molecular ones as well as complex networks, which include a vast range of complex systems embracing biological, social, ecological, infrastructural and technological ones. Diffusion-like processes, such as diffusion, reaction-diffusion, synchronization, epidemic spreading, etc., are ubiquitous in those previously mentioned systems [6]. Apart from the normal diffusive processes, where the mean square displacement (MSD) of the diffusive particle scales linearly with time, there are many real-world examples where anomalous diffusion takes place. In these anomalous diffusive processes, MSD scales nonlinearly with time giving rise to subdiffusive and superdiffusive processes [29]. In Part I [15] of this series we introduced a new theoretical framework to study superdiffusive processes on graphs. In that work we considered transformations of the so-called k-path Laplace operators Lk. The latter are defined in a similar way as the standard graph Laplacian, but they take only nodes into account whose distance is equal to k; here the distance is measured as the length of the shortest path connecting two nodes. Hence Lk describes hops to nodes at distance k. The above mentioned transformations of Lk are combinations of the form ∑∞ k=1 ckLk with some non-negative coefficients ck. This combination describes interactions with all nodes where different strengths are used for nodes at different distances. In general, one uses a sequence ck that is decreasing in k. In particular, in [15] we considered the Mellin transform of Lk, which is obtained by choosing ck = k −s with some positive parameter s. The choice of the transformation has proved to be crucial in determining the diffusive behaviour. In [15] we studied, in particular, the one-dimensional path graph. We proved that superdiffusion appears when a Mellin transform of the k-path Laplace operators is considered with s satisfying 1 < s < 3, while for s > 3 normal diffusion is obtained; the latter occurs also if one considers different transformations of Lk like the Laplace and factorial transforms. This new method adds new values to the already existing ones for modelling anomalous diffusion. Among such existing methods we should mention the use of random walks with Lévy flights (RWLF) [12, 34, 38, 30] and the use of the fractional