Ring correlations in random networks

Abstract

We examine the correlations between rings in random network glasses in two dimensions as a function of their separation. Initially, we use the topological separation (measured by the number of intervening rings), but this leads to pseudo-long-range correlations due to a lack of topological charge neutrality in the shells surrounding a central ring. This effect is associated with the noncircular nature of the shells. It is, therefore, necessary to use the geometrical distance between ring centers. Hence we find a generalization of the Aboav-Weaire law out to larger distances, with the correlations between rings decaying away when two rings are more than about three rings apart.

Realizations of Isostatic Material Frameworks

“…The network has rings of many sizes, but the mean ring size is six. [ 26 ] Therefore, we propose a hexagon as a toy model: Trihex ( Figure 1 ).…”

“…While Trihex is essentially a ring of triangles forming a hexagon, in experimental samples ring size varies from 4 to 8 [ 65 ] which are distributed nonrandomly on a plane. [ 26 ]…”

“…While Trihex is essentially a ring of triangles forming a hexagon, in experimental samples ring size varies from 4 to 8 [65] which are distributed nonrandomly on a plane. [26] Second, 2D glasses are generally much larger than Trihex and it might not be computationally feasible to apply the techniques directly. In fact, solving the set of edge length equations (Equation (1)) exactly is practically impossible for systems as large as 2D glasses with N ≥ Oð10 2 Þ which, in turn, means for larger systems the complete set of solutions and their evolving on branches are inaccessible.…”

Persistence Diagrams to Visualise Damage to Biological Networks

“…The rings in a 2D network can be conceived of as being the next level up in a hierarchy of network features: a node at an interaction site is a zerodimensional feature, an edge between two nodes is a one-dimensional feature, and the rings formed edges are two-dimensional features as shown in Figure 1. A ring representation of a 2D network allows for analysis of medium range order, including how those rings are correlated to one another (for example, are rings with many sides adjacent to rings with few sides or vice versa as measured by Sadjadi and Thorpe (2016)) or characterising networks by the distribution of the rings seen in them (referred to by authors such as Kumar et al (2014) or Le Roux and Jund (2010) as 'ring statistics'). Ring statistics are also used in studies of 3D networks, although the definition of a ring is less well defined and the ring structure is considerably harder to visualise.…”

Refining glass structure in two dimensions

“…However, simulation models can greatly aid the interpretation of these data as the atom positions are known unequivocally. As a result, information such as the ring statistics, which is in many ways a natural language for discussing network structure [4][5][6], is directly accessible. While this work has been very informative and clearly established the correctness of the CRN model for materials like vitreous silica, it is not accurate enough to distinguish between different models with varying ring statistics etc.…”