On Hierarchical Extensions of Large-Scale 4-regular Grid Network Structures

It is studied how the introduction of ordered hierarchies in 4-regular grid network structures decreases distances remarkably, while at the same time allowing for simple topological routing schemes. Both meshes and tori are considered; in both cases non-hierarchical structures have power law dependencies between the number of nodes and the distances in the structures. The perfect square mesh is introduced for hierarchies, and it is shown that applying ordered hierarchies in this way results in logarithmic dependencies between the number of nodes and the distances, resulting in better scaling structures. For example, in a mesh of 391876 nodes the average distance is reduced from 417.33 to 17.32 by adding hierarchical lines. This is gained by increasing the number of lines by 4.20% compared to the non-hierarchical structure. A similar hierarchical extension of the torus structure also results in logarithmic dependencies, the relative difference between performance of mesh and torus structures being less significant than for non-hierarchical structures, especially for large structures. The skew and extended meshes are introduced as variants of the perfect square mesh and their performances studied, and it is shown that while they allow for more flexibility in design and construction of structures supporting topological routing, their performances are comparable to the performance of the perfect square mesh. Finally suggestions for further research within the field are given.