Abstract
We introduce an informative labelling method for vertices in a family of Farey graphs, and deduce a routing algorithm on all the shortest paths between any two vertices in Farey graphs. The label of a vertex is composed of the precise locating position in graphs and the exact time linking to graphs. All the shortest paths routing between any pair of vertices, which number is exactly the product of two Fibonacci numbers, are determined only by their labels, and the time complexity of the algorithm is O(n). It is the first algorithm to figure out all the shortest paths between any pair of vertices in a kind of deterministic graphs. For Farey networks, the existence of an efficient routing protocol is of interest to design practical communication algorithms in relation to dynamical processes (including synchronization and structural controllability) and also to understand the underlying mechanisms that have shaped their particular structure.
Highlights
We introduce an informative labelling method for vertices in a family of Farey graphs, and deduce a routing algorithm on all the shortest paths between any two vertices in Farey graphs
We have provided a labelling and routing algorithm for a wide family of Farey graphs, including the model created by edge iterations, evolving networks with geographical attachment preference, general geometric growth model for pseudofractal scale-free web and the networks with multidimensional growth
Our labelling method is simpler than that of refs 12, 15 and 16, because it is easier to deduce the labels of the new vertices which are linked to graphs at step t, by our labelling method
Summary
We introduce an informative labelling method for vertices in a family of Farey graphs, and deduce a routing algorithm on all the shortest paths between any two vertices in Farey graphs. All the shortest paths routing between any pair of vertices, which number is exactly the product of two Fibonacci numbers, are determined only by their labels, and the time complexity of the algorithm is O(n). It is the first algorithm to figure out all the shortest paths between any pair of vertices in a kind of deterministic graphs. Finding shortest paths in networks is a well-studied and important problem with many applications. The labelling and the routing in several deterministic models, on the basis of the relationship between vertices labels and the shortest paths, have been pioneered by Comellas and Zhang[12,15,16]. The graphs of which have some important properties as empirical networks, for example, the expanded Apollonian networks www.nature.com/scientificreports/
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