Abstract
We introduce an analytic model for directed Watts-Strogatz small-world graphs and deduce an algebraic expression of its defining adjacency matrix. The latter is then used to calculate the small-world digraph's asymmetry index and clustering coefficient in an analytically exact fashion, valid nonasymptotically for all graph sizes. The proposed approach is general and can be applied to all algebraically well-defined graph-theoretical measures, thus allowing for an analytical investigation of finite-size small-world graphs.
Highlights
In recent years, small-world graphs have gained considerable interest as models of real-world systems [1], which often display features residing between regularity and randomness
We introduce a generative model of small worlds that can be viewed as a directed version of a canonical Watts-Strogatz graph and provide an algebraic expression for the adjacency matrix defining the graph
We have introduced a generative model of directed small-world graphs GSW, a canonical model of WattsStrogatz digraphs, and have proposed an approach that yields the graph’s defining adjacency matrix in algebraic terms, with the goal to provide mathematically rigorous access to the study of finite-size small-world graphs
Summary
We start by describing the algorithmic construction of a non-self-looped directed small-world graph. Strogatz small-world [2] digraphs The latter are constructed from an undirected non-self-looped ring graph GRG with NN nodes, degree k, and subsequent rewiring of all edges with probability q. For each edge aij , a new target node j with aij = 0 is chosen and with probability q the original edge is removed (aij = 1 → aij = 0) and a new edge added (aij = 0 → aij = 1), leading to a strict conservation of the graph’s total adjacency This algorithmic approach is equivalent to constructing a “reduced” ring graph with 2kNN − q2kNN (NN − 2k − 1)/(NN − 1) edges uniformly distributed randomly across the 2kNN edges of GRG and distributing the remaining q2kNN (NN − 2k − 1)/(NN − 1) edges outside GRG. Strogatz small-world digraph model remains elusive, we will present an analytically exact consideration of GSW in the remainder of this study
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