Abstract
The Newman-Watts model is given by taking a cycle graph of n vertices and then adding each possible edge $(i,j), |i-j|\neq 1 \mod n$ with probability $\rho/n$ for some $\rho>0$ constant. In this paper we add i.i.d. exponential edge weights to this graph, and investigate typical distances in the corresponding random metric space given by the least weight paths between vertices. We show that typical distances grow as $\frac1\lambda \log n$ for a $\lambda>0$ and determine the distribution of smaller order terms in terms of limits of branching process random variables. We prove that the number of edges along the shortest weight path follows a Central Limit Theorem, and show that in a corresponding epidemic spread model the fraction of infected vertices follows a deterministic curve with a random shift.
Highlights
Determine the distribution of smaller order terms in terms of limits of branching process random variables
We work on the Newman–Watts small world model [31] with independent random edge weights: we take a cycle Cn on n vertices, that we denote by [n] := {1, 2, . . . , n}, and each edge (i, j) ∈ [n], |i − j| = 1 mod n is present
Janson [25] studied typical distances and the corresponding hopcount, flooding times as well as diameter of First passage percolation (FPP) on the complete graph. He showed that typical distances, the flooding time and diameter converge to 1, 2, and 3 times log n/n, respectively, while the hopcount is of order log n
Summary
The Newman–Watts small world model, often referred to as “small world” in short, is one of the first random graph models created to model real-life networks. In [8], they studied the discrete model, with all edge lengths equal to 1. They showed that typical distances in both models scale as log n. Durrett [20] showed that the order of the mixing time is between (log n) and (log n), Addario-Berry and Lei [1] proved that Durett’s lower bound is sharp
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