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- W63073375 abstract "We prove additional results about Kleinberg’s Small-World (KSW) model and its extensions. Kleinberg adds a number of directed long-range random links to an n ◊ n lattice network (vertices as nodes of a grid, undirected edges between any two adjacent nodes) under a specific non-uniform distribution. He shows that the following phenomenon occurs: between almost any two nodes there exits a short path with length at most O(log 2 n) which can be found using a simple greedy algorithm which has no global knowledge of long-range links. Kleinberg proves that the expected delivery time using his algorithm is O(log 2 n). We show that indeed Kleinberg’s algorithm achieves (log 2 n) delivery time. Moreover, we show that the expected diameter of the graph is (logn), so a logn factor smaller. These results are proved for the general k-dimensional model and our diameter result extends traditional work on the diameter of random graphs which largely focuses on uniformly distributed arcs. Using little additional knowledge of the graph, we show that we can find paths with expected length O(log 3/2 n) in the basic 2-dimensional model and O(log 1+1/k n) in the general k-dimensional model (for k 1). Finally, we suggest open problems and initiate the study of further generalized models where gridrelated factors (e.g. the use of lattice distance) get weaker roles or are dismissed, and constraints (such as the uniformness of degree distribution) are relaxed." @default.
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- W63073375 date "2004-01-01" @default.
- W63073375 modified "2023-09-24" @default.
- W63073375 title "The Complexity of Message Delivery in Kleinberg's Small-world Model" @default.
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