Graphical evolution: an introduction to the theory of random graphs
Graphical evolution: an introduction to the theory of random graphs
The small-world phenomenon: an algorithmic perspective
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Regular Article: The Diameter of Sparse Random Graphs
Advances in Applied Mathematics
The cover time of sparse random graphs.
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Sharp thresholds For monotone properties in random geometric graphs
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Distance estimation and object location via rings of neighbors
Proceedings of the twenty-fourth annual ACM symposium on Principles of distributed computing
Random geometric graphs: an algorithmic perspective
Random geometric graphs: an algorithmic perspective
Small worlds as navigable augmented networks: model, analysis, and validation
ESA'07 Proceedings of the 15th annual European conference on Algorithms
On the cover time of random geometric graphs
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Do neighbor-avoiding walkers walk as if in a small-world network?
INFOCOM'09 Proceedings of the 28th IEEE international conference on Computer Communications Workshops
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We introduce and study random distance graph. A random distance, D(n,g) results from placing n points uniformly at random on the unit area disk and connecting every two points independently with probability g(d), where d is the distance between the nodes and g is the connection function. We give a connection function g(r,a,d) with parameters r and a and show the following: (a) D(n,g(r,a)) captures as special cases both the standard random geometric graph G(n,r) and the Bernoulli random graph B(n,p) (a.k.a. Erdos-Renyi graph). (b) Using results from continuum percolation we are able to bound the connectivity threshold of D(n,g(r,a)), with G(n,r) and B(n,p) as special (previously known) cases. (c) Contrary to G(n,r) and B(n,p), for a wide range of r and α a is, in fact, a "Small World" graph with high clustering coefficient of about 0.5865a and diameter of Θ({log n}/{log log n}). As opposed to previous Small World models that rely on deterministic sub-structures to grantee connectivity, random distance graphs offer a completely randomized model with a proved bounds for connectivity threshold, clustering coefficient and diameter.