Discrete small world networks

Small world models are networks consisting of many local links and fewer long range `shortcuts', used to model networks with a high degree of local clustering but relatively small diameter. Here, we concern ourselves with the distribution of typical inter-point network distances. We establish approximations to the distribution of the graph distance in a discrete ring network with extra random links, and compare the results to those for simpler models, in which the extra links have zero length and the ring is continuous. E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Vol. 11 (2006), Paper no. 47, pages 1234–1283. Journal URL http://www.math.washington.edu/~ejpecp/ Discrete small world networks A. D. Barbour∗ Angewandte Mathematik Universität Zürich Winterthurertrasse 190 CH-8057 ZÜRICH G. Reinert† Universität Zürich and University of Oxford Department of Statistics 1 South Parks Road OXFORD OX1 3TG, UK Abstract Small world models are networks consisting of many local links and fewer long range ‘shortcuts’, used to model networks with a high degree of local clustering but relatively small diameter. Here, we concern ourselves with the distribution of typical inter-point network distances. We establish approximations to the distribution of the graph distance in a discrete ring network with extra random links, and compare the results to those for simpler models, in which the extra links have zero length and the ring is continuousSmall world models are networks consisting of many local links and fewer long range ‘shortcuts’, used to model networks with a high degree of local clustering but relatively small diameter. Here, we concern ourselves with the distribution of typical inter-point network distances. We establish approximations to the distribution of the graph distance in a discrete ring network with extra random links, and compare the results to those for simpler models, in which the extra links have zero length and the ring is continuous