Atomic object services for mobile and dynamic networks

Small-world graphs have a relatively low diameter, with respect to an structured graph with a similar number of vertices and edges, and a large clustering (vertices have many mutual neighbors) in relation to a random graph also with the same number of vertices and edges. Small-world graphs are of considerable interest because they model important real life networks as the WWW, transportation and communication networks, or even biological and social networks. Most previous research on small-world graphs has used simulations, probabilistic techniques, and random replacements of edges to study the main parameters and behaviour of these graphs. However, a deterministic approach has been considered in [F. Comellas, J. Ozón and J.G. Peters, Inf. Proc. Lett., 76, 83 (2000)] by starting with a circulant graph with n vertices, large diameter and high clustering and choosing a small number h of evenly-spaced vertices to be hubs, and interconnecting the hubs. With this approach we obtain deterministic small-world graphs (including regular graphs), that match former simulations. Small-world graphs occur when approximately 1% of the vertices are chosen to be hubs. Then the diameter is approximately 20% of the value in the original circulant graph, but 95% of the clustering remains. On the other hand, the knowledge of the spectrum of a graph is important for the relation which the eigenvalues and their multiplicities have with relevant graph invariants and topological and communication properties such as diameter, bisection width, distances, connectivity, expansion, partitions, edge-loading distribution etc. Here we determine completely the spectra of several families of deterministic small-world graphs based on the technique described before. Joint work with: Silvia Gago Full version available at: http://www-mat.upc.es/∼comellas/papers/CoGa03-IWIN.djvu