These pages contain links and notes on Small-World Networks and Graphs and our research activities towards understanding them.
Pages Under development. See Technical Notes:
Small-world networks are graph based structures that have various "shortcut" links applied so that it is in various quantifiable ways easier to reach a given node from a given start point. These ideas can be applied to all sorts of graphs and networks, but some of the most interesting cases are where we have a network embedded in "normal" space in 2 or 3 dimensions and where we add shortcuts that act like "wormholes". We can measure dramatic changes in the statistical properties of models or phenomena that live on such small-worlded networks.
The core idea can be seen in the following illustration - generated by my GraViz graph algorithm vizualization program.
A one-dimensional lattice with periodic boundaries has been generated. This graph is drawn as a ring of vertices numbered 0 to 35. Each is conncted by a di-arc to its nearest neighbours. Vertex zero is shown highjlighted in red and vertex 8 is highlighted in purple. The bold blue line is the shortest path betwene these two highlighted vertices. You can see that the "shortcut arcs" drawn as chords across the circle have changed the shortest path from its natural route. It is now shorter to follow the path 0-1-21-20-19-10-9-8.
There are various student projects on offer in this area. Contact me for further information.