Small-world graphs are examples of random graphs which mimic empirically observed features of social networks. We propose an intrinsic definition of small-world graphs, based on a probabilistic formulation of scaling properties of the graph, which does not rely on any particular construction. Our definition is shown to encompass existing models of small-world graphs, proposed by Watts (1999) and studied by Barbour and Reinert (2001), which are based on random perturbations of a regular lattice. We also propose alternative constructions of small-world graphs which are not based on lattices and study their scaling properties.

We consider two different models of small-world graphs on nodes whose locations are modelled by a stochastic point process. In the first model each node is connected to a fixed number of its nearest neighbours, while in the second model each node is connected to all nodes located within some fixed distance. In both models, nodes are additionally connected via shortcuts to other nodes chosen uniformly at random. We obtain sufficient conditions for connectivity in the first model, and necessary conditions in the second model. Thereby, we show that connectivity is achieved at a smaller value of total degree (nearest neighbours plus shortcuts) in the first model. We also obtain bounds on the diameter of the graph in this model.