We study spatial embeddings of random graphs in which nodes are randomly distributed in geographical space. We let the edge probability between any two nodes to be dependent on the spatial distance between them and demonstrate that this model captures many generic properties of social networks, including the ``small-world'' properties, skewed degree distribution, and most distinctively the existence of community structures.Comment: To be published in Physica A (2005

We use the finite lattice method to count the number of punctured staircase and self-avoiding polygons with up to three holes on the square lattice. New or radically extended series have been derived for both the perimeter and area generating functions. We show that the critical point is unchanged by a finite number of punctures, and that the critical exponent increases by a fixed amount for each puncture. The increase is 1.5 per puncture when enumerating by perimeter and 1.0 when enumerating by area. A refined estimate of the connective constant for polygons by area is given. A similar set of results is obtained for finitely punctured polyominoes. The exponent increase is proved to be 1.0 per puncture for polyominoes.

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In two dimensions the universality classes of self-avoiding walks on the square lattice, restricted by allowing only certain two-step configurations to occur within each walk, has been argued to be determined primarily by the symmetry of the set of allowed two-step configurations. In a recent paper, primarily tackling the three-dimensional analogues of these models, a novel two-dimensional model was discovered that seemed either to break the classification of the models into universality classes according to microscopic symmetry or was itself a member of a novel universality class. This was supported by series analysis of exact enumeration data. Here we provide conclusive evidence that this model, known as 'anti-spiral walks', is in the directed walk universality class. We arrive at these conclusions from Monte Carlo simulations of these walks using a PERM algorithm modified for this problem. We point out that the behaviour of this model is unusual in that other models in the directed walk universality class remain directed when the self-avoidance condition is removed, whereas the behaviour of anti-spiral walks becomes that of a isotropic simple random walk. We also remark that the symmetry classification of walk models can be kept by adding a natural condition to the scheme that disallows models, all of whose configurations avoid some infinite region of the plane by virtue of their microscopic constraints.