A heuristic for the distribution of point counts for random curves over a finite field.

How many rational points are there on a random algebraic curve of large genus g over a given finite field Fq? We propose a heuristic for this question motivated by a (now proven) conjecture of Mumford on the cohomology of moduli spaces of curves; this heuristic suggests a Poisson distribution with mean q+1+1/(q-1). We prove a weaker version of this statement in which g and q tend to infinity, with q much larger than g.

Using network properties to predict disease dynamics on human contact networks.

Recent studies have increasingly turned to graph theory to model more realistic contact structures that characterize disease spread. Because of the computational demands of these methods, many researchers have sought to use measures of network structure to modify analytically tractable differential equation models. Several of these studies have focused on the degree distribution of the contact network as the basis for their modifications. We show that although degree distribution is sufficient to predict disease behaviour on very sparse or very dense human contact networks, for intermediate density networks we must include information on clustering and path length to accurately predict disease behaviour. Using these three metrics, we were able to explain more than 98 per cent of the variation in endemic disease levels in our stochastic simulations.

Pair statistics clarify percolation properties of spatially explicit simulations.

Dispersal is a fundamental control on the spatial structure of a population. We investigate the precise mechanism by which a mixed strategy of short- and long-distance dispersal affects spatial patterning. Using techniques from pair approximation and percolation theory, we demonstrate that dispersal controls the extent to which a population is completely connected by modulating the proportion of neighboring sites which are simultaneously occupied. We show that near the percolation threshold this pair statistic, rather than other metrics proposed earlier, best explains clustering, and we suggest more general circumstances under which this may hold.

Comment on "Families and clustering in a natural numbers network".

Corso [Phys. Rev. E 69, 036106 (2004)] constructs a family of graphs from subsets of the natural numbers, and numerically estimates diameter, degree and clustering. We give exact asymptotic formulas for these quantities, and thereby argue that number theory is a more appropriate tool than simulation.