We investigate spatial random graphs defined on the points of a Poisson process in d-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point is assigned an independent weight. Given the weight and position of the points, we form an edge between any pair of points independently with a probability depending on the two weights of the points and their distance. Preference is given to short edges and connections to vertices with large weights. We characterize the parameter regime where there is a non-trivial percolation phase transition and show that it depends not only on the power-law exponent of the degree distribution but also on a geometric model parameter. We apply this result to characterize robustness of age-based spatial preferential attachment networks.

Consider the following random spatial network: in a large disk, construct a network using a stationary and isotropic Poisson line process of unit intensity. Connect pairs of points using the network, with initial/final segments of the connecting path formed by travelling off the network in the opposite direction to that of the destination/source. Suppose further that connections are established using ‘near geodesics’, constructed between pairs of points using the perimeter of the cell containing these two points and formed using only the Poisson lines not separating them. If each pair of points generates an infinitesimal amount of traffic divided equally between the two connecting near geodesics, and if the Poisson line pattern is conditioned to contain a line through the centre, then what can be said about the total flow through the centre? In Kendall (2011) it was shown that a scaled version of this flow has asymptotic distribution given by the 4-volume of a region in 4-space, constructed using an improper anisotropic Poisson line process in an infinite planar strip. Here we construct a more amenable representation in terms of two ‘seminal curves’ defined by the improper Poisson line process, and establish results which produce a framework for effective simulation from this distribution up to an L1 error which tends to 0 with increasing computational effort.

The model of memory process we propose is based on two assumptions. First, spatial or adresses network models in economics can be easily adapted to describe a significative part of the episodic memory mechanism as defined by Tulving (1983). Second, brain viewed as a network behaves as a decision-maker who arbitrates between two economic dimensions of recollection: the reward—i.e., the satisfaction for recovering old informations located in mnesic traces—and the cost—i.e., the price for stimulating the traces network. Indeed, the two results exhibited in the paper—and devoted to a formal and appealing characterization of true and false recollections—are directly derived from the idea of a rational brain. Finally, this paper aims at showing that it could be relevant to model memory processes in a pure symbolic way—contrary to most of the neuroeconomics contributions which are generally experimental—and also that such an attempt for an abstract and analogical representation of the episodic memory process based on a spatial microeconomics methodology seems to be specially efficient and illustrative of Hintzman (1986) and recent Doeller et al. (2010) intuitions and features.

A navigation on a set of points S is a rule for choosing which point to move to from the present point in order to progress toward a specified target. We study some navigations in the plane where S is a nonuniform Poisson point process (in a finite domain) with intensity going to +∞. We show the convergence of the traveller's path lengths, and give the number of stages and the geometry of the traveller's trajectories, uniformly for all starting points and targets, for several navigations of geometric nature. Other costs are also considered. This leads to asymptotic results on the stretch factors of random Yao graphs and random θ-graphs.

In designing a network to link n points in a square of area n, we might be guided by the following two desiderata. First, the total network length should not be much greater than the length of the shortest network connecting all points. Second, the average route length (taken over source-destination pairs) should not be much greater than the average straight-line distance. How small can we make these two excesses? Speaking loosely, for a nondegenerate configuration, the total network length must be at least of order n and the average straight-line distance must be at least of order n1/2, so it seems implausible that a single network might exist in which the excess over the first minimum is o(n) and the excess over the second minimum is o(n1/2). But in fact we can do better: for an arbitrary configuration, we can construct a network where the first excess is o(n) and the second excess is almost as small as O(log n). The construction is conceptually simple and uses stochastic methods: over the minimum-length connected network (Steiner tree) superimpose a sparse stationary and isotropic Poisson line process. Together with a few additions (required for technical reasons), the mean values of the excess for the resulting random network satisfy the above asymptotics; hence, a standard application of the probabilistic method guarantees the existence of deterministic networks as required (speaking constructively, such networks can be constructed using simple rejection sampling). The key ingredient is a new result about the Poisson line process. Consider two points a distance r apart, and delete from the line process all lines which separate these two points. The resulting pattern of lines partitions the plane into cells; the cell containing the two points has mean boundary length approximately equal to 2r + constant(log r). Turning to lower bounds, consider a sequence of networks in satisfying a weak equidistribution assumption. We show that if the first excess is O(n) then the second excess cannot be