We investigate a class of growing graphs embedded into the d-dimensional torus where new vertices arrive according to a Poisson process in time, are randomly placed in space and connect to existing vertices with a probability depending on time, their spatial distance and their relative ages. This simple model for a scale-free network is called the age-based spatial preferential attachment network and is based on the idea of preferential attachment with spatially induced clustering. We show that the graphs converge weakly locally to a variant of the random connection model, which we call the age-dependent random connection model. This is a natural infinite graph on a Poisson point process where points are marked by a uniformly distributed age and connected with a probability depending on their spatial distance and both ages. We use the limiting structure to investigate asymptotic degree distribution, clustering coefficients and typical edge lengths in the age-based spatial preferential attachment network.MSc classification (2010): Primary 05C80 Secondary 60K35.

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 graph induced by Z d , equipped with uniformly elliptic random conductances. At time 0, place a Poisson point process of particles on Z d and let them perform independent simple random walks. Tessellate the graph into cubes indexed by i ∈ Z d and tessellate time into intervals indexed by τ . Given a local event E(i, τ ) that depends only on the particles inside the space time region given by the cube i and the time interval τ , we prove the existence of a Lipschitz connected surface of cells (i, τ ) that separates the origin from infinity on which E(i, τ ) holds. This gives a directly applicable and robust framework for proving results in this setting that need a multi-scale argument. For example, this allows us to prove that an infection spreads with positive speed among the particles. the space-time region. In [7], the effect of empty regions was controlled via an intricate multi-scale argument.The problem of spread of infection among Poisson random walks is just one example where longrange dependences give rise to serious mathematical challenges, and where multi-scale arguments have been applied to great success. In fact, multi-scale arguments have proved to be very useful in the analysis of several models, including the solution of several important questions regarding Poisson random walks [7,8,9,13], activated random walks [12], random interlacements [11,14], multi-particle diffusion limited aggregation [10] and more general dependent percolation [3,15].However, the main problem in developing a multi-scale analysis is that the argument is quite involved and can become very technical. Also, in each of the examples above, the involved multi-scale argument had to be developed from scratch and be tailored to the specific question being analyzed. Our main goal in this paper is to develop a more robust and systematic framework that can be applied to solve questions in the model of Poisson random walks without the need of carrying out a whole multi-scale argument each time. We do this by showing that given a local event which is translation invariant and whose probability of occurrence is large enough, we can find a special percolating structure in space-time where this event holds.We now explain our idea in a high-level way, deferring precise statements and definitions to Section 2. We tesselate space into cubes, indexed by i ∈ Z d , and tessellate time into intervals indexed by τ ∈ Z. Thus (i, τ ) denotes the space-time cell of the tessellation consisting of the cube i and the time interval τ . Given any increasing, translation invariant event E(i, τ ) that is local (i.e., measurable with respect to the particles that get within some fixed distance to the space-time cell (i, τ )), if the marginal distribution P(E(i, τ )) is large enough, our main result gives the existence of a two-sided Lipschitz surface of space-time cells where E(i, τ ) holds for all cells in the surface.Once we obtain such a Lipschitz surface, instead of having to carry out a whole multi-scale analysis from scratch to analyze som...

We investigate a large class of random graphs on the points of a Poisson process in d-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point carries an independent random weight and given weight and position of the points we form an edge between two points independently with a probability depending on the two weights and the distance of the points. In dimensions d ∈ {1, 2} we completely characterise recurrence vs transience of random walks on the infinite cluster. In d ≥ 3 we prove transience in all cases except for a regime where we conjecture that scale-free and long-range effects play no role. Our results are particularly interesting for the special case of the age-dependent random connection model recently introduced in [8].

Let (G, µ) be a uniformly elliptic random conductance graph on Z d with a Poisson point process of particles at time t = 0 that perform independent simple random walks. We show that inside a cube Q K of side length K, if all subcubes of side length ℓ < K inside Q K have sufficiently many particles, the particles return to stationarity after cℓ 2 time with a probability close to 1. We also show this result for percolation clusters on locally finite graphs. Using this mixing result and the results of [6], we show that in this setup, an infection spreads with positive speed in any direction. Our framework is robust enough to allow us to also extend the result to infection with recovery, where we show positive speed and that the infection survives indefinitely with positive probability.