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The voter model is a classical interacting particle system modelling how consensus is formed across a network. We analyse the time to consensus for the voter model when the underlying graph is a subcritical scale-free random graph. Moreover, we generalise the model to include a `temperature parameter. The interplay between the temperature and the structure of the random graph leads to a very rich phase diagram, where in the different phases different parts of the underlying geometry dominate the time to consensus. Finally, we also consider a discursive voter model, where voters discuss their opinions with their neighbours. Our proofs rely on the well-known duality to coalescing random walks and a detailed understanding of the structure of the random graphs.
A bootstrap percolation process on a graph G is an infection process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round every uninfected node which has at least r infected neighbours becomes infected
Consider a collection of random variables attached to the vertices of a graph. The reconstruction problem requires to estimate one of them given `far away observations. Several theoretical results (and simple algorithms) are available when their join
Consider a set of $n$ vertices, where each vertex has a location in $mathbb{R}^d$ that is sampled uniformly from the unit cube in $mathbb{R}^d$, and a weight associated to it. Construct a random graph by placing edges independently for each vertex pa
Motivated by limits of critical inhomogeneous random graphs, we construct a family of sequences of measured metric spaces that we call continuous multiplicative graphs, that are expected to be the universal limit of graphs related to the multiplicati
We study the typical behavior of a generalized version of Googles PageRank algorithm on a large family of inhomogeneous random digraphs. This family includes as special cases direct