ﻻ يوجد ملخص باللغة العربية
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 pair with a probability that is a function of the distance between the locations, and the vertex weights. Under appropriate integrability assumptions on the edge probabilities that imply sparseness of the model, after appropriately blowing up the locations, we prove that the local limit of this random graph sequence is the (countably) infinite random graph on $mathbb{R}^d$ with vertex locations given by a homogeneous Poisson point process, having weights which are i.i.d. copies of limiting vertex weights. Our setup covers many sparse geometric random graph models from the literature, including Geometric Inhomogeneous Random Graphs (GIRGs), Hyperbolic Random Graphs, Continuum Scale-Free Percolation and Weight-dependent Random Connection Models. We prove that the limiting degree distribution is mixed Poisson, and the typical degree sequence is uniformly integrable, and obtain convergence results on various measures of clustering in our graphs as a consequence of local convergence. Finally, as a by-product of our argument, we prove a doubly logarithmic lower bound on typical distances in this general setting.
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 consider a natural model of inhomogeneous random graphs that extends the classical ErdH os-Renyi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous [AOP 1997]. In this model, the vertices are assigne
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
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 gener
Local convergence techniques have become a key methodology to study random graphs in sparse settings where the average degree remains bounded. However, many random graph properties do not directly converge when the random graph converges locally. A n