Random walk on the high-dimensional IIC

We study the asymptotic behavior the exit times of random walk from Euclidean balls around the origin of the incipient infinite cluster in a manner inspired by [26]. We do this by obtaining bounds on the effective resistance between the origin and the boundary of these Euclidean balls. We show that the geometric properties of long-range percolation clusters are significantly different from those of finite-range clusters. We also study the behavior of random walk on the backbone of the IIC and we prove that the Alexander-Orbach conjecture holds for the incipient infinite cluster in high dimensions, both for long-range percolation and for finite-range percolation

High-dimensional incipient infinite clusters revisited

The incipient infinite cluster (IIC) measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite. Using the lace expansion, we construct the IIC measure for high-dimensional percolation models in three different ways, extending previous work by the second author and Jarai. We show that each construction yields the same measure, indicating that the IIC is a robust object. Furthermore, our constructions apply to spread-out versions of both finite-range and long-range percolation models. We also obtain estimates on structural properties of the IIC, such as the volume of the intersection between the IIC and Euclidean balls

Two more ways of spelling Gini Coefficient with Applications

In this paper, we draw attention to a promising yet slightly underestimated measure of variability - the Gini coefficient. We describe two new ways of defining and interpreting this parameter. Using our new representations, we compute the Gini index for a few probability distributions and describe it in more detail for the negative binomial distribution. We also suggest the latter as a tool to measure overdispersion in epidemiology

Dynamic random intersection graph: Dynamic local convergence and giant structure.

Random intersection graphs containing an underlying community structure are a popular choice for modelling real-world networks. Given the group memberships, the classical random intersection graph is obtained by connecting individuals when they share at least one group. We extend this approach and make the communities dynamic by letting them alternate between an active and inactive phase. We analyse the new model, delivering results on degree distribution, local convergence, giant component, and maximum group size, paying particular attention to the dynamic description of these properties. We also describe the connection between our model and the bipartite configuration model, which is of independent interest

Switch chain mixing times and triangle counts in simple random graphs with given degrees

Sampling uniform simple graphs with power-law degree distributions with degree exponent τ∈(2,3) is a non-trivial problem. Firstly, we propose a method to sample uniform simple graphs that uses a constrained version of the configuration model together with a Markov Chain switching method. We test the convergence of this algorithm numerically in the context of the presence of small subgraphs and we estimate the mixing time to be at most O(n log2 n)⁠. Secondly, we compare the number of triangles in uniform random graphs with the number of triangles in the erased configuration model where double edges and self-loops of the configuration model are removed. Using simulations and heuristic arguments, we conjecture that the number of triangles in the erased configuration model is larger than the number of triangles in the uniform random graph, provided that the graph is sufficiently large. Lastly, we argue that certain switch-chain-based proof methods can not be used in the regime τ∈(2,3) due to the possibility of creating m