6

6

3332

The Sizes of Large Hierarchical Long-Range Percolation Clusters

We study a long-range percolation model in the hierarchical
lattice ΩN of order N where probability of connection between
two nodes separated by distance k is of the form min{αβ−k, 1},
α ≥ 0 and β > 0. The parameter α is the percolation parameter,
while β describes the long-range nature of the model. The ΩN is
an example of so called ultrametric space, which has remarkable
qualitative difference between Euclidean-type lattices. In this paper,
we characterize the sizes of large clusters for this model along the
line of some prior work. The proof involves a stationary embedding
of ΩN into Z. The phase diagram of this long-range percolation is
well understood.

percolation, component, hierarchical lattice, phase transition.

5

6474

Topological Properties of an Exponential Random Geometric Graph Process

In this paper we consider a one-dimensional random
geometric graph process with the inter-nodal gaps evolving according
to an exponential AR(1) process. The transition probability matrix
and stationary distribution are derived for the Markov chains concerning
connectivity and the number of components. We analyze the
algorithm for hitting time regarding disconnectivity. In addition to
dynamical properties, we also study topological properties for static
snapshots. We obtain the degree distributions as well as asymptotic
precise bounds and strong law of large numbers for connectivity
threshold distance and the largest nearest neighbor distance amongst
others. Both exact results and limit theorems are provided in this
paper.

random geometric graph, autoregressive process, degree,connectivity, Markovian, wireless network.

4

7330

Likelihood Estimation for Stochastic Epidemics with Heterogeneous Mixing Populations

We consider a heterogeneously mixing SIR stochastic
epidemic process in populations described by a general graph.
Likelihood theory is developed to facilitate statistic inference for the
parameters of the model under complete observation. We show that
these estimators are asymptotically Gaussian unbiased estimates by
using a martingale central limit theorem.

statistic inference, maximum likelihood, epidemicmodel, heterogeneous mixing.

3

8522

The Giant Component in a Random Subgraph of a Weak Expander

In this paper, we investigate the appearance of the giant component in random subgraphs G(p) of a given large finite graph family Gn = (Vn, En) in which each edge is present independently with probability p. We show that if the graph Gn satisfies a weak isoperimetric inequality and has bounded degree, then the probability p under which G(p) has a giant component of linear order with some constant probability is bounded away from zero and one. In addition, we prove the probability of abnormally large order of the giant component decays exponentially. When a contact graph is modeled as Gn, our result is of special interest in the study of the spread of infectious diseases or the identification of community in various social networks.

subgraph, expander, random graph, giant component, percolation.

2

15186

On the Central Limit Theorems for Forward and Backward Martingales

Let {Xi}i≥1 be a martingale difference sequence with
Xi = Si - Si-1. Under some regularity conditions, we show that
(X2
1+· · ·+X2N
n)-1/2SNn is asymptotically normal, where {Ni}i≥1
is a sequence of positive integer-valued random variables tending
to infinity. In a similar manner, a backward (or reverse) martingale
central limit theorem with random indices is provided.

central limit theorem, martingale difference sequence, backward martingale.

1

9996769

On the Hierarchical Ergodicity Coefficient

In this paper, we deal with the fundamental concepts and properties of ergodicity coefficients in a hierarchical sense by making use of partition. Moreover, we establish a hierarchial Hajnal’s inequality improving some previous results.

Stochastic matrix, ergodicity coefficient, partition.