This volume contains the proceedings of the AMS Special Session on Unimodularity in Randomly Generated Graphs, held from October 8-9, 2016, in Denver, Colorado. Unimodularity, a term initially used ...in locally compact topological groups, is one of the main examples in which the generalization from groups to graphs is successful. The "randomly generated graphs", which include percolation graphs, random Erdős-Rényi graphs, and graphings of equivalence relations, are much easier to describe if they result as random objects in the context of unimodularity, with respect to either a vertex-transient "host"-graph or a probability measure. This volume tries to give an impression of the various fields in which the notion currently finds strong development and application: percolation theory, point processes, ergodic theory, and dynamical systems.
The author studies continuous processes indexed by a special family of graphs. Processes indexed by vertices of graphs are known as probabilistic graphical models. In 2011, Burdzy and Pal proposed a ...continuous version of graphical models indexed by graphs with an embedded time structure-- so-called time-like graphs. The author extends the notion of time-like graphs and finds properties of processes indexed by them. In particular, the author solves the conjecture of uniqueness of the distribution for the process indexed by graphs with infinite number of vertices. The author provides a new result showing the stochastic heat equation as a limit of the sequence of natural Brownian motions on time-like graphs. In addition, the author's treatment of time-like graphical models reveals connections to Markov random fields, martingales indexed by directed sets and branching Markov processes.
In this article, we introduce the concept of
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η
1
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-convex stochastic processes on coordinates and establish Hermite-Hadamard-type inequality for these stochastic processes. Moreover, we ...prove new integral inequality of Hermite-Hadamard-Fejér type for newly defined coordinated
η
-convex stochastic processes on a rectangle. The results presented in this article would provide extensions of those given in earlier works.
The mutualistic symbiosis relationship between the gut microbiome and their insect hosts has attracted much scientific attention. The native woodwasp, Sirex nitobei, and the invasive European ...woodwasp, Sirex noctilio, are two pests that infest pines in northeastern China. Following its encounter with the native species, however, there is a lack of research on whether the gut microbiome of S. noctilio changed, what causes contributed to these alterations, and whether these changes were more conducive to invasive colonization. We used high-throughput and metatranscriptomic sequencing to investigate S. noctilio larval gut and frass from four sites where only S. noctilio and both two Sirex species and investigated the effects of environmental factors, biological interactions, and ecological processes on S. noctilio gut microbial community assembly. Amplicon sequencing of two Sirex species revealed differential patterns of bacterial and fungal composition and functional prediction. S. noctilio larval gut bacterial and fungal diversity was essentially higher in coexistence sites than in separate existence sites, and most of the larval gut bacterial and fungal community functional predictions were significantly different as well. Moreover, temperature and precipitation positively correlate with most of the highly abundant bacterial and fungal genera. Source-tracking analysis showed that S. noctilio larvae at coexistence sites remain dependent on adult gut transmission (vertical transmission) or recruitment to frass (horizontal transmission). Meanwhile, stochastic processes of drift and dispersal limitation also have important impacts on the assembly of S. noctilio larval gut microbiome, especially at coexistence sites. In summary, our results reveal the potential role of changes in S. noctilio larval gut microbiome in the successful colonization and better adaptation of the environment.
In this paper, we consider a stochastic process that may experience random reset events which relocate the system to its starting position. We focus our attention on a one-dimensional, monotonic ...continuous-time random walk with a constant drift: the process moves in a fixed direction between the reset events, either by the effect of the random jumps, or by the action of a deterministic bias. However, the orientation of its motion is randomly determined after each restart. As a result of these alternating dynamics, interesting properties do emerge. General formulas for the propagator as well as for two extreme statistics, the survival probability and the mean first-passage time, are also derived. The rigor of these analytical results is verified by numerical estimations, for particular but illuminating examples.
We show that, under a sparsity scenario, the Lasso estimator and the Dantzig selector exhibit similar behavior. For both methods, we derive, in parallel, oracle inequalities for the prediction risk ...in the general nonparametric regression model, as well as bounds on the $\ell_{p}$ estimation loss for 1 ≤ p ≤ 2 in the linear model when the number of variables can be much larger than the sample size.
We consider the problem of estimating the graph associated with a binary Ising Markov random field. We describe a method based on ℓ₁-regularized logistic regression, in which the neighborhood of any ...given node is estimated by performing logistic regression subject to an ℓ₁-constraint. The method is analyzed under high-dimensional scaling in which both the number of nodes p and maximum neighborhood size d are allowed to grow as a function of the number of observations n. Our main results provide sufficient conditions on the triple (n, p, d) and the model parameters for the method to succeed in consistently estimating the neighborhood of every node in the graph simultaneously. With coherence conditions imposed on the population Fisher information matrix, we prove that consistent neighborhood selection can be obtained for sample sizes n = Ω(d³ log p) with exponentially decaying error. When these same conditions are imposed directly on the sample matrices, we show that a reduced sample size of n = Ω(d² log p) suffices for the method to estimate neighborhoods consistently. Although this paper focuses on the binary graphical models, we indicate how a generalization of the method of the paper would apply to general discrete Markov random fields.
Continuously indexed Gaussian fields (GFs) are the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive ...interpretation of the field properties. On the computational side, GFs are hampered with the big n problem, since the cost of factorizing dense matrices is cubic in the dimension. Although computational power today is at an all time high, this fact seems still to be a computational bottleneck in many applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the precision matrix involved sparse, which enables the use of numerical algorithms for sparse matrices, that for fields in only use the square root of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parameterization. We show that, using an approximate stochastic weak solution to (linear) stochastic partial differential equations, we can, for some GFs in the Matérn class, provide an explicit link, for any triangulation of , between GFs and GMRFs, formulated as a basis function representation. The consequence is that we can take the best from the two worlds and do the modelling by using GFs but do the computations by using GMRFs. Perhaps more importantly, our approach generalizes to other covariance functions generated by SPDEs, including oscillating and non-stationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a non-stationary model defined on a sphere.
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