Infinite random geometric graphs

We introduce a new class of countably infinite random geometric graphs, whose vertices are points in a metric space, and vertices are adjacent independently with probability p if the metric distance between the vertices is below a given threshold. If the vertex set is a countable dense set in R^n equipped with the metric derived from the L_{\infty}-norm, then it is shown that with probability 1 such infinite random geometric graphs have a unique isomorphism type. The isomorphism type, which we call GR_n, is characterized by a geometric analogue of the existentially closed adjacency property, and we give a deterministic construction of GR_n. In contrast, we show that infinite random geometric graphs in R^2 with the Euclidean metric are not necessarily isomorphic.Comment: 17 pages, 4 figure

Hierarchical Random Graphs Based on Motifs

Network motifs are characteristic patterns which occur in the networks essentially more frequently than the other patterns. For five motifs found in S. Itzkovitz, U. Alon, Phys. Rev.~E, 2005, 71, 026117-1, hierarchical random graphs are proposed in which the motifs appear at each hierarchical level. A rigorous construction of such graphs is given and a number of their structural properties are analyzed. This includes degree distribution, amenability, clustering, and the small world property. For one of the motifs, annealed phase transitions in the Ising model based on the corresponding graph are also studied.Comment: 8 figures. arXiv admin note: substantial text overlap with arXiv:1106.439

Evolving Shelah-Spencer Graphs

An \emph{evolving Shelah-Spencer process} is one by which a random graph grows, with at each time $\tau \in {\bf N}$ a new node incorporated and attached to each previous node with probability $\tau^{-\alpha}$, where $\alpha \in (0,1) \setminus {\bf Q}$ is fixed. We analyse the graphs that result from this process, including the infinite limit, in comparison to Shelah-Spencer sparse random graphs discussed in [Spencer, J., 2013. The strange logic of random graphs (Vol. 22). Springer Science & Business Media.] and throughout the model-theoretic literature. The first order axiomatisation for classical Shelah-Spencer graphs comprises a 'Generic Extension' axiom and a 'No Dense Subgraphs' axiom. We show that in our context 'Generic Extension' continues to hold. While 'No Dense Subgraphs' fails, a weaker 'Few Rigid Subgraphs' property holds

Strong classification of extensions of classifiable C*-algebras

We show that certain extensions of classifiable C*-algebra are strongly classified by the associated six-term exact sequence in K-theory together with the positive cone of K_{0}-groups of the ideal and quotient. We apply our result to give a complete classification of graph C*-algebras with exactly one ideal.Comment: 29 page

A Preferential Attachment Process Approaching the Rado Graph

We consider a simple Preferential Attachment graph process, which begins with a finite graph, and in which a new $(t+1)$st vertex is added at each subsequent time step $t$, and connected to each previous vertex $u \leq t$ with probability $\frac{d_u(t)}{t}$ where $d_u(t)$ is the degree of $u$ at time $t$. We analyse the graph obtained as the infinite limit of this process, and show that so long as the initial finite graph is neither edgeless nor complete, with probability 1 the outcome will be a copy of the Rado graph augmented with a finite number of either isolated or universal vertices.Comment: arXiv admin note: text overlap with arXiv:1502.0561

Monte Carlo simulations with a generalized detailed balance using the quantum-classical isomorphism

The main idea of this work is that the quantum-classical isomorphism is a suitable framework for a generalization of the notion of detailed balance. The quantum-classical isomorphism is used in order to develop a Monte Carlo simulation with controlled deviation from detailed balance, that is with a generalized detailed balance and known relative entropy with respect to the reference process at each point. In order to apply this method to molecular simulations a new algorithm for realization of a partial chirotope, based on linear programming, a new distance geometry algorithm and a new all-atom off-lattice Monte Carlo method are proposed

Spin Networks in Nonperturbative Quantum Gravity

A spin network is a generalization of a knot or link: a graph embedded in space, with edges labelled by representations of a Lie group, and vertices labelled by intertwining operators. Such objects play an important role in 3-dimensional topological quantum field theory, functional integration on the space A/G of connections modulo gauge transformations, and the loop representation of quantum gravity. Here, after an introduction to the basic ideas of nonperturbative canonical quantum gravity, we review a rigorous approach to functional integration on A/G in which L^2(A/G) is spanned by states labelled by spin networks. Then we explain the `new variables' for general relativity in 4-dimensional spacetime and describe how canonical quantization of gravity in this formalism leads to interesting applications of these spin network states.Comment: 41 pages in LaTe

Local weak convergence for PageRank

PageRank is a well-known algorithm for measuring centrality in networks. It was originally proposed by Google for ranking pages in the World-Wide Web. One of the intriguing empirical properties of PageRank is the so-called `power-law hypothesis': in a scale-free network the PageRank scores follow a power law with the same exponent as the (in-)degrees. Up to date, this hypothesis has been confirmed empirically and in several specific random graphs models. In contrast, this paper does not focus on one random graph model but investigates the existence of an asymptotic PageRank distribution, when the graph size goes to infinity, using local weak convergence. This may help to identify general network structures in which the power-law hypothesis holds. We start from the definition of local weak convergence for sequences of (random) undirected graphs, and extend this notion to directed graphs. To this end, we define an exploration process in the directed setting that keeps track of in- and out-degrees of vertices. Then we use this to prove the existence of an asymptotic PageRank distribution. As a result, the limiting distribution of PageRank can be computed directly as a function of the limiting object. We apply our results to the directed configuration model and continuous-time branching processes trees, as well as preferential attachment models.Comment: 32 pages, 5 figure

Amenability versus non-exactness of dense subgroups of a compact group

Given a countable residually finite group, we construct a compact group K and two elements w and u of K with the following properties: The group generated by w and the cube of u is amenable, the group generated by w and u contains a copy of the given group, and these two groups are dense in K. By combining it with a construction of non-exact groups that are LEF by Osajda and Arzhantseva--Osajda and formation of diagonal products, we construct an example for which the latter dense group is non-exact. Our proof employs approximations in the space of marked groups of LEF ("Locally Embeddable into Finite groups") groups.Comment: 32 pages, explanation brushed up; 28 pages, can take $t=u^3$ in the main theorem (Theorem A); 27 pages, results are optimized; 19 pages, no figur

Random geometry on the sphere

We introduce and study a universal model of random geometry in two dimensions. To this end, we start from a discrete graph drawn on the sphere, which is chosen uniformly at random in a certain class of graphs with a given size $n$, for instance the class of all triangulations of the sphere with $n$ faces. We equip the vertex set of the graph with the usual graph distance rescaled by the factor $n^{-1/4}$. We then prove that the resulting random metric space converges in distribution as $n\to\infty$, in the Gromov-Hausdorff sense, toward a limiting random compact metric space called the Brownian map, which is universal in the sense that it does not depend on the class of graphs chosen initially. The Brownian map is homeomorphic to the sphere, but its Hausdorff dimension is equal to $4$. We obtain detailed information about the structure of geodesics in the Brownian map. We also present the infinite-volume variant of the Brownian map called the Brownian plane, which arises as the scaling limit of the uniform infinite planar quadrangulation. Finally, we discuss certain open problems. This study is motivated in part by the use of random geometry in the physical theory of two-dimensional quantum gravity.Comment: To appear in the Proceedings of ICM 2014, Seou