scholarly journals Unimodular random trees

2013 ◽  
Vol 35 (2) ◽  
pp. 359-373 ◽  
Author(s):  
ITAI BENJAMINI ◽  
RUSSELL LYONS ◽  
ODED SCHRAMM
Keyword(s):  
Weak Convergence ◽  
Degree Distribution ◽  
Local Convergence ◽  
Hyperbolic Plane ◽  
Random Trees ◽  
Uniform Integrability ◽  
Ideal Boundary ◽  
Bounded Degree ◽  
Regular Tree ◽  
Finite Graphs

AbstractWe consider unimodular random rooted trees (URTs) and invariant forests in Cayley graphs. We show that URTs of bounded degree are the same as the law of the component of the root in an invariant percolation on a regular tree. We use this to give a new proof that URTs are sofic, a result of Elek. We show that ends of invariant forests in the hyperbolic plane converge to ideal boundary points. We also note that uniform integrability of the degree distribution of a family of finite graphs implies tightness of that family for local convergence, also known as random weak convergence.

Approximation of periodic queues

1987 ◽  
Vol 19 (03) ◽  
pp. 691-707 ◽  
Author(s):  
Tomasz Rolski
Keyword(s):  
Weak Convergence ◽  
Time Of Day ◽  
Uniform Integrability ◽  
Queue Size ◽  
Approximation Theorems ◽  
Mean Delay ◽  
Periodic Queues ◽  
Poisson Arrivals ◽  
Markov Modulated

In this paper we demonstrate how some characteristics of queues with the periodic Poisson arrivals can be approximated by the respective characteristics in queues with Markov modulated input. These Markov modulated queues were recently studied by Regterschot and de Smit (1984). The approximation theorems are given in terms of the weak convergence of some characteristics and their uniform integrability. The approximations are applicable for the following characteristics: mean workload, mean workload at the time of day, mean delay, mean queue size.

scholarly journals Random graphs, weak coarse embeddings, and higher index theory

2015 ◽  
Vol 07 (03) ◽  
pp. 361-388 ◽  
Author(s):  
Rufus Willett
Keyword(s):  
Geometric Property ◽  
Random Sequence ◽  
Index Theory ◽  
Weak Form ◽  
Bounded Degree ◽  
Random Sequences ◽  
Finite Graphs ◽  
Coarse Embedding ◽  
Property T ◽  
Assembly Map

This paper studies higher index theory for a random sequence of bounded degree, finite graphs with diameter tending to infinity. We show that in a natural model for such random sequences the following hold almost surely: the coarse Baum–Connes assembly map is injective; the coarse Baum–Connes assembly map is not surjective; the maximal coarse Baum–Connes assembly map is an isomorphism. These results are closely tied to issues of expansion in graphs: in particular, we also show that such random sequences almost surely do not have geometric property (T), a strong form of expansion.The key geometric ingredients in the proof are due to Mendel and Naor: in our context, their results imply that a random sequence of graphs almost surely admits a weak form of coarse embedding into Hilbert space.

Approximation of periodic queues

1987 ◽  
Vol 19 (3) ◽  
pp. 691-707 ◽  
Author(s):  
Tomasz Rolski
Keyword(s):  
Weak Convergence ◽  
Time Of Day ◽  
Uniform Integrability ◽  
Queue Size ◽  
Approximation Theorems ◽  
Mean Delay ◽  
Periodic Queues ◽  
Poisson Arrivals ◽  
Markov Modulated

In this paper we demonstrate how some characteristics of queues with the periodic Poisson arrivals can be approximated by the respective characteristics in queues with Markov modulated input. These Markov modulated queues were recently studied by Regterschot and de Smit (1984). The approximation theorems are given in terms of the weak convergence of some characteristics and their uniform integrability. The approximations are applicable for the following characteristics: mean workload, mean workload at the time of day, mean delay, mean queue size.

scholarly journals Flows on measurable spaces

2021 ◽  
Author(s):  
László Lovász
Keyword(s):  
Directed Graphs ◽  
Flow Theory ◽  
Space Structure ◽  
Borel Space ◽  
Bounded Degree ◽  
Counting Measure ◽  
Finite Graphs ◽  
Graph Limits ◽  
Dense Graphs ◽  
Bounded Degree Graphs

AbstractThe theory of graph limits is only understood to a somewhat satisfactory degree in the cases of dense graphs and of bounded degree graphs. There is, however, a lot of interest in the intermediate cases. It appears that one of the most important constituents of graph limits in the general case will be Markov spaces (Markov chains on measurable spaces with a stationary distribution). This motivates our goal to extend some important theorems from finite graphs to Markov spaces or, more generally, to measurable spaces. In this paper, we show that much of flow theory, one of the most important areas in graph theory, can be extended to measurable spaces. Surprisingly, even the Markov space structure is not fully needed to get these results: all we need a standard Borel space with a measure on its square (generalizing the finite node set and the counting measure on the edge set). Our results may be considered as extensions of flow theory for directed graphs to the measurable case.

scholarly journals Degree distribution of random Apollonian network structures and Boltzmann sampling

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Alexis Darrasse ◽  
Michèle Soria
Keyword(s):  
Power Law ◽  
Degree Distribution ◽  
Random Trees ◽  
Network Structures ◽  
Higher Dimension ◽  
Random Generation ◽  
International Audience ◽  
Boltzmann Model

International audience Random Apollonian networks have been recently introduced for representing real graphs. In this paper we study a modified version: random Apollonian network structures (RANS), which preserve the interesting properties of real graphs and can be handled with powerful tools of random generation. We exhibit a bijection between RANS and ternary trees, that transforms the degree of nodes in a RANS into the size of particular subtrees. The distribution of degrees in RANS can thus be analysed within a bivariate Boltzmann model for the generation of random trees, and we show that it has a Catalan form which reduces to a power law with an exponential cutoff: $α ^k k^{-3/2}$, with $α = 8/9$. We also show analogous distributions for the degree in RANS of higher dimension, related to trees of higher arity.

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