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Eternal Family Trees and dynamics on unimodular random graphs
Abstract
This paper is centered on covariant dynamics on unimodular random graphs and random networks (marked graphs), namely maps from the set of vertices to itself which are preserved by graph or network isomorphisms. Such dynamics are referred to as vertex-shifts here. The first result of the paper is a classification of vertex-shifts on unimodular random networks. Each such vertex-shift partitions the vertices into a collection of connected components and foils. The latter are discrete analogues the stable manifold of the dynamics. The classification is based on the cardinality of the connected components and foils. Up to an event of zero probability, there are three classes of foliations in a connected component: F/F (with finitely many finite foils), I/F (infinitely many finite foils), and I/I (infinitely many infinite foils). An infinite connected component of the graph of a vertex-shift on a random network forms an infinite tree with one selected end which is referred to as an Eternal Family Tree. Such trees can be seen as stochastic extensions of branching processes. Unimodular Eternal Family Trees can be seen as extensions of critical branching processes. The class of offspring-invariant Eternal Family Trees, which is introduced in the paper, allows one to analyze dynamics on networks which are not necessarily unimodular. These can be seen as extensions of not necessarily critical branching processes. Several construction techniques of Eternal Family Trees are proposed, like the joining of trees or moving the root to a far descendant. Eternal Galton-Watson Trees and Eternal Multitype Galton-Watson Trees are also introduced as special cases of Eternal Family Trees satisfying additional independence properties. These examples allow one to show that the results on Eternal Family Trees unify and extend to the dependent case several well known theorems of the literature on branching processes.
... By the explicit construction of EGW trees in [7], Y n is a critical Galton-Watson tree up to generation n. Also, for 0 ≤ i < n, Y i has the same structure up to generation i, except that the distribution of the first generation is size-biased minus one (i.e., (np n+1 ) n with the notation of Subsection 3.2.1). ...
... By considering the partition Π n by cubes, one can construct a b n -covering R n as in Theorem 2.16. This covering satisfies [7] for more details on eternal family trees). Therefore, Lemma 5.7 of [7] implies that ...
... This covering satisfies [7] for more details on eternal family trees). Therefore, Lemma 5.7 of [7] implies that ...
The notions of unimodular Minkowski and Hausdorff dimensions are defined in [F. Baccelli, M.-O. Haji-Mirsadeghi, A. Khezeli, preprint (2018)] for unimodular random discrete metric spaces. The present paper is focused on the connections between these notions and the polynomial growth rate of the underlying space. It is shown that bounding the dimension is closely related to finding suitable equivariant weight functions (i.e., measures) on the underlying discrete space. The main results are unimodular versions of the mass distribution principle, Billingsley's lemma and Frostman's lemma, which allow one to derive upper bounds on the unimodular Hausdorff dimension from the growth rate of suitable equivariant weight functions. These results allow one to compute or bound both types of unimodular dimensions in a large set of examples in the theory of point processes, unimodular random graphs, and self-similarity. Further results of independent interest are also presented, like a version of the max-flow min-cut theorem for unimodular one-ended trees and a weak form of pointwise ergodic theorems for all unimodular discrete spaces.
... In terms of unimodular dimensions, it will be shown that unimodular one-ended trees are the richest class of unimodular trees. First, the following notation is borrowed from [6]. Every one-ended tree T can be regarded as a family tree as follows. ...
... Nevertheless, the same lemma implies that the connected component T of T containing o is a unimodular tree. Since it is one-ended, Theorem 3.9 of [6] implies that the foils T ∩ Φ i are infinite a.s. By noting that the edges do not cross (as segments in the plane), one obtains that T ∩ Φ i should be the whole Φ i ; hence, T = T . ...
... Eternal Galton-Watson (EGW) trees are defined in [6]. Unimodular EGW trees (in the nontrivial case) can be characterized as unimodular one-ended trees in which the descendants of the root constitute a Galton-Watson tree. ...
This work introduces two new notions of dimension, namely the unimodular Minkowski and Hausdorff dimensions, which are inspired from the classical analogous notions. These dimensions are defined for unimodular discrete spaces, introduced in this work, which provide a common generalization to stationary point processes under their Palm version and unimodular random rooted graphs. The use of unimodularity in the definitions of dimension is novel. Also, a toolbox of results is presented for the analysis of these dimensions. In particular, analogues of Billingsley's lemma and Frostman's lemma are presented. These last lemmas are instrumental in deriving upper bounds on dimensions, whereas lower bounds are obtained from specific coverings. The notions of unimodular Hausdorff size, which is a discrete analogue of the Hausdorff measure, and unimodular dimension function are also introduced. This toolbox allows one to connect the unimodular dimensions to other notions such as volume growth rate, discrete dimension and scaling limits. It is also used to analyze the dimensions of a set of examples pertaining to point processes, branching processes, random graphs, random walks, and self-similar discrete random spaces. Further results of independent interest are also presented, like a version of the max-flow min-cut theorem for unimodular one-ended trees and a weak form of pointwise ergodic theorems for all unimodular discrete spaces.
... Before presenting the definitions and results, Subsection 3.1 provides a motivation and some basic examples of additional structures on compact metric spaces. The proofs of the results are postponed to Subsection 3. 6. Further examples are provided in Subsection 3.4 and also in Section 5. ...
... Also, we study the set of boundedly-compact pointed metric spaces equipped with a closed subset of ends. A similar set is considered in [6] for trees without going into details. ...
In this paper, a general approach is presented for generalizing the Gromov-Hausdorff metric to consider metric spaces equipped with some additional structure. A special case is the Gromov-Hausdorff-Prokhorov metric which considers measured metric spaces. This abstract framework also unifies several existing generalizations which consider metric spaces equipped with a measure, a point, a closed subset, a curve or a tuple of such structures. It can also be useful for studying new examples of additional structures. The framework is provided both for compact metric spaces and for boundedly-compact pointed metric spaces. In addition, completeness and separability of the metric is proved under some conditions. This enables one to study random metric spaces equipped with additional structures, which is the main motivation of this work.
... Decades of Software Engineering, Programing Language Theory etc have been dedicated to produce rigorous programming frames, in order to avoid bugs and inconsistencies and to invent methods for proving partial correctness -see the broad literature on Model Checking (Sifakis 2011), on the Abstract Interpretation (Cousot 2016), on Type Theory (Girard et al. 1989) and a lot more. Two of the many amazing successes of the Science of Programming are … one, the Internet, whose functioning is also made possible by geometric tools in Concurrency (Aceto et al. 2003) and in networks (Baccelli, 2016), and, the other, Embedded Computing and their Nets, such as Aircraft and Flight Control, following, among others, the approach in (Sifakis 2011). ...
This text will first discuss the great and productive “linguistic turn” that originated in Logic, at the end of the nineteenth century. After Turing’s work on the Logical Computing Machine (1936–1950) and Shannon’s theory (1948), Brillouin attributed information content to Maxwell’s demon’s action (1956), with a scientific invention and, in my opinion, a touch of humor, thus granting to inert matter an intelligent activity capable of inverting entropy. Moreover, since the 1950s, many identified the digital elaboration of data with human intelligence and some projected the processing of information onto inert matter. In the same years, DNA was identified as the complete information carrier of phylogenesis and ontogenesis. In these approaches, an ambiguity has been arbitrarily played out, and a threshold has been sometimes crossed, sometimes not; first, in Logic and Computing, between “formal principles of proof” or of computation and their “semantics” (their mathematical meaning); then in Biology, between elaboration of information and biological dynamics in their context. Today, well beyond classical Artificial Intelligence on discrete data types, Brillouin’s invention and Deep Learning on continuous mathematical structures are the new loci of confusion between knowledge construction and transmission or processing of information. Yet, the construction of human knowledge is even more radically by-passed and science dehumanized when information is directly embedded in a world that is viewed as a massive information processor. The remarkable scientific and economic productivity of our new observable, information, must instead be seen in the context of our human, ecosystemic and historical activities, in order to subordinate it to interpretation and meaningful action that enrich and do not strip the sense from our forms of knowledge and life. To this purpose, we must strictly distinguish between information, as formal elaboration and transmission of signs, and information as production of “meaning” in our active friction on reality.
... Decades of Software Engineering, Programing Language Theory etc have been dedicated to produce rigorous programming frames, in order to avoid bugs and inconsistencies and to invent methods for proving partial correctness -see the broad literature on Model Checking (Sifakis 2011), on the Abstract Interpretation (Cousot 2016), on Type Theory (Girard et al. 1989) and a lot more. Two of the many amazing successes of the Science of Programming are … one, the Internet, whose functioning is also made possible by geometric tools in Concurrency (Aceto et al. 2003) and in networks (Baccelli, 2016), and, the other, Embedded Computing and their Nets, such as Aircraft and Flight Control, following, among others, the approach in (Sifakis 2011). ...
This text will first discuss the great and productive "linguistic turn" that originated in Logic, at the end of the XIX century. After Turing's work on the Logical Computing Machine (1936-1950) and Shannon's theory (1948), Brillouin attributed information content to Maxwell's demon's action (1956), with a scientific invention and, in my opinion, a touch of humor, thus granting to inert matter an intelligent activity capable of inverting entropy. Moreover, since the 1950s, many identified the digital elaboration of data with human intelligence and some projected the processing of information onto inert matter. In the same years, DNA was identified as the complete information carrier of phylogenesis and ontogenesis. In these approaches, an ambiguity has been arbitrarily played out, and a threshold has been sometimes crossed, sometimes not; first, in Logic and Computing, between "formal principles of proof" or of computation and their "semantics" (their mathematical meaning); then in Biology, between elaboration of information and biological dynamics in their context. Today, well beyond classical Artificial Intelligence on discrete data types, Brillouin's invention and Deep Learning on continuous mathematical structures are the new loci of confusion between knowledge construction and transmission or processing of information. Yet, the construction of human knowledge is even more radically bypassed and science dehumanized when information is directly embedded in a world that is viewed as a massive information processor. The remarkable scientific and economic productivity of our new observable, information, must instead be seen in the context of our human, ecosystemic and historical activities, in order to subordinate it to interpretation and meaningful action that enrich and do not strip the sense from our forms of knowledge and life. To this purpose, we must strictly distinguish between information, as formal elaboration and transmission of signs, and information as production of "meaning" in our active friction on reality.
... In computer networks, distributed in space-time continua, this presents some peculiar difficulties that are adequately dealt with by the difficult mathematics of concurrent and network programming. These analysis are also based on continuous dynamics in complex network structures (Baccelli, 2016a;2016b), whose nodes are digital computers (Aceto, 2003). Moreover, if one looks closely into today's computer's hardware, the instructions that modify a discrete (possibly digital) data-type actually work by variations in electric tension's levels in continuous fields and/or by driving electric currents in continua into two stable states, through discrete thresholds. ...
Theorizing and measuring radically change in physics when using discrete vs. continuous mathematical spaces. We first recall that, in the 20 th century, discrete structures were brought into the limelight by Quantum and Information Theories. In these theories, the reference to discrete observables and parameters and to digital information is far from neutral in knowledge construction; it leads to dramatic consequences when biological dynamics are identified with information processing. Following an early debate in physics, we briefly analyze the origin and the nature of the bias thus induced in life sciences. We show how strong consequences have been derived from vague, common sense hypotheses and then stress their role in cancer biology. Finally, we summarize new theoretical frames that propose different directions as for the organizing principles for biological thinking and experimenting, including in cancer research. Cancer is then viewed as an organismal, tissue-based issue, according to the perspective proposed in (Sonnenschein & Soto, 1999; Baker, 2015).
... In computer networks, distributed in space-time continua, this presents some peculiar difficulties that are adequately dealt with by the difficult mathematics of concurrent and network programming. These analysis are also based on continuous dynamics in complex network structures (Baccelli, 2016a;2016b), whose nodes are digital computers (Aceto, 2003). Moreover, if one looks closely into today's computer's hardware, the instructions that modify a discrete (possibly digital) data-type actually work by variations in electric tension's levels in continuous fields and/or by driving electric currents in continua into two stable states, through discrete thresholds. ...
The role of continua has been clear since antiquity in the mathematical approaches to physics, while discrete manifolds were brought to the limelight mostly by Quantum and Information Theories, in the XX century. We first recall how theorizing and measuring radically change in physics when using discrete vs. continuous mathematical manifolds. It will follow that the reference to discrete structures and digital information is far from neutral in knowledge construction. In biology, in particular, the introduction of information as a new observable on discrete data types has been promoting a dramatic reorganization of the tools for knowledge. We briefly analyze the origin and the nature, then some consequences of the bias thus induced in life sciences, with particular emphasis on research on cancer. We finally summarize new theoretical frames that propose different directions as for the organizing principles for biological thinking and experimenting, including in cancer research. Cancer is now viewed as an organismal, tissue based issue, according to the perspective proposed in (Sonnenschein, Soto, 1999; Baker, 2015).
This work develops a novel approach toward performance guarantees for all links in arbitrarily large wireless networks. It introduces a
spatial network calculus
, consisting of spatial regulation properties for stationary point processes and the first steps of a calculus for this regulation, which can be seen as an extension to space of the classical network calculus. Specifically, two classes of regulations are defined: one includes ball regulation and shot-noise regulation, which are shown to be equivalent and upper constraint interference; the other one includes void regulation, which lower constraints the signal power. These regulations are defined both in the strong and weak sense: the former requires the regulations to hold everywhere in space, whereas the latter only requires the regulations to hold as observed by a jointly stationary point process. Using this approach, we derive performance guarantees in device-to-device, ad hoc, and cellular networks under proper regulations. We give universal bounds on the SINR for all links, which gives link service guarantees based on information-theoretic achievability. They are combined with classical network calculus to provide end-to-end latency guarantees for all packets in wireless queuing networks. Such guarantees do not exist in networks that are not spatially regulated, e.g., Poisson networks.
The blockchain paradigm provides a mechanism for content dissemination and distributed consensus on Peer-to-Peer (P2P) networks. While this paradigm has been widely adopted in industry, it has not been carefully analyzed in terms of its network scaling with respect to the number of peers. Applications for blockchain systems, such as cryptocurrencies and IoT, require this form of network scaling. In this paper, we propose a new stochastic network model for a blockchain system. We identify a structural property called one-endedness, which we show to be desirable in any blockchain system as it is directly related to distributed consensus among the peers. We show that the stochastic stability of the network is sufficient for the one-endedness of a blockchain. We further establish that our model belongs to a class of network models, called monotone separable models. This allows us to establish upper and lower bounds on the stability region. The bounds on stability depend on the connectivity of the P2P network through its conductance and allow us to analyze the scalability of blockchain systems on large P2P networks. We verify our theoretical insights using both synthetic data and real data from the Bitcoin network.
Graphings are special bounded-degree graphs on probability
spaces, representing limits of graph sequences that are convergent in a local or
local-global sense. A graphing is compact, if the underlying space is a compact
metric space, the edge set is closed and nearby points have nearby graph neighborhoods.
We prove some simple properties of compact graphings. Our main result
is a procedure for embedding an arbitrary graphing into an essentially equivalent
compact graphing.
Based on a simple object, an i.i.d. sequence of positive integer-valued random variables { a n } n ∊ℤ , we introduce and study two random structures and their connections. First, a population dynamics, in which each individual is born at time n and dies at time n + a n . This dynamics is that of a D/GI/∞ queue, with arrivals at integer times and service times given by { a n } n ∊ℤ . Second, the directed random graph T f on ℤ generated by the random map f ( n ) = n + a n . Assuming only that E [ a0 ] < ∞ and P [ a0 = 1] > 0, we show that, in steady state, the population dynamics is regenerative, with one individual alive at each regeneration epoch. We identify a unimodular structure in this dynamics. More precisely, T f is a unimodular directed tree, in which f ( n ) is the parent of n . This tree has a unique bi-infinite path. Moreover, T f splits the integers into two categories: ephemeral integers, with a finite number of descendants of all degrees, and successful integers, with an infinite number. Each regeneration epoch is a successful individual such that all integers less than it are its descendants of some order. Ephemeral, successful, and regeneration integers form stationary and mixing point processes on ℤ.
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