Summary
Statistical network modelling has focused on representing the graph as a discrete structure, namely the adjacency matrix. When assuming exchangeability of this array—which can aid in modelling, computations and theoretical analysis—the Aldous–Hoover theorem informs us that the graph is necessarily either dense or empty. We instead consider representing the graph as an exchangeable random measure and appeal to the Kallenberg representation theorem for this object. We explore using completely random measures (CRMs) to define the exchangeable random measure, and we show how our CRM construction enables us to achieve sparse graphs while maintaining the attractive properties of exchangeability. We relate the sparsity of the graph to the Levy measure defining the CRM. For a specific choice of CRM, our graphs can be tuned from dense to sparse on the basis of a single parameter. We present a scalable Hamiltonian Monte Carlo algorithm for posterior inference, which we use to analyse network properties in a range of real data sets, including networks with hundreds of thousands of nodes and millions of edges.

Sparse graphs using exchangeable random measures

This paper presents an opportunity for the uncertainty that has plagued the novel's criticism to appear as absences in the body of historical knowledge, particularly regarding the notion of life after death. Taking appearance (eg. proof of existence), as opposed to disappearance, as a universally accepted value allows this analysis to interrogate the novel's logic in relation to a variety of conventional systems whose very existence depends on the reproduction of their systems. The ineffectuality of Foucauldian disciplinary institutions in the novel establishes the threat of nonexistence. A significant relationship to Dante's Inferno is rendered, lending the appearance of language an 'enchanted' value through allusions to Dante's intentional invocation of Augustinian corporeal vision. The novel's metalanguage appears enchanted by the body of historical knowledge, particularly as the product of capitalism, discipline and Judeo-Christianity, and programmed by literary precursors William S. Burroughs, Gertrude Stein and Ernest Hemingway. Foregrounded by this complex network, an analysis of the novel’s first chapter demonstrates how an attention to appearance brings the language to life and draws the narrator, equally invested in appearance, into its realm of representation.

Will To Appear

We endow the $\sqrt{8/3}$-Liouville quantum gravity sphere with a metric space structure and show that the resulting metric measure space agrees in law with the Brownian map Recall that a Liouville quantum gravity sphere is a priori naturally parameterized by the Euclidean sphere ${\mathbf S}^2$ Previous work in this series used quantum Loewner evolution (QLE) to construct a metric $d_{\mathcal Q}$ on a countable dense subset of ${\mathbf S}^2$ Here we show that $d_{\mathcal Q}$ as extends uniquely and continuously to a metric $\bar{d}_{\mathcal Q}$ on all of ${\mathbf S}^2$ Letting $d$ denote the Euclidean metric on ${\mathbf S}^2$, we show that the identity map between $({\mathbf S}^2, d)$ and $({\mathbf S}^2, \bar{d}_{\mathcal Q})$ is as Holder continuous in both directions We establish several other properties of $({\mathbf S}^2, \bar{d}_{\mathcal Q})$, culminating in the fact that (as a random metric measure space) it agrees in law with the Brownian map We establish analogous results for the Brownian disk and plane Our proofs involve new estimates on the size and shape of QLE balls and related quantum surfaces, as well as a careful analysis of $({\mathbf S}^2, \bar{d}_{\mathcal Q})$ geodesics

Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding

Previous works in this series have shown that an instance of a $\sqrt{8/3}$-Liouville quantum gravity (LQG) sphere has a well-defined distance function, and that the resulting metric measure space (mm-space) agrees in law with the Brownian map (TBM). In this work, we show that given just the mm-space structure, one can a.s. recover the LQG sphere. This implies that there is a canonical way to parameterize an instance of TBM by the Euclidean sphere (up to Mobius transformation). In other words, an instance of TBM has a canonical conformal structure. 
The conclusion is that TBM and the $\sqrt{8/3}$-LQG sphere are equivalent. They ultimately encode the same structure (a topological sphere with a measure, a metric, and a conformal structure) and have the same law. From this point of view, the fact that the conformal structure a.s. determines the metric and vice-versa can be understood as a property of this unified law. The results of this work also imply that the analogous facts hold for Brownian and $\sqrt{8/3}$-LQG surfaces with other topologies.

Liouville quantum gravity and the Brownian map III: the conformal structure is determined

In this work, we study asymptotics of the genealogy of Galton--Watson processes conditioned on the total progeny. We consider a fixed, aperiodic and critical offspring distribution such that the rescaled Galton--Watson processes converges to a continuous-state branching process (CSBP) with a stable branching mechanism of index $\alpha \in (1, 2]$. We code the genealogy by two different processes: the contour process and the height process that Le Gall and Le Jan recently introduced \cite{LGLJ1, LGLJ1}. We show that the rescaled height process of the corresponding Galton--Watson family tree, with one ancestor and conditioned on the total progeny, converges in a functional sense, to a new process: the normalized excursion of the continuous height process associated with the $\alpha $-stable CSBP. We deduce from this convergence an analogous limit theorem for the contour process. In the Brownian case $\alpha =2$, the limiting process is the normalized Brownian excursion that codes the continuum random tree: the result is due to Aldous who used a different method.

A limit theorem for the contour process of conditioned Galton-Watson trees

We survey the theory and applications of mating-of-trees bijections for random planar maps and their continuum analog: the mating-of-trees theorem of Duplantier, Miller, and Sheffield (2014). The latter theorem gives an encoding of a Liouville quantum gravity (LQG) surface decorated by a Schramm-Loewner evolution (SLE) curve in terms of a pair of correlated linear Brownian motions. We assume minimal familiarity with the theory of SLE and LQG. 
Mating-of-trees theory enables one to reduce problems about SLE and LQG to problems about Brownian motion and leads to deep rigorous connections between random planar maps and LQG. Applications discussed in this article include scaling limit results for various functionals of decorated random planar maps, estimates for graph distances and random walk on (not necessarily uniform) random planar maps, computations of the Hausdorff dimensions of sets associated with SLE, scaling limit results for random planar maps conformally embedded in the plane, and special symmetries for $\sqrt{8/3}$-LQG which allow one to prove its equivalence with the Brownian map.

Mating of trees for random planar maps and Liouville quantum gravity: a survey

The insertion-deletion channel takes as input a bit string ${\bf x}\in\{0,1\}^{n}$, and outputs a string where bits have been deleted and inserted independently at random. The trace reconstruction problem is to recover $\bf x$ from many independent outputs (called "traces") of the insertion-deletion channel applied to $\bf x$. We show that if $\bf x$ is chosen uniformly at random, then $\exp(O(\log^{1/3} n))$ traces suffice to reconstruct $\bf x$ with high probability. For the deletion channel with deletion probability $q < 1/2$ the earlier upper bound was $\exp(O(\log^{1/2} n))$. The case of $q\geq 1/2$ or the case where insertions are allowed has not been previously analyzed, and therefore the earlier upper bound was as for worst-case strings, i.e., $\exp(O( n^{1/3}))$. We also show that our reconstruction algorithm runs in $n^{1+o(1)}$ time. 
A key ingredient in our proof is a delicate two-step alignment procedure where we estimate the location in each trace corresponding to a given bit of $\bf x$. The alignment is done by viewing the strings as random walks and comparing the increments in the walk associated with the input string and the trace, respectively.

/pdf/subpolynomial-trace-reconstruction-for-random-strings-and-isg35qn8lf.pdf

Subpolynomial trace reconstruction for random strings \{and arbitrary deletion probability

The peanosphere construction of Duplantier, Miller and Sheffield provides a means of representing a $\gamma $-Liouville quantum gravity (LQG) surface, $\gamma \in (0,2)$, decorated with a space-filling form of Schramm’s $\mathrm{SLE}_{\kappa }$, $\kappa =16/\gamma^{2}\in (4,\infty)$, $\eta $ as a gluing of a pair of trees which are encoded by a correlated two-dimensional Brownian motion $Z$. We prove a KPZ-type formula which relates the Hausdorff dimension of any Borel subset $A$ of the range of $\eta $, which can be defined as a function of $\eta $ (modulo time parameterization) to the Hausdorff dimension of the corresponding time set $\eta^{-1}(A)$. This result serves to reduce the problem of computing the Hausdorff dimension of any set associated with an $\mathrm{SLE}$, $\mathrm{CLE}$ or related processes in the interior of a domain to the problem of computing the Hausdorff dimension of a certain set associated with a Brownian motion. For many natural examples, the associated Brownian motion set is well known. As corollaries, we obtain new proofs of the Hausdorff dimensions of the $\mathrm{SLE}_{\kappa}$ curve for $\kappa \neq4$; the double points and cut points of $\mathrm{SLE}_{\kappa }$ for $\kappa >4$; and the intersection of two flow lines of a Gaussian free field. We obtain the Hausdorff dimension of the set of $m$-tuple points of space-filling $\mathrm{SLE}_{\kappa }$ for $\kappa >4$ and $m\geq 3$ by computing the Hausdorff dimension of the so-called $(m-2)$-tuple $\pi /2$-cone times of a correlated planar Brownian motion.

An almost sure KPZ relation for SLE and Brownian motion

We introduce a general technique for proving estimates for certain random planar maps which belong to the 
$$\gamma $$

-Liouville quantum gravity (LQG) universality class for 
$$\gamma \in (0,2)$$

. The family of random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d. increments via a mating-of-trees bijection, and includes the uniform infinite planar triangulation (UIPT; 
$$\gamma =\sqrt{8/3}$$

); and planar maps weighted by the number of different spanning trees (
$$\gamma =\sqrt{2}$$

), bipolar orientations (
$$\gamma =\sqrt{4/3}$$

), or Schnyder woods (
$$\gamma =1$$

) that can be put on the map. Using our technique, we prove estimates for graph distances in the above family of random planar maps. In particular, we obtain non-trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the Watabiki (Prog Theor Phys Suppl 114:1–17, 1993) prediction for the Hausdorff dimension of 
$$\gamma $$

-LQG and we establish the existence of an exponent for certain distances in the map. The basic idea of our approach is to compare a given random planar map M to a mated-CRT map—a random planar map constructed from a correlated two-dimensional Brownian motion—using a strong coupling (Zaitsev in ESAIM Probab Stat 2:41–108, 1998) of the encoding walk for M and the Brownian motion used to construct the mated-CRT map. This allows us to deduce estimates for graph distances in M from the estimates for graph distances in the mated-CRT map which we proved (using continuum theory) in a previous work. In the special case when 
$$\gamma =\sqrt{8/3}$$

, we instead deduce estimates for the 
$$\sqrt{8/3}$$

-mated-CRT map from known results for the UIPT. The arguments of this paper do not directly use SLE/LQG, and can be read without any knowledge of these objects.

/pdf/a-mating-of-trees-approach-for-graph-distances-in-random-59gu19t1mx.pdf

A mating-of-trees approach for graph distances in random planar maps

Let $$\gamma \in (0,2)$$
 and let h be the random distribution on $$\mathbb C$$
 which describes a $$\gamma $$
 -Liouville quantum gravity (LQG) cone. Also let $$\kappa = 16/\gamma ^2 >4$$
 and let $$\eta $$
 be a whole-plane space-filling SLE
 $$_\kappa $$
 curve sampled independent from h and parametrized by $$\gamma $$
 -quantum mass with respect to h. We study a family $$\{\mathcal G^\epsilon \}_{\epsilon >0}$$
 of planar maps associated with $$(h, \eta )$$
 called the LQG structure graphs (a.k.a. mated-CRT maps) which we conjecture converge in probability in the scaling limit with respect to the Gromov–Hausdorff topology to a random metric space associated with $$\gamma $$
 -LQG. In particular, $$\mathcal G^\epsilon $$
 is the graph whose vertex set is $$\epsilon \mathbb Z$$
 , with two such vertices $$x_1,x_2\in \epsilon \mathbb Z$$
 connected by an edge if and only if the corresponding curve segments $$\eta ([x_1-\epsilon , x_1])$$
 and $$\eta ([x_2-\epsilon ,x_2])$$
 share a non-trivial boundary arc. Due to the peanosphere description of SLE-decorated LQG due to Duplantier et al. (Liouville quantum gravity as a mating of trees, 2014), the graph $$\mathcal G^\epsilon $$
 can equivalently be expressed as an explicit functional of a correlated two-dimensional Brownian motion, so can be studied without any reference to SLE or LQG. We prove non-trivial upper and lower bounds for the cardinality of a graph-distance ball of radius n in $$\mathcal G^\epsilon $$
 which are consistent with the prediction of Watabiki (Prog Theor Phys Suppl 114:1–17, 1993) for the Hausdorff dimension of LQG. Using subadditivity arguments, we also prove that there is an exponent $$\chi > 0$$
 for which the expected graph distance between generic points in the subgraph of $$\mathcal G^\epsilon $$
 corresponding to the segment $$\eta ([0,1])$$
 is of order $$\epsilon ^{-\chi + o_\epsilon (1)}$$
 , and this distance is extremely unlikely to be larger than $$\epsilon ^{-\chi + o_\epsilon (1)}$$
 .

Nina Holden

Papers

Mating of trees for random planar maps and Liouville quantum gravity: a survey

Subpolynomial trace reconstruction for random strings \{and arbitrary deletion probability

An almost sure KPZ relation for SLE and Brownian motion

A mating-of-trees approach for graph distances in random planar maps

A distance exponent for Liouville quantum gravity