Universality for the directed configuration model:Metric space convergence of the strongly connected components at criticality

We consider the strongly connected components (SCCs) of a uniform directed graph on n vertices with i.i.d. in-and out-degree pairs distributed as (D−, D+), with E[D+ ] = E[D− ] = µ, conditioned on equal total in-and out-degree. A phase transition for the emergence of a giant SCC is known to occur when E[D− D+ ] is at the critical value µ. We study the model at this critical value and, additionally, require E[(D−)3 ], E[(D+)3 ], E[D− (D+)3 ] and E[(D−)3 D+ ] to be finite. Under these conditions, we show that the SCCs ranked by decreasing number of edges with distances rescaled by n−1/3 converge in distribution to a sequence of finite strongly connected directed multigraphs with edge lengths, and that these are either 3-regular or loops. The limit objects lie in a 3-parameter family, which contains the scaling limit of the SCCs in the directed Erdős-Rényi model at criticality as found by Goldschmidt and Stephenson (2019). This is the first universality result for the scaling limit of a critical directed graph model and the first quantitative result on the directed configuration model at criticality. As a direct consequence, the largest SCCs at criticality contain Θ(n1/3) vertices and edges in probability, and the diameter of the directed graph at criticality is Ω(n1/3) in probability. We use a metric on the space of weighted multigraphs in which two multigraphs are close if there are compatible isomorphisms between their vertex and edge sets which roughly preserve the edge lengths. The topology used is the product topology on the sequence of multigraphs. Our method of proof involves a depth-first exploration of the directed graph, resulting in a spanning forest with additional identifications, of which we study the limit under rescaling.</p

Deep Learning Classification of Building Types in Northern Cyprus

Among the areas where AI studies centered on developing models that provide real-time solutions for the real estate industry are real estate price forecasting, building age, and types and design of the building (villa, apartment, floor number). Nevertheless, within the ML sector, DL is an emerging region with an Interest increases every year. As a result, a growing number of DL research are in conferences and papers, models for real estate have begun to emerge. In this study, we present a deep learning method for classification of houses in Northern Cyprus using Convolutional neural network. This work proposes the use of Convolutional neural networks in the classification of houses images. The classification will be based on the house age, house price, number of floors in the house, house type i.e. Villa and Apartment. The first category is Villa versus Apartments class; based on the training dataset of 362 images the class result shows the overall accuracy of 96.40%. The second category is split into two classes according to age of the buildings, namely 0 to 5 years Apartments 6 to 10 years Apartments. This class is to classify the building based on their age and the result shows the accuracy of 87.42%. The third category is villa with roof versus Villa without roof apartments class which also shows the overall accuracy of 87.60%. The fourth category is Villa Price from 10,000 euro to 200,000 Versus Villa Price from 200,000 Euro to above and the result shows the accuracy of 81.84%. The last category consists of three classes namely 2 floor Apartment versus 3 floor Apartment, 2 floor Apartment versus 4 floor Apartment and 2 floor Apartment versus 5 floor Apartment which all shows the accuracy of 83.54%, 82.48% and 84.77% respectively. From the experiments carried out in this thesis and the results obtained we conclude that the main aims and objectives of this thesis which is to used Deep learning in Classification and detection of houses in Northern Cyprus and to test the performance of AlexNet for houses classification was successful. This study will be very significant in creation of smart cities and digitization of real estate sector as the world embrace the used of the vast power of Artificial Intelligence, machine learning and machine vision

Asymptotics for Sina\u{i} excursions

We study a class of random polymers, introduced by Sina\u{i}, which are related to persistence probabilities in integrated simple random walk bridges. We find the precise asymptotics of these probabilities, and describe their combinatorics, using limit theory for infinitely divisible distributions, and the number-theoretic subset counting formulas of von Sterneck from the early 1900s. Our results sharpen estimates by Aurzada, Dereich and Lifshits, and respond to a conjecture of Caravenna and Deuschel, which arose in their study of the pinning/wetting models, for random linear chains exhibiting entropic repulsion. Our key combinatorial result is an analogue of Sparre Andersen's classical formula.Comment: v3: fixed typo in (1.3

Tournaments and random walks

We study the relationship between tournaments and random walks. This connection was first observed by Erd\H{o}s and Moser. Winston and Kleitman came close to showing that $S_n=\Theta(4^n/n^{5/2})$. Building on this, and works by Tak\'acs, these asymptotic bounds were confirmed by Kim and Pittel. In this work, we verify Moser's conjecture that $S_n\sim C4^n/n^{5/2}$, using limit theory for integrated random walk bridges. Moreover, we show that $C$ can be described in terms of random walks. Combining this with a recent proof and number-theoretic description of $C$ by the second author, we obtain an analogue of Louchard's formula, for the Laplace transform of the squared Brownian excursion/Airy area measure. Finally, we describe the scaling limit of random score sequences, in terms of the Kolmogorov excursions, studied recently by B\"{a}r, Duraj and Wachtel. Our results can also be interpreted as answering questions related to a class of random polymers, which began with influential work of Sina\u{i}. From this point of view, our methods yield the precise asymptotics of a persistence probability, related to the pinning/wetting models from statistical physics, that was estimated up to constants by Aurzada, Dereich and Lifshits, as conjectured by Caravenna and Deuschel.Comment: v3: added ref #25 + corrected typo in Prop 2

Graphical sequences and plane trees

A sequence $d_1\le\cdots\le d_n$ is graphical if it is the degree sequence of a graph. Balister, the second author, Groenland, Johnston and Scott showed that there are asymptotically $C4^n/n^{3/4}$ such sequences. However, the constant $C$ involves a probability that is only approximated. Using random walks and limit theory for infinitely divisible probability distributions, we describe $C$ in terms of Walkup's formula for the number of rooted, unlabelled and cyclically distinct plane trees

Tournament score sequences, Erd\H{o}s-Ginzburg-Ziv numbers, and the L\'evy-Khintchine method

We give a short proof of a recent result of Claesson, Dukes, Frankl\'in and Stef\'ansson, that connects tournament score sequences and the Erd\H{o}s-Ginzburg-Ziv numbers from additive number theory. We show that this connection is, in fact, an instance of the L\'evy-Khintchine formula from probability theory, and highlight how such formulas can be useful in enumerative combinatorics. Our proof combines renewal theory with the representation of score sequences as lattice paths, due to Erd\H{o}s and Moser in the 1960s. These probabilistic and geometric points of view lead to a simpler proof. A key idea in the original proof and ours is to consider cyclic shifts of score sequences. We observe, however, that this idea is already present in Kleitman's remarks added to one of Moser's final articles in 1968, and in subsequent works by Kleitman. In the same article, Moser conjectured that there are asymptotically $C4^n/n^{5/2}$ many score sequences of length $n$. Combining the arguments in the current work with those in a recent work by the third author, we demonstrate the utility of the L\'evy-Khintchine method, by giving a short proof of Moser's conjecture

Critical trees are neither too short nor too fat

We establish lower tail bounds for the height, and upper tail bounds for the width, of critical size-conditioned Bienaym\'e trees. Our bounds are optimal at this level of generality. We also obtain precise asymptotics for offspring distributions within the domain of attraction of a Cauchy distribution, under a local regularity assumption. Finally, we pose some questions on the possible asymptotic behaviours of the height and width of critical size-conditioned Bienaym\'e trees.Comment: 30 page

Refined Horton-Strahler numbers I: a discrete bijection

The Horton-Strahler number of a rooted tree $T$ is the height of the tallest complete binary tree that can be homeomorphically embedded in $T$. The number of full binary trees with $n$ internal vertices and Horton-Strahler number $s$ is known to be the same as the number of Dyck paths of length $2n$ whose height $h$ satisfies $\lfloor \log_2(1+h)\rfloor=s$. In this paper, we present a new bijective proof of the above result, that in fact strengthens and refines it as follows. We introduce a sequence of trees $(\tau_i,i \ge 0)$ which "interpolates" the complete binary trees, in the sense that $\tau_{2^h-1}$ is the complete binary tree of height $h$ for all $h \ge 0$, and $\tau_{i+1}$ strictly contains $\tau_i$ for all $i \ge 0$. Defining $\mathcal{S}(T)$ to be the largest $i$ for which $\tau_i$ can be homeomorphically embedded in $T$, we then show that the number of full binary trees $T$ with $n$ internal vertices and with $\mathcal{S}(T)=h$ is the same as the number of Dyck paths of length $2n$ with height $h$. (We call $\mathcal{S}(T)$ the refined Horton-Strahler number of $T$.) Our proof is bijective and relies on a recursive decomposition of binary trees (resp. Dyck paths) into subtrees with strictly smaller refined Horton-Strahler number (resp. subpaths with strictly smaller height). In a subsequent paper, we will show that the bijection has a continuum analogue, which transforms a Brownian continuum random tree into a Brownian excursion and under which (a continuous analogue of) the refined Horton-Strahler number of the tree becomes the height of the excursion.Comment: 12 pages, 5 figure

Random friend trees

We study a random recursive tree model featuring complete redirection called the random friend tree and introduced by Saram\"aki and Kaski. Vertices are attached in a sequential manner one by one by selecting an existing target vertex and connecting to one of its neighbours (or friends), chosen uniformly at random. This model has interesting emergent properties, such as a highly skewed degree sequence. In contrast to the preferential attachment model, these emergent phenomena stem from a local rather than a global attachment mechanism. The structure of the resulting tree is also strikingly different from both the preferential attachment tree and the uniform random recursive tree: every edge is incident to a macro-hub of asymptotically linear degree, and with high probability all but at most $n^{9/10}$ vertices in a tree of size $n$ are leaves. We prove various results on the neighbourhood of fixed vertices and edges, and we study macroscopic properties such as the diameter and the degree distribution, providing insights into the overall structure of the tree. We also present a number of open questions on this model and related models.Comment: 36 pages, 4 figure