1,344 research outputs found

    A Look at Financial Dependencies by Means of Econophysics and Financial Economics

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    This is a review about financial dependencies which merges efforts in econophysics and financial economics during the last few years. We focus on the most relevant contributions to the analysis of asset markets' dependencies, especially correlational studies, which in our opinion are beneficial for researchers in both fields. In econophysics, these dependencies can be modeled to describe financial markets as evolving complex networks. In particular we show that a useful way to describe dependencies is by means of information filtering networks that are able to retrieve relevant and meaningful information in complex financial data sets. In financial economics these dependencies can describe asset comovement and spill-overs. In particular, several models are presented that show how network and factor model approaches are related to modeling of multivariate volatility and asset returns respectively. Finally, we sketch out how these studies can inspire future research and how they contribute to support researchers in both fields to find a better and a stronger common language

    Zero-temperature stochastic Ising model on planar quasi-transitive graphs

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    We study the zero-temperature stochastic Ising model on some connected planar quasi-transitive graphs, which are invariant under rotation and translation. The initial spin configuration is distributed according to a Bernoulli product measure with parameter p(0,1) p\in(0,1) . In particular, we prove that if p=1/2 p=1/2 and the graph underlying the model satisfies the planar shrink property (which causes each finite cluster to shrink to a site and then vanish with positive probability) then all vertices flip infinitely often almost surely

    Jake pozicione igre

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    In this thesis, we study 2-player combinatorial games on graphs. We devote a lot of attention to strong positional games, where both players have the same goal. First, we consider the so-called fixed graph strong Avoider-Avoider game in which two players called Red and Blue alternately claim edges of the complete graph Kn, and the player who first completes a copy of a fixed graph F loses the game. If neither of the players claimed a copy of F in his graph and all the elements of the board are claimed, the game is declared a draw. Even though these games have been studied for decades, there are very few known results. We make a step forward by proving that Blue has a winning strategy it two different games of this kind. Furthermore, we introduce strong CAvoiderCAvoider F games where the claimed edges of each player must form a connected graph throughout the game. This is a natural extension of the strong Avoider-Avoider games, with a connectedness constraint. We prove that Blue can win in three standard CAvoider-CAvoider F games. Next, we study strong Maker-Maker F games, where now, the player who first occupies a copy of F is the winner. It is well-known that the outcome of these games when both players play optimally can be either the first player's win or a draw. We are interested in finding the achievement number a(F) of a strong Maker-Maker F game, that is, the smallest n for which Red has a winning strategy. We can find the exact value a(F) for several graphs F, including paths, cycles, perfect matchings, and a subclass of trees on n vertices. We also give the upper and lower bounds for the achievement number of stars and trees. Finally, we introduce generalized saturation games as a natural extension of two different types of combinatorial games, saturation games and Constructor-Blocker games. In the generalized saturation game, two graphs H and F are given in advance. Two players called Max and Mini alternately claim unclaimed edges of the complete graph Kn and together gradually building the game graph G, the graph that consists of all edges claimed by both players. The graph G must never contain a copy of F, and the game ends when there are no more moves, i.e. when G is a saturated F-free graph. We are interested in the score of this game, that is, the number of copies of the graph H in G at the end of the game. Max wants to maximize this score, whereas Mini tries to minimize it. The game is played under the assumption that both players play optimally. We study several generalized saturation games for natural choices of F and H, in an effort to locate the score of the game as precisely as possible.У овој тези проучавамо комбинаторне игре на графовима које играју 2 играча. Посебну пажњу посвећујемо јаким позиционим играма, у којима оба играча имају исти циљ. Прво, посматрамо такозвану јаку Авојдер-Авојдер игру са задатим фиксним графом у којој два играча, Црвени и Плави наизменично селектују гране комплетног графа Kn, а играч који први селектује копију фиксног графа F губи игру. Ако ниједан од играча не садржи копију од F у свом графу и сви елементи табле су селектовани, игра се проглашава нерешеном. Иако су ове игре проучаване деценијама, врло је мало познатих резултата. Ми смо направили корак напред доказавши да Плави има победничку стратегију у две различите игре ове врсте. Такође, уводимо јаке ЦАвојдер-ЦАвојдер F игре у којима граф сваког играча мора остати повезан током игре. Ово је природно проширење јаких Авојдер-Авојдер игара, са ограничењем повезаности. Доказујемо да Плави може да победи у три стандардне ЦАвојдер-ЦАвојдер F игре. Затим проучавамо јаке Мејкер-Мејкер F игре, у којима је играч који први селектује копију од F победник. Познато је да исход ових игара уколико оба играча играју оптимално може бити или победа првог играча или нерешено. Циљ нам је да пронађемо ачивмент број а(F) јаке Мејкер-Мејкер F игре, односно најмање n за које Црвени има победничку стратегију. Дајемо тачну вредност a(F) за неколико графова F, укључујући путеве, циклусе, савршене мечинге и поткласу стабала са n чворова. Такође, дајемо горње и доње ограничење ачивмент броја за звезде и стабла. Коначно, уводимо уопштене игре сатурације као природно проширење две различите врсте комбинаторних игара, игара сатурације и Конструктор-Блокер игара. У уопштеној игри сатурације унапред су дата два графа H и F. Два играча по имену Макс и Мини наизменично селектују слободне гране комплетног графа Kn и заједно постепено граде граф игре G, који се састоји од свих грана које су селектовала оба играча. Граф G не сме да садржи копију од F, а игра се завршава када више нема потеза, односно када је G сатуриран граф који не садржи F. Занима нас резултат ове игре, односно, број копија графа H у G на крају игре. Макс жели да максимизира овај резултат, док Мини покушава да га минимизира. Игра се под претпоставком да оба играча играју оптимално. Проучавамо неколико уопштених игара сатурације за природне изборе F и H, у настојању да што прецизније одредимо резултат игре.U ovoj tezi proučavamo kombinatorne igre na grafovima koje igraju 2 igrača. Posebnu pažnju posvećujemo jakim pozicionim igrama, u kojima oba igrača imaju isti cilj. Prvo, posmatramo takozvanu jaku Avojder-Avojder igru sa zadatim fiksnim grafom u kojoj dva igrača, Crveni i Plavi naizmenično selektuju grane kompletnog grafa Kn, a igrač koji prvi selektuje kopiju fiksnog grafa F gubi igru. Ako nijedan od igrača ne sadrži kopiju od F u svom grafu i svi elementi table su selektovani, igra se proglašava nerešenom. Iako su ove igre proučavane decenijama, vrlo je malo poznatih rezultata. Mi smo napravili korak napred dokazavši da Plavi ima pobedničku strategiju u dve različite igre ove vrste. Takođe, uvodimo jake CAvojder-CAvojder F igre u kojima graf svakog igrača mora ostati povezan tokom igre. Ovo je prirodno proširenje jakih Avojder-Avojder igara, sa ograničenjem povezanosti. Dokazujemo da Plavi može da pobedi u tri standardne CAvojder-CAvojder F igre. Zatim proučavamo jake Mejker-Mejker F igre, u kojima je igrač koji prvi selektuje kopiju od F pobednik. Poznato je da ishod ovih igara ukoliko oba igrača igraju optimalno može biti ili pobeda prvog igrača ili nerešeno. Cilj nam je da pronađemo ačivment broj a(F) jake Mejker-Mejker F igre, odnosno najmanje n za koje Crveni ima pobedničku strategiju. Dajemo tačnu vrednost a(F) za nekoliko grafova F, uključujući puteve, cikluse, savršene mečinge i potklasu stabala sa n čvorova. Takođe, dajemo gornje i donje ograničenje ačivment broja za zvezde i stabla. Konačno, uvodimo uopštene igre saturacije kao prirodno proširenje dve različite vrste kombinatornih igara, igara saturacije i Konstruktor-Bloker igara. U uopštenoj igri saturacije unapred su data dva grafa H i F. Dva igrača po imenu Maks i Mini naizmenično selektuju slobodne grane kompletnog grafa Kn i zajedno postepeno grade graf igre G, koji se sastoji od svih grana koje su selektovala oba igrača. Graf G ne sme da sadrži kopiju od F, a igra se završava kada više nema poteza, odnosno kada je G saturiran graf koji ne sadrži F. Zanima nas rezultat ove igre, odnosno, broj kopija grafa H u G na kraju igre. Maks želi da maksimizira ovaj rezultat, dok Mini pokušava da ga minimizira. Igra se pod pretpostavkom da oba igrača igraju optimalno. Proučavamo nekoliko uopštenih igara saturacije za prirodne izbore F i H, u nastojanju da što preciznije odredimo rezultat igre

    The Distributed Complexity of Locally Checkable Labeling Problems Beyond Paths and Trees

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    We consider locally checkable labeling LCL problems in the LOCAL model of distributed computing. Since 2016, there has been a substantial body of work examining the possible complexities of LCL problems. For example, it has been established that there are no LCL problems exhibiting deterministic complexities falling between ω(logn)\omega(\log^* n) and o(logn)o(\log n). This line of inquiry has yielded a wealth of algorithmic techniques and insights that are useful for algorithm designers. While the complexity landscape of LCL problems on general graphs, trees, and paths is now well understood, graph classes beyond these three cases remain largely unexplored. Indeed, recent research trends have shifted towards a fine-grained study of special instances within the domains of paths and trees. In this paper, we generalize the line of research on characterizing the complexity landscape of LCL problems to a much broader range of graph classes. We propose a conjecture that characterizes the complexity landscape of LCL problems for an arbitrary class of graphs that is closed under minors, and we prove a part of the conjecture. Some highlights of our findings are as follows. 1. We establish a simple characterization of the minor-closed graph classes sharing the same deterministic complexity landscape as paths, where O(1)O(1), Θ(logn)\Theta(\log^* n), and Θ(n)\Theta(n) are the only possible complexity classes. 2. It is natural to conjecture that any minor-closed graph class shares the same complexity landscape as trees if and only if the graph class has bounded treewidth and unbounded pathwidth. We prove the "only if" part of the conjecture. 3. In addition to the well-known complexity landscapes for paths, trees, and general graphs, there are infinitely many different complexity landscapes among minor-closed graph classes

    Pearcey universality at cusps of polygonal lozenge tiling

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    We study uniformly random lozenge tilings of general simply connected polygons. Under a technical assumption that is presumably generic with respect to polygon shapes, we show that the local statistics around a cusp point of the arctic curve converge to the Pearcey process. This verifies the widely predicted universality of edge statistics in the cusp case. Together with the smooth and tangent cases proved in Aggarwal-Huang and Aggarwal-Gorin, these are believed to be the three types of edge statistics that can arise in a generic polygon. Our proof is via a local coupling of the random tiling with non-intersecting Bernoulli random walks (NBRW). To leverage this coupling, we establish an optimal concentration estimate for the tiling height function around the cusp. As another step and also a result of potential independent interest, we show that the local statistics of NBRW around a cusp converge to the Pearcey process when the initial configuration consists of two parts with proper density growth, via careful asymptotic analysis of the determinantal formula.Comment: 59 pages, 9 figure

    WL meet VC

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    Recently, many works studied the expressive power of graph neural networks (GNNs) by linking it to the 11-dimensional Weisfeiler--Leman algorithm (1-WL1\text{-}\mathsf{WL}). Here, the 1-WL1\text{-}\mathsf{WL} is a well-studied heuristic for the graph isomorphism problem, which iteratively colors or partitions a graph's vertex set. While this connection has led to significant advances in understanding and enhancing GNNs' expressive power, it does not provide insights into their generalization performance, i.e., their ability to make meaningful predictions beyond the training set. In this paper, we study GNNs' generalization ability through the lens of Vapnik--Chervonenkis (VC) dimension theory in two settings, focusing on graph-level predictions. First, when no upper bound on the graphs' order is known, we show that the bitlength of GNNs' weights tightly bounds their VC dimension. Further, we derive an upper bound for GNNs' VC dimension using the number of colors produced by the 1-WL1\text{-}\mathsf{WL}. Secondly, when an upper bound on the graphs' order is known, we show a tight connection between the number of graphs distinguishable by the 1-WL1\text{-}\mathsf{WL} and GNNs' VC dimension. Our empirical study confirms the validity of our theoretical findings.Comment: arXiv admin note: text overlap with arXiv:2206.1116

    Fast Algorithms for Separable Linear Programs

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    In numerical linear algebra, considerable effort has been devoted to obtaining faster algorithms for linear systems whose underlying matrices exhibit structural properties. A prominent success story is the method of generalized nested dissection~[Lipton-Rose-Tarjan'79] for separable matrices. On the other hand, the majority of recent developments in the design of efficient linear program (LP) solves do not leverage the ideas underlying these faster linear system solvers nor consider the separable structure of the constraint matrix. We give a faster algorithm for separable linear programs. Specifically, we consider LPs of the form minAx=b,lxucx\min_{\mathbf{A}\mathbf{x}=\mathbf{b}, \mathbf{l}\leq\mathbf{x}\leq\mathbf{u}} \mathbf{c}^\top\mathbf{x}, where the graphical support of the constraint matrix ARn×m\mathbf{A} \in \mathbb{R}^{n\times m} is O(nα)O(n^\alpha)-separable. These include flow problems on planar graphs and low treewidth matrices among others. We present an O~((m+m1/2+2α)log(1/ϵ))\tilde{O}((m+m^{1/2 + 2\alpha}) \log(1/\epsilon)) time algorithm for these LPs, where ϵ\epsilon is the relative accuracy of the solution. Our new solver has two important implications: for the kk-multicommodity flow problem on planar graphs, we obtain an algorithm running in O~(k5/2m3/2)\tilde{O}(k^{5/2} m^{3/2}) time in the high accuracy regime; and when the support of A\mathbf{A} is O(nα)O(n^\alpha)-separable with α1/4\alpha \leq 1/4, our algorithm runs in O~(m)\tilde{O}(m) time, which is nearly optimal. The latter significantly improves upon the natural approach of combining interior point methods and nested dissection, whose time complexity is lower bounded by Ω(m(m+mαω))=Ω(m3/2)\Omega(\sqrt{m}(m+m^{\alpha\omega}))=\Omega(m^{3/2}), where ω\omega is the matrix multiplication constant. Lastly, in the setting of low-treewidth LPs, we recover the results of [DLY,STOC21] and [GS,22] with significantly simpler data structure machinery.Comment: 55 pages. To appear at SODA 202

    A machine learning approach to constructing Ramsey graphs leads to the Trahtenbrot-Zykov problem.

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    Attempts at approaching the well-known and difficult problem of constructing Ramsey graphs via machine learning lead to another difficult problem posed by Zykov in 1963 (now commonly referred to as the Trahtenbrot-Zykov problem): For which graphs F does there exist some graph G such that the neighborhood of every vertex in G induces a subgraph isomorphic to F? Chapter 1 provides a brief introduction to graph theory. Chapter 2 introduces Ramsey theory for graphs. Chapter 3 details a reinforcement learning implementation for Ramsey graph construction. The implementation is based on board game software, specifically the AlphaZero program and its success learning to play games from scratch. The chapter ends with a description of how computing challenges naturally shifted the project towards the Trahtenbrot-Zykov problem. Chapter 3 also includes recommendations for continuing the project and attempting to overcome these challenges. Chapter 4 defines the Trahtenbrot-Zykov problem and outlines its history, including proofs of results omitted from their original papers. This chapter also contains a program for constructing graphs with all neighborhood-induced subgraphs isomorphic to a given graph F. The end of Chapter 4 presents constructions from the program when F is a Ramsey graph. Constructing such graphs is a non-trivial task, as Bulitko proved in 1973 that the Trahtenbrot-Zykov problem is undecidable. Chapter 5 is a translation from Russian to English of this famous result, a proof not previously available in English. Chapter 6 introduces Cayley graphs and their relationship to the Trahtenbrot-Zykov problem. The chapter ends with constructions of Cayley graphs Γ in which the neighborhood of every vertex of Γ induces a subgraph isomorphic to a given Ramsey graph, which leads to a conjecture regarding the unique extremal Ramsey(4, 4) graph

    Two-sets cut-uncut on planar graphs

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    We study the following Two-Sets Cut-Uncut problem on planar graphs. Therein, one is given an undirected planar graph GG and two sets of vertices SS and TT. The question is, what is the minimum number of edges to remove from GG, such that we separate all of SS from all of TT, while maintaining that every vertex in SS, and respectively in TT, stays in the same connected component. We show that this problem can be solved in time 2S+TnO(1)2^{|S|+|T|} n^{O(1)} with a one-sided error randomized algorithm. Our algorithm implies a polynomial-time algorithm for the network diversion problem on planar graphs, which resolves an open question from the literature. More generally, we show that Two-Sets Cut-Uncut remains fixed-parameter tractable even when parameterized by the number rr of faces in the plane graph covering the terminals STS \cup T, by providing an algorithm of running time 4r+O(r)nO(1)4^{r + O(\sqrt r)} n^{O(1)}.Comment: 22 pages, 5 figure
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