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Bassler K.E.,University of Houston | Bassler K.E.,Max Planck Institute for the Physics of Complex Systems | Genio C.I.D.,University of Warwick | Genio C.I.D.,Max Planck Institute for the Physics of Complex Systems | And 5 more authors.
New Journal of Physics | Year: 2015

Many real-world networks exhibit correlations between the node degrees. For instance, in socialnetworks nodes tend to connect to nodes of similar degree and conversely, in biological and technological networks, high-degree nodes tend to be linked with low-degree nodes. Degree correlations also affect the dynamics of processes supported by a network structure, such as the spread of opinions or epidemics. The proper modelling of these systems, i.e., without uncontrolled biases, requires the sampling of networks with a specified set of constraints.Wepresent a solution to the sampling problem when the constraints imposed are the degree correlations. In particular, we develop an exact method to construct and sample graphs with a specified joint-degree matrix, which is a matrix providing the number of edges between all the sets of nodes of a given degree, for all degrees, thus completely specifying all pairwise degree correlations, and additionally, the degree sequence itself. Our algorithm always produces independent samples without backtracking. The complexity of the graph construction algorithmis ⊙(NM)whereNis the number of nodes andMis the number of edges. © 2015 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.

Gyori E.,MTA Alfred Renyi Institute of Mathematics | Gyori E.,Central European University | Mezei T.R.,Central European University
Computational Geometry: Theory and Applications | Year: 2016

We prove that every simply connected orthogonal polygon of n vertices can be partitioned into ⌊3n+416⌋ (simply connected) orthogonal polygons of at most 8 vertices. It yields a new and shorter proof of the theorem of A. Aggarwal that ⌊3n+416⌋ mobile guards are sufficient to control the interior of an n-vertex orthogonal polygon. Moreover, we strengthen this result by requiring combinatorial guards (visibility is only needed at the endpoints of patrols) and prohibiting intersecting patrols. This yields positive answers to two questions of O'Rourke [7, Section 3.4]. Our result is also a further example of the “metatheorem” that (orthogonal) art gallery theorems are based on partition theorems. © 2016 Elsevier B.V.

Balint P.,Budapest University of Technology and Economics | Nemedy Varga A.,MTA Alfred Renyi Institute of Mathematics
Nonlinearity | Year: 2016

In this paper we study the system of two falling balls in continuous time. We model the system by a suspension flow over a two dimensional, hyperbolic base map. By detailed analysis of the geometry of the system we identify special periodic points and show that the ratio of certain periods in continuous time is Diophantine for almost every value of the mass parameter in an interval. Using results of Melbourne (2007 Trans. Am. Math. Soc. 359 2421-41) and our previous achievements (Bálint et al 2012 Chaos 22 026104) we conclude that for these values of the parameter the flow mixes faster than any polynomial. Even though the calculations are presented for the specific physical system, the method is quite general and can be applied to other suspension flows, too. © 2016 IOP Publishing Ltd & London Mathematical Society.

Backhausz A.,MTA Alfred Renyi Institute of Mathematics | Backhausz A.,Eötvös Loránd University | Szegedy B.,MTA Alfred Renyi Institute of Mathematics | Virag B.,MTA Alfred Renyi Institute of Mathematics | Virag B.,University of Toronto
Random Structures and Algorithms | Year: 2015

Let G be a d-regular graph of sufficiently large-girth (depending on parameters k and r) and μ be a random process on the vertices of G produced by a randomized local algorithm of radius r. We prove the upper bound (k+1-2k/d)(1d-1)k for the (absolute value of the) correlation of values on pairs of vertices of distance k and show that this bound is optimal. The same results hold automatically for factor of i.i.d processes on the d-regular tree. In that case we give an explicit description for the (closure) of all possible correlation sequences. Our proof is based on the fact that the Bernoulli graphing of the infinite d-regular tree has spectral radius 2d-1. Graphings with this spectral gap are infinite analogues of finite Ramanujan graphs and they are interesting on their own right. © 2015 Wiley Periodicals, Inc.

Berczi K.,MTA ELTE Egervary Research Group | Bernath A.,MTA ELTE Egervary Research Group | Vizer M.,MTA Alfred Renyi Institute of Mathematics
Electronic Journal of Combinatorics | Year: 2015

An undirected simple graph G = (V,E) is called antimagic if there exists an injective function f: E → {1,…|E|} such that (formula presented) for any pair of different nodes u, v ∈ V. In this note we prove - with a slight modification of an argument of Cranston et al. - that k-regular graphs are antimagic for k ≥ 2. © 2015, Australian National University. All rights reserved.

Toth G.F.,MTA Alfred Renyi Institute of Mathematics | Fodor F.,University of Szeged | Fodor F.,University of Calgary | Vigh V.,University of Szeged
Discrete and Computational Geometry | Year: 2015

The packing density of the regular cross-polytope in Euclidean $$n$$n-space is unknown except in dimensions $$2$$2 and $$4$$4 where it is 1. The only non-trivial upper bound is due to Gravel et al. [Discrete Comput Geom 46(4):799–818, 2011], who proved that for $$n=3$$n=3 the packing density of the regular octahedron is at most $$1-1.4\ldots \times 10^{-12}$$1-1.4…×10-12. In this paper, we prove upper bounds for the packing density $$\delta (X^n)$$δ(Xn) of the $$n$$n-dimensional regular cross-polytope $$X^n$$Xn. It turns out that $$\delta (X^n)$$δ(Xn) approaches zero exponentially fast with growing dimension. Our bound is non-trivial, that is, less than 1, for $$n\ge 7$$n≥7. © 2015, Springer Science+Business Media New York.

Carassus L.,University of Reims Champagne Ardenne | Rasonyi M.,MTA Alfred Renyi Institute of Mathematics
Mathematics of Operations Research | Year: 2016

This paper investigates the problem of maximizing expected terminal utility in a (generically incomplete) discrete-time financial market model with finite time horizon. By contrast to the standard setting, a possibly nonconcave utility function U is considered, with domain of definition equal to the whole real line. Simple conditions are presented that guarantee the existence of an optimal strategy for the problem. In particular, the asymptotic elasticity of U plays a decisive role: Existence can be shown when it is strictly greater at -∞ than at C +∞. © 2016 INFORMS.

Katona G.O.H.,MTA Alfred Renyi Institute of Mathematics | Nagy D.T.,Eötvös Loránd University
Order | Year: 2015

Let (Formula presented.) be the poset generated by the subsets of [n] with the inclusion relation and let (Formula presented.) be a finite poset. We want to embed (Formula presented.) as many times as possible such that the subsets in different copies are incomparable. The maximum number of such embeddings is asymptotically determined for all finite posets(Formula presented.), where (Formula presented.) denotes the minimal size of the convex hull of a copy of (Formula presented.). We discuss both weak and strong (induced) embeddings. © 2014, Springer Science+Business Media Dordrecht.

Blomer V.,Mathematisches Institute | Maga P.,MTA Alfred Renyi Institute of Mathematics
Selecta Mathematica, New Series | Year: 2016

Let F be an L2-normalized Hecke Maaß cusp form for Γ 0(N) ⊆ SL n(Z) with Laplace eigenvalue λF. If Ω is a compact subset of Γ 0(N) \ PGL n/ PO n, we show the bound ‖F|Ω‖∞≪ΩNελFn(n-1)/8-δ for some constant δ= δn> 0 depending only on n. © 2016, Springer International Publishing.

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