Random Planar Graphs Research Articles

Central loops in random planar graphs.

Random planar graphs appear in a variety of contexts and it is important for many different applications to be able to characterize their structure. Local quantities fail to give interesting information and it seems that path-related measures are able to convey relevant information about the organization of these structures. In particular, nodes with a large betweenness centrality (BC) display nontrivial patterns, such as central loops. We first discuss empirical results for different random planar graphs and we then propose a toy model which allows us to discuss the condition for the emergence of nontrivial patterns such as central loops. This toy model is made of a star network with N_{b} branches of size n and links of weight 1, superimposed to a loop at distance ℓ from the center and with links of weight w. We estimate for this model the BC at the center and on the loop and we show that the loop can be more central than the origin if w<w_{c} where the threshold of this transition scales as w_{c}∼n/N_{b}. In this regime, there is an optimal position of the loop that scales as ℓ_{opt}∼N_{b}w/4. This simple model sheds some light on the organization of these random structures and allows us to discuss the effect of randomness on the centrality of loops. In particular, it suggests that the number and the spatial extension of radial branches are the crucial ingredients that control the existence of central loops.

Critical behaviour of spanning forests on random planar graphs**Dedicated to Tony Guttmann on the occasion of his 70th birthday.

As a follow-up of previous work of the authors, we analyse the statistical mechanics model of random spanning forests on random planar graphs. Special emphasis is given to the analysis of the critical behaviour.Exploiting an exact relation with a model of -loops and dimers, previously solved by Kostov and Staudacher, we identify critical and multicritical loci, and find them consistent with recent results of Bousquet-Mélou and Courtiel. This is also consistent with the KPZ relation, and the Berker–Kadanoff phase in the anti-ferromagnetic regime of the Potts Model on periodic lattices, predicted by Saleur. To our knowledge, this is the first known example of KPZ appearing explicitly to work within a Berker–Kadanoff phase.We set up equations for the generating function, at the value t = −1 of the fugacity, which is of combinatorial interest, and we investigate the resulting numerical series, a favourite problem of Tony Guttmann’s.

Random cubic planar graphs revisited

The goal of our work is to analyze random cubic planar graphs according to the uniform distribution. More precisely, let G be the class of labelled cubic planar graphs and let gn be the number of graphs with n vertices. Then each graph in G with n vertices has the same probability 1/gn. This model was analyzed first by Bodirsky et al. [1], and here we revisit and extend their work. The motivation for this revision is twofold. First, some proofs in [1] where incomplete with respect to the singularity analysis and we aim at providing full proofs. Secondly, we obtain new results that considerably strengthen those in [1] and shed more light on the structure of random cubic planar graphs. We present a selection of our results on asymptotic enumeration and on limit laws for parameters of random graphs.

Scaling Limits in Models of Statistical Mechanics

The emphasis of the workshop was on the deep relations between, on the one hand, recent advances in probabilistic investigation of statistical mechanical models and spatial stochastic processes and, on the other hand, rigorous field-theoretic and analytic methods of mathematical physics. There were 52 participants, including 6 postdocs and graduate students, working in diverse intertwining areas of probability, statistical mechanics and field theory. Specific topics addressed during the 24 talks include: Universality and critical phenomena, disordered models, Gaussian free field (GFF), stochastic representation of classical and quantum-mechanical models and related random interchange and permutation processes, random planar graphs and unimodular planar maps, random walks on critical graphs and the Alexander-Orbach conjecture, reinforced random walks and non-linear -models, metastability, aging, equilibrium and dynamics for continuum particles with hard core interactions, non-equilibrium dynamics and Toom’s interfaces.

Triangles in random cubic planar graphs

In this extended abstract we determine a normal limiting distribution for the number of triangles in a uniformly at random 3-connected cubic planar graph, as well as the precise expectation and variance values. Further comments towards the more complicated problem of studying both the limiting distribution of triangles in random cubic planar graphs, and the (asymptotic) number of triangle-free cubic planar graphs are discussed as well.

Universal measurement-based quantum computation with spin-2 Affleck-Kennedy-Lieb-Tasaki states

We demonstrate that the spin-2 Affleck-Kennedy-Lieb-Tasaki (AKLT) state on the square lattice is a universal resource for the measurement-based quantum computation. Our proof is done by locally converting the AKLT to two-dimensional random planar graph states and by certifying that with high probability the resulting random graphs are in the supercritical phase of percolation using Monte Carlo simulations. One key enabling point is the exact weight formula that we derive for arbitrary measurement outcomes according to a spin-2 POVM on all spins. We also argue that the spin-2 AKLT state on the three-dimensional diamond lattice is a universal resource, the advantage of which would be the possibility of implementing fault-tolerant quantum computation with topological protection. In addition, as we deform the AKLT Hamiltonian, there is a finite region that the ground state can still support a universal resource before making a transition in its quantum computational power.

The maximum degree of random planar graphs

McDiarmid and Reed [‘On the maximum degree of a random planar graph’, Combin. Probab. Comput. 17 (2008) 591–601] showed that the maximum degree Δ n of a random labeled planar graph with n vertices satisfies with high probability (w.h.p.) c 1 log n < Δ n < c 2 log n , for suitable constants . In this paper, we determine the precise asymptotics, showing in particular that w.h.p. | Δ n − c log n | = O ( log log n ) , for a constant c ≈ 2 . 52946 that we determine explicitly. The proof combines tools from analytic combinatorics and Boltzmann sampling techniques.

Chasing a Fast Robber on Planar Graphs and Random Graphs

We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, that is, can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let denote the number of cops needed to capture the robber in a graph G in this variant, and let denote the treewidth of G. We show that if G is planar then , and there is a polynomial-time constant-factor approximation algorithm for computing . We also determine, up to constant factors, the value of of the Erdős–Rényi random graph for all admissible values of p, and show that when the average degree is ω(1), is typically asymptotic to the domination number.

A completely random [formula omitted]-tessellation model and Gibbsian extensions

In their 1993 paper, Arak, Clifford and Surgailis discussed a new model of random planar graph. As a particular case, that model yields tessellations with only T-vertices (T-tessellations). Using a similar approach involving Poisson lines, a new model of random T-tessellations is proposed. Campbell measures, Papangelou kernels and formulae of Georgii–Nguyen–Zessin type are translated from point process theory to random T-tessellations. It is shown that the new model shows properties similar to the Poisson point process and can therefore be considered as a completely random T-tessellation. Gibbs variants are introduced leading to models of random T-tessellations where selected features are controlled. Gibbs random T-tessellations are expected to better represent observed tessellations. As numerical experiments are a key tool for investigating Gibbs models, we derive a simulation algorithm of the Metropolis–Hastings–Green family.

Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm

Many practical problems in almost all scientific and technological disciplines have been classified as computationally hard (NP-hard or even NP-complete). In life sciences, combinatorial optimization problems frequently arise in molecular biology, e.g., genome sequencing; global alignment of multiple genomes; identifying siblings or discovery of dysregulated pathways. In almost all of these problems, there is the need for proving a hypothesis about certain property of an object that can be present if and only if it adopts some particular admissible structure (an NP-certificate) or be absent (no admissible structure), however, none of the standard approaches can discard the hypothesis when no solution can be found, since none can provide a proof that there is no admissible structure. This article presents an algorithm that introduces a novel type of solution method to “efficiently” solve the graph 3-coloring problem; an NP-complete problem. The proposed method provides certificates (proofs) in both cases: present or absent, so it is possible to accept or reject the hypothesis on the basis of a rigorous proof. It provides exact solutions and is polynomial-time (i.e., efficient) however parametric. The only requirement is sufficient computational power, which is controlled by the parameter . Nevertheless, here it is proved that the probability of requiring a value of to obtain a solution for a random graph decreases exponentially: , making tractable almost all problem instances. Thorough experimental analyses were performed. The algorithm was tested on random graphs, planar graphs and 4-regular planar graphs. The obtained experimental results are in accordance with the theoretical expected results.

Two critical periods in the evolution of random planar graphs

Let P (n;M) be a graph chosen uniformly at random from the family of all labeled planar graphs with n vertices and M edges. In the paper we study the component structure of P (n;M). Combining counting arguments with analytic techniques, we show that there are two critical periods in the evolution of P (n;M). The first one, of width ( n 2=3 ), is analogous to the phase transition observed in the standard random graph models and takes place for M = n=2 +O(n 2=3 ), when the largest complex component is formed. Then, for M = n + O(n 3=5 ), when the complex components covers nearly all vertices, the second critical period of width n 3=5 occurs. Starting from that moment increasing of M mostly affects the density of the complex components, not its size.

PERFORMANCE ANALYSIS AND EVALUATION OF RANDOM WALK ALGORITHMS ON WIRELESS NETWORKS

We propose a model of dynamically evolving random networks and give an analytical result of the cover time of the simple random walk algorithm on a dynamic random symmetric planar point graph. Our dynamic network model considers random node distribution and random node mobility. We analyze the cover time of the parallel random walk algorithm on a complete network and show by numerical data that k parallel random walks reduce the cover time by almost a factor of k. We present simulation results for four random walk algorithms on random asymmetric planar point graphs. These algorithms include the simple random walk algorithm, the intelligent random walk algorithm, the parallel random walk algorithm, and the parallel intelligent random walk algorithm. Our random network model considers random node distribution and random battery transmission power. Performance measures include normalized cover time, probability distribution of the length of random walks, and load distribution.

Graph classes with given 3‐connected components: Asymptotic enumeration and random graphs

AbstractConsider a family \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\mathcal{T}\end{align*}\end{document} of 3‐connected graphs of moderate growth, and let \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\mathcal{G}\end{align*}\end{document} be the class of graphs whose 3‐connected components are graphs in \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\mathcal{T}\end{align*}\end{document}. We present a general framework for analyzing such graphs classes based on singularity analysis of generating functions, which generalizes previously studied cases such as planar graphs and series‐parallel graphs. We provide a general result for the asymptotic number of graphs in \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\mathcal{G}\end{align*}\end{document}, based on the singularities of the exponential generating function associated to \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\mathcal{T}\end{align*}\end{document}. We derive limit laws, which are either normal or Poisson, for several basic parameters, including the number of edges, number of blocks and number of components. For the size of the largest block we find a fundamental dichotomy: classes similar to planar graphs have almost surely a unique block of linear size, while classes similar to series‐parallel graphs have only sublinear blocks. This dichotomy was already observed by Panagiotou and Steger [25], and we provide a finer description. For some classes under study both regimes occur, because of a critical phenomenon as the edge density in the class varies. Finally, we analyze the size of the largest 3‐connected component in random planar graphs. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 438–479, 2013

Algorithms for Self-Organized Resource Allocation in Wireless Networks

We consider the problem of allocating orthogonal resources in a self-organized manner on conflict graphs that describe discretized interference couplings of wireless networks. The target is to find a global optimum, i.e., a conflict-free allocation. We consider both random planar and nonplanar graphs that result from a path-loss model in a wireless network. Two algorithms for the self-organized coloring of arbitrary graphs are devised, and their performances are compared with each other and a rule-based reasoning algorithm that is known from the literature. The semigreedy distributed local search (SDLS) algorithm, which is a particularly simple algorithm that is proposed here, is shown to outperform other algorithms in several cases. In a cellular system setting, we consider a negotiated worst coupling procedure to symmetrize the interference coupling between cells, targeting an improvement of the channel quality of cell-edge users. We compare this approach with an interference coupling based on the path loss that was experienced between base stations and see significant gains in cell-edge performance. In addition, we compare the channel quality that was experienced by users in a cellular network that employs SDLS with the channel quality in a network that employs an interference-reducing network algorithm which finds a local optimum in terms of real-valued interference couplings. In some cases, attempting global optimization based on a conflict graph interpretation outperforms local real-valued optimization.

Degree distribution in random planar graphs

We prove that for each k ⩾ 0 , the probability that a root vertex in a random planar graph has degree k tends to a computable constant d k , so that the expected number of vertices of degree k is asymptotically d k n , and moreover that ∑ k d k = 1 . The proof uses the tools developed by Giménez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of the generating functions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function p ( w ) = ∑ k d k w k . From this we can compute the d k to any degree of accuracy, and derive the asymptotic estimate d k ∼ c ⋅ k − 1 / 2 q k for large values of k, where q ≈ 0.67 is a constant defined analytically.

Combining Speedup Techniques based on Landmarks and Containers with parallelised preprocessing in Random and Planar Graphs

The Dijkstra’s algorithm is applied in many real world problems like mobile routing, road maps, railway networks, etc,. There are many techniques available to speedup the algorithm while guaranteeing the optimality of the solution. Almost all of the speedup techniques have a substantial amount of parallelism that can be exploited to decrease its running time. By suitably modifying portions of the existing system various degrees of parallelism can be achieved. The rapidly growing field of multiprocessing systems and multi-core processors provide many opportunities for such improvements. In these techniques there’s always a demand for the running time and the time required for pre-processing. Space requirements for the pre-processing also have a major influence on the running time of the algorithm. The main focus of the work is to implement landmark technique and to identify the segment of the code in landmark pre-processing which can be parallelized to obtain better speedup. The results are applied to the combined speedup technique which is based on landmarks and containers. The experimental results were compared and analysed for determining better performance improvements in random graphs and planar graphs.

3-Connected Cores In Random Planar Graphs

The study of the structural properties of large random planar graphs has become in recent years a field of intense research in computer science and discrete mathematics. Nowadays, a random planar graph is an important and challenging model for evaluating methods that are developed to study properties of random graphs from classes with structural side constraints.In this paper we focus on the structure of random 2-connected planar graphs regarding the sizes of their 3-connected building blocks, which we callcores. In fact, we prove a general theorem regarding random biconnected graphs from various classes. IfBnis a graph drawn uniformly at random from a suitable classof labelled biconnected graphs, then we show that with probability 1 −o(1) asn→ ∞,Bnbelongs to exactly one of the following categories:(i)either there is a uniquegiantcore inBn, that is, there is a 0 &lt;c=c() &lt; 1 such that the largest core contains ~cnvertices, and every other core contains at mostnαvertices, where 0 &lt; α = α() &lt; 1;(ii)or all cores ofBncontainO(logn) vertices.Moreover, we find the critical condition that determines the category to whichBnbelongs, and also provide sharp concentration results for the counts of cores of all sizes between 1 andn. As a corollary, we obtain that a random biconnected planar graph belongs to category (i), where in particularc= 0.765. . . and α = 2/3.

On properties of random dissections and triangulations

In the past decades the Gn,p model of random graphs, introduced by Erdős and Renyi in the 60's, has led to numerous beautiful and deep theorems. A key feature that is used in basically all proofs is that edges in Gn,p appear independently. The independence of the edges allows, for example, to obtain extremely tight bounds on the number of edges of Gn,p and its degree sequence by straightforward applications of Chernoff bounds. This situation changes dramatically if one considers graph classes with structural side constraints. For example, in a random planar graph Rn (a graph drawn uniformly at random from the class of all labeled planar graphs on n vertices) the edges are obviously far from being independent. Consequently, so far basically all results about properties of random graphs with structural side constraints rely on completely different methods, mostly from analytic combinatorics. In this paper we show that recent progress in the construction of so-called Boltzmann samplers by Duchon, Flajolet, Louchard, and Schaeffer [3] and Fusy [6] can be used to reduce the study of degree sequences and subgraph counts to properties of sequences of independent and identically distributed random variables -- to which we can then again apply Chernoff bounds to obtain extremely tight results. We elaborate our ideas by studying random dissections and triangulations of a labeled convex n-gon. For both we obtain the degree sequence and the number of induced copies of given fixed graphs. The degree sequence for triangulations was already obtained previously by Gao and Wormald [8] using deep methods from analytic combinatorics. We do, however, get better bounds for the tails of the probability distribution.

A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings

Combining tree decomposition and transfer matrix techniques provides a very general algorithm for computing exact partition functions of statistical models defined on arbitrary graphs. The algorithm is particularly efficient in the case of planar graphs. We illustrate it by computing the Potts model partition functions and chromatic polynomials (the number of proper vertex colourings using Q colours) for large samples of random planar graphs with up to N = 100 vertices. In the latter case, our algorithm yields a sub-exponential average running time of , a substantial improvement over the exponential running time ∼exp (0.245N) provided by the hitherto best-known algorithm. We study the statistics of chromatic roots of random planar graphs in some detail, comparing the findings with results for finite pieces of a regular lattice.

Random Planar Graphs with Bounds on the Maximum and Minimum Degrees

Let P n,d,D denote the graph taken uniformly at random from the set of all labelled planar graphs on {1, 2, . . . , n} with minimum degree at least d(n) and maximum degree at most D(n). We use counting arguments to investigate the probability that P n,d,D will contain given components and subgraphs, showing exactly when this is bounded away from 0 and 1 as n → ∞.