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Nick Wormald

Publications and source records attributed to Nick Wormald.

At least 19 recordsLinked to original sources
arXiv2024

Asymptotic normality of pattern occurrences in random maps

The purpose of this paper is to study the limiting distribution of special {\it additive functionals} on random planar maps, namely the number of occurrences of a given {\it pattern}. The main result is a central limit theorem for these pattern counts in the case of pattern with a simple boundary. The proof relies on a combination of analytic and combinatorial methods together with a moment method due to Gao and Wormald~\cite{GaoWormald}. It is an important issue to handle the overlap structure of two pattern which is the main difficulty in the proof.

math.CO↗
arXiv2023

Multivariate asymptotic normality determined by high moments

We extend a general result showing that the asymptotic behavior of high moments, factorial or standard, of random variables, determines the asymptotically normality, from the one dimensional to the multidimensional setting. This approach differs from the usual moment method which requires that the moments of each fixed order converge. We illustrate our results by considering a joint distribution of the numbers of bins (having the same, finite, capacity) containing a prescribed number of balls in a classical allocation scheme.

math.PR↗
arXiv2023

Sharper analysis of the random graph $d$-process via a balls-in-bins model

A graph $d$-process starts with an empty graph on $n$ vertices, and adds one edge at each time step, chosen uniformly at random from those pairs which are not yet edges and whose both vertices have current degree less than $d$. If, in the final graph, at most one vertex has degree $d-1$ and all other have degree $d$, we call the process saturated. We present a new approach to analysing this process based on random allocation of balls in bins. This allows us to get improved results on the degree distribution throughout the process and, consequently, to determine the asymptotic probability of non-saturation of the process.

math.CO↗
arXiv2022

Asymptotic enumeration of hypergraphs by degree sequence

We prove an asymptotic formula for the number of $k$-uniform hypergraphs with a given degree sequence, for a wide range of parameters. In particular, we find a formula that is asymptotically equal to the number of $d$-regular $k$-uniform hypergraphs on $n$ vertices provided that $dn\le c\binom{n}{k}$ for a constant $c>0$, and $3 \leq k < n^C$ for any $C<1/9.$ Our results relate the degree sequence of a random $k$-uniform hypergraph to a simple model of nearly independent binomial random variables, thus extending the recent results for graphs due to the second and third author.

math.CO↗
arXiv2021

Engineering Uniform Sampling of Graphs with a Prescribed Power-law Degree Sequence

We consider the following common network analysis problem: given a degree sequence $\mathbf{d} = (d_1, \dots, d_n) \in \mathbb N^n$ return a uniform sample from the ensemble of all simple graphs with matching degrees. In practice, the problem is typically solved using Markov Chain Monte Carlo approaches, such as Edge-Switching or Curveball, even if no practical useful rigorous bounds are known on their mixing times. In contrast, Arman et al. sketch Inc-Powerlaw, a novel and much more involved algorithm capable of generating graphs for power-law bounded degree sequences with $γ\gtrapprox 2.88$ in expected linear time. For the first time, we give a complete description of the algorithm and add novel switchings. To the best of our knowledge, our open-source implementation of Inc-Powerlaw is the first practical generator with rigorous uniformity guarantees for the aforementioned degree sequences. In an empirical investigation, we find that for small average-degrees Inc-Powerlaw is very efficient and generates graphs with one million nodes in less than a second. For larger average-degrees, parallelism can partially mitigate the increased running-time.

cs.DS↗
arXiv2020

Asymptotic enumeration of digraphs and bipartite graphs by degree sequence

We provide asymptotic formulae for the numbers of bipartite graphs with given degree sequence, and of loopless digraphs with given in- and out-degree sequences, for a wide range of parameters. Our results cover medium range densities and close the gaps between the results known for the sparse and dense ranges. In the case of bipartite graphs, these results were proved by Greenhill, McKay and Wang in 2006 and by Canfield, Greenhill and McKay in 2008, respectively. Our method also essentially covers the sparse range, for which much less was known in the case of loopless digraphs. For the range of densities which our results cover, they imply that the degree sequence of a random bipartite graph with m edges is accurately modelled by a sequence of independent binomial random variables, conditional upon the sum of variables in each part being equal to m. A similar model also holds for loopless digraphs.

math.CO↗
arXiv2019

Asymptotic enumeration of graphs by degree sequence, and the degree sequence of a random graph

In this paper we relate a fundamental parameter of a random graph, its degree sequence, to a simple model of nearly independent binomial random variables. This confirms a conjecture made in 1997. As a result, many interesting functions of the joint distribution of graph degrees, such as the distribution of the median degree, become amenable to estimation. Our result is established by proving an asymptotic formula conjectured in 1990 for the number of graphs with given degree sequence. In particular, this gives an asymptotic formula for the number of $d$-regular graphs for all $d$, as $n\to\infty$.

math.CO↗
arXiv2019

A Weight-based Information Filtration Algorithm for Stock-Correlation Networks

Several algorithms have been proposed to filter information on a complete graph of correlations across stocks to build a stock-correlation network. Among them the planar maximally filtered graph (PMFG) algorithm uses $3n-6$ edges to build a graph whose features include a high frequency of small cliques and a good clustering of stocks. We propose a new algorithm which we call proportional degree (PD) to filter information on the complete graph of normalised mutual information (NMI) across stocks. Our results show that the PD algorithm produces a network showing better homogeneity with respect to cliques, as compared to economic sectoral classification than its PMFG counterpart. We also show that the partition of the PD network obtained through normalised spectral clustering (NSC) agrees better with the NSC of the complete graph than the corresponding one obtained from PMFG. Finally, we show that the clusters in the PD network are more robust with respect to the removal of random sets of edges than those in the PMFG network.

q-fin.ST↗
arXiv2018

Full rainbow matchings in graphs and hypergraphs

Let $G$ be a simple graph that is properly edge coloured with $m$ colours and let $\M=\{M_1,\ldots, M_m\}$ be the set of $m$ matchings induced by the colours in $G$. Suppose that $m\le n-n^{c}$, where $c>9/10$, and every matching in $\M$ has size $n$. Then $G$ contains a full rainbow matching, i.e.\ a matching that contains exactly one edge from $M_i$ for each $1\le i\le m$. This answers an open problem of Pokrovskiy and gives an affirmative answer to a generalisation of a special case of a conjecture of Aharoni and Berger. Related results are also found for multigraphs with edges of bounded multiplicity, and for hypergraphs. Finally, we provide counterexamples to several conjectures on full rainbow matchings made by Aharoni and Berger.

math.CO↗
arXiv2016

The number of satisfying assignments of random regular k-SAT formulas

Let $Φ$ be a random $k$-SAT formula in which every variable occurs precisely $d$ times positively and $d$ times negatively. Assuming that $k$ is sufficiently large and that $d$ is slightly below the critical degree where the formula becomes unsatisfiable with high probability, we determine the limiting distribution of the logarithm of the number of satisfying assignments.

math.CO↗
arXiv2016

The probability of nonexistence of a subgraph in a moderately sparse random graph

We develop a general procedure that finds recursions for statistics counting isomorphic copies of a graph $G_0$ in the common random graph models ${\cal G}(n,m)$ and ${\cal G}(n,p)$. Our results apply when the average degrees of the random graphs are below the threshold at which each edge is included in a copy of $G_0$. This extends an argument given earlier by the second author for $G_0=K_3$ with a more restricted range of average degree. For all strictly balanced subgraphs $G_0$, our results gives much information on the distribution of the number of copies of $G_0$ that are not in large "clusters" of copies. The probability that a random graph in ${\cal G}(n,p)$ has no copies of $G_0$ is shown to be given asymptotically by the exponential of a power series in $n$ and $p$, over a fairly wide range of $p$. A corresponding result is also given for ${\cal G}(n,m)$, which gives an asymptotic formula for the number of graphs with $n$ vertices, $m$ edges and no copies of $G_0$, for the applicable range of $m$. An example is given, computing the asymptotic probability that a random graph has no triangles for $p=o(n^{-7/11})$ in ${\cal G}(n,p)$ and for $m=o(n^{15/11})$ in ${\cal G}(n,m)$, extending results of the second author.

math.CO↗
arXiv2015

On the push&pull protocol for rumour spreading

The asynchronous push&pull protocol, a randomized distributed algorithm for spreading a rumour in a graph $G$, works as follows. Independent Poisson clocks of rate 1 are associated with the vertices of $G$. Initially, one vertex of $G$ knows the rumour. Whenever the clock of a vertex $x$ rings, it calls a random neighbour $y$: if $x$ knows the rumour and $y$ does not, then $x$ tells $y$ the rumour (a push operation), and if $x$ does not know the rumour and $y$ knows it, $y$ tells $x$ the rumour (a pull operation). The average spread time of $G$ is the expected time it takes for all vertices to know the rumour, and the guaranteed spread time of $G$ is the smallest time $t$ such that with probability at least $1-1/n$, after time $t$ all vertices know the rumour. The synchronous variant of this protocol, in which each clock rings precisely at times $1,2,\dots$, has been studied extensively. We prove the following results for any $n$-vertex graph: In either version, the average spread time is at most linear even if only the pull operation is used, and the guaranteed spread time is within a logarithmic factor of the average spread time, so it is $O(n\log n)$. In the asynchronous version, both the average and guaranteed spread times are $Ω(\log n)$. We give examples of graphs illustrating that these bounds are best possible up to constant factors. We also prove theoretical relationships between the guaranteed spread times in the two versions. Firstly, in all graphs the guaranteed spread time in the asynchronous version is within an $O(\log n)$ factor of that in the synchronous version, and this is tight. Next, we find examples of graphs whose asynchronous spread times are logarithmic, but the synchronous versions are polynomially large. Finally, we show for any graph that the ratio of the synchronous spread time to the asynchronous spread time is $O(n^{2/3})$.

cs.DC↗
arXiv2015

Almost all 5-regular graphs have a 3-flow

Tutte conjectured in 1972 that every 4-edge connected graph has a nowhere-zero 3-flow. This has long been known to be equivalent to the conjecture that every 5-regular 4-edge-connected graph has an edge orientation in which every out-degree is either 1 or 4. We show that the assertion of the conjecture holds asymptotically almost surely for random 5-regular graphs. It follows that the conjecture holds for almost all 4-edge connected 5-regular graphs.

math.CO↗
arXiv2014

Meyniel's conjecture holds for random graphs

In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph $G$ is called the cop number of $G$. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant $C$, the cop number of every connected graph $G$ is at most $C \sqrt{|V(G)|}$. In this paper, we show that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph. We do this by first showing that the conjecture holds for a general class of graphs with some specific expansion-type properties. This will also be used in a separate paper on random $d$-regular graphs, where we show that the conjecture holds asymptotically almost surely when $d = d(n) \ge 3$.

math.CO↗
arXiv2014

It's a Small World for Random Surfers

We prove logarithmic upper bounds for the diameters of the random-surfer Webgraph model and the PageRank-based selection Webgraph model, confirming the small world phenomenon holds for them. In the special case when the generated graph is a tree, we provide close lower and upper bounds for the diameters of both models.

cs.DM↗
arXiv2014

Longest paths in random Apollonian networks and largest $r$-ary subtrees of random $d$-ary recursive trees

Let $r$ and $d$ be positive integers with $r<d$. Consider a random $d$-ary tree constructed as follows. Start with a single vertex, and in each time-step choose a uniformly random leaf and give it $d$ newly created offspring. Let ${\mathcal T}_t$ be the tree produced after $t$ steps. We show that there exists a fixed $δ<1$ depending on $d$ and $r$ such that almost surely for all large $t$, every $r$-ary subtree of ${\mathcal T}_t$ has less than $t^δ$ vertices. The proof involves analysis that also yields a related result. Consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. In this way, one face is destroyed and three new faces are created. After $t$ steps, we obtain a random triangulated plane graph with $t+3$ vertices, which is called a random Apollonian network. We prove that there exists a fixed $δ<1$, such that eventually every path in this graph has length less than $t^δ$, which verifies a conjecture of Cooper and Frieze.

math.PR↗
arXiv2014

Asymptotic enumeration of sparse connected 3-uniform hypergraphs

We derive an asymptotic formula for the number of connected 3-uniform hypergraphs with vertex set $[N]$ and $M$ edges for $M=N/2+R$ as long as $R$ satisfies $R = o(N)$ and $R=ω(N^{1/3}\ln^{2} N)$. This almost completely fills the gap in the range of $M$ for which the formula is known. We approach the problem using an `inside-out' approach of an earlier paper of Pittel and the second author, for connected graphs. A key part of the method uses structural components of connected hypergraphs called cores and kernels. These are structural components of connected hypergraphs. Our results also give information on the numbers of them with a given number of vertices and edges, and hence their typical size in random connected $3$-uniform hypergraphs with $N$ vertices and $M$ edges, for the range of $M$ we consider.

math.CO↗