arXiv subjects

Paul Balister

Publications and source records attributed to Paul Balister.

At least 37 records · Page 2Linked to original sources

arXiv2018

Topological Phase Transitions in Spatial Networks

Most social, technological and biological networks are embedded in a finite dimensional space, and the distance between two nodes influences the likelihood that they link to each other. Indeed, in social systems, the chance that two individuals know each other drops rapidly with the distance between them; in the cell, proteins predominantly interact with proteins in the same cellular compartment; in the brain, neurons mainly link to nearby neurons. Most modeling frameworks that aim to capture the empirically observed degree distributions tend to ignore these spatial constraints. In contrast, models that account for the role of the physical distance often predict bounded degree distributions, in disagreement with the empirical data. Here we address a long-standing gap in the spatial network literature by deriving several key network characteristics of spatial networks, from the analytical form of the degree distribution to path lengths and local clustering. The mathematically exact results predict the existence of two distinct phases, each governed by a different dynamical equation, with distinct testable predictions. We use empirical data to offer direct evidence for the practical relevance of each of these phases in real networks, helping better characterize the properties of spatial networks.

physics.soc-ph↗

arXiv2018

Dense Subgraphs in Random Graphs

For a constant $γ\in[0,1]$ and a graph $G$, let $ω_γ(G)$ be the largest integer $k$ for which there exists a $k$-vertex subgraph of $G$ with at least $γ\binom{k}{2}$ edges. We show that if $0<p<γ<1$ then $ω_γ(G_{n,p})$ is concentrated on a set of two integers. More precisely, with $α(γ,p)=γ\log\fracγ{p}+(1-γ)\log\frac{1-γ}{1-p}$, we show that $ω_γ(G_{n,p})$ is one of the two integers closest to $\frac{2}{α(γ,p)}\big(\log n-\log\log n+\log\frac{eα(γ,p)}{2}\big)+\frac{1}{2}$, with high probability. While this situation parallels that of cliques in random graphs, a new technique is required to handle the more complicated ways in which these "quasi-cliques" may overlap.

math.CO↗

arXiv2017

A note on Linnik's Theorem on quadratic non-residues

We present a short, self-contained, and purely combinatorial proof of Linnik's theorem: for any $\varepsilon > 0$ there exists a constant $C_\varepsilon$ such that for any $N$, there are at most $C_\varepsilon$ primes $p \leqslant N$ such that the least positive quadratic non-residue modulo $p$ exceeds $N^\varepsilon$.

math.NT↗

arXiv2017

Coalescence on the real line

We study a geometrically constrained coalescence model derived from spin systems. Given two probability distributions $\mathbb{P}_R$ and $\mathbb{P}_B$ on the positive reals with finite means, colour the real line alternately with red and blue intervals so that the lengths of the red intervals have distribution $\mathbb{P}_R$, the lengths of the blue intervals have distribution $\mathbb{P}_B$, and distinct intervals have independent lengths. Now, iteratively update this colouring of the line by coalescing intervals: change the colour of any interval that is surrounded by longer intervals so that these three consecutive intervals subsequently form a single monochromatic interval. We say that a colour (either red or blue) wins if every point of the line is eventually of that colour. Holroyd, in 2011, asked the following question: under what natural conditions on the initial distributions is one of the colours almost surely guaranteed to win? It turns out that the answer to this question can be quite counter-intuitive due to the non-monotone dynamics of the model. In this paper, we investigate various notions of advantage one of the colours might initially possess, and in the course of doing so, we determine which of the two colours emerges victorious for various nontrivial pairs of distributions.

math.PR↗

arXiv2017

Reconstructing random jigsaws

A colouring of the edges of an $n \times n$ grid is said to be \emph{reconstructible} if the colouring is uniquely determined by the multiset of its $n^2$ \emph{tiles}, where the tile corresponding to a vertex of the grid specifies the colours of the edges incident to that vertex in some fixed order. In 2015, Mossel and Ross asked the following question: if the edges of an $n \times n$ grid are coloured independently and uniformly at random using $q=q(n)$ different colours, then is the resulting colouring reconstructible with high probability? From below, Mossel and Ross showed that such a colouring is not reconstructible when $q = o(n^{2/3})$ and from above, Bordenave, Feige and Mossel and Nenadov, Pfister and Steger independently showed, for any fixed $ε> 0$, that such a colouring is reconstructible when $q \ge n^{1+ε}$. Here, we improve on these results and prove the following: there exist absolute constants $C, c > 0$ such that, as $n \to \infty$, the probability that a random colouring as above is reconstructible tends to $1$ if $q \ge Cn$ and to $0$ if $q \le cn$.

math.CO↗

arXiv2017

Line percolation

We study a new geometric bootstrap percolation model, line percolation, on the $d$-dimensional integer grid $[n]^d$. In line percolation with infection parameter $r$, infection spreads from a subset $A\subset [n]^d$ of initially infected lattice points as follows: if there exists an axis-parallel line $L$ with $r$ or more infected lattice points on it, then every lattice point of $[n]^d$ on $L$ gets infected, and we repeat this until the infection can no longer spread. The elements of the set $A$ are usually chosen independently, with some density $p$, and the main question is to determine $p_c(n,r,d)$, the density at which percolation (infection of the entire grid) becomes likely. In this paper, we determine $p_c(n,r,2)$ up to a multiplicative factor of $1+o(1)$ and $p_c(n,r,3)$ up to a multiplicative constant as $n\rightarrow \infty$ for every fixed $r\in \mathbb{N}$. We also determine the size of the minimal percolating sets in all dimensions and for all values of the infection parameter.

math.PR↗

arXiv2017

Catching a fast robber on the grid

We study the problem of cops and robbers on the grid where the robber is allowed to move faster than the cops. It is well known that two cops are necessary and sufficient to catch the robber on any finite grid when the robber has unit speed. Here, we prove that when the speed of the robber is a sufficiently large constant, the number of cops needed to catch the robber on an $n \times n$ grid is $\exp(Ω(\log n / \log \log n))$.

math.CO↗

arXiv2016

The sharp threshold for making squares

Consider a random sequence of $N$ integers, each chosen uniformly and independently from the set $\{1,\dots,x\}$. Motivated by applications to factorisation algorithms such as Dixon's algorithm, the quadratic sieve, and the number field sieve, Pomerance in 1994 posed the following problem: how large should $N$ be so that, with high probability, this sequence contains a subsequence, the product of whose elements is a perfect square? Pomerance determined asymptotically the logarithm of the threshold for this event, and conjectured that it in fact exhibits a sharp threshold in $N$. More recently, Croot, Granville, Pemantle and Tetali determined the threshold up to a factor of $4/π+ o(1)$ as $x \to \infty$, and made a conjecture regarding the location of the sharp threshold. In this paper we prove both of these conjectures, by determining the sharp threshold for making squares. Our proof combines techniques from combinatorics, probability and analytic number theory; in particular, we use the so-called method of self-correcting martingales in order to control the size of the 2-core of the random hypergraph that encodes the prime factors of our random numbers. Our method also gives a new (and completely different) proof of the upper bound in the main theorem of Croot, Granville, Pemantle and Tetali.

math.NT↗

arXiv2016

Positive independence densities of finite rank countable hypergraphs are achieved by finite hypergraphs

The independence density of a finite hypergraph is the probability that a subset of vertices, chosen uniformly at random contains no hyperedges. Independence densities can be generalized to countable hypergraphs using limits. We show that, in fact, every positive independence density of a countably infinite hypergraph with hyperedges of bounded size is equal to the independence density of some finite hypergraph whose hyperedges are no larger than those in the infinite hypergraph. This answers a question of Bonato, Brown, Kemkes, and Prałat about independence densities of graphs. Furthermore, we show that for any $k$, the set of independence densities of hypergraphs with hyperedges of size at most $k$ is closed and contains no infinite increasing sequences.

math.CO↗

arXiv2015

The time of bootstrap percolation in two dimensions

We study the distribution of the percolation time $T$ of two-neighbour bootstrap percolation on $[n]^2$ with initial set $A\sim\mathrm{Bin}([n]^2,p)$. We determine $T$ with high probability up to a constant factor for all $p$ above the critical probability for percolation, and to within a $1+o(1)$ factor for a large range of $p$.

math.PR↗

arXiv2015

Consistent random vertex-orderings of graphs

Given a hereditary graph property $\mathcal{P}$, consider distributions of random orderings of vertices of graphs $G\in\mathcal{P}$ that are preserved under isomorphisms and under taking induced subgraphs. We show that for many properties $\mathcal{P}$ the only such random orderings are uniform, and give some examples of non-uniform orderings when they exist.

math.PR↗

arXiv2015

Random Geometric Graphs and Isometries of Normed Spaces

Given a countable dense subset $S$ of a finite-dimensional normed space $X$, and $0<p<1$, we form a random graph on $S$ by joining, independently and with probability $p$, each pair of points at distance less than $1$. We say that $S$ is `Rado' if any two such random graphs are (almost surely) isomorphic. Bonato and Janssen showed that in $l_\infty^d$ almost all $S$ are Rado. Our main aim in this paper is to show that $l_\infty^d$ is the unique normed space with this property: indeed, in every other space almost all sets $S$ are non-Rado. We also determine which spaces admit some Rado set: this turns out to be the spaces that have an $l_\infty$ direct summand. These results answer questions of Bonato and Janssen. A key role is played by the determination of which finite-dimensional normed spaces have the property that every bijective step-isometry (meaning that the integer part of distances is preserved) is in fact an isometry. This result may be of independent interest.

math.FA↗

arXiv2015

Subtended Angles

We consider the following question. Suppose that $d\ge2$ and $n$ are fixed, and that $θ_1,θ_2,\dots,θ_n$ are $n$ specified angles. How many points do we need to place in $\mathbb{R}^d$ to realise all of these angles? A simple degrees of freedom argument shows that $m$ points in $\mathbb{R}^2$ cannot realise more than $2m-4$ general angles. We give a construction to show that this bound is sharp when $m\ge 5$. In $d$ dimensions the degrees of freedom argument gives an upper bound of $dm-\binom{d+1}{2}-1$ general angles. However, the above result does not generalise to this case; surprisingly, the bound of $2m-4$ from two dimensions cannot be improved at all. Indeed, our main result is that there are sets of $2m-3$ of angles that cannot be realised by $m$ points in any dimension.

math.MG↗

arXiv2014

Subcritical $\mathcal{U}$-bootstrap percolation models have non-trivial phase transitions

We prove that there exist natural generalizations of the classical bootstrap percolation model on $\mathbb{Z}^2$ that have non-trivial critical probabilities, and moreover we characterize all homogeneous, local, monotone models with this property. Van Enter (in the case $d=r=2$) and Schonmann (for all $d \geq r \geq 2$) proved that $r$-neighbour bootstrap percolation models have trivial critical probabilities on $\mathbb{Z}^d$ for every choice of the parameters $d \geq r \geq 2$: that is, an initial set of density $p$ almost surely percolates $\mathbb{Z}^d$ for every $p>0$. These results effectively ended the study of bootstrap percolation on infinite lattices. Recently Bollobás, Smith and Uzzell introduced a broad class of percolation models called $\mathcal{U}$-bootstrap percolation, which includes $r$-neighbour bootstrap percolation as a special case. They divided two-dimensional $\mathcal{U}$-bootstrap percolation models into three classes -- subcritical, critical and supercritical -- and they proved that, like classical 2-neighbour bootstrap percolation, critical and supercritical $\mathcal{U}$-bootstrap percolation models have trivial critical probabilities on $\mathbb{Z}^2$. They left open the question as to what happens in the case of subcritical families. In this paper we answer that question: we show that every subcritical $\mathcal{U}$-bootstrap percolation model has a non-trivial critical probability on $\mathbb{Z}^2$. This is new except for a certain `degenerate' subclass of symmetric models that can be coupled from below with oriented site percolation. Our results re-open the study of critical probabilities in bootstrap percolation on infinite lattices, and they allow one to ask many questions of subcritical bootstrap percolation models that are typically asked of site or bond percolation.

math.PR↗

arXiv2014

Essential enhancements revisited

In 1991 Aizenman and Grimmett claimed that any `essential enhancement' of site or bond percolation on a lattice lowers the critical probability, an important result with many implications, such as strict inequalities between critical probabilities on suitable pairs of lattices. Their proof has two parts, one probabilistic and one combinatorial. In this paper we point out that a key combinatorial lemma, for which they provide only a figure as proof, is false. We prove an alternative form of the lemma, and thus the enhancement result, in the special cases of site percolation on the square, triangular and cubic lattices, and for bond percolation on $Z^d$, $d\ge 2$. The general case remains open, even for site percolation on $Z^d$, $d\ge 4$.

math.PR↗

arXiv2013

Minimal symmetric differences of lines in projective planes

Let q be an odd prime power and let f(r) be the minimum size of the symmetric difference of r lines in the Desarguesian projective plane PG(2,q). We prove some results about the function f(r), in particular showing that there exists a constant C>0 such that f(r)=O(q) for Cq^{3/2}<r<q^2 - Cq^{3/2}.

math.CO↗

arXiv2011

Multi-path Routing Metrics for Reliable Wireless Mesh Routing Topologies

Several emerging classes of applications that run over wireless networks have a need for mathematical models and tools to systematically characterize the reliability of the network. We propose two metrics for measuring the reliability of wireless mesh routing topologies, one for flooding and one for unicast routing. The Flooding Path Probability (FPP) metric measures the end-to-end packet delivery probability when each node broadcasts a packet after hearing from all its upstream neighbors. The Unicast Retransmission Flow (URF) metric measures the end-to-end packet delivery probability when a relay node retransmits a unicast packet on its outgoing links until it receives an acknowledgement or it tries all the links. Both metrics rely on specific packet forwarding models, rather than heuristics, to derive explicit expressions of the end-to-end packet delivery probability from individual link probabilities and the underlying connectivity graph. We also propose a distributed, greedy algorithm that uses the URF metric to construct a reliable routing topology. This algorithm constructs a Directed Acyclic Graph (DAG) from a weighted, undirected connectivity graph, where each link is weighted by its success probability. The algorithm uses a vector of decreasing reliability thresholds to coordinate when nodes can join the routing topology. Simulations demonstrate that, on average, this algorithm constructs a more reliable topology than the usual minimum hop DAG.

cs.NI↗

arXiv2011

Energy-Latency Tradeoff for In-Network Function Computation in Random Networks

The problem of designing policies for in-network function computation with minimum energy consumption subject to a latency constraint is considered. The scaling behavior of the energy consumption under the latency constraint is analyzed for random networks, where the nodes are uniformly placed in growing regions and the number of nodes goes to infinity. The special case of sum function computation and its delivery to a designated root node is considered first. A policy which achieves order-optimal average energy consumption in random networks subject to the given latency constraint is proposed. The scaling behavior of the optimal energy consumption depends on the path-loss exponent of wireless transmissions and the dimension of the Euclidean region where the nodes are placed. The policy is then extended to computation of a general class of functions which decompose according to maximal cliques of a proximity graph such as the $k$-nearest neighbor graph or the geometric random graph. The modified policy achieves order-optimal energy consumption albeit for a limited range of latency constraints.

cs.NI↗