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Peter Hegarty

Publications and source records attributed to Peter Hegarty.

At least 19 recordsLinked to original sources
arXiv2018

The "No Justice in the Universe" phenomenon: why honesty of effort may not be rewarded in tournaments

In 2000 Allen Schwenk, using a well-known mathematical model of matchplay tournaments in which the probability of one player beating another in a single match is fixed for each pair of players, showed that the classical single-elimination, seeded format can be "unfair" in the sense that situations can arise where an indisputibly better (and thus higher seeded) player may have a smaller probability of winning the tournament than a worse one. This in turn implies that, if the players are able to influence their seeding in some preliminary competition, situations can arise where it is in a player's interest to behave "dishonestly", by deliberately trying to lose a match. This motivated us to ask whether it is possible for a tournament to be both honest, meaning that it is impossible for a situation to arise where a rational player throws a match, and "symmetric" - meaning basically that the rules treat everyone the same - yet unfair, in the sense that an objectively better player has a smaller probability of winning than a worse one. After rigorously defining our terms, our main result is that such tournaments exist and we construct explicit examples for any number n >= 3 of players. For n=3, we show (Theorem 3.6) that the collection of win-probability vectors for such tournaments form a 5-vertex convex polygon in R^3, minus some boundary points. We conjecture a similar result for any n >= 4 and prove some partial results towards it.

math.PR↗
arXiv2015

Permutations destroying arithmetic progressions in finite cyclic groups

A permutation πof an abelian group G is said to destroy arithmetic progressions (APs) if, whenever (a,b,c) is a non-trivial 3-term AP in G, that is c-b=b-a and a,b,c are not all equal, then (π(a),π(b),π(c)) is not an AP. In a paper from 2004, the first author conjectured that such a permutation exists of Z/nZ, for all n except 2,3,5 and 7. Here we prove, as a special case of a more general result, that such a permutation exists for all n >= n_0, for some explcitly constructed number n_0 \approx 1.4 x 10^{14}. We also construct such a permutation of Z/pZ for all primes p > 3 such that p = 3 (mod 8).

math.NT↗
arXiv2014

The Hegselmann-Krause dynamics on the circle converge

We consider the Hegselmann-Krause dynamics on a one-dimensional torus and provide the first proof of convergence of this system. The proof requires only fairly minor modifications of existing methods for proving convergence in Euclidean space.

eess.SY↗
arXiv2014

The Hegselmann-Krause dynamics for equally spaced agents

We consider the Hegselmann-Krause bounded confidence dynamics for n equally spaced opinions on the real line, with gaps equal to the confidence bound r, which we take to be 1. We prove rigorous results on the evolution of this configuration, which confirm hypotheses previously made based on simulations for small values of n. Namely, for every n, the system evolves as follows: after every 5 time steps, a group of 3 agents become disconnected at either end and collapse to a cluster at the subsequent step. This continues until there are fewer than 6 agents left in the middle, and these finally collapse to a cluster, if n is not a multiple of 6. In particular, the final configuration consists of 2*[n/6] clusters of size 3, plus one cluster in the middle of size n (mod 6), if n is not a multiple of 6, and the number of time steps before freezing is 5n/6 + O(1). We also consider the dynamics for arbitrary, but constant, inter-agent spacings d \in [0, 1] and present three main findings. Firstly we prove that the evolution is periodic also at some other, but not all, values of d, and present numerical evidence that for all d something "close" to periodicity nevertheless holds. Secondly, we exhibit a value of d at which the behaviour is periodic and the time to freezing is n + O(1), hence slower than that for d = 1. Thirdly, we present numerical evidence that, as d --> 0, the time to freezing may be closer, in order of magnitude, to the diameter d(n-1) of the configuration rather than the number of agents n.

physics.soc-ph↗
arXiv2014

On the existence of accessible paths in various models of fitness landscapes

We present rigorous mathematical analyses of a number of well-known mathematical models for genetic mutations. In these models, the genome is represented by a vertex of the $n$-dimensional binary hypercube, for some $n$, a mutation involves the flipping of a single bit, and each vertex is assigned a real number, called its fitness, according to some rules. Our main concern is with the issue of existence of (selectively) accessible paths; that is, monotonic paths in the hypercube along which fitness is always increasing. Our main results resolve open questions about three such models, which in the biophysics literature are known as house of cards (HoC), constrained house of cards (CHoC) and rough Mount Fuji (RMF). We prove that the probability of there being at least one accessible path from the all-zeroes node $\mathbf {v}^0$ to the all-ones node $\mathbf {v}^1$ tends respectively to 0, 1 and 1, as $n$ tends to infinity. A crucial idea is the introduction of a generalization of the CHoC model, in which the fitness of $\mathbf {v}^0$ is set to some $α=α_n\in[0,1]$. We prove that there is a very sharp threshold at $α_n=\frac{\ln n}{n}$ for the existence of accessible paths from $\mathbf {v}^0$ to $\mathbf {v}^1$. As a corollary we prove significant concentration, for $α$ below the threshold, of the number of accessible paths about the expected value (the precise statement is technical; see Corollary 1.4). In the case of RMF, we prove that the probability of accessible paths from $\mathbf {v}^0$ to $\mathbf {v}^1$ existing tends to $1$ provided the drift parameter $θ=θ_n$ satisfies $nθ_n\rightarrow\infty$, and for any fitness distribution which is continuous on its support and whose support is connected.

math.PR↗
arXiv2013

A variant of the multi-agent rendezvous problem

The classical multi-agent rendezvous problem asks for a deterministic algorithm by which $n$ points scattered in a plane can move about at constant speed and merge at a single point, assuming each point can use only the locations of the others it sees when making decisions and that the visibility graph as a whole is connected. In time complexity analyses of such algorithms, only the number of rounds of computation required are usually considered, not the amount of computation done per round. In this paper, we consider $Ω(n^2 \log n)$ points distributed independently and uniformly at random in a disc of radius $n$ and, assuming each point can not only see but also, in principle, communicate with others within unit distance, seek a randomised merging algorithm which asymptotically almost surely (a.a.s.) runs in time O(n), in other words in time linear in the radius of the disc rather than in the number of points. Under a precise set of assumptions concerning the communication capabilities of neighboring points, we describe an algorithm which a.a.s. runs in time O(n) provided the number of points is $o(n^3)$. Several questions are posed for future work.

cs.MA↗
arXiv2012

On the notion of balance in social network analysis

The notion of "balance" is fundamental for sociologists who study social networks. In formal mathematical terms, it concerns the distribution of triad configurations in actual networks compared to random networks of the same edge density. On reading Charles Kadushin's recent book "Understanding Social Networks", we were struck by the amount of confusion in the presentation of this concept in the early sections of the book. This confusion seems to lie behind his flawed analysis of a classical empirical data set, namely the karate club graph of Zachary. Our goal here is twofold. Firstly, we present the notion of balance in terms which are logically consistent, but also consistent with the way sociologists use the term. The main message is that the notion can only be meaningfully applied to undirected graphs. Secondly, we correct the analysis of triads in the karate club graph. This results in the interesting observation that the graph is, in a precise sense, quite "unbalanced". We show that this lack of balance is characteristic of a wide class of starlike-graphs, and discuss possible sociological interpretations of this fact, which may be useful in many other situations.

cs.SI↗
arXiv2012

Can connected commuting graphs of finite groups have arbitrarily large diameter ?

We present a family of finite, non-abelian groups and propose that there are members of this family whose commuting graphs are connected and of arbitrarily large diameter. If true, this would disprove a conjecture of Iranmanesh and Jafarzadeh. While unable to prove our claim, we present a heuristic argument in favour of it. We also present the results of simulations which yielded explicit examples of groups whose commuting graphs have all possible diameters up to and including 10. Previously, no finite group whose commuting graph had diameter greater than 6 was known.

math.GR↗
arXiv2012

Limit points in the range of the commuting probability function on finite groups

If G is a finite group, then Pr(G) denotes the fraction of ordered pairs of elements of G which commute. We show that, if l \in (2/9,1] is a limit point of the function Pr on finite groups, then l \in \Q and there exists an e = e_l > 0 such that Pr(G) \not\in (l - e_l, l) for any finite group G. These results lend support to some old conjectures of Keith Joseph.

math.GR↗
arXiv2012

Permutations all of whose patterns of a given length are distinct

For each integer k >= 2, let F(k) denote the largest n for which there exists a permutation σ\in S_n, all of whose patterns of length k are distinct. We prove that F(k) = k + \lfloor \sqrt{2k-3} \rfloor + e_k, where e_k \in {-1,0} for every k. Suggestions for further investigations along these lines are discussed.

math.CO↗
arXiv2012

Why should one expect to find long runs of (non)-Ramanujan primes ?

Sondow et al have studied Ramanujan primes (RPs) and observed numerically that, while half of all primes are RPs asymptotically, one obtains runs of consecutives RPs (resp. non-RPs) which are statistically significantly longer than one would expect if one was tossing an unbiased coin. In this discussion paper we attempt a heuristic explanation of this phenomenon. Our heuristic follows naturally from the Prime Number Theorem, but seems to be only partly satisfactory. It motivates why one should obtain long runs of both RPs and non-RPs, and also longer runs of non-RPs than of RPs. However, it also suggests that one should obtain longer runs of RPs than have so far been observed in the data, and this issue remains puzzling.

math.NT↗
arXiv2010

Independent sets in almost-regular graphs and the Cameron-Erdos problem for non-invariant linear equations

We propose a generalisation of the Cameron-Erdos conjecture for sum-free sets to arbitrary non-translation invariant linear equations over Z in three or more variables and, using well-known methods from graph theory, prove a weak form of the conjecture for a class of equations where the structure of the maximum-size sets avoiding solutions to the equation has been previously obtained.

math.NT↗
arXiv2010

On m-covering families of Beatty sequences with irrational moduli

We generalise Uspensky's theorem characterising eventual exact (e.e.) covers of the positive integers by homogeneous Beatty sequences, to e.e. m-covers, for any m \in \N, by homogeneous sequences with irrational moduli. We also consider inhomogeneous sequences, again with irrational moduli, and obtain a purely arithmetical characterisation of e.e. m-covers. This generalises a result of Graham for m = 1, but when m > 1 the arithmetical description is more complicated. Finally we speculate on how one might make sense of the notion of an exact m-cover when m is not an integer, and present a "fractional version" of Beatty's theorem.

math.NT↗
arXiv2010

Invariant and dual subtraction games resolving the Duchê-Rigo conjecture

We prove a recent conjecture of Duchêne and Rigo, stating that every complementary pair of homogeneous Beatty sequences represents the solution to an \emph{invariant} impartial game. Here invariance means that each available move in a game can be played anywhere inside the game-board. In fact, we establish such a result for a wider class of pairs of complementary sequences, and in the process generalize the notion of a \emph{subtraction game}. Given a pair of complementary sequences $(a_n)$ and $(b_n)$ of positive integers, we define a game $G$ by setting $\{\{a_n, b_n\}\}$ as invariant moves. We then introduce the invariant game $G^\star $, whose moves are all non-zero $P$-positions of $G$. Provided the set of non-zero $P$-positions of $G^\star$ equals $\{\{a_n,b_n\}\}$, this \emph{is} the desired invariant game. We give sufficient conditions on the initial pair of sequences for this 'duality' to hold.

math.CO↗
arXiv2009

A Cauchy-Davenport type result for arbitrary regular graphs

Motivated by the Cauchy-Davenport theorem for sumsets, and its interpretation in terms of Cayley graphs, we prove the following main result : There is a universal constant e > 0 such that, if G is a connected, regular graph on n vertices, then either every pair of vertices can be connected by a path of length at most 3, or the number of pairs of such vertices is at least 1+e times the number of edges in G. We discuss a range of further questions to which this result gives rise.

math.CO↗
arXiv2009

Answers to two questions posed by Farhi concerning additive bases

Let A be an asymptotic basis for N and X a finite subset of A such that A\X is still an asymptotic basis. Farhi recently proved a new batch of upper bounds for the order of A\X in terms of the order of A and a variety of parameters related to the set X. He posed two questions concerning possible improvements to his bounds. In this note, we answer both questions.

math.NT↗