An improved energy argument for the Hegselmann-Krause model

We show that the freezing time of the $d$-dimensional Hegselmann-Krause model is $O(n^4)$ where $n$ is the number of agents. This improves the best known upper bound whenever $d\geq 2$

First-passage percolation on Cartesian power graphs

We consider first-passage percolation on the class of "high-dimensional" graphs that can be written as an iterated Cartesian product $G\square G \square \dots \square G$ of some base graph $G$ as the number of factors tends to infinity. We propose a natural asymptotic lower bound on the first-passage time between $(v, v, \dots, v)$ and $(w, w, \dots, w)$ as $n$, the number of factors, tends to infinity, which we call the critical time $t^*_G(v, w)$. Our main result characterizes when this lower bound is sharp as $n\rightarrow\infty$. As a corollary, we are able to determine the limit of the so-called diagonal time-constant in $\mathbb{Z}^n$ as $n\rightarrow\infty$ for a large class of distributions of passage times.Comment: 30 pages, 1 figur

Permutations destroying arithmetic progressions in finite cyclic groups

A permutation \pi of 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 (\pi(a),\pi(b),\pi(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).Comment: 11 pages, no figure

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 $\alpha=\alpha_n\in[0,1]$. We prove that there is a very sharp threshold at $\alpha_n=\frac{\ln n}{n}$ for the existence of accessible paths from $\mathbf {v}^0$ to $\mathbf {v}^1$. As a corollary we prove significant concentration, for $\alpha$ 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 $\theta=\theta_n$ satisfies $n\theta_n\rightarrow\infty$, and for any fitness distribution which is continuous on its support and whose support is connected.Comment: Published in at http://dx.doi.org/10.1214/13-AAP949 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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 $\Omega(n^2 \log n)$ 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.Comment: 18 pages, 3 figures. None of the authors has any previous experience in this area of research (multi-agent systems), hence we welcome any feedback from specialist

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.Comment: 9 pages, 2 figures. Version 2: A small error in the proof of Theorem 1.1 is corrected and an acknowledgement added. Bibliography update

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.Comment: 26 pages, 2 figure