arXiv subjects

Tom Johnston

Publications and source records attributed to Tom Johnston.

At least 19 recordsLinked to original sources

arXiv2026

Game connectivity and adaptive dynamics in many-action games

We study the typical structure of games in terms of their connectivity properties. A game is `connected' if it has a pure Nash equilibrium and there is a best-response path from every action profile which is not a pure Nash equilibrium to every pure Nash equilibrium; a game is generic if it has no indifferences. In previous work we showed that, among all $n$-player $k$-action generic games that admit a pure Nash equilibrium, the fraction that are connected tends to $1$ as $n$ gets sufficiently large relative to $k$. Here, we consider the large-$k$ regime, which behaves differently: we show that the connected fraction tends to $1-\zeta_n$ as $k$ gets large, where $\zeta_n>0$ is an explicit constant. Thus, a constant fraction of many-action games are \emph{not} connected. However, for $n\geq3$, $\zeta_n$ is small and tends to $0$ rapidly with $n$, so as $n$ increases all but a vanishingly small fraction of many-player-many-action games are connected. Since connectedness is conducive to equilibrium convergence, we find a simple adaptive dynamic that is guaranteed to converge to a pure Nash equilibrium in all but a vanishingly small fraction of generic games that have one. We rely on new probabilistic and combinatorial arguments to tackle the large-$k$ regime.

econ.TH↗

arXiv2025

Recurrence, transience and anti-concentration of Rademacher random walks

The Rademacher random walk associated with a deterministic sequence $(a_n)_{n \geq 1}$ is the walk which starts at zero and, at step $i$, independently steps either up or down by $a_i$ with equal probability. We continue the study begun by Bhattacharya and Volkov in 2023 of the transience or recurrence of one-dimensional Rademacher random walks. In particular, we show that if the sequence of step sizes is bounded, the walk is weakly recurrent, meaning that it returns infinitely often to a random finite interval, while if the step sizes tend to infinity arbitrarily slowly, the walk may be transient. On the other hand, using a construction with integer step sizes, we show that the step sizes may grow arbitrarily fast and still give a weakly recurrent random walk. We also show, using a construction with non-integer step sizes, that the same conclusion holds even if we restrict to strictly increasing step sizes. However, we prove that if $a_n = n^{\alpha + o(1)}$ for some $\alpha > 1/2$, then the walk is transient. We show that the bound on $\alpha$ is tight by giving an example where $a_n = \Theta(n^{1/2})$ and the walk is weakly recurrent.

math.PR↗

arXiv2025

Infinite induced-saturated graphs

A graph $G$ is $H$-induced-saturated if $G$ is $H$-free but deleting any edge or adding any edge creates an induced copy of $H$. There are non-trivial graphs $H$, such as $P_4$, for which no finite $H$-induced-saturated graph $G$ exists. We show that for every finite graph $H$ that is not a clique or an independent set, there always exists a countable $H$-induced-saturated graph. In fact, we show that a far stronger property can be achieved: there is a countably infinite $H$-free graph $G$ such that any graph $G'\ne G$ obtained by making a locally finite set of changes to $G$ contains a copy of $H$.

math.CO↗

arXiv2024

A note on inverting the dijoin of oriented graphs

For an oriented graph $D$ and a set $X\subseteq V(D)$, the inversion of $X$ in $D$ is the graph obtained from $D$ by reversing the orientation of each edge that has both endpoints in $X$. Define the inversion number of $D$, denoted $\mathrm{inv}(D)$, to be the minimum number of inversions required to obtain an acyclic oriented graph from $D$. The dijoin, denoted $D_1\rightarrow D_2$, of two oriented graphs $D_1$ and $D_2$ is constructed by taking vertex-disjoint copies of $D_1$ and $D_2$ and adding all edges from $D_1$ to $D_2$. We show that $\mathrm{inv}({D_1 \rightarrow D_2}) > \mathrm{inv}(D_1)$, for any oriented graphs $D_1$ and $D_2$ such that $\mathrm{inv}(D_1) = \mathrm{inv}(D_2) \ge 1$. This resolves a question of Aubian, Havet, H\"orsch, Klingelhoefer, Nisse, Rambaud and Vermande. Our proof proceeds via a natural connection between the graph inversion number and the subgraph complementation number.

math.CO↗

arXiv2023

Reconstruction from smaller cards

The $\ell$-deck of a graph $G$ is the multiset of all induced subgraphs of $G$ on $\ell$ vertices. We say that a graph is reconstructible from its $\ell$-deck if no other graph has the same $\ell$-deck. In 1957, Kelly showed that every tree with $n\ge3$ vertices can be reconstructed from its $(n-1)$-deck, and Giles strengthened this in 1976, proving that trees on at least 6 vertices can be reconstructed from their $(n-2)$-decks. Our main theorem states that trees are reconstructible from their $(n-r)$-decks for all $r\le n/{9}+o(n)$, making substantial progress towards a conjecture of Nýdl from 1990. In addition, we can recognise the connectedness of a graph from its $\ell$-deck when $\ell\ge 9n/10$, and reconstruct the degree sequence when $\ell\ge\sqrt{2n\log(2n)}$. All of these results are significant improvements on previous bounds.

math.CO↗

arXiv2023

A Polynomial Upper Bound for Poset Saturation

Given a finite poset $\mathcal P$, we say that a family $\mathcal F$ of subsets of $[n]$ is $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced copy of $\mathcal P$. The induced saturation number of $\mathcal P$, denoted by $\text{sat}^*(n,\mathcal P)$, is the size of the smallest $\mathcal P$-saturated family with ground set $[n]$. In this paper we prove that the saturation number for any given poset grows at worst polynomially. More precisely, we show that $\text{sat}^*(n, \mathcal P)=O(n^c)$, where $c\leq|\mathcal{P}|^2/4+1$ is a constant depending on $\mathcal P$ only. We obtain this result by bounding the VC-dimension of our family.

math.CO↗

arXiv2023

Game Connectivity and Adaptive Dynamics

We analyse the typical structure of games in terms of the connectivity properties of their best-response graphs. Our central result shows that, among games that are `generic' (without indifferences) and that have a pure Nash equilibrium, all but a small fraction are \emph{connected}, meaning that every action profile that is not a pure Nash equilibrium can reach every pure Nash equilibrium via best-response paths. This has important implications for dynamics in games. In particular, we show that there are simple, uncoupled, adaptive dynamics for which period-by-period play converges almost surely to a pure Nash equilibrium in all but a small fraction of generic games that have one (which contrasts with the known fact that there is no such dynamic that leads almost surely to a pure Nash equilibrium in \emph{every} generic game that has one). We build on recent results in probabilistic combinatorics for our characterisation of game connectivity.

econ.TH↗

arXiv2023

Zero-sum squares in $\{-1, 1\}$-matrices with low discrepancy

Given a matrix $M = (a_{i,j})$ a square is a $2 \times 2$ submatrix with entries $a_{i,j}$, $a_{i, j+s}$, $a_{i+s, j}$, $a_{i+s, j +s}$ for some $s \geq 1$, and a zero-sum square is a square where the entries sum to $0$. Recently, Arévalo, Montejano and Roldán-Pensado proved that all large $n \times n$ $\{-1,1\}$-matrices $M$ with discrepancy $|\sum a_{i,j}| \leq n$ contain a zero-sum square unless they are split. We improve this bound by showing that all large $n \times n$ $\{-1,1\}$-matrices $M$ with discrepancy at most $n^2/4$ are either split or contain a zero-sum square. Since zero-sum square free matrices with discrepancy at most $n^2/2$ are already known, this bound is asymptotically optimal.

math.CO↗

arXiv2023

Exact antichain saturation numbers via a generalisation of a result of Lehman-Ron

For given positive integers $k$ and $n$, a family $\mathcal{F}$ of subsets of $\{1,\dots,n\}$ is $k$-antichain saturated if it does not contain an antichain of size $k$, but adding any set to $\mathcal{F}$ creates an antichain of size $k$. We use sat$^*(n, k)$ to denote the smallest size of such a family. For all $k$ and sufficiently large $n$, we determine the exact value of sat$^*(n, k)$. Our result implies that sat$^*(n, k)=n(k-1)-Θ(k\log k)$, which confirms several conjectures on antichain saturation. Previously, exact values for sat$^*(n,k)$ were only known for $k$ up to $6$. We also prove a generalisation of a result of Lehman-Ron which may be of independent interest. We show that given $m$ disjoint chains in the Boolean lattice, we can create $m$ disjoint skipless chains that cover the same elements (where we call a chain skipless if any two consecutive elements differ in size by exactly one).

math.CO↗

arXiv2023

Decomposing random permutations into order-isomorphic subpermutations

Two permutations $s$ and $t$ are $k$-similar if they can be decomposed into subpermutations $s^1, \ldots, s^k$ and $t^1, \ldots, t^k$ such that $s^i$ is order-isomorphic to $t^i$ for all $i$. Recently, Dudek, Grytczuk and Ruciński posed the problem of determining the minimum $k$ for which two permutations chosen independently and uniformly at random are $k$-similar. We show that two such permutations are $O(n^{1/3}\log^{11/6}(n))$-similar with high probability, which is tight up to a polylogarithmic factor. Our result also generalises to simultaneous decompositions of multiple permutations.

math.CO↗

arXiv2023

Counting graphic sequences via integrated random walks

Given an integer $n$, let $G(n)$ be the number of integer sequences $n-1\ge d_1\ge d_2\ge\dotsb\ge d_n\ge 0$ that are the degree sequence of some graph. We show that $G(n)=(c+o(1))4^n/n^{3/4}$ for some constant $c>0$, improving both the previously best upper and lower bounds by a factor of $n^{1/4}$ (up to polylog-factors). Additionally, we answer a question of Royle, extend the values of $n$ for which the exact value of $G(n)$ is known from $n\le290$ to $n\le 1651$ and determine the asymptotic probability that the integral of a (lazy) simple symmetric random walk bridge remains non-negative.

math.CO↗

arXiv2022

Shotgun assembly of random graphs

In the graph shotgun assembly problem, we are given the balls of radius $r$ around each vertex of a graph and asked to reconstruct the graph. We study the shotgun assembly of the Erd\H{o}s-R\'enyi random graph $\mathcal G(n,p)$ for a wide range of values of $r$. We determine the threshold for reconstructibility for each $r\geq 3$, extending and improving substantially on results of Mossel and Ross for $r=3$. For $r=2$, we give upper and lower bounds that improve on results of Gaudio and Mossel by polynomial factors. We also give a sharpening of a result of Huang and Tikhomirov for $r=1$.

math.CO↗

arXiv2022

The rainbow saturation number is linear

Given a graph $H$, we say that an edge-coloured graph $G$ is $H$-rainbow saturated if it does not contain a rainbow copy of $H$, but the addition of any non-edge in any colour creates a rainbow copy of $H$. The rainbow saturation number $\text{rsat}(n,H)$ is the minimum number of edges among all $H$-rainbow saturated edge-coloured graphs on $n$ vertices. We prove that for any non-empty graph $H$, the rainbow saturation number is linear in $n$, thus proving a conjecture of Gir\~{a}o, Lewis, and Popielarz. In addition, we also give an improved upper bound on the rainbow saturation number of the complete graph, disproving a second conjecture of Gir\~{a}o, Lewis, and Popielarz.

math.CO↗

arXiv2022

Perfect shuffling with fewer lazy transpositions

A lazy transposition $(a,b,p)$ is the random permutation that equals the identity with probability $1-p$ and the transposition $(a,b)\in S_n$ with probability $p$. How long must a sequence of independent lazy transpositions be if their composition is uniformly distributed? It is known that there are sequences of length $\binom{n}2$, but are there shorter sequences? This was raised by Fitzsimons in 2011, and independently by Angel and Holroyd in 2018. We answer this question negatively by giving a construction of length $\frac23 \binom{n}2+O(n\log n)$, and consider some related questions.

math.CO↗

arXiv2022

Short reachability networks

We investigate the following generalisation of permutation networks. We say a sequence $T=(T_1,\dots,T_\ell)$ of transpositions in $S_n$ forms a $t$-reachability network if, for every choice of $t$ distinct points $x_1, \dots, x_t\in \{1,\dots,n\}$, there is a subsequence of $T$ whose composition maps $j$ to $x_j$ for every $1\leq j\leq t$. When $t=n$, any permutation in $S_n$ can be created and $T$ is a permutation network. Waksman [JACM, 1968] showed that the shortest permutation networks have length about $n \log_2(n)$. In this paper, we investigate the shortest $t$-reachability networks for other values of $t$. Our main result settles the case of $t=2$: the shortest $2$-reachability network has length $\lceil 3n/2\rceil-2 $. For fixed $t \geq 3$, we give a simple randomised construction which shows that there exist $t$-reachability networks with $(2+o_t(1))n$ transpositions. We also study the effect of restricting to star-transpositions, i.e. restricting all transpositions to have the form $(1, \cdot)$.

math.CO↗

arXiv2022

Reconstructing the degree sequence of a sparse graph from a partial deck

The deck of a graph $G$ is the multiset of cards $\{G-v:v\in V(G)\}$. Myrvold (1992) showed that the degree sequence of a graph on $n\geq7$ vertices can be reconstructed from any deck missing one card. We prove that the degree sequence of a graph with average degree $d$ can reconstructed from any deck missing $O(n/d^3)$ cards. In particular, in the case of graphs that can be embedded on a fixed surface (e.g. planar graphs), the degree sequence can be reconstructed even when a linear number of the cards are missing.

math.CO↗

arXiv2022

Improved bounds for 1-independent percolation on $\mathbb{Z}^n$

A 1-independent bond percolation model on a graph $G$ is a probability distribution on the spanning subgraphs of $G$ in which, for all vertex-disjoint sets of edges $S_1$ and $S_2$, the states of the edges in $S_1$ are independent of the states of the edges in $S_2$. Such a model is said to percolate if the random subgraph has an infinite component with positive probability. In 2012 the first author and Bollob\'as defined $p_{\max}(G)$ to be the supremum of those $p$ for which there exists a 1-independent bond percolation model on $G$ in which each edge is present in the random subgraph with probability at least $p$ but which does not percolate. A fundamental and challenging problem in this area is to determine the value of $p_{\max}(G)$ when $G$ is the lattice graph $\mathbb{Z}^2$. Since $p_{\max}(\mathbb{Z}^n)\leq p_{\max}(\mathbb{Z}^{n-1})$, it is also of interest to establish the value of $\lim_{n\to\infty} p_{\max}(\mathbb{Z}^n)$. In this paper we significantly improve the best known upper bound on this limit and obtain better upper and lower bounds on $p_{\max}(\mathbb{Z}^2)$. In proving these results, we also give an upper bound on the critical probability for a 1-independent model on the hypercube graph to contain a giant component asymptotically almost surely.

math.PR↗

arXiv2022

The lengths for which bicrucial square-free permutations exist

A square is a factor $S = (S_1; S_2)$ where $S_1$ and $S_2$ have the same pattern, and a permutation is said to be square-free if it contains no non-trivial squares. The permutation is further said to be bicrucial if every extension to the left or right contains a square. We completely classify for which $n$ there exists a bicrucial square-free permutation of length $n$.

math.CO↗