arXiv subjects

Fiona Skerman

Publications and source records attributed to Fiona Skerman.

28 records · Page 2Linked to original sources

arXiv2019

The parameterised complexity of computing the maximum modularity of a graph

The maximum modularity of a graph is a parameter widely used to describe the level of clustering or community structure in a network. Determining the maximum modularity of a graph is known to be NP-complete in general, and in practice a range of heuristics are used to construct partitions of the vertex-set which give lower bounds on the maximum modularity but without any guarantee on how close these bounds are to the true maximum. In this paper we investigate the parameterised complexity of determining the maximum modularity with respect to various standard structural parameterisations of the input graph G. We show that the problem belongs to FPT when parameterised by the size of a minimum vertex cover for G, and is solvable in polynomial time whenever the treewidth or max leaf number of G is bounded by some fixed constant; we also obtain an FPT algorithm, parameterised by treewidth, to compute any constant-factor approximation to the maximum modularity. On the other hand we show that the problem is W[1]-hard (and hence unlikely to admit an FPT algorithm) when parameterised simultaneously by pathwidth and the size of a minimum feedback vertex set.

cs.DS↗

arXiv2019

Modularity of Erdős-Rényi random graphs

For a given graph $G$, each partition of the vertices has a modularity score, with higher values indicating that the partition better captures community structure in $G$. The modularity $q^*(G)$ of the graph $G$ is defined to be the maximum over all vertex partitions of the modularity score, and satisfies $0\leq q^*(G) < 1$. Modularity is at the heart of the most popular algorithms for community detection. We investigate the behaviour of the modularity of the Erdős-Rényi random graph $G_{n,p}$ with $n$ vertices and edge-probability $p$. Two key findings are that the modularity is $1+o(1)$ with high probability (whp) for $np$ up to $1+o(1)$ and no further; and when $np \geq 1$ and $p$ is bounded below 1, it has order $(np)^{-1/2}$ whp, in accord with a conjecture by Reichardt and Bornholdt in 2006. We also show that the modularity of a graph is robust to changes in a few edges, in contrast to the sensitivity of optimal vertex partitions.

math.CO↗

arXiv2019

Random tree recursions: which fixed points correspond to tangible sets of trees?

Let $\mathcal{B}$ be the set of rooted trees containing an infinite binary subtree starting at the root. This set satisfies the metaproperty that a tree belongs to it if and only if its root has children $u$ and $v$ such that the subtrees rooted at $u$ and $v$ belong to it. Let $p$ be the probability that a Galton-Watson tree falls in $\mathcal{B}$. The metaproperty makes $p$ satisfy a fixed-point equation, which can have multiple solutions. One of these solutions is $p$, but what is the meaning of the others? In particular, are they probabilities of the Galton-Watson tree falling into other sets satisfying the same metaproperty? We create a framework for posing questions of this sort, and we classify solutions to fixed-point equations according to whether they admit probabilistic interpretations. Our proofs use spine decompositions of Galton-Watson trees and the analysis of Boolean functions.

math.PR↗

arXiv2019

Embedding small digraphs and permutations in binary trees and split trees

We investigate the number of permutations that occur in random labellings of trees. This is a generalisation of the number of subpermutations occurring in a random permutation. It also generalises some recent results on the number of inversions in randomly labelled trees. We consider complete binary trees as well as random split trees a large class of random trees of logarithmic height introduced by Devroye in 1998. Split trees consist of nodes (bags) which can contain balls and are generated by a random trickle down process of balls through the nodes. For complete binary trees we show that asymptotically the cumulants of the number of occurrences of a fixed permutation in the random node labelling have explicit formulas. Our other main theorem is to show that for a random split tree, with high probability the cumulants of the number of occurrences are asymptotically an explicit parameter of the split tree. For the proof of the second theorem we show some results on the number of embeddings of digraphs into split trees which may be of independent interest.

math.PR↗

arXiv2019

K-cut on paths and some trees

We define the (random) $k$-cut number of a rooted graph to model the difficulty of the destruction of a resilient network. The process is as the cut model of Meir and Moon except now a node must be cut $k$ times before it is destroyed. The first order terms of the expectation and variance of $\mathcal{X}_{n}$, the $k$-cut number of a path of length $n$, are proved. We also show that $\mathcal{X}_{n}$, after rescaling, converges in distribution to a limit $\mathcal{B}_{k}$, which has a complicated representation. The paper then briefly discusses the $k$-cut number of some trees and general graphs. We conclude by some analytic results which may be of interest.

math.PR↗

arXiv2018

Inversions in split trees and conditional Galton--Watson trees

We study $I(T)$, the number of inversions in a tree $T$ with its vertices labeled uniformly at random, which is a generalization of inversions in permutations. We first show that the cumulants of $I(T)$ have explicit formulas involving the $k$-total common ancestors of $T$ (an extension of the total path length). Then we consider $X_n$, the normalized version of $I(T_n)$, for a sequence of trees $T_n$. For fixed $T_{n}$'s, we prove a sufficient condition for $X_n$ to converge in distribution. As an application, we identify the limit of $X_n$ for complete $b$-ary trees. For $T_n$ being split trees, we show that $X_n$ converges to the unique solution of a distributional equation. Finally, when $T_n$'s are conditional Galton--Watson trees, we show that $X_n$ converges to a random variable defined in terms of Brownian excursions. By exploiting the connection between inversions and the total path length, we are able to give results that are stronger and much broader compared to previous work by Panholzer and Seitz.

math.PR↗

arXiv2017

Modularity of regular and treelike graphs

Clustering algorithms for large networks typically use modularity values to test which partitions of the vertex set better represent structure in the data. The modularity of a graph is the maximum modularity of a partition. We consider the modularity of two kinds of graphs. For $r$-regular graphs with a given number of vertices, we investigate the minimum possible modularity, the typical modularity, and the maximum possible modularity. In particular, we see that for random cubic graphs the modularity is usually in the interval $(0.666, 0.804)$, and for random $r$-regular graphs with large $r$ it usually is of order $1/\sqrt{r}$. These results help to establish baselines for statistical tests on regular graphs. The modularity of cycles and low degree trees is known to be close to 1: we extend these results to `treelike' graphs, where the product of treewidth and maximum degree is much less than the number of edges. This yields for example the (deterministic) lower bound $0.666$ mentioned above on the modularity of random cubic graphs.

math.CO↗

arXiv2016

Guessing Numbers of Odd Cycles

For a given number of colours, $s$, the guessing number of a graph is the base $s$ logarithm of the size of the largest family of colourings of the vertex set of the graph such that the colour of each vertex can be determined from the colours of the vertices in its neighbourhood. An upper bound for the guessing number of the $n$-vertex cycle graph $C_n$ is $n/2$. It is known that the guessing number equals $n/2$ whenever $n$ is even or $s$ is a perfect square \cite{Christofides2011guessing}. We show that, for any given integer $s\geq 2$, if $a$ is the largest factor of $s$ less than or equal to $\sqrt{s}$, for sufficiently large odd $n$, the guessing number of $C_n$ with $s$ colours is $(n-1)/2 + \log_s(a)$. This answers a question posed by Christofides and Markström in 2011 \cite{Christofides2011guessing}. We also present an explicit protocol which achieves this bound for every $n$. Linking this to index coding with side information, we deduce that the information defect of $C_n$ with $s$ colours is $(n+1)/2 - \log_s(a)$ for sufficiently large odd $n$. Our results are a generalisation of the $s=2$ case which was proven in \cite{bar2011index}.

cs.IT↗

arXiv2015

Avoider-Enforcer star games

In this paper, we study $(1 : b)$ Avoider-Enforcer games played on the edge set of the complete graph on $n$ vertices. For every constant $k\geq 3$ we analyse the $k$-star game, where Avoider tries to avoid claiming $k$ edges incident to the same vertex. We analyse both versions of Avoider-Enforcer games -- the strict and the monotone -- and for each provide explicit winning strategies for both players. We determine the order of magnitude of the threshold biases $f^{mon}_\mathcal{F}$, $f^-_\mathcal{F}$ and $f^+_\mathcal{F}$, where $\mathcal{F}$ is the hypergraph of the game.

math.CO↗

arXiv2015

Degree sequences of random digraphs and bipartite graphs

We investigate the joint distribution of the vertex degrees in three models of random bipartite graphs. Namely, we can choose each edge with a specified probability, choose a specified number of edges, or specify the vertex degrees in one of the two colour classes. This problem can alternatively be described in terms of the row and sum columns of random binary matrix or the in-degrees and out-degrees of a random digraph, in which case we can optionally forbid loops. It can also be cast as a problem in random hypergraphs, or as a classical occupancy, allocation, or coupon collection problem. In each case, provided the two colour classes are not too different in size nor the number of edges too low, we define a probability space based on independent binomial variables and show that its probability masses asymptotically equal those of the degrees in the graph model almost everywhere. The accuracy is sufficient to asymptotically determine the expectation of any joint function of the degrees whose maximum is at most polynomially greater than its expectation.

math.CO↗