# Fiona Skerman

#### Uppsala University / Simon's Institute

For Fall 2021 I am a Simons-Berkeley Research Fellow in the program on Computational Complexity of Statistical Inference at UC Berkeley's Simons Institute.

I am tenure-track researcher (biträdande lektor) at Uppsala University. My previous position was a Heilbronn Research Fellow at Bristol University. Before this I was a postdoc with Dan Kráľ at Masaryk University, with Cecilia Holmgren at Uppsala University and with the Heilbronn Institute at Bristol University.

My DPhil with Colin McDiarmid researched the modularity of networks and for which I was awarded the Corcoran Memorial Prize for the best thesis in the Oxford Statistics Department [thesis]. Before coming to Oxford, I did my honours thesis with Brendan McKay at the Australian National University, looking at likely degree sequences in random bipartite graphs [thesis].

### ALEA Young Workshop Normandy 2019

Here are the lecture notes and the exercises for course in Random Graphs, Thresholds and Phase transitions.

### Preprints

Click on arrows to expand.

From flip processes to dynamical systems on graphons with Jan Hladky, Frederik Garbe and Matas Šileikis;
Preprint. [arXiv] [pdf]

We introduce a class of random graph processes, which we call \emph{flip processes}. Each such process is given by a \emph{rule} which is a function $\mathcal{R}:\mathcal{H}_k\rightarrow \mathcal{H}_k$ from all labelled $k$-vertex graphs into itself ($k$ is fixed). The process starts with a given $n$-vertex graph $G_0$. In each step, the graph $G_i$ is obtained by sampling $k$ random vertices $v_1,\ldots,v_k$ of $G_{i-1}$ and replacing the induced graph $F:=G_{i-1}[v_1,\ldots,v_k]$ by $\mathcal{R}(F)$. This class contains several previously studied processes including the Erd\H{o}s--R\'enyi random graph process and the triangle removal process. Actually, our definition of flip processes is more general, in that $\mathcal{R}(F)$ is a probability distribution on $\mathcal{H}_k$, thus allowing randomised replacements.

Given a flip process with a rule $\mathcal{R}$, we construct time-indexed trajectories $\Phi: \mathcal{W}_{0} \times [0,\infty)\rightarrow \mathcal{W}_{0}$ in the space of graphons. We prove that for any $T > 0$ starting with a large finite graph $G_0$ which is close to a graphon $W_0$ in the cut norm, with high probability the flip process will stay in a thin sausage around the trajectory $(\Phi(W_0,t))_{t=0}^T$ (after rescaling).

These graphon trajectories are then studied from the perspective of dynamical systems. Among others topics, we study continuity properties of these trajectories with respect to time and initial graphon, existence and stability of fixed points and speed of convergence (whenever the infinite time limit exists). We give an example of a flip process with a periodic trajectory.

Modularity and edge sampling with Colin McDiarmid;
Submitted. [arXiv] [pdf]

Suppose that there is an unknown underlying graph $G$ on a large vertex set, and we can test only a proportion of the possible edges to check whether they are present in $G$. If $G$ has high modularity, is the observed graph $G'$ likely to have high modularity? We see that this is indeed the case under a mild condition, in a natural model where we test edges at random. We find that $q^*(G') \geq q^*(G)-\varepsilon$ with probability at least $1-\varepsilon$, as long as the expected number edges in $G'$ is large enough. Similarly, $q^*(G') \leq q^*(G)+\varepsilon$ with probability at least $1-\varepsilon$, under the stronger condition that the expected average degree in $G'$ is large enough. Further, under this stronger condition, finding a good partition for $G'$ helps us to find a good partition for $G$. Modularity introduced by Newman and Girvan (in this paper) is a popular metric in community detection.

A branching process with deletions and mergers that matches the threshold for hypercube percolation with Laura Eslava and Sarah Penington;
Submitted. [arXiv] [pdf]

We define a graph process $\mathcal{G}(p,q)$ based on a discrete branching process with deletions and mergers, which is inspired by the 4-cycle structure of both the hypercube $Q_d$ and the lattice $\mathbb{Z}^d$ for large $d$. Individuals have Poisson offspring distribution with mean $1+p$ and certain deletions and mergers occur with probability $q$; these parameters correspond to the mean number of edges discovered from a given vertex in an exploration of a percolation cluster and to the probability that a non-backtracking path of length four closes a cycle, respectively.

We prove survival and extinction under certain conditions on $p$ and $q$ that heuristically match the known expansions of the critical probabilities for bond percolation on the lattice $\mathbb{Z}^d$ and the hypercube $Q_d$. These expansions have been rigorously established by Hara and Slade in 1995, and van der Hofstad and Slade in 2006, respectively. We stress that our method does not constitute a branching process proof for the percolation threshold.

The analysis of the graph process survival is considerably more challenging than for branching processes in discrete time, due to the interdependence between the descendants of different individuals in the same generation. In fact, it is left open whether the survival probability of $\mathcal{G}(p,q)$ is monotone in $p$ or $q$; we discuss this and some other open problems regarding the new graph process.

Finding large expanders in graphs: from topological minors to induced subgraphs with Baptiste Louf;
Submitted. [arXiv] [pdf]

A structural and geometric property of graphs is the presence of large expanders. The problem of finding such structures was considered by Krivelevich in this paper. Here, we show that the problem of finding a large induced subgraph that is an expander can be reduced to the simpler problem of finding a topological minor that is an expander. Our proof is constructive, which is helpful in an algorithmic setting.
We also show that every large subgraph of an expander graph contains a large subgraph which is itself an expander.

No additional tournaments are quasirandom-forcing with Robert Hancock, Adam Kabela, Dan Kráľ, Taísa Martins, Roberto Parente and Jan Volec;
Submitted. [arXiv] [pdf] [slides]

A tournament $H$ is quasirandom-forcing if the following holds for every sequence $(G_n)_{n}$ of tournaments of growing orders: if the density of $H$ in $G_n$ converges to the expected density of $H$ in a random tournament, then $(G_n)_n$ is quasirandom. Every transitive tournament with at least 4 vertices is quasirandom-forcing (link), and Coregliano et al. showed that there is also a non-transitive 5-vertex tournament with the property (link). We show that no additional tournament has this property.

### Publications and accepted manuscripts

Do not hesitate to email if you have trouble accessing any of the manuscripts listed below.

The modularity of random graphs on the hyperbolic plane with Jordan Chellig and Nikolaos Fountoulakis;
Journal of Complex Networks, to appear. [arXiv] [pdf]

Modularity is a quantity which has been introduced in the context of complex networks in order to quantify how close a network is to an ideal modular network in which the nodes form small interconnected communities that are joined together with relatively few edges. In this paper, we consider this quantity on a recent probabilistic model of complex networks introduced by Krioukov et al. (Phys. Rev. E 2010).

This model views a complex network as an expression of hidden hierarchies, encapsulated by an underlying hyperbolic space. For certain parameters, this model was proved to have typical features that are observed in complex networks such as power law degree distribution, bounded average degree, clustering coefficient that is asymptotically bounded away from zero, and ultra-small typical distances. In the present work, we investigate its modularity and we show that, in this regime, it converges to 1 in probability.

Assigning times to minimise reachability in temporal graphs with Jess Enright and Kitty Meeks;
Journal of Computer and System Sciences 115 (2021). [journal] [arXiv] [pdf]

Temporal graphs, in which edges are active at specified times, are of relevance for spreading processes on graphs. In our setting the number of active times for each edge is fixed, but we can change the relative order in which (sets of) edges are active. We investigate the problem of determining an ordering of edges that minimises the maximum number of vertices reachable from any single starting vertex. Epidemiologically, this corresponds to the worst-case number of vertices infected in a single disease outbreak.

The parameterised complexity of computing the maximum modularity of a graph with Kitty Meeks;
Algorithmica 82 8 (2020) + IPEC (2018). [journal] [IPEC] [arXiv] [pdf]

Determining the maximum modularity of a graph is known to be NP-complete in general, even to find a multiplicative approximation, but we show it has reduced complexity with respect to some structural parameterisations of the input graph.

On the other hand we prove modularity 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.

Permutations in binary trees and split trees with Michael Albert, Cecilia Holmgren and Tony Johansson;
Algorithmica 82 3 (2020). [journal] [arXiv] [pdf] [slides]

Given a tree on $n$ vertices, we randomly label the vertices 1 to $n$. An occurence of a length $k$ permutation $\sigma$ is a sequence of vertices on a common descending path in the tree whose labels, when read from root to leaf, are ordered according to $\sigma$. We calculate the number of occurences of fixed length permutations in binary trees and split trees.
For the trees considered, this generalizes the inversion results below from $\sigma=21$ to more general permutations $\sigma$.
Conference version: AofA

Modularity of Erdos-Renyi random graphs with Colin McDiarmid;
Random Structures and Algorithms: 57 1 (2020) [journal] [arXiv] [pdf]

Modularity introduced by Newman and Girvan (in this paper) is a popular metric in community detection.
Two key features which we find for Erdos-Renyi random graphs 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 (and disproving another conjecture from the physics literature).
Conference version: AofA

Random tree recursions: which fixed points correspond to tangible sets of trees? with Toby Johnson and Moumanti Podder;
Random Structures and Algorithms 56 3 (2020) [journal] [arXiv] [pdf]
We answer the following question of Joel Spencer:
Say that a set of trees $\mathcal{B}$ follows the at-least-two rule if $t \in \mathcal{B}$ if and only if the root of $t$ has two children $u$ and $v$ such that the subtrees rooted at $u$ and $v$ are also in $\mathcal{B}$. Suppose that $\lambda > \lambda_{crit}$. Does there exist a set of trees $\mathcal{B}$ with this metaproperty such that for random tree $T_\lambda$, we have that $\mathbb{P}[T_\lambda \in \mathcal{B}]$ is the middle solution of the classical fixed point equation?

$k$ cuts on paths and some trees with Xing Shi Cai, Cecilia Holmgren and Luc Devroye;
Electronic Journal of Probability 24 (2019) [journal] [arXiv] [pdf]

We define a new cutting process on trees the `blunt scissors' or $k$-cut model where each node must be cut $k$ times before it is destroyed. For $k=1$ this is the model of Meir and Moon. This paper studies the distribution of the $k$-cut number of a path on $n$ vertices.

Inversions in split trees and Galton-Watson trees with Xing Shi Cai, Cecilia Holmgren, Svante Janson and Tony Johansson.
Combinatorics, Probability and Computing 28:3 (2019). [journal] [arXiv] [pdf]

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). For three sequence of trees $T_n$ we consider $X_n$ the normalized version of $I(T_n)$.
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 and 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 extend earlier results by Panholzer and Seitz. Conference version: [AofA]

Modularity of regular and treelike graphs with Colin McDiarmid;
Journal of Complex Networks (6) 4 (2018) [journal] [arXiv] [pdf]

Modularity introduced by Newman and Girvan (in this paper) is a popular metric in community detection.
We show that random cubic graphs usually have modularity in the interval (0.666, 0.804); and random $r$-regular graphs for large $r$ usually have modularity $\Theta(1/\sqrt{r})$. Our results give thresholds for the statistical significance of clustering found in large regular networks.
The modularity of cycles and low degree trees is known to be asymptotically 1. We extend these results to all graphs whose product of treewidth and maximum degree is much less than the number of edges. This shows for example that random planar graphs typically have modularity close to 1.

Guessing Numbers of Odd Cycles with Ross Atkins and Puck Rombach;
Electronic Journal of Combinatorics 24:1 (2017) [journal] [arXiv] [pdf]

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. 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 the cycle $C_n$ with $s$ colours is $(n -1)/2 + \log_s (a)$. This answers a question posed in this paper by Christofides and Markstrom in 2011.

Degree sequences of random digraphs and bipartite graphs with Brendan McKay;
Journal of Combinatorics 7:1 (2016) [journal] [arXiv] [pdf]

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.

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.

Avoider-Enforcer Star Games with Andrzej Grzesik, Mirjana Mikalački, Zoltán Nagy, Alon Naor, Balázs Patkós;
Discrete Mathematics and Theoretical Computer Science 17:1 (2015) [journal] [arXiv] [pdf]

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 consider 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 $f_\mathcal{F}^{mon}$, $f_\mathcal{F}^{-}$ and $f_\mathcal{F}^{+}$, where $\mathcal{F}$ is the hypergraph of the game.

Modularity in random regular graphs and lattices with Colin McDiarmid;
Electronic Notes in Discrete Mathematics 43 (2013) [journal]

Given a graph $G$, the modularity of a partition of the vertex set measures the extent to which edge density is higher within parts than between parts; and the modularity of $G$ is the maximum modularity of a partition.
We give an upper bound on the modularity of r-regular graphs as a function of the edge expansion (or isoperimetric number) under the restriction that each part in our partition has a sub-linear numbers of vertices. This leads to results for random $r$-regular graphs. In particular we show the modularity of a random cubic graph partitioned into sub-linear parts is almost surely in the interval $(0.66, 0.88)$.