Random Networks Workshop
18th May, 2022
The University of Sheffield Probability Group invites you to attend a one-day workshop on random networks due to take place in person on May 18th, 2022. Please see below for details of this workshop as well as information about registration.
Programme, Titles, and Abstracts
09:00-09:30, Registration (tea/coffee + Pastries, catered by PJ Taste.)
09:30-10:20, Dr Minmin Wang, Scaling limits of critical inhomogeneous random graphs
Dr Minmin Wang, University of Sussex, Website
The so-called continuum multiplicative graphs are a family of random metric spaces that arise naturally in the scaling limits of various inhomogeneous random graphs, including the Poissonian model and the configuration model. I'll present a construction of these graphs with an emphasis on its connections with more familiar objects from the branching world: Bienaymé-Galton-Watson tree, Lévy process and Lévy trees. Thanks to these connections, we can identify near optimal conditions for the convergence of discrete graphs in the case of Poissonian model. I’ll also comment on the universality of the limit graphs, in light of some latest developments.
Based on joint works with Nicolas Broutin and Thomas Duquesne.
10:20-10:50, Refreshments (tea/coffee + biscuits, catered by PJ Taste.)
10:50-11:40, Dr John Haslegrave, A tale of two randomly dividing graphs
Dr John Haslegrave, Oxford University, Website
I will discuss two versions of a random multigraph process that evolve by division of vertices. When a vertex divides, it is replaced by two new vertices with its existing connections distributed randomly between them, and some new connections form between these offspring.
One version evolves by synchronised divisions in discrete time and the other by asynchronous divisions in continuous time. The former is closely related to long-range percolation on an infinitely-generated group; the latter is perhaps more natural as a model for real-world networks, but significantly harder to analyse.
In both versions, we prove that the component of the root converges in distribution to a unique (but different) invariant random multigraph. We consider the expected size of this component, and the threshold for connectedness.
This is joint work with Agelos Georgakopoulos (Warwick).
11:50-12:40, Dr John Sylvester, Tangled Paths: a new random graph model from Mallows permutations.
Dr John Sylvester, University of Glasgow, Website
The main idea of this talk will be to explore what happens when you take the union of two graphs on {1,...,n}, and the vertex set of one graph has been labelled by a random permutation. To begin we shall see that combining simple structures (e.g. paths, matchings) using a uniformly random permutation often results in graphs with complex structure (e.g. high treewidth). Is then natural to ask what happens under non-uniform models for random permutations.
We introduce a new random graph model, the tangled path, which results from taking the union of two paths where the vertices of one have been relabelled according to a (non-uniform) random permutation sampled from the Mallows distribution with real parameter 0 < q(n) < 1. Increasing the parameter q in the Mallows distribution has the effect of increasing the number of inverted pairs of elements in the permutation. This has the following effect on the resulting graph: if q is close to 0 the tangled path bears resemblance to a path (if q<1 is fixed then the diameter is linear) and as q tends to 1 it becomes an expander.
In order to further understand the effect of the parameter q on the structure we obtain bounds on the treewidth in terms of q. We also give a sharp threshold for the property of having a balanced separator of size one.
The talk is based on joint work with Jessica Enright, Kitty Meeks and William Pettersson.
12:50-14:10, Lunch (BBQ buffet style, catered by Interval Bar and Kitchen.)
14:10-15:30, Contributed talks by PhD students
14:10-14:30, Alastair Haig, Clique Asymptotics in Scale-Free Social Networks
Alastair Haig, Heriot-Watt University
How does the number of friend groups change as more people join a social network? Are individuals with a mutual popular friend more likely to be friends with each other? How can we apply this to non-social settings?
During this talk we will investigate results for the asymptotic expected number of cliques in the Chung-Lu Inhomogeneous Random Graph Model (in which nodes are independently assigned "popularity" weights with tail probabilities h1−al(h), where a>2 and l is a slowly varying function) as the graph grows, the probability of triangles given one node's popularity, and how to interpret those results in a social context.
14:30-14:50, Erin Russell, Playing with Fire: The Necessary Evil of Self-organised Criticality
Erin Russell, University of Bristol
Consider a mean field Erdős-Rényi random digraph process on n vertices. Let each possible directed edge arrive with rate 1/2n. Without opposition, this process is guaranteed to result in the n-complete graph. Hence we introduce a Poisson rain of “lightning strikes” to each vertex with rate λ(n) which propagates across its out-graph, burning all edges of an affected vertex. Subsequently, the system continues to fluctuate to no end in a battle between creation and destruction.
14:50-15:10, Minwei Sun, Interlacement limit of a stopped random walk trace on a torus
Minwei Sun, University of Bath
This talk will consider a simple random walk on the d-dimensional integer lattice Z^d to start at the origin and stop on its first exit time from (-L, L)^d. Write L in the form L = mN with m = m(N) and N, an integer going to infinity in such a way that L^2 ~ AN^d for some real constant A > 0. The main result is that for dimensions higher than and equal to 3, the projection of the stopped trajectory to the N-torus locally converges, away from the origin, to an interlacement process at level Ad\sigma_1, where \sigma_1 is the exit time of a Brownian motion from (-1, 1)^d that is independent of the interlacement process. The talk is based on joint work with Antal A. Jarai.
15:10-15:30, Qasem Tawhari, Random Spatial Graphs
Qasem Tawhari, Durham University
We study random spatial graphs constructed on the points of a Poisson point process in the unit square, in which each point is joined by an edge to its nearest neighbour in a given direction specified by a cone. The unrestricted case is the ordinary nearest-neighbour graph; the restricted case is a version of the minimal directed spanning forest (MDSF) introduced by Bhatt & Roy. For the ordinary nearest-neighbour graph, Avram & Bertsimas obtained a central limit theorem for the total edge length. For the MDSF, the limit theory is known (Penrose & Wade) in two special cases, namely the `south' and `south-west' versions.
In this talk, I will describe ongoing work to extend the limit theory to the case of general cones; depending on the parameters, the limit distribution may be normal, or the convolution of a normal distribution with a non-normal element due to boundary effects.
15:30-16:00, Refreshments (tea/coffee + biscuits, catered by PJ Taste.)
16:00-16:50, Prof Codina Cotar, Disorder relevance for non-convex random gradient Gibbs measures in d=2
Professor Codina Cotar, University College London, Website
It is a famous result of statistical mechanics that, at low enough temperature, the random field Ising model is disorder relevant for d=2, i.e. the phase transition between uniqueness/non-uniqueness of Gibbs measures disappears, and disorder irrelevant otherwise (Aizenman-Wehr 1990). Generally speaking, adding disorder to a model tends to destroy the non-uniqueness of Gibbs measures.
In this talk we consider - in non-convex potential regime - a random gradient model with disorder in which the interface feels like a bulk term of random fields. We show that this model is disorder relevant with respect to the question of uniqueness of gradient Gibbs measures for a class of non-convex potentials and disorders. No previous knowledge of gradient models will be assumed in the talk
This is joint work with Simon Buchholz.
Location
All talks will take place in Lecture Theatre C of the School of Mathematics and Statistics located in Hicks Building. This room is located at the south entrance to the Hicks Building by the Harley Pub (down the hill from the main entrance.) Helpful images are below
The red box on the right hand side is Sheffield's train station. The red circle on the left is the School of Mathematics and Statistics. Distance between the two is a 15-20 minute walk.
Here you can see the School of Mathematics and Statistics (Hicks Building) and a red arrow indicating the entrance where you will find Lecture Theatre C. The Black pentagon is where lunch will be served.
Address:
Hicks Building,
Hounsfield Road
Sheffield,
S3 7RH
Registration
Registration has closed.
Hotels in the area
Below are a few hotels in the area for those of you who intend to stay overnight in Sheffield.
Funding bodies
This meeting is generously supported by a grant from the Heilbronn Institute for Mathematical Research, the Engineering and Physical Sciences Research Council. The University of Sheffield has supplied physical and online facilities for this workshop. The Applied Probability Trust is providing administrative and organisational support for this meeting.
