Midwinter meeting in discrete probability
January 18-19, 2017, Umeå
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Wednesday
All talks are in room MC313 in the MIT-building
9.15-9.55 Carl Johan Casselgren: Coloring graphs from random lists
The topic of this talk is list coloring of graphs. In this model each vertex of
a graph is assigned a list (set) of colors and the task is then to construct a
proper coloring of the graph such that each vertex gets a color from its list. I
will review some basic facts about list coloring and then discuss a relatively new
variation on list coloring where each vertex receives a random list: letG = G(n)
be a graph on n vertices, and assign to each vertex v of G a list L(v) of colors
by choosing each list independently and uniformly at random from all k-subsets
of a color set of size σ = σ(n). I will discuss various conditions which imply
that with probability tending to 1 as n goes to infinity, G has a proper coloring
from the random lists. .
Coffee
10.20-11.00 Cecilia Holmgren: Using Polya urns to show
normal limit laws for fringe subtrees in preferential attachment trees
11.10-11.50 Xing Shi Cai: Large fringe and non-fringe subtrees in conditional Galton-Watson trees
One particularly attractive random tree model is the tree chosen uniformly at
random from a collection of trees. Many of these models are equivalent to
the Galton-Watson tree conditional on its size – these trees, in turn, go back
to the model proposed by Bienaym, Watson and Galton for the evolution of
populations.
We study the conditions for families of subtrees to exist with high probability
(whp) in a Galton-Walton tree of size n. We first give a Poisson approximation
of fringe subtree counts, which yields the height of the maximal complete rary fringe subtree. Then we determine the maximal Kn such that every tree
of size at most Kn appears as a fringe subtree whp. Finally, we study nonfringe subtree counts and determine the height of the maximal complete r-ary
non-fringe subtree.
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12.00-13.30 Lunch
13.30-14.10 Roland Häggkvst: A Random Scheduling problem
14.20-15.00 Markus Jonsson: An exponential limit shape
of random q-Bulgarian solitaire
We introduce pn -random qn -Bulgarian solitaire (0 < pn , qn ≤ 1), played on n
cards distributed in piles. In each pile, a number of cards equal to the proportion
qn of the pile size rounded upward to the closest integer are candidates to be
picked. Each candidate card is picked with probability pn , independently of
other candidate cards. This generalizes Popov’s random Bulgarian solitaire,
in which there is a single candidate card in each pile. Popov showed that a
triangular limit shape is obtained for a fixed p as n tends to infinity. Here we let
both pn and qn vary with n. We show that under the conditions qn2 pn n/log n →
∞ and pn qn → 0 as n → ∞, the pn -random qn -Bulgarian solitaire has an
exponential limit shape.
Coffee
15.30–16.10 Lan Anh Pham: Avoiding/completing precoloring of hypercubes
A latin square is an nxn array filled with n different symbols, each occurring
exactly once in each row and exactly once in each column. A hypercube in
n dimensions, or an n- cube, is the n dimensional analog of a cube. Some
research before solved the problem of avoiding and completing partial Latin
square. We try to apply these techniques to solve similar problems on avoiding
and completing precolourings of a hypercube.
16.20-17.00 Joel Larsson: Biased random K-SAT
’The random k-SAT problem is as follows: We have a set of n boolean variables
and pick m = m(n) clauses of size k uniformly at random from the set of all
such clauses on our variables, and then ask whether the conjunction of these
clauses satisfiable.
It is known that there is a sharp threshold for satisfiability, but the threshold
function is only known for large k and for k at most 2. We consider a variation
of the problem where there is a bias towards variables occurring pure rather
than negated, i.e. variables occur pure w.p. 1/2+b for some b¿0, and study
how the satisfiability threshold depends on the parameter b
19.00 Dinner at Rex
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Thursday
9-9.40 Tatyana Turova: Random Distance Graphs
We consider random graphs on the set of vertices placed on the discrete ddimentional torus. The edges between pairs of vertices are independent, and
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their probabilities depend on the distance between the vertices, typically nonincreasing with the distance. Hence, the probabilities of connections are intrinsically coupled and scaled with the number of vertices via distance.
We identify a general class of such models which exhibit phase transition
similar to the one in the classical random graph model. We derive also another type of phase transitions considering some local properties, as, e.g., the
clustering coefficient.
Coffee
10-10.40 Sebastian Rosengren: A Dynamic Erdős-Renyi
Graph Model
In this article we introduce a dynamic Erdős-Rényi graph model, in which,
independently for each vertex pair, edges appear and disappear according to a
Markov on-off process. In studying the dynamic graph we present two results.
The first being on how long it takes for the graph to reach stationarity. We give
an explicit expression for this time, as well as proving that this is the fastest
time to reach stationarity among all strong stationary times. The main result
concerns the time it takes for the dynamic graph to reach a certain number of
edges. We give an explicit expression for the expected value of such a time, as
well as study its asymptotic behavior. This time is related to the first time the
dynamic Erdős-Rényi graph contains a cluster exceeding a certain size.
10.50-11.30 Fiona Skerman: Modularity of Random Graphs
An important problem in network analysis is to identify highly connected components or ‘communities’. Most popular clustering algorithms work by approximately optimising modularity. 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 maximum modularity q*(G) of G is the maximum
of the modularity over all partitions of V(G) and takes a value in the interval
[0,1) where larger values indicates a more clustered graph.
Knowledge of the maximum modularity of random graphs helps determine
the significance of a division into communities/vertex partition of a real network.
We investigate the maximum modularity of Erdos-Renyi random graphs and find
there are three different phases of the likely maximum modularity. This is joint
work with Colin McDiarmid.
11.40-12.20 Victor Falgas-Ravry: Problems and results for
random subcube intersection graphs
Random subcube intersection graphs are graphs obtained by selecting a random
collection of subcubes of a fixed hypercube Qd to serve as the vertices, and
setting an edge between a pair of subcubes if their intersection is non-empty.
In this talk I shall discuss two models of random subcube intersection graphs
and the motivation for studying them. I shall sketch results on their clique
and covering thresholds, before discussing a number of open but — I believe—
accessible further problems.
?(Joint work with Klas Markström.)
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2.1
Lunch 12.30-13.30
13.30-14.10 Tom Britton: An epidemic in a dynamic population with importation of infectives
14.20-15.00 X: Y
Coffee
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