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April 141516, 2015
Random Walks on Random Graphs and Applications
Discrete random walks and related processes. Analysis on random graph models, and algorithmic applications to large networks
SUMMARY The workshop focuses on: 1) The analysis of discrete random processes on random graph models. In particular: random walks, voting, information dissemination by randomized broadcasting, bootstrap percolation and interacting particle systems. Also of interest are selfmodifying random networks and algorithmic aspects thereof. 2) The practical application of these methods to the algorithmic aspects of networks, such as organization of, and search and ranking in large networks.
ORGANISERS
INVITED SPEAKERS
Tuesday April 14
Wednesday April 15
Thursday April 16
***************************************************************************************************************************************** ABSTRACTS Omer Angel Random walks on unimodular planar networks We study random hyperbolic planar graphs by using their circle packing
embedding to connect their geometry to that of the hyperbolic plane. This leads
to several results: Identification of the Poisson and Konstantin Avratchenkov Petra Berenbrink The Moran Process The Moran process was suggested by Lieberman, Hauert and Nowak (Nature, 433:312316, 2005). Each node of a finite, connected (directed or undirected) graph models one individual. It is assumed that the initial population consists of a single mutant of fitness r>0 placed uniformly at random, with every other vertex occupied by an individual of fitness 1. At each time step, an individual (node) is chosen at random with probability proportional to its assigned ``fitness'' value. It reproduces by placing a copy of itself on a neighbouring node that is chosen uniformly at random, replacing the individual that was there. The main quantities of interest are the probabilities that the descendants of the initial mutant come to occupy the whole graph (fixation) and that they die out (extinction), and the time that it takes to reach fixation or extinction. In this talk I will give an overview over results bounding the fixation and extinction probabilities, as well as results analysing the convergence time. Jiří Černý Macroscopic phase transition for the vacant set of random walk on a large torus It was conjectured that the vacant set of random walk on (some) finite graphs exhibits a phase transition similar to the ErdösRényi random graph. Up to now, this conjecture was proved only on the graphs that are "locally treelike". The most interesting graph, the large ddimensional discrete torus, remains open. In the talk, I will present the recent work with A. Teixeira (IMPA) proving the phase transition for certain specific observables in this case, and explain the softlocal times coupling technique that is the main ingredient of the proof. Moez Draief Viral Processes by Random Walks on Random Regular Graphs We study the SIR epidemic model with
infections carried by \(k\) particles making independent random walks on a
random regular graph. We give an edgeweighted graph reduction of the dynamics
of the process that allows us to apply standard results of ErdosRenyi random
graphs on the particle set. In particular, we show how the parameters of the
model produce the following phase transitions: In the subcritical regime, \(O(ln
k)\) particles are infected. In the supercritical regime, for a constant \(c \in
(0,1)\) determined by the parameters of the model, \(c.\) \(k\) get infected
with probability \(c\), and \(O(ln k)\) get infected with probability \((1c)\).
Finally, there is a regime in which all \(k\) particles are infected.
Furthermore, the edge weights give information about when a particle becomes
infected. We demonstrate how this can be exploited to determine the completion
time of the process by applying a results on ﬁrstpassage percolation. Martin Dyer Random regular graphs and peertopeer networks Random regular graphs have been proposed as a topology for peertopeer networks. A useful property for such networks is to be “selforganising”, in the sense that they retain their properties as clients join and leave the network. We will discuss Markov chain approaches to the construction and maintenance of these networks. Robert Elsaesser Randomized broadcasting  general results and recent advances Information dissemination is a fundamental problem in parallel and distributed computing. In its simplest variant, known as the broadcasting problem, a message has to be spread among all nodes of a graph. A prominent communication protocol for this problem is the socalled random phone call model (Karp et al., FOCS 2000). In each step, every node opens a communication channel to a randomly chosen neighbor, which can then be used for bidirectional communication. It is known that in a complete graph, the broadcasting problem can be solved in time O(log n) by using O(log log n) message transmissions per node. Unfortunately, this performance cannot be achieved in sparse graphs even with best expansion and connectivity properties. We will present some simple modifications of the standard model, which allow us to significantly improve on the efficiency of randomized broadcasting in a number of sparse graphs. Dick Epema Exploiting random walks in decentralized reputation systems In the BitTorrentbased peertopeer filesharing system Tribler developed at
Delft University of Technology, we have designed and implemented the BarterCast
reputation system, which is based the contributions of the peers. Using a
gossiping protocol, every peer builds its own local view of the system, which is
a directed graph with peers as nodes and data transfers as edges. Peers compute
the reputations of other peers by means of random walks in their local views
that are either unbiased, or biased with local or global properties. In this
talk the whole reputation mechanism will be explained and performance results of Nikolaos Fountoulakis Bootstrap percolation and the geometry of complex networks We consider a geometric framework for
complex networks that was proposed recently by Krioukov et al. Alan Frieze Random Walks on Random Graphs: Covertime; Vacant Sets and Vacant Nets We survey some results on the title
topics. Catherine Greenhill The switch Markov chain for sampling irregular graphs The problem of efficiently sampling from a set of (undirected) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the sampling. The switch chain is known to be rapidly mixing for regular degree sequences. We prove that the switch chain is rapidly mixing for any degree sequence with minimum degree at least 1 and with maximum degree \(d_{\max}\) which satisfies \(3\leq d_{\max}\leq \frac{1}{4}\sqrt{M}\), where \(M\) is the sum of the degrees. The mixing time bound obtained is only an order of \(n\) larger than that established in the regular case, where \(n\) is the number of vertices. Karen Gunderson Bootstrap percolation with recovery on the grid Bootstrap processes are a type of cellular automaton in which vertices of a graph are in one of two states: ‘healthy’ or ‘infected’ and from an initial configuration of states, healthy vertices become infected by local rules. While the usual bootstrap processes are monotone in the sets of infected vertices, in this talk, I shall describe a model in which infected vertices can return to a healthy state. I will present a process on a square grid in which healthy vertices with at least two infected neighbours become infected and simultaneously any infected vertices with zero infected neighbours become healthy. These updates are repeated and the central question is whether all vertices eventually become infected and stay infected, in which case percolation is said to occur. I will give some background and describe sharp thresholds for the critical probability for percolation when vertices are initially infected independently at random. Johannes Lengler Bootstrap Percolation with Inhibition The Peach Company has a new fantastic innovation (SmartSocks) and releases it to the market. To get a start, the company gives SmartSocks to a small random set of people. A third of the people love it, while two thirds hate it. Of course, the people who love it recommend to buy it to everyone they meet (they meet their friends at random time intervals), while those who hate it warn everyone they see. In general, people are not so quick to buy new stuff, but if they get at least two more recommendations than warnings then they buy it. Then they also either love or hate SmartSocks (1/3 vs. 2/3) and start with recommendations or warnings. How many people will eventually buy Smart Socks? This seems like a ludicrous question as the answer must clearly depend on the structure of the social network. But in fact, it does not. Rather, under some mild assumptions exactly 25%+o(1) of all people will eventually buy SmartSocks. Nelly Litvak Ranking algorithms on directed configuration networks We study the distribution of a family
of rankings, which includes Google's PageRank, on a directed configuration model
(DCM). Our main result is that the rank of a randomly chosen node in the graph
converges in distribution to a finite random variable that can be written as a
linear combination of i.i.d. copies of the endogenous solution to a stochastic
fixed point equation. For the first time in the literature, this result
establishes a limiting behavior for a complete PageRank distribution. This
provides a very accurate approximation for the PageRank distribution on the DCM
but also on a complete English Wikipedia graph. The essence of the proof is in
coupling of the DCM with a specially constructed tree. To obtain the main
result, we show that the ranking in the graph converges with any given precision
before the coupling breaks. Ryuhei Mori Analysis of the peeling algorithm on random hypergraph When we try to solve a system of
linear equations, we can consider a simple iterative algorithm in which an
equation including only one variable is chosen at each step, and the variable is
fixed to the value satisfying the equation. The dynamics of this algorithm is
captured by the peeling algorithm. Analyses of the peeling algorithm on random
hypergraphs are required for many problems, e.g., the decoding threshold of
lowdensity parity check codes, the inverting threshold of Goldreich's
pseudorandom generator, the load threshold of cuckoo hashing, etc. On the other
hand, the peeling algorithm gives natural generalization of the concepts of
connectivity and connected components for hypergraphs. In this work, we deal
with random hypergraphs including polynomially many hyperedges, and derive the
tight threshold for the succeeding of the peeling algorithm. For the analysis,
Wormald's method of differential equations, which is commonly used for analyses
of the peeling algorithm, cannot be used due to the polynomially many
hyperedges.A new method called the evolution of the generating function is used
in this work. Tobias Muller A hyperbolic model for complex networks In this talk, I will discuss a model for complex
networks that was introduced recently by Krioukov et al. In this model, N points
are chosen randomly on the hyperbolic plane and any two of them are joined by an
edge if they are within a certain hyperbolic distance. The model turns out to
have characteristics usually associated with complex networks. In particular, it
exhibits a powerlaw degree sequence, small distances and clustering. I will
present some results on the component structure of the graph model, and on the
probability that it is connected. James Norris Robust nonconcentration for hitting times of Markov chains Uniform estimates for a class of
random processes are of value when the underlying model is random or in some way
unknown. I will describe some upper bounds for `surprise probabilities' for
finite statespace Markov chains. By a `surprise probability' we mean the
probability that, after t steps, the chain arrives at a given state y for the
first time, or simply at some state never seen before. These estimates depend on
the size of the statespace and on qualitative properties of the chain, such as
reversibility, or being the random walk on a simple graph, but are otherwise
uniform. Alexandre Stauffer Random walks on dynamic graphs given by dynamical percolation We study the behavior of random walk on
dynamical percolation. In this model, the edges of a graph \( G\) are either
open or closed, and refresh their status at rate \(\mu\). At the same time a
random walker moves on \(G\) at rate 1 but only along edges which are open. The
regime of interest here is when \(\mu\) goes to zero as the number of vertices
of \(G\) goes to infinity, since this creates longrange dependencies on the
model. When G is the \(d\)dimensional torus of side length \(n\), we prove that
in the subcritical regime, the mixing times is of order \(n^2/\mu\). We also
obtain results concerning mean squared displacement and hitting times. He Sun Heat kernel in graphs Heat kernel is one of the most
fundamental concepts in physics and mathematics. In physics, heat kernel is a
fundamental solution of the heat equation and, through heat kernel, the
Laplacian is associated with the rate of dissipation of heat. In spectral
geometry, many of the most basic techniques involve heat kernels. In finite
Markov chain theory, heat kernel corresponds to continuoustime random walks and
becomes one of the most powerful techniques in estimating mixing time, etc. Andrew Wade Hypercycles in sparse random
hypergraphs
PRACTICAL INFORMATION ● VenueEurandom, Mathematics and Computer Science Dept, TU Eindhoven, Den Dolech 2, 5612 AZ EINDHOVEN, The Netherlands
Eurandom is located on the campus of
Eindhoven University of
Technology, in the
Metaforum building
(4th floor) (about
the building). The university is
located at 10 minutes walking distance from Eindhoven main railway station (take
the exit north side and walk towards the tall building on the right with the
sign TU/e).
● RegistrationRegistration is free, but compulsory for speakers and participants. Please follow the link: REGISTRATION PAGE
● AccommodationFor invited participants, we will take care of accommodation. Other attendees will have to make their own arrangements. We have a preferred hotel, which can be booked at special rates. Please email Patty Koorn for instructions on how to make use of this special offer. For other hotels around the university, please see: Hotels (please note: prices listed are "best available"). More hotel options can be found on the webpages of the Tourist Information Eindhoven, Postbus 7, 5600 AA Eindhoven.
● TravelFor those arriving by plane, there is a convenient direct train connection between Amsterdam Schiphol airport and Eindhoven. This trip will take about one and a half hour. For more detailed information, please consult the NS travel information pages or see Eurandom web page location. Many low cost carriers also fly to Eindhoven Airport. There is a bus connection to the Eindhoven central railway station from the airport. (Bus route number 401) For details on departure times consult http://www.9292ov.nl The University can be reached easily by car from the highways leading to Eindhoven (for details, see our route descriptions or consult our map with highway connections.
● Conference facilities : Conference room, Metaforum Building MF11&12The meetingroom is equipped with a data projector, an overhead projector, a projection screen and a blackboard. Please note that speakers and participants making an oral presentation are kindly requested to bring their own laptop or their presentation on a memory stick.
● Conference SecretariatUpon arrival, participants should register with the workshop officer, and collect their name badges. The workshop officer will be present for the duration of the conference, taking care of the administrative aspects and the daytoday running of the conference: registration, issuing certificates and receipts, etc.
● CancellationShould you need to cancel your participation, please contact Patty Koorn, the Workshop Officer. There is no registration fee, but should you need to cancel your participation after January 2, 2014, we will be obliged to charge a noshow fee of 30 euro.
● ContactMrs. Patty Koorn, Workshop Officer, Eurandom/TU Eindhoven, koorn@eurandom.tue.nl SPONSORSThe organisers acknowledge the financial support/sponsorship of:
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