MATRIX-MFO Tandem Workshop: Stochastic Reinforcement Processes and Graphs

Jan de Gier

  •  6 Mar 2023 - 11 Mar 2023
     8:00 am - 5:00 pm

Mark Holmes (The University of Melbourne)
Viktor Kleptsyn (Université de Rennes)

Markus Heydenreich (Universität München)
Cécile Mailler (University of Bath)

Program Description:
This workshop brings together many of the leading international researchers in the area of reinforcement processes and graphs.

Polya-type urn models are random processes where balls are repeatedly sampled from an urn, and additional balls are added depending on the colour of the sampled ball. Since their introduction in 1931, generalisations of Polya urn models have spurred a rich variety of mathematical research activity. They are basic building blocks of competition-type probabilistic models in the fields of economics, biology and neuroscience.
A single urn is often insufficient to capture the complexity inherent in real-world applications, and consequently systems of interacting urns have gained popularity. In the field of neuroscience, when a neuron fires, only synapses that are connected to this neuron can be chosen to transmit the signal. Hence, Polya models with graph-based interactions are a natural starting point for addressing one of the mechanisms of neuroplasticity: synapses that have been identified as useful in the past are more likely to be chosen in the future.

In recent decades various random walk models involving reinforcement have become central objects of study in the probability literature. This includes (linearly)-reinforced random walks (linearly-RRW), as introduced by Coppersmith and Diaconis, and once-RRW as introduced by Davis (this was introduced as a simpler model, but in many ways this is harder to study). Other variants have included strongly RRW, non-backtracking RRW, and RRW with finite or decaying memory. Often such models are well-understood on certain graphs with special properties such as trees or complete graphs. Connections with other fields of mathematics, such as random Schrodinger operators and stochastic dynamical systems have been successfully exploited but there
remain many important and elegant open problems, as well as some (apparently) embarrassingly simple ones.

Program Structure: 

The workshop will be focused around one joint session Monday-Thursday: at 9-11 MFO time, which is 19-21 MATRIX time. During each of these sessions between 1 and 2 participants (chosen by the organisers) will organise an open problem session where they will:
(i) introduce an appropriate stochastic model, and
(ii) present known results about the model, and
(iii) specify one or more important open problems related to that model, and
(iv) initiate discussion between the participants.
Topics for these open problems sessions could include the ants process, WARMs, the extremal linkage model and preferential attachment, self-reinforced random walks.
Each two-hour session will consist of two 45 minute talks covering (i)-(iii) above followed by 15 minutes for (iv). In addition there will be one recorded 1-hour talk per day at each institution (morning at MATRIX and afternoon at MFO). The talk recorded on day n in the morning at MATRIX will be played on day n in the afternoon at MFO; the talk recorded on day n in the afternoon at MFO will be played on day n+1 in the morning at MATRIX.

MFO webpage:

Participants List:
Mark Holmes (University of Melbourne)
Victor Kleptsyn (CNRS, University of Rennes 1)
Omer Angel (University of British Columbia)
Andrea Collevecchio (Monash University)
Daniel Kious (University of Bath)
Tuan Minh Nguyen (Monash University)
Nathan Ross (University of Melbourne)
Kais Hamza (Monash University)
Antar Bandyopadyay (Indian Statistical Institute, Delhi Centre)
Giacomo Aletti (U. Milan)
Andrea Ghiglietti (Università degli Studi Milano-Bicocca)
Tim Banova (University of Melbourne)

Registration is closed



MATRIX Wine and Cheese Afternoon 7 March 2023
On the first Tuesday of each program, MATRIX provides a pre-dinner wine and cheese afternoon. Produce is locally-sourced to showcase delicacies from the region.