Frank den Hollander received his PhD at the University of Leiden in 1985. From 1985 to 2005 he worked at the universities of Delft, Utrecht, Nijmegen and Eindhoven. He is currently professor of probability theory at the University of Leiden. His research covers probability theory, statistical physics, ergodic theory, population genetics and complex networks. His main focus has been on interacting particle systems, phase transitions and disordered media. He was visiting professor in Bonn, Erlangen, Gottingen, Heidelberg, Toronto and Vancouver. Frank was elected to the Royal Netherlands Academy of Sciences in 2005. In 2016 he became Knight in the Order of the Dutch Lion. In 2018 he received a Humboldt Research Award. He has served on national and international advisory boards, and has co-organised 55 national and international workshops. He is the author of 170 research papers and 3 monographs. He has supervised 27 PhD students and 34 postdocs. Frank has received multiple grants from the Netherlands Organisation for Scientific Research, as well as an Advanced Grant from the European Research Council. He is also one of the Principle Investigators of the Dutch Gravitation program NETWORKS.
Networks with Constraints
Complex systems, like social structures, economic trades, genetic populations or
wildlife interactions, can be analysed with the help of network theory. Large networks
are so complex that they can only be described in terms of probability distributions,
also called ‘statistical ensembles”. The details of the network remain unspecified and it is assumed that the network is in one of many possible configurations compatible with known local constraints, like the number of links in the network or the degrees of the nodes. There are two approaches for incorporating local constraints: they can be either ‘hard’ (= satisfied by every realisation of the network) or ‘soft’ (=satisfied on average in the network) These two approaches are referred to as micro-canonical’
and `canonical’, respectively.
For many years it was assumed that for the analysis of large complex systems it
does not matter which approach is being taken: both types of local constraints,
hard or soft, should lead to the same global behaviour, because fluctuations
should average out. However, this assumption has turned out to be false. If the
local constraints are either many or are in some sense competing with each other,
then it actually does matter which statistical ensemble is chosen.
This so-called ‘breaking of ensemble equivalence’ has important practical implications. For instance, in a social network we do not know for every person in the network how many connections he or she has with other people in the network. So, we need to make an educated guess. In the micro-canonical ensemble, we would pick a fixed number of connections for each person. But if we make a wrong guess for one person, then this error will lead to a correlated error for another person. In the canonical ensemble, however, we automatically allow for such uncertainties, and a wrong guess does not set off a chain of errors.
In this talk I will describe a mathematical framework for handling such situations, based on so-called “exponential families of probability distributions’. I will
illustrate this framework by discussing several examples, and by showing how the
choice of local constraints may affect the global behaviour of the network. The
goal of the talk is to provide an introduction to the proper way of thinking about
networks with constraints.