Maximum entropy network models have become a major tool in the analysis of complex systems, including ecological ones. In these models, entropy is a fundamental concept related to ideas borrowed from statistical mechanics and information theory. In particular, network entropy relates both to the configurations the system can take and the uncertainty in the probability distribution describing those configurations. As for the latter one, the key idea is that the probability distribution needs be chosen such to maximize the entropy and so ensure maximal randomness, apart from the imposed constraints. Here, I will mostly focus on the configurational aspect, and whether the metric of entropy itself could measure aspects of the complexity of the network that may help understand its structure and dynamics. In the case of the microcanonical ensemble, for binary configuration models with hard constraints on the degree sequence, one obtains the uniform distribution and a simple interpretation of the entropy, that is the count of the number of allowed configurations (all equally likely). In most applications, however, we have no access to that number, neither are hard constraints reasonable. The soft constraints case is much more reasonable for ecological datasets, and in principle all configurations are possible but not equally probable, with the observed one assumed to be the average one, and the averaged one also assumed to be the most probable one. This allows a new point of view: the system configurations can now be seen as fluctuations around the most typical configuration, the average one. As a concept, this vision suits ecological system at equilibrium very well and, in principle, it allows measuring other properties that can be predicted through the constraints, including their statistical fluctuations. In the seminar, I will push this idea to the limit to propose experimental research lines on systems such as food webs and mutualistic networks around concepts such as stability vs. entropy, fluctuations vs. induced variance (perturbations), and realistic vs. null network models.
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