#### This workshop is organized by Leonhard Horstmeyer (Medical University of Vienna), Christian Kühn (Technical University of Munich) and CSH President Stefan Thurner.

#### Abstract:

Adaptive co-evolving dynamic networks play a key role in ecological, epidemiological, social or financial systems. To understand the efficiency, resilience and systemic risks of such networks one needs a much deeper mathematical understanding of their critical behavior than is currently available. What remains largely unknown about these systems is their phase-structure, their critical transitions, if they can be classified into universality classes and whether it is possible to derive network-based early-warning signals. In particular in this workshop we want to discuss to what extent the network topology captures information about the critical transition and whether this allows us to explore the limits of predicting regime shifts and catastrophic events.

#### Workshop Philosophy:

For this workshop we envisage a mindset that does not rest in the past, a mindset that does not look for greedy increments, but rather one that is prone to leaps and to visions and to the big problems or challenges ahead of us. The idea of the workshop is not for individuals to talk only about their work, get inspiration only for their work and get back to their work, but rather for the participants to come together at eye level, discuss perspectives that are beyond themselves and to join forces to tackle them. For this reason we focus on discussions rather than talks and for this reason we don’t want to surpass a critical number of participants.

*Workshop Summary*

*by Leonhard Horstmeyer (Nov. 10, 2017)*

*On the 2nd and 3rd of November 2017 the Complexity Science Hub in Vienna hosted a workshop on adaptive co-evolving networks and catastrophes. These types of networks play a key role in ecological, epidemiological, social or financial systems. Yet we are still far away from an indepth understanding of their critical behaviour and a comprehensive precursor theory.*

*The objective of this workshop has been to map out the dimensions along which these models differ and, consequently, along which they may be classified. The working hypothesis has been that one may divide them essentially into two classes of models: Those models whose critical behaviour can be described at the level of aggregated macroscopic variables and those models whose critical behaviour can only be described by means of microscopic information at the network level.*

*It has been pointed out that this is precisely the distinction between models that are amenable to a mean field description and those that aren’t. Thilo Gross made the case with the fragmentation transition in the adaptive voter model, which is not obtainable by a moment closure ( i.e. higher mean field) approximation.*

*We have encountered two prototypical adaptive network models that are in this sense diametrically opposed. We recall them here respectively and summarise comments, observations and arguments put forth during the workshop.*

*Sanjay Jain introduced and discussed the model, that is referred to as the Jain and Krishna model (JK). A network of catalytic interactions evolves in time according to a simple dynamical rule: The unfittest node is replaced by a new node that forms new catalytic ties at random. The fitness is determined by the node’s relative abundance after an underlying fast catalytic reaction dynamics has reached equilibrium. Formally the model is a Markov chain on the space of di-graphs. It exhibits growth phases, seemingly stable phases and crashes. To date it has eluded any mean field description. Scaling laws have been observed numerically for the number of paths and the duration of the stable phase, btw with similar exponents. These however are exponential laws and not power laws. Nothing in the model appears to behave in a power law fashion. The two parameters of the model are the size of the network and the average number of links added during node replacement. Crashes appear irrespective of these parameters.*

*On the other side we have been exposed to the adaptive SIS model (adpt-SIS) in the talks of Thilo Gross and Peter Simon in different contexts and flavours. This is a continuous time binary-state stochastic network model where nodes can be infected or susceptible and whose dynamics is driven by recovery, infection and rewiring rates. The model exhibits a disease-free phase, an epidemic phase, a bistable phase and an oscillatory phase. The various critical transitions are characterised by power laws, which can in many instances be obtained from mean field considerations. A slow variation of the exogenous parameters can carry the dynamics across a phase transition and trigger a collapse. Relaxation times and volatilities rise close to the transition. They may serve as early warning signals and can be obtained from mean field predictions.*

*Thus the major differences in the critical behaviour of these two models, the adpt-SIS and the JK model, has been summarised as follows: Crashes in the adpt-SIS occur when an equilibrium looses stability as the exogenous parameters are varied adiabatically. All relevant critical functions are power laws and the respective critical quantities can be obtained from mean field methods. In contrast to this, crashes in the JK model appear irrespective of exogenous parameters. The type of stability that is seemingly present in the JK model is different from the one in the adpt-SIS model. Most quantities follow exponential laws and are prima facie not derivable from mean field approximations.*

*Regarding the working hypothesis it has been criticized that one may not know whether a suitable mean field description exists or not. A good choice is often problem dependent. Instead of looking at a strict dichotomy it has then been suggested to find a convex space of models spanned by some extremal ones, so that every model finds its place somewhere in this space.*

*In this direction we raised the following question: How does one know whether the driving mechanism behind a collapse is for instance of the Jain-Krishna type, the tipping point type or the saddle point type? So the convex space of models might be spanned by the mechanisms that lead to a crash. During the workshop we have encountered at least three or four such mechanisms: A crash in the adaptive SIS model can be caused by carrying the dynamics across a phase transition so that certain equilibria loose stability. However at finite system sizes crashes can already occur long before the transition by stochastic exploration beyond the basin of attraction. A crash in the JK model occurs when rivaling auto-catalytic systems build up outside of the dominant auto-catalytic system. Finally we discussed crashes that occur in homoclinics or heteroclinics: A saddle point in a high dimensional space may appear stable, because the unstable direction has such a small probability of being hit and if it is hit a stochastic perturbation can bring it back to effective stability. During the workshop it has been suggested that this type of mechanism leads to the alternation of cooperative and non-cooperative phases in the model presented by Matteo Cavaliere.*

*Apart from classifying crashes we have also looked into control mechanisms of networks through rewiring. Peter Simon introduced control variables to steer the adaptive SIS model into a disease-free state, whilst Thilo Gross discussed how the SIS model self-organises to criticality through rewiring.*

*In contrast to the adaptive SIS model we have heard talks about the adaptive voter model in various flavours. Jonathan Donges looked at the effect of zealots on the fragmentation transition whilst Marc Wiederman discussed an adaptive voter model coupled to an underlying population dynamics that models the harvesting behaviour of villages with respect to their stock of fish.*

*Sandeep Ameta presented experimental work on catalytic systems of enzymes in which he managed to find experimentally the catalytic networks for many combinations of reacting enzymes. Despite the simplicity of the JK model, some features seem to be observable in this experiment. We have discussed ways to change the experimental setup, accommodating for an in- and outflux of species.*

*Thilo Gross also presented in his talk the translation of a stochastic network dynamics, or more precisely a heterogeneous moment closure derived from it, into a PDE. Christian Kühn suggested a classification of the network dynamics induced by the classification of their respective PDEs. Maybe PDE methods will play a role in classifying the critical behaviour of adaptive co-evolving networks.*

*Finally, Christian Kühn suggested to compile a collection of benchmark adaptive network models and discuss their main features. This could be in the form of a review article and the idea was well received.*