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Highlights and Prospects in the Mathematics of Complexity Science

Thursday 22 May, 15.00-17.30, D1.07 Zeeman Building

Organiser: Robert MacKay



14.30-15.00 Tea/Coffee and biscuits, Complexity common room
15.00-15.25 Samuel Johnson (DNV GL, London) Learning from food webs: The role of trophic coherence in complex systems1
15.30-15.55 Young-Ho Eom (Physique Théorique, Université de Toulouse III) Mathematical understanding of complex systems: complex network and data science approach2
16.00-16.25 Ahmet Sensoy (Borsa Istanbul, Turkey) Dynamic spanning trees in stock market networks: the case of Asia-Pacific3
16.30-16.55 Tiago Pereira (Imperial College London) Dynamics in heterogeneous networks: emergence at various scales4
17.00-17.25 Dario Bauso (Universita di Palermo) Robust mean-field games5
17.30-18.00 Drinks and nibbles, Complexity common room



1. The existence of large, complex ecosystems has posed a paradox ever since, over forty years ago, Robert May proved that randomly interconnected systems of dynamical elements should become more unstable with increasing size and link density. We have recently shown that "trophic coherence", a hitherto ignored network property, can account for most of the variation in stability observed in empirical food webs. Moreover, a simple model capturing this feature not only outperforms existing food-web models according to several measures, but predicts that stability can increase with size and link density. As well as suggesting a key to May's Paradox, this result is relevant for a variety of complex systems. For instance, ongoing research shows that directed networks can fall into either of two regimes depending on their trophic coherence: one with a diverging number of feedback cycles, or one in which there is a vanishing probability of cycles occurring at all. This might explain the existence of "qualitatively stable" genetic and metabolic networks. We also find that spreading phenomena such as epidemics or opinions depend crucially on trophic coherence. To conclude, I shall mention some potential applications of these results, such as predicting ecological tipping points and mitigating industrial risk.

2. Mathematical understanding of complex systems and complexity is crucial for modern science. In this talk, I will talk about my recent research on complex systems with complex network and data science approach and highlight generalised friendship paradox in complex network. In addition, I will talk about my research plan on understanding roles of node characteristics in structure and dynamics of complex networks.

3. We propose a new empirical procedure called Dynamic Spanning Trees (DST) to evaluate the time-varying stock market interconnections. It combines the concepts of minimal spainning tree from network theory and dynamic conditional correlation model from econometrics. The idea is applied to stock markets in the Asia-Pacific region. The findings and implications are discussed.

4. Recent results reveal that typical real-world networks have various levels of connectivity. These networks exhibit emergent behaviour at various levels. Striking examples are found in the brain, where synchronisation between highly connected neurons coordinate and shape the network development. These phenomena remain a major challenge. I will discuss a probabilistic dimension reduction principle to describe the network dynamics. I show that, at large levels of connectivity, the high-dimensional network dynamics can be reduced to a few macroscopic equations. The strategy is to describe ensembles of random networks, and the dynamics almost every initial state. This reduction provides the opportunity to explore the coherent properties at various network connectivity scales. This is a joint work with Sebastian van Strien and Jeroen Lamb.

5.Within the realm of mean field games under uncertainty, we study a population of players with individual states driven by a standard Brownian motion and a disturbance term. The contribution is three-fold: First, we establish a mean field system for such robust games. Second, we apply the methodology to an exhaustible resource production. Third, we discuss solution techniques based on state space extension and polynomial approximation.