Abstract: Classical rich-get-richer models have found much success in being able to broadly reproduce the statistics and dynamics of diverse real complex systems. These rich-get-richer models are based on classical urn models and unfold step-by-step in discrete time. Here, we consider a natural variation acting on a temporal continuum in the form of a partial differential equation (PDE). We first show that the continuum version of Herbert Simon's canonical preferential attachment model exhibits an identical size distribution. In relaxing Simon's assumption of a linear growth mechanism, we consider the case of an arbitrary growth kernel and find the general solution to the resultant PDE. We then extend the PDE to multiple spatial dimensions, again determining the general solution. Finally, we apply the model to size and wealth distributions of firms. We obtain power law scaling for both to be concordant with simulations as well as observational data.
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Bongard's work focuses on understanding the general nature of cognition, regardless of whether it is found in humans, animals or robots. This unique approach focuses on the role that morphology and evolution plays in cognition. Addressing these questions has taken him into the fields of biology, psychology, engineering and computer science.
Danforth is an applied mathematician interested in modeling a variety of physical, biological, and social phenomenon. He has applied principles of chaos theory to improve weather forecasts as a member of the Mathematics and Climate Research Network, and developed a real-time remote sensor of global happiness using messages from Twitter: the Hedonometer. Danforth co-runs the Computational Story Lab with Peter Dodds, and helps run UVM's reading group on complexity.
Laurent studies the interaction of structure and dynamics. His research involves network theory, statistical physics and nonlinear dynamics along with their applications in epidemiology, ecology, biology, and sociology. Recent projects include comparing complex networks of different nature, the coevolution of human behavior and infectious diseases, understanding the role of forest shape in determining stability of tropical forests, as well as the impact of echo chambers in political discussions.
Hines' work broadly focuses on finding ways to make electric energy more reliable, more affordable, with less environmental impact. Particular topics of interest include understanding the mechanisms by which small problems in the power grid become large blackouts, identifying and mitigating the stresses caused by large amounts of electric vehicle charging, and quantifying the impact of high penetrations of wind/solar on electricity systems.
Bagrow's interests include: Complex Networks (community detection, social modeling and human dynamics, statistical phenomena, graph similarity and isomorphism), Statistical Physics (non-equilibrium methods, phase transitions, percolation, interacting particle systems, spin glasses), and Optimization(glassy techniques such as simulated/quantum annealing, (non-gradient) minimization of noisy objective functions).