TBA

Complex networks usually exhibit a rich architecture organized over multiple intertwined scales. Information pathways are expected to pervade these scales reflecting structural insights that are not manifest from analyses of the network topology. Moreover, small-world effects correlate with the different network hierarchies complicating identifying coexisting mesoscopic structures and functional cores. We present a communicability analysis of effective information pathways throughout complex networks based on information diffusion to shed further light on these issues. This leads us to formulate a new renormalization group scheme for heterogeneous networks. The Renormalization Group is the cornerstone of the modern theory of universality and phase transitions, a powerful tool to scrutinize symmetries and organizational scales in dynamical systems. However, its network counterpart is particularly challenging due to correlations between intertwined scales. The Laplacian RG picture for complex networks defines both the supernodes concept à la Kadanoff, and the equivalent momentum space procedure à la Wilson for graphs.

What do football passes and financial transactions have in common? Both are observable events in some real-world walk process that is happening over some network that is, however, not directly observable. In both cases, the basis for record-keeping is that these events move something tangible from one node to another. Here we explore process-driven approaches towards analyzing such data, with the goal of answering domain-specific research questions. First, we consider transaction data from a digital community currency recorded over 16 months. Because these are records of a real-world walk process, we know that the time-aggregated network is a flow network. Flow-based network analysis techniques let us concisely describe where and among whom this community currency was circulating. Second, we use a technique called trajectory extraction to transform football match event data into passing sequence data. This allows us to replicate classic results from sports science about possessions and uncover intriguing dynamics of play in five first-tier domestic leagues in Europe during the 2017-18 club season. Taken together, these two applied examples demonstrate the interpretability of process-driven approaches as opposed to, e.g., temporal network analysis, when the data are records of a real-world walk processes.

A conjecture due to Zilber predicts that the complex exponential field is quasiminimal: that is, that all subsets of the complex numbers that are definable in the language of rings expanded by a symbol for the complex exponential function are countable or cocountable.
Zilber showed that this conjecture would follow from Schanuel's Conjecture and an existential closedness type property asserting that certain systems of exponential-polynomial equations can be solved in the complex numbers; later on, Bays and Kirby were able to remove the dependence on Schanuel's Conjecture, shifting all the focus to the existence of solutions. In this talk, I will discuss recent work about the quasiminimality of a reduct of the complex exponential field, that is, the complex numbers expanded by multivalued power functions. This is joint work with Jonathan Kirby.