Feigenbaum graphs: a complex network perspective of chaos

The recently formulated theory of horizontal visibility graphs transforms time series into graphs and allows the possibility of studying dynamical systems through the characterization of their associated networks. This method leads to a natural graph-theoretical description of nonlinear systems with qualities in the spirit of symbolic dynamics. We support our claim via the case study of the period-doubling and band-splitting attractor cascades that characterize unimodal maps. We provide a universal analytical description of this classic scenario in terms of the horizontal visibility graphs associated with the dynamics within the attractors, that we call Feigenbaum graphs, independent of map nonlinearity or other particulars. We derive exact results for their degree distribution and related quantities, recast them in the context of the renormalization group and find that its fixed points coincide with those of network entropy optimization. Furthermore, we show that the network entropy mimics the Lyapunov exponent of the map independently of its sign, hinting at a Pesin-like relation equally valid out of chaos.Comment: Published in PLoS ONE (Sep 2011

Setting expectations for educational innovations

This paper considers the problematic enactment of instructional innovations. We examine how different interpretations of “success” might be explained within a frame of reference that confronts the complexities of and uncovers the contingencies relating to educational policy implementation in schools. Based on the detailed description and comparison of three different educational innovations developed and implemented in the same educational context — Singapore — we show how the intricate and delicate interrelationships that exist within and across adopters, innovators and environments influence what might be reasonably expected and achieved from specific innovation initiatives. By doing so, we hope not only to test Cohen and Ball’s framework and conjectures but also lay the groundwork for future comparative work on innovation design and evaluation, moving the research agenda forward by critically examining reasonable expectations for educational innovation