# Peter Grassberger

Complexity Science Group, University of Calgary, Calgary, Canada

Forschungszentrum Jülich, Germany

Peter Grassberger is a honorary director of the Complex Systems Research Group at the John-von-Neumann Institute, Forschungszentrum Jülich and holds an iCORE visiting research professorship at the University of Calgary. Grassberger made outstanding contributions to the fields of statistical and particle physics. He is most famous for his contributions to chaos theory, where he introduced the idea of correlation dimension, a means of measuring a type of fractal dimension of the strange attractor. With early work on particle phenomenology, his career-long scientific contributions include reaction-diffusion systems, cellular automata and lattice gases, fractals, Griffiths phases, self-organized criticality, and percolation. His current interests concern generalized percolation type transitions and the inference of causal connections in networks.

## The many faces of percolation

Although percolation theory was considered a mature subject several years ago, recent progress has changed this radically. While "ordinary" percolation (OP) is a second order phase transition that establishes long range connectivity on diluted regular lattices or random graphs, examples have now been found where this transition can range from infinite order to first order. The latter is of particular importance in social sciences, where first order percolation transitions show up as a consequence of synergistic effects, and I will point out analogies with the relationship between percolation and rough interfaces in physics. Another case where first order percolation transitions show up is interdependent networks, although first claims about this have to be substantially modified -- in some cases of interdependent networks the transition is second order but in a new universality class. A similar but even more unexpected result holds for so-called "Achleoptas processes" that were originally claimed to show first order transitions, but which actually show second order transitions with a completely new type of finite size scaling. Finally, I will present "agglomerative percolation" (AP), a model originally introduced to understand claims that network renormalization can demonstrate the fractality of some small world networks. Due to a spontaneously broken symmetry on bipartite graphs, AP leads e.g. to different scaling behaviors on square and triangular 2-d lattices, in flagrant violations of universality.