Networks arise in wide variety of contexts across disciplines, from social networks in sociology and transport networks in engineering, to networks for disease spreading in epidemiology and metabolic networks in biology. In this essay we provide an introduction to the systematic study of networks. We discuss a few important network models, focusing particularly on features that are similar to t...

To be definite we shall consider the problem of critical fluctuations with the purpose of illustrating their relevance in the analysis of other difficult problems like field theory.

It was well known that strong enough initial configurations of matter field would result in gravitational collapse to black holes, whereas weak enough initial data become dispersed to infinity. However, what happens in the intermediate region was not clear until the discovery of the critical phenomena in gravitational collapse by Choptuik in 1993 [1]. He examined numerically the spacetime evolu...

We investigate the effect of self-propulsion on a mean-field order-disorder transition. Starting from a φ^{4} scalar field theory subject to an exponentially correlated noise, we exploit the unified colored-noise approximation to map the nonequilibrium active dynamics onto an effective equilibrium one. This allows us to follow the evolution of the second-order critical point as a function of th...

As first discovered by Choptuik, the black hole threshold in the space of initial data for general relativity shows both surprising structure and surprising simplicity. Universality, power-law scaling of the black hole mass, and scale echoing have given rise to the term "critical phenomena". They are explained by the existence of exact solutions which are attractors within the black hole thresh...

Entanglement, one of the most intriguing features of quantum theory and a main resource in quantum information science, is expected to play a crucial role also in the study of quantum phase transitions, where it is responsible for the appearance of long-range correlations. We investigate, through a microscopic calculation, the scaling properties of entanglement in spin chain systems, both near ...

The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, researchers have made important steps toward understanding the qualitatively new critical phenomena in complex networks. We review the results, concepts, an...

We investigate the gravitational collapse of a spherically symmetric, perfect fluid with equation of state P = (Γ − 1)ρ. We restrict attention to the ultrarelativistic (“kinetic-energy-dominated”, “scale-free”) limit where black hole formation is anticipated to turn on at infinitesimal black hole mass (Type II behavior). Critical solutions (those which sit at the threshold of black hole formati...

Traditionally, the participants of the “Ahrenshoop International Symposium on the Theory of Elementary Particles” form two nearly disjoint subsets, consisting of string theorists and lattice gauge theorists. So, for a plenary speaker the question arises: is it possible to give a talk which addresses both of these species? I don’t have an answer, but this contribution is meant as an attempt. Con...