We refer to this tendency as the temporal rich club phenomenon, characterized by a coefficient defined as the maximal value of the density of links between nodes with a minimal required degree, which remain stable for a certain duration. Here, we propose a measure to quantitatively investigate the tendency of well-connected nodes to form simultaneous and stable structures in a temporal network. In temporal networks, the simultaneity of connections is crucial for temporally stable structures to arise. To this end, uncovering the structures involving the prominent nodes in a network is an effective approach. Identifying the hidden organizational principles and relevant structures of complex networks is fundamental to understand their properties. Below four spatial dimensions, an unconventional dynamical universality class can emerge as thermal fluctuations destroy hydrodynamic behaviour. Fractons are particles that can only move in tandem, which substantially affects their thermalization. Our theory provides a controlled example of how kinematic constraints lead to a rich landscape of dynamical universality classes in high-dimensional models. We numerically simulate many-body classical dynamics in one- and two-dimensional models with dipole and momentum conservation, and find evidence for the predicted breakdown of hydrodynamics. In one spatial dimension, our construction is reminiscent of the well-established renormalization group flow of the stochastic NavierвЂ“Stokes equations however, the fixed point we find possesses subdiffusive scaling rather than the superdiffusive scaling of the KardarвЂ“ParisiвЂ“Zhang universality class. This hydrodynamic effective theory is unstable below four spatial dimensions: dipole-conserving fluids at rest are unstable to fluctuations, which drive the system to a dynamical universality class with qualitatively distinct features from conventional fluids. Here we present the nonlinear fluctuating hydrodynamics of models with simultaneous charge/mass, dipole/centre of mass and momentum conservation. Although the NavierвЂ“Stokes equations can describe classical liquids and gases, quantum fluids of ultracold atoms or quarkвЂ“gluon plasma, they cannot yet describe the phases of matter where particle motion is kinematically constrained. Hydrodynamics is a universal effective theory that describes the thermalization of chaotic many-body systems, and depends only on the symmetries of the underlying theory.

Nature physics