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As a promising computational paradigm, occurrence of critical states in artificial and biological neural networks has attracted wide-spread attention. An often-made explicit or implicit assumption is that one single critical state is…
The critical behavior of the XY model on small-world network is investigated by means of dynamic Monte Carlo simulations. We use the short-time relaxation scheme, i.e., the critical behavior is studied from the nonequilibrium relaxation to…
Continuous phase transitions are studied in a two dimensional nonequilibrium model with an infinite number of absorbing configurations. Spreading from a localized source is characterized by nonuniversal critical exponents, which vary…
Scale-free dynamics in physical and biological systems can arise from a variety of causes. Here, we explore a branching process which leads to such dynamics. We find conditions for the appearance of power laws and study quantitatively what…
In this paper we present rigorous results on the critical behavior of the Activated Random Walk model. We conjecture that on a general class of graphs, including $\mathbb{Z}^d$, and under general initial conditions, the system at the…
In active matter, the lack of momentum conservation makes non-reciprocal interactions the rule rather than the exception. They lead to a rich set of emerging behaviors that are hard to account for and to predict starting from the…
We study the small-world network model, which mimics the transition between regular-lattice and random-lattice behavior in social networks of increasing size. We contend that the model displays a normal continuous phase transition with a…
We analyze the entropic equation of state for a many-particle interacting system in a scale-free network. The analysis is performed in terms of scaling functions which are of fundamental interest in the theory of critical phenomena and have…
We consider critical phenomena on heterogeneous small-world networks having a scale-free character but also arbitrary short-loops. After deriving the self-consistent equation for the order parameter and the critical surface, we prove that…
Condensation phenomena in particle systems typically occur as one of two distinct types: either as a spontaneous symmetry breaking in a homogeneous system, in which particle interactions enforce condensation in a randomly located site, or…
As a function of connectivity, spring networks exhibit a critical transition between floppy and rigid phases at an isostatic threshold. For connectivity below this threshold, fiber networks were recently shown theoretically to exhibit a…
Non-reciprocal systems have been shown to sustain time-dependent patterns, most prominently travelling waves. The transition into these time-dependent states generally breaks time-translational invariance, representing a clear deviation…
We report on parallel observations in two seemingly unrelated areas of dynamical network research. The one is the so-called small world phenomenon and/or the observation of scale freeness in certain types of large (empirical) networks and…
Spatially extended chaotic systems with power-law decaying interactions are considered. Two coupled replicas of such systems synchronize to a common spatio-temporal chaotic state above a certain coupling strength. The synchronization…
In clean and weakly disordered systems, topological and trivial phases having a finite bulk energy gap can transit to each other via a quantum critical point. In presence of strong disorder, both the nature of the phases and the associated…
We demonstrate a new type of non-Hermitian phase transition in open systems far from thermal equilibrium, which takes place in coupled systems interacting with reservoirs at different temperatures. The frequency of the maximum in the…
Scale-free networks are ubiquitous in social, biological and technological networked systems. Dynamic Scale-free networks and their synchronizations are important to understand and predict the behavior of social, biological and…
Transition points mark qualitative changes in the macroscopic properties of large complex systems. Explosive transitions, exhibiting properties of both continuous and discontinuous phase transitions, have recently been uncovered in network…
The critical behavior of Ising model on a one-dimensional network, which has long-range connections at distances $l>1$ with the probability $\Theta(l)\sim l^{-m}$, is studied by using Monte Carlo simulations. Through studying the Ising…
We study the collective behavior of an Ising system on a small-world network with the interaction $J(r) \propto r^{-\alpha}$, where $r$ represents the Euclidean distance between two nodes. In the case of $\alpha = 0$ corresponding to the…