Microscopic Intervention Yields Abrupt Transition in Interdependent Ferromagnetic Networks.

The study of interdependent networks has recently experienced a boost with the development of experimentally testable materials that physically realize their novel critical behaviors, calling for systematic studies that go beyond the percolation paradigm. Here we study the critical kinetics and phase transitions of a model of interdependent spatial ferromagnetic networks where dependency couplings between networks are realized by a thermal interaction having a tunable spatial range. We show how the critical phenomena and the phase diagram of this realistic model are highly affected by the range of thermal dissipation and how the latter influences the microscopic kinetics of the model. Furthermore, we show the existence of a new phase where localized microscopic interventions by heating or magnetic fields yield a macroscopic phase transition. Our results unveil rich phenomena and realistic protocols for controlling the macroscopic phases of interdependent materials by means of microscopic interventions.

Dynamics of cascades in spatial interdependent networks.

The dynamics of cascading failures in spatial interdependent networks significantly depends on the interaction range of dependency couplings between layers. In particular, for an increasing range of dependency couplings, different types of phase transition accompanied by various cascade kinetics can be observed, including mixed-order transition characterized by critical branching phenomena, first-order transition with nucleation cascades, and continuous second-order transition with weak cascades. We also describe the dynamics of cascades at the mutual mixed-order resistive transition in interdependent superconductors and show its similarity to that of percolation of interdependent abstract networks. Finally, we lay out our perspectives for the experimental observation of these phenomena, their phase diagrams, and the underlying kinetics, in the context of physical interdependent networks. Our studies of interdependent networks shed light on the possible mechanisms of three known types of phase transitions, second order, first order, and mixed order as well as predicting a novel fourth type where a microscopic intervention will yield a macroscopic phase transition.

Fractal Fluctuations at Mixed-Order Transitions in Interdependent Networks.

We study the critical features of the order parameter's fluctuations near the threshold of mixed-order phase transitions in randomly interdependent spatial networks. Remarkably, we find that although the structure of the order parameter is not scale invariant, its fluctuations are fractal up to a well-defined correlation length ξ^{'} that diverges when approaching the mixed-order transition threshold. We characterize the self-similar nature of these critical fluctuations through their effective fractal dimension d_{f}^{'}=3d/4, and correlation length exponent ν^{'}=2/d, where d is the dimension of the system. By analyzing percolation and magnetization, we demonstrate that d_{f}^{'} and ν^{'} are the same for both, i.e., independent of the symmetry of the process for any d of the underlying networks.

Critical Stretching of Mean-Field Regimes in Spatial Networks.

We study a spatial network model with exponentially distributed link lengths on an underlying grid of points, undergoing a structural crossover from a random, Erdos-Rényi graph, to a d-dimensional lattice at the characteristic interaction range ζ. We find that, whilst far from the percolation threshold the random part of the giant component scales linearly with ζ, close to criticality it extends in space until the universal length scale ζ^{6/(6-d)}, for d<6, before crossing over to the spatial one. We demonstrate the universal behavior of the spatiotemporal scales characterizing this critical stretching phenomenon of mean-field regimes in percolation and in dynamical processes on d=2 networks, and we discuss its general implications to real-world phenomena, such as neural activation, traffic flows or epidemic spreading.

Physical networks as network-of-networks.

Physical networks are made of nodes and links that are physical objects embedded in a geometric space. Understanding how the mutual volume exclusion between these elements affects the structure and function of physical networks calls for a suitable generalization of network theory. Here, we introduce a network-of-networks framework where we describe the shape of each extended physical node as a network embedded in space and these networks are bound together by physical links. Relying on this representation, we introduce a minimal model of network growth and we show for a general class of physical networks that volume exclusion induces heterogeneity in both node volume and degree, with the two becoming correlated. These emergent properties strongly affect the dynamics on physical networks: by calculating their Laplacian spectrum as a function of the coupling strength between the nodes we show that degree-volume correlations suppress the role of hubs as early spreaders in diffusive dynamics. We apply the network-of-networks framework to describe several real systems and find properties analog to the minimal model networks. The prevalence of these properties points towards general growth mechanisms that do not depend on the specifics of the systems.

Rhythmic synchronization and hybrid collective states of globally coupled oscillators.

Macroscopic rhythms are often signatures of healthy functioning in living organisms, but they are still poorly understood on their microscopic bases. Globally interacting oscillators with heterogeneous couplings are here considered. Thorough theoretical and numerical analyses indicate the presence of multiple phase transitions between different collective states, with regions of bi-stability. Novel coherent phases are unveiled, and evidence is given of the spontaneous emergence of macroscopic rhythms where oscillators' phases are always found to be self-organized as in Bellerophon states, i.e. in multiple clusters with quantized values of their average frequencies. Due to their rather unconditional appearance, the circumstance is paved that the Bellerophon states grasp the microscopic essentials behind collective rhythms in more general systems of interacting oscillators.

Spontaneous repulsion in the A+B→0 reaction on coupled networks.

We study the transient dynamics of an A+B→0 process on a pair of randomly coupled networks, where reactants are initially separated. We find that, for sufficiently small fractions q of cross couplings, the concentration of A (or B) particles decays linearly in a first stage and crosses over to a second linear decrease at a mixing time t_{x}. By numerical and analytical arguments, we show that for symmetric and homogeneous structures t_{x}∝(〈k〉/q)log(〈k〉/q) where 〈k〉 is the mean degree of both networks. Being this behavior is in marked contrast with a purely diffusive process, where the mixing time would go simply like 〈k〉/q, we identify the logarithmic slowing down in t_{x} to be the result of a spontaneous mechanism of repulsion between the reactants A and B due to the interactions taking place at the networks' interface. We show numerically how this spontaneous repulsion effect depends on the topology of the underlying networks.

Synchronization in networks with multiple interaction layers.

The structure of many real-world systems is best captured by networks consisting of several interaction layers. Understanding how a multilayered structure of connections affects the synchronization properties of dynamical systems evolving on top of it is a highly relevant endeavor in mathematics and physics and has potential applications in several socially relevant topics, such as power grid engineering and neural dynamics. We propose a general framework to assess the stability of the synchronized state in networks with multiple interaction layers, deriving a necessary condition that generalizes the master stability function approach. We validate our method by applying it to a network of Rössler oscillators with a double layer of interactions and show that highly rich phenomenology emerges from this. This includes cases where the stability of synchronization can be induced even if both layers would have individually induced unstable synchrony, an effect genuinely arising from the true multilayer structure of the interactions among the units in the network.

Synchronization and Bellerophon states in conformist and contrarian oscillators.

The study of synchronization in generalized Kuramoto models has witnessed an intense boost in the last decade. Several collective states were discovered, such as partially synchronized, chimera, π or traveling wave states. We here consider two populations of globally coupled conformist and contrarian oscillators (with different, randomly distributed frequencies), and explore the effects of a frequency-dependent distribution of the couplings on the collective behaviour of the system. By means of linear stability analysis and mean-field theory, a series of exact solutions is extracted describing the critical points for synchronization, as well as all the emerging stationary coherent states. In particular, a novel non-stationary state, here named as Bellerophon state, is identified which is essentially different from all other coherent states previously reported in the Literature. A robust verification of the rigorous predictions is supported by extensive numerical simulations.