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Register a new userPhase transitions and stability of dynamical processes on hypergraphs
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Hypergraphs naturally represent higher-order interactions, which persistently appear from social interactions to neural networks and other natural systems. Although their importance is well recognized, a theoretical framework to describe general dynamical processes on hypergraphs is not available yet. In this paper, we bridge this gap and derive expressions for the stability of dynamical systems defined on an arbitrary hypergraph. The framework allows us to reveal that, near the fixed point, the relevant structure is the graph-projection of the hypergraph and that it is possible to identify the role of each structural order for a given process. We also analytically solve two dynamics of general interest, namely, social contagion and diffusion processes, and show that the stability conditions can be decoupled in structural and dynamical components. Our results show that in social contagion processes, only pairwise interactions play a role in the stability of the absorbing state, while for the diffusion dynamics, the order of the interactions plays a differential role. Ours is the first attempt to provide a general framework for further exploration of dynamical processes on hypergraphs.
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Our understanding of the dynamics of complex networked systems has increased significantly in the last two decades. However, most of our knowledge is built upon assuming pairwise relations among the systems components. This is often an oversimplification, for instance, in social interactions that occur frequently within groups. To overcome this limitation, here we study the dynamics of social contagion on hypergraphs. We develop an analytical framework and provide numerical results for arbitrary hypergraphs, which we also support with Monte Carlo simulations. Our analyses show that the model has a vast parameter space, with first and second-order transitions, bi-stability, and hysteresis. Phenomenologically, we also extend the concept of latent heat to social contexts, which might help understanding oscillatory social behaviors. Our work unfolds the research line of higher-order models and the analytical treatment of hypergraphs, posing new questions and paving the way for modeling dynamical processes on these networks.
We study in this paper the structure of solutions in the random hypergraph coloring problem and the phase transitions they undergo when the density of constraints is varied. Hypergraph coloring is a constraint satisfaction problem where each constraint includes $K$ variables that must be assigned one out of $q$ colors in such a way that there are no monochromatic constraints, i.e. there are at least two distinct colors in the set of variables belonging to every constraint. This problem generalizes naturally coloring of random graphs ($K=2$) and bicoloring of random hypergraphs ($q=2$), both of which were extensively studied in past works. The study of random hypergraph coloring gives us access to a case where both the size $q$ of the domain of the variables and the arity $K$ of the constraints can be varied at will. Our work provides explicit values and predictions for a number of phase transitions that were discovered in other constraint satisfaction problems but never evaluated before in hypergraph coloring. Among other cases we revisit the hypergraph bicoloring problem ($q=2$) where we find that for $K=3$ and $K=4$ the colorability threshold is not given by the one-step-replica-symmetry-breaking analysis as the latter is unstable towards more levels of replica symmetry breaking. We also unveil and discuss the coexistence of two different 1RSB solutions in the case of $q=2$, $K ge 4$. Finally we present asymptotic expansions for the density of constraints at which various phase transitions occur, in the limit where $q$ and/or $K$ diverge.
Higher order interactions are increasingly recognised as a fundamental aspect of complex systems ranging from the brain to social contact networks. Hypergraph as well as simplicial complexes capture the higher-order interactions of complex systems and allow to investigate the relation between their higher-order structure and their function. Here we establish a general framework for assessing hypergraph robustness and we characterize the critical properties of simple and higher-order percolation processes. This general framework builds on the formulation of the random multiplex hypergraph ensemble where each layer is characterized by hyperedges of given cardinality. We reveal the relation between higher-order percolation processes in random multiplex hypergraphs, interdependent percolation of multiplex networks and K-core percolation. The structural correlations of the random multiplex hypergraphs are shown to have a significant effect on their percolation properties. The wide range of critical behaviors observed for higher-order percolation processes on multiplex hypergraphs elucidates the mechanisms responsible for the emergence of discontinuous transition and uncovers interesting critical properties which can be applied to the study of epidemic spreading and contagion processes on higher-order networks.
Phase transitions have recently been formulated in the time domain of quantum many-body systems, a phenomenon dubbed dynamical quantum phase transitions (DQPTs), whose phenomenology is often divided in two types. One refers to distinct phases according to long-time averaged order parameters, while the other is focused on the non-analytical behavior emerging in the rate function of the Loschmidt echo. Here we show that such DQPTs can be found in systems with few degrees of freedom, i.e. they can take place without resorting to the traditional thermodynamic limit. We illustrate this by showing the existence of the two types of DQPTs in a quantum Rabi model -- a system involving a spin-$frac{1}{2}$ and a bosonic mode. The dynamical criticality appears in the limit of an infinitely large ratio of the spin frequency with respect to the bosonic one. We determine its dynamical phase diagram and study the long-time averaged order parameters, whose semiclassical approximation yields a jump at the transition point. We find the critical times at which the rate function becomes non-analytical, showing its associated critical exponent as well as the corrections introduced by a finite frequency ratio. Our results open the door for the study of DQPTs without the need to scale up the number of components, thus allowing for their investigation in well controllable systems.
The quadratic contact process (QCP) is a natural extension of the well studied linear contact process where infected (1) individuals infect susceptible (0) neighbors at rate $lambda$ and infected individuals recover ($1 longrightarrow 0$) at rate 1. In the QCP, a combination of two 1s is required to effect a $0 longrightarrow 1$ change. We extend the study of the QCP, which so far has been limited to lattices, to complex networks. comment{as a model for the change in a population through sexual reproduction and death.} We define t
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