Dependence on the thermodynamic state of self-diffusion of pseudo-hard-sphere and Lennard-Jones potentials.

Self-diffusion D in a system of particles that interact with a pseudo-hard-sphere or a Lennard-Jones potential is analyzed. Coupling with a solvent is represented by a Langevin thermostat, characterized by the damping time t_{d}. The hypotheses that D=D_{0}φ is proposed, where D_{0} is the small concentration diffusivity and φ is a thermodynamic function that represents the effects of interactions as concentration is increased. Molecular dynamics simulations show that different values of the noise intensity modify D_{0}, but do not have an effect on φ. This result is consistent with the assumption that φ is a thermodynamic function since the thermodynamic state is not altered by the presence of damping and noise.

Diffusion on a lattice: Transition rates, interactions, and memory effects.

We analyze diffusion of particles on a two-dimensional square lattice. Each lattice site contains an arbitrary number of particles. Interactions affect particles only in the same site, and are macroscopically represented by the excess chemical potential. In a recent work, a general expression for transition rates between neighboring cells as functions of the excess chemical potential was derived. With transition rates, the mean-field tracer diffusivity, D^{MF}, is immediately obtained. The tracer diffusivity, D=D^{MF}f, contains the correlation factor f, representing memory effects. An analysis of the joint probability of having given numbers of particles at different sites when a force is applied to a tagged particle allows an approximate expression for f to be derived. The expression is applied to soft core interaction (different values for the maximum number of particles in a site are considered) and extended hard core.

Application of the Widom insertion formula to transition rates in a lattice.

We consider diffusion of particles on a lattice in the so-called dynamical mean-field regime (memory effects are neglected). Interactions are local, that is, only among particles at the same lattice site. It is shown that a statistical mechanics analysis that combines detailed balance and Widom's insertion formula allows for the derivation of an expression for transition rates in terms of the excess chemical potential. The rates reproduce the known dependence of self-diffusivity as the inverse of the thermodynamic factor. Soft-core interactions and general forms of the excess chemical potential (linear, quadratic, and cubic with the density) are considered.

Dynamic vaccination in partially overlapped multiplex network.

In this work we propose and investigate a strategy of vaccination which we call "dynamic vaccination." In our model, susceptible people become aware that one or more of their contacts are infected and thereby get vaccinated with probability ω, before having physical contact with any infected patient. Then the nonvaccinated individuals will be infected with probability ß. We apply the strategy to the susceptible-infected-recovered epidemic model in a multiplex network composed by two networks, where a fraction q of the nodes acts in both networks. We map this model of dynamic vaccination into bond percolation model and use the generating functions framework to predict theoretically the behavior of the relevant magnitudes of the system at the steady state. We find a perfect agreement between the solutions of the theoretical equations and the results of stochastic simulations. In addition, we find an interesting phase diagram in the plane ß-ω, which is composed of an epidemic and a nonepidemic phase, separated by a critical threshold line ß_{c}, which depends on q. As q decreases, ß_{c} increases, i.e., as the overlap decreases, the system is more disconnected, and therefore more virulent diseases are needed to spread epidemics. Surprisingly, we find that, for all values of q, a region in the diagram where the vaccination is so efficient that, regardless of the virulence of the disease, it never becomes an epidemic. We compare our strategy with random immunization and find that, using the same amount of vaccines for both scenarios, we obtain that the spread of disease is much lower in the case of dynamic vaccination when compared to random immunization. Furthermore, we also compare our strategy with targeted immunization and we find that, depending on ω, dynamic vaccination will perform significantly better and in some cases will stop the disease before it becomes an epidemic.

Cascading Failures in Interdependent Networks with Multiple Supply-Demand Links and Functionality Thresholds.

Various social, financial, biological and technological systems can be modeled by interdependent networks. It has been assumed that in order to remain functional, nodes in one network must receive the support from nodes belonging to different networks. So far these models have been limited to the case in which the failure propagates across networks only if the nodes lose all their supply nodes. In this paper we develop a more realistic model for two interdependent networks in which each node has its own supply threshold, i.e., they need the support of a minimum number of supply nodes to remain functional. In addition, we analyze different conditions of internal node failure due to disconnection from nodes within its own network. We show that several local internal failure conditions lead to similar nontrivial results. When there are no internal failures the model is equivalent to a bipartite system, which can be useful to model a financial market. We explore the rich behaviors of these models that include discontinuous and continuous phase transitions. Using the generating functions formalism, we analytically solve all the models in the limit of infinitely large networks and find an excellent agreement with the stochastic simulations.

Cascading failures in interdependent networks with finite functional components.

We present a cascading failure model of two interdependent networks in which functional nodes belong to components of size greater than or equal to s. We find theoretically and via simulation that in complex networks with random dependency links the transition is first order for s≥3 and continuous for s=2. We also study interdependent lattices with a distance constraint r in the dependency links and find that increasing r moves the system from a regime without a phase transition to one with a second-order transition. As r continues to increase, the system collapses in a first-order transition. Each regime is associated with a different structure of domain formation of functional nodes.

Recovery of Interdependent Networks.

Recent network research has focused on the cascading failures in a system of interdependent networks and the necessary preconditions for system collapse. An important question that has not been addressed is how to repair a failing system before it suffers total breakdown. Here we introduce a recovery strategy for nodes and develop an analytic and numerical framework for studying the concurrent failure and recovery of a system of interdependent networks based on an efficient and practically reasonable strategy. Our strategy consists of repairing a fraction of failed nodes, with probability of recovery γ, that are neighbors of the largest connected component of each constituent network. We find that, for a given initial failure of a fraction 1 - p of nodes, there is a critical probability of recovery above which the cascade is halted and the system fully restores to its initial state and below which the system abruptly collapses. As a consequence we find in the plane γ - p of the phase diagram three distinct phases. A phase in which the system never collapses without being restored, another phase in which the recovery strategy avoids the breakdown, and a phase in which even the repairing process cannot prevent system collapse.

Publisher's Note: Cascading failures in interdependent networks with finite functional components [Phys. Rev. E 94, 042304 (2016)].

This corrects the article DOI: 10.1103/PhysRevE.94.042304.