Two golden times in two-step contagion models: A nonlinear map approach.

The two-step contagion model is a simple toy model for understanding pandemic outbreaks that occur in the real world. The model takes into account that a susceptible person either gets immediately infected or weakened when getting into contact with an infectious one. As the number of weakened people increases, they eventually can become infected in a short time period and a pandemic outbreak occurs. The time required to reach such a pandemic outbreak allows for intervention and is often called golden time. Understanding the size-dependence of the golden time is useful for controlling pandemic outbreak. Using an approach based on a nonlinear mapping, here we find that there exist two types of golden times in the two-step contagion model, which scale as O(N^{1/3}) and O(N^{ζ}) with the system size N on Erdos-Rényi networks, where the measured ζ is slightly larger than 1/4. They are distinguished by the initial number of infected nodes, o(N) and O(N), respectively. While the exponent 1/3 of the N-dependence of the golden time is universal even in other models showing discontinuous transitions induced by cascading dynamics, the measured ζ exponents are all close to 1/4 but show model-dependence. It remains open whether or not ζ reduces to 1/4 in the asymptotically large-N limit. Our method can be applied to several models showing a hybrid percolation transition and gives insight into the origin of the two golden times.

Universal mechanism for hybrid percolation transitions.

Hybrid percolation transitions (HPTs) induced by cascading processes have been observed in diverse complex systems such as k-core percolation, breakdown on interdependent networks and cooperative epidemic spreading models. Here we present the microscopic universal mechanism underlying those HPTs. We show that the discontinuity in the order parameter results from two steps: a durable critical branching (CB) and an explosive, supercritical (SC) process, the latter resulting from large loops inevitably present in finite size samples. In a random network of N nodes at the transition the CB process persists for O(N 1/3) time and the remaining nodes become vulnerable, which are then activated in the short SC process. This crossover mechanism and scaling behavior are universal for different HPT systems. Our result implies that the crossover time O(N 1/3) is a golden time, during which one needs to take actions to control and prevent the formation of a macroscopic cascade, e.g., a pandemic outbreak.

Epidemic spreading on evolving signed networks.

Most studies of disease spreading consider the underlying social network as obtained without the contagion, though epidemic influences people's willingness to contact others: A "friendly" contact may be turned to "unfriendly" to avoid infection. We study the susceptible-infected disease-spreading model on signed networks, in which each edge is associated with a positive or negative sign representing the friendly or unfriendly relation between its end nodes. In a signed network, according to Heider's theory, edge signs evolve such that finally a state of structural balance is achieved, corresponding to no frustration in physics terms. However, the danger of infection affects the evolution of its edge signs. To describe the coupled problem of the sign evolution and disease spreading, we generalize the notion of structural balance by taking into account the state of the nodes. We introduce an energy function and carry out Monte Carlo simulations on complete networks to test the energy landscape, where we find local minima corresponding to the so-called jammed states. We study the effect of the ratio of initial friendly to unfriendly connections on the propagation of disease. The steady state can be balanced or a jammed state such that a coexistence occurs between susceptible and infected nodes in the system.

Hybrid phase transition into an absorbing state: Percolation and avalanches.

Interdependent networks are more fragile under random attacks than simplex networks, because interlayer dependencies lead to cascading failures and finally to a sudden collapse. This is a hybrid phase transition (HPT), meaning that at the transition point the order parameter has a jump but there are also critical phenomena related to it. Here we study these phenomena on the Erdos-Rényi and the two-dimensional interdependent networks and show that the hybrid percolation transition exhibits two kinds of critical behaviors: divergence of the fluctuations of the order parameter and power-law size distribution of finite avalanches at a transition point. At the transition point global or "infinite" avalanches occur, while the finite ones have a power law size distribution; thus the avalanche statistics also has the nature of a HPT. The exponent ß_{m} of the order parameter is 1/2 under general conditions, while the value of the exponent γ_{m} characterizing the fluctuations of the order parameter depends on the system. The critical behavior of the finite avalanches can be described by another set of exponents, ß_{a} and γ_{a}. These two critical behaviors are coupled by a scaling law: 1-ß_{m}=γ_{a}.

Small but slow world: how network topology and burstiness slow down spreading.

While communication networks show the small-world property of short paths, the spreading dynamics in them turns out slow. Here, the time evolution of information propagation is followed through communication networks by using empirical data on contact sequences and the susceptible-infected model. Introducing null models where event sequences are appropriately shuffled, we are able to distinguish between the contributions of different impeding effects. The slowing down of spreading is found to be caused mainly by weight-topology correlations and the bursty activity patterns of individuals.

Structure and tie strengths in mobile communication networks.

Electronic databases, from phone to e-mails logs, currently provide detailed records of human communication patterns, offering novel avenues to map and explore the structure of social and communication networks. Here we examine the communication patterns of millions of mobile phone users, allowing us to simultaneously study the local and the global structure of a society-wide communication network. We observe a coupling between interaction strengths and the network's local structure, with the counterintuitive consequence that social networks are robust to the removal of the strong ties but fall apart after a phase transition if the weak ties are removed. We show that this coupling significantly slows the diffusion process, resulting in dynamic trapping of information in communities and find that, when it comes to information diffusion, weak and strong ties are both simultaneously ineffective.

Critical packing in granular shear bands.

In a realistic three-dimensional setup, we simulate the slow deformation of idealized granular media composed of spheres undergoing an axisymmetric triaxial shear test. We follow the self-organization of the spontaneous strain localization process leading to a shear band and demonstrate the existence of a critical packing density inside this failure zone. The asymptotic criticality arising from the dynamic equilibrium of dilation and compaction is found to be restricted to the shear band, while the density outside of it keeps the memory of the initial packing. The critical density of the shear band depends on friction (and grain geometry) and in the limit of infinite friction it defines a specific packing state, namely the dynamic random loose packing.

Shear zones in granular materials: optimization in a self-organized random potential.

We introduce a model to describe the wide shear zones observed in modified Couette cell experiments with granular material. The model is a generalization of the recently proposed approach based on a variational principle. The instantaneous shear band is identified with the surface that minimizes the dissipation in a random potential that is biased by the local velocity difference and pressure. The apparent shear zone is the ensemble average of the instantaneous shear bands. The numerical simulation of this model matches excellently with experiments and has measurable predictions.

Morphologies of three-dimensional shear bands in granular media.

We present numerical results on spontaneous symmetry breaking strain localization in axisymmetric triaxial shear tests of granular materials. We simulated shear band formation using the three-dimensional distinct element method with spherical particles. We demonstrate that the local shear intensity, the angular velocity of the grains, the coordination number, and the local void ratio are correlated and any of them can be used to identify shear bands; however, the latter two are less sensitive. The calculated shear band morphologies are in good agreement with those found experimentally. We show that boundary conditions play an important role. We discuss the formation mechanism of shear bands in the light of our observations and compare the results with experiments. At large strains, with enforced symmetry, we found strain hardening.

Piling and avalanches of magnetized particles.

We performed computer simulations based on a two-dimensional distinct element method to study granular systems of magnetized spherical particles. We measured the angle of repose and the surface roughness of particle piles, and we studied the effect of magnetization on avalanching. We report linear dependence of both angle of repose and surface roughness on the ratio f of the magnetic dipole interaction and the gravitational force (interparticle force ratio). There is a difference in avalanche formation at small and at large interparticle force ratios. The transition is at f(c) approximately 7. For f < f(c) small vertical chains follow each other at short times (granular regime), while for f > f(c) the avalanches are typically formed by one single large particle-cluster (correlated regime). The transition is not sharp. We give plausible estimates for f(c) based on stability criteria.

Shear band formation in granular media as a variational problem.

Strain in sheared dense granular material is often localized in a narrow region called the shear band. Recent experiments in a modified Couette cell provided localized shear flow in the bulk away from the confining walls. The nontrivial shape of the shear band was measured as the function of the cell geometry. First, we present a geometric argument for narrow shear bands that connects the function of their surface position with the shape in the bulk. Assuming a simple dissipation mechanism, we show that the principle of minimum dissipation of energy provides a good description of the shape function. Furthermore, we discuss the possibility and behavior of shear bands that are detached from the free surface and are entirely covered in the bulk.

Dynamics of market correlations: taxonomy and portfolio analysis.

The time dependence of the recently introduced minimum spanning tree description of correlations between stocks, called the "asset tree" has been studied in order to reflect the financial market taxonomy. The nodes of the tree are identified with stocks and the distance between them is a unique function of the corresponding element of the correlation matrix. By using the concept of a central vertex, chosen as the most strongly connected node of the tree, an important characteristic is defined by the mean occupation layer. During crashes, due to the strong global correlation in the market, the tree shrinks topologically, and this is shown by a low value of the mean occupation layer. The tree seems to have a scale-free structure where the scaling exponent of the degree distribution is different for "business as usual" and "crash" periods. The basic structure of the tree topology is very robust with respect to time. We also point out that the diversification aspect of portfolio optimization results in the fact that the assets of the classic Markowitz portfolio are always located on the outer leaves of the tree. Technical aspects such as the window size dependence of the investigated quantities are also discussed.

Two-dimensional array of magnetic particles: the role of an interaction cutoff.

Based on theoretical results and simulations, in two-dimensional arrangements of a dense dipolar particle system, there are two relevant local dipole arrangements: (1) a ferromagnetic state with dipoles organized in a triangular lattice and (2) an antiferromagnetic state with dipoles organized in a square lattice. In order to accelerate simulation algorithms, we search for the possibility of cutting off the interaction potential. Simulations on a dipolar two-line system lead to the observation that the ferromagnetic state is much more sensitive to the interaction cutoff R than the corresponding antiferromagnetic state. For R approximately > 8 (measured in particle diameters) there is no substantial change in the energetical balance of the ferromagnetic and antiferromagnetic state and the ferromagnetic state slightly dominates over the antiferromagnetic state, while the situation is changed rapidly for lower interaction cutoff values, leading to the disappearance of the ferromagnetic ground state. We studied the effect of bending ferromagnetic and antiferromagnetic two-line systems and observed that the cutoff has a major impact on the energetical balance of the ferromagnetic and the antiferromagnetic state for R approximately < 4. Based on our results we argue that R approximately 5 is a reasonable choice for dipole-dipole interaction cutoff in two-dimensional dipolar hard sphere systems, if one is interested in local ordering.

Time-dependent cross-correlations between different stock returns: a directed network of influence.

We study the time-dependent cross-correlations of stock returns, i.e., we measure the correlation as the function of the time shift between pairs of stock return time series using tick-by-tick data. We find a weak but significant effect showing that in many cases the maximum correlation appears at nonzero time shift, indicating directions of influence between the companies. Due to the weakness of this effect and the shortness of the characteristic time (of the order of a few minutes), our findings are compatible with market efficiency. The interaction of companies defines a directed network of influence.

Elastic behavior in contact dynamics of rigid particles.

The systematic errors due to the practical implementation of the contact dynamics method for simulation of dense granular media are examined. It is shown that, using the usual iterative solver to simulate a chain of rigid particles, effective elasticity and sound propagation with a finite velocity occur. The characteristics of these phenomena are investigated analytically and numerically in order to assess the limits of applicability of this simulation method and to compare it with soft particle molecular dynamics.

Scaling of random spreading in small world networks.

In this study we have carried out computer simulations of random walks on Watts-Strogatz-type small world networks and measured the mean number of visited sites and the return probabilities. These quantities were found to obey scaling behavior with intuitively reasoned exponents as long as the probability p of having a long range bond was sufficiently low.

Preferential growth: exact solution of the time-dependent distributions.

We consider a preferential growth model where particles are added one by one to the system consisting of clusters of particles. A new particle can either form a new cluster (with probability q) or join an already existing cluster with a probability proportional to the size thereof. We calculate exactly the probability Pi(k,t) that the size of the ith cluster at time t is k. We analyze the asymptotics, the scaling properties of the size distribution and of the mean size, as well as the relation of our system to recent network models.

Stochastic boundary conditions in the deterministic Nagel-Schreckenberg traffic model.

We consider open systems where cars move according to the deterministic Nagel-Schreckenberg rules [K. Nagel and M. Schreckenberg, J. Phys. I 2, 2221 (1992)] and with maximum velocity v(max)>1, which is an extension of the asymmetric exclusion process (ASEP). It turns out that the behavior of the system is dominated by two features: (a) the competition between the left and the right boundary, (b) the development of so-called "buffers" due to the hindrance that an injected car feels from the front car at the beginning of the system. As a consequence, there is a first-order phase transition between the free flow and the congested phase accompanied by the collapse of the buffers, and the phase diagram essentially differs from that for v(max)=1 (ASEP).

Nondeterministic Nagel-Schreckenberg traffic model with open boundary conditions.

We study the phases of the Nagel-Schreckenberg traffic model with open boundary conditions as a function of the randomization probabilities p>0 and the maximum velocity v(max)>1. Due to the existence of "buffer sites" which enhance the free-flow region, the behavior is much richer than that of the related, parallel updated asymmetric exclusion process [(ASEP), v(max)=1]. Such sites exist for v(max)> or =3 and pp(c) an additional maximum current phase separated by second-order transitions occurs like for the ASEP. The density profile decays in the maximum current phase algebraically with an exponent gamma approximately 2 / 3 for all v(max)> or =2 indicating that these models belong to another universality class than the ASEP where gamma=1 / 2.

Effect of anisotropy on the instability of crack propagation

Dynamics of fracture is investigated in an anisotropic two-dimensional Born-Maxwell model by numerical simulations. From previous studies it is known that the isotropic model shows crack branching and velocity oscillations of the propagating main crack above a critical velocity, similarly with experimental findings in some brittle materials. Here we present studies in which anisotropy has been introduced to the model system. Anisotropy is found to have significant effects on crack propagation and on the pattern it forms. In the case of symmetric anisotropy (relative to the crack direction) we found changes in velocity oscillation and side branching properties. In the case of asymmetric anisotropy two kinds of periodicities occur and strong anisotropy causes different branch patterns to form at two sides of the main crack. In addition, the role of disorder through distributed spring constants has been studied for both types of anisotropy. Finally a simple exactly solvable model for investigating the initial stages of crack branching has been developed and analyzed.