Multifractal Properties of the Random Resistor Network

We study the multifractal spectrum of the current in the two-dimensional random resistor network at the percolation threshold. We consider two ways of applying the voltage difference: (i) two parallel bars, and (ii) two points. Our numerical results suggest that in the infinite system limit, the probability distribution behaves for small current i as P(i) ~ 1/i. As a consequence, the moments of i of order q less than q_c=0 do not exist and all current of value below the most probable one have the fractal dimension of the backbone. The backbone can thus be described in terms of only (i) blobs of fractal dimension d_B and (ii) high current carrying bonds of fractal dimension going from $1/\nu$ to d_B.Comment: 4 pages, 6 figures; 1 reference added; to appear in Phys. Rev. E (Rapid Comm

The Emergence of El-Ni\~{n}o as an Autonomous Component in the Climate Network

We construct and analyze a climate network which represents the interdependent structure of the climate in different geographical zones and find that the network responds in a unique way to El-Ni\~{n}o events. Analyzing the dynamics of the climate network shows that when El-Ni\~{n}o events begin, the El-Ni\~{n}o basin partially loses its influence on its surroundings. After typically three months, this influence is restored while the basin loses almost all dependence on its surroundings and becomes \textit{autonomous}. The formation of an autonomous basin is the missing link to understand the seemingly contradicting phenomena of the afore--noticed weakening of the interdependencies in the climate network during El-Ni\~{n}o and the known impact of the anomalies inside the El-Ni\~{n}o basin on the global climate system.Comment: 5 pages,10 figure

Assortativity and leadership emergence from anti-preferential attachment in heterogeneous networks

Many real-world networks exhibit degree-assortativity, with nodes of similar degree more likely to link to one another. Particularly in social networks, the contribution to the total assortativity varies with degree, featuring a distinctive peak slightly past the average degree. The way traditional models imprint assortativity on top of pre-defined topologies is via degree-preserving link permutations, which however destroy the particular graph's hierarchical traits of clustering. Here, we propose the first generative model which creates heterogeneous networks with scale-free-like properties and tunable realistic assortativity. In our approach, two distinct populations of nodes are added to an initial network seed: one (the followers) that abides by usual preferential rules, and one (the potential leaders) connecting via anti-preferential attachments, i.e. selecting lower degree nodes for their initial links. The latter nodes come to develop a higher average degree, and convert eventually into the final hubs. Examining the evolution of links in Facebook, we present empirical validation for the connection between the initial anti-preferential attachment and long term high degree. Thus, our work sheds new light on the structure and evolution of social networks

Recurrence intervals between earthquakes strongly depend on history

We study the statistics of the recurrence times between earthquakes above a certain magnitude M$in California. We find that the distribution of the recurrence times strongly depends on the previous recurrence time$\tau_0$. As a consequence, the conditional mean recurrence time$\hat \tau(\tau_0)$between two events increases monotonically with$\tau_0$. For$\tau_0$well below the average recurrence time$\ov{\tau}, \hat\tau(\tau_0)$is smaller than$\ov{\tau}$, while for$\tau_0>\ov{\tau}$,$\hat\tau(\tau_0)$is greater than$\ov{\tau}$. Also the mean residual time until the next earthquake does not depend only on the elapsed time, but also strongly on$\tau_0$. The larger$\tau_0$ is, the larger is the mean residual time. The above features should be taken into account in any earthquake prognosis.Comment: 5 pages, 3 figures, submitted to Physica

Robustness of interdependent networks under targeted attack

When an initial failure of nodes occurs in interdependent networks, a cascade of failure between the networks occurs. Earlier studies focused on random initial failures. Here we study the robustness of interdependent networks under targeted attack on high or low degree nodes. We introduce a general technique and show that the {\it targeted-attack} problem in interdependent networks can be mapped to the {\it random-attack} problem in a transformed pair of interdependent networks. We find that when the highly connected nodes are protected and have lower probability to fail, in contrast to single scale free (SF) networks where the percolation threshold $p_c=0$, coupled SF networks are significantly more vulnerable with $p_c$ significantly larger than zero. The result implies that interdependent networks are difficult to defend by strategies such as protecting the high degree nodes that have been found useful to significantly improve robustness of single networks.Comment: 11 pages, 2 figure

Dynamics of Surface Roughening with Quenched Disorder

We study the dynamical exponent $z$ for the directed percolation depinning (DPD) class of models for surface roughening in the presence of quenched disorder. We argue that $z$ for $(d+1)$ dimensions is equal to the exponent $d_{\rm min}$ characterizing the shortest path between two sites in an isotropic percolation cluster in $d$ dimensions. To test the argument, we perform simulations and calculate $z$ for DPD, and $d_{\rm min}$ for percolation, from $d = 1$ to $d = 6$.Comment: RevTex manuscript 3 pages + 6 figures (obtained upon request via email [email protected]

A lattice Boltzmann model with random dynamical constraints

In this paper we introduce a modified lattice Boltzmann model (LBM) with the capability of mimicking a fluid system with dynamic heterogeneities. The physical system is modeled as a one-dimensional fluid, interacting with finite-lifetime moving obstacles. Fluid motion is described by a lattice Boltzmann equation and obstacles are randomly distributed semi-permeable barriers which constrain the motion of the fluid particles. After a lifetime delay, obstacles move to new random positions. It is found that the non-linearly coupled dynamics of the fluid and obstacles produces heterogeneous patterns in fluid density and non-exponential relaxation of two-time autocorrelation function.Comment: 10 pages, 9 figures, to be published in Eur. Phys. J.

On the Tomography of Networks and Multicast Trees

In this paper we model the tomography of scale free networks by studying the structure of layers around an arbitrary network node. We find, both analytically and empirically, that the distance distribution of all nodes from a specific network node consists of two regimes. The first is characterized by rapid growth, and the second decays exponentially. We also show that the nodes degree distribution at each layer is a power law with an exponential cut-off. We obtain similar results for the layers surrounding the root of multicast trees cut from such networks, as well as the Internet. All of our results were obtained both analytically and on empirical Interenet data

Generalized Shortest Path Kernel on Graphs

We consider the problem of classifying graphs using graph kernels. We define a new graph kernel, called the generalized shortest path kernel, based on the number and length of shortest paths between nodes. For our example classification problem, we consider the task of classifying random graphs from two well-known families, by the number of clusters they contain. We verify empirically that the generalized shortest path kernel outperforms the original shortest path kernel on a number of datasets. We give a theoretical analysis for explaining our experimental results. In particular, we estimate distributions of the expected feature vectors for the shortest path kernel and the generalized shortest path kernel, and we show some evidence explaining why our graph kernel outperforms the shortest path kernel for our graph classification problem.Comment: Short version presented at Discovery Science 2015 in Banf