Impact of embedding on predictability of failure-recovery dynamics in networks

Failure, damage spread and recovery crucially underlie many spatially embedded networked systems ranging from transportation structures to the human body. Here we study the interplay between spontaneous damage, induced failure and recovery in both embedded and non-embedded networks. In our model the network's components follow three realistic processes that capture these features: (i) spontaneous failure of a component independent of the neighborhood (internal failure), (ii) failure induced by failed neighboring nodes (external failure) and (iii) spontaneous recovery of a component.We identify a metastable domain in the global network phase diagram spanned by the model's control parameters where dramatic hysteresis effects and random switching between two coexisting states are observed. The loss of predictability due to these effects depend on the characteristic link length of the embedded system. For the Euclidean lattice in particular, hysteresis and switching only occur in an extremely narrow region of the parameter space compared to random networks. We develop a unifying theory which links the dynamics of our model to contact processes. Our unifying framework may help to better understand predictability and controllability in spatially embedded and random networks where spontaneous recovery of components can mitigate spontaneous failure and damage spread in the global network.Comment: 22 pages, 20 figure

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

Comment on "Scaling of atmosphere and ocean temperature correlations in observations and climate models"

In a recent letter [K. Fraedrich and R. Blender, Phys. Rev. Lett. 90, 108501 (2003)], Fraedrich and Blender studied the scaling of atmosphere and ocean temperature. They analyzed the fluctuation functions F(s) ~ s^alpha of monthly temperature records (mostly from grid data) by using the detrended fluctuation analysis (DFA2) and claim that the scaling exponent alpha over the inner continents is equal to 0.5, being characteristic of uncorrelated random sequences. Here we show that this statement is (i) not supported by their own analysis and (ii) disagrees with the analysis of the daily observational data from which the grid monthly data have been derived. We conclude that also for the inner continents, the exponent is between 0.6 and 0.7, similar as for the coastline-stations.Comment: 1 page with 2 figure

The robustness of interdependent clustered networks

It was recently found that cascading failures can cause the abrupt breakdown of a system of interdependent networks. Using the percolation method developed for single clustered networks by Newman [Phys. Rev. Lett. {\bf 103}, 058701 (2009)], we develop an analytical method for studying how clustering within the networks of a system of interdependent networks affects the system's robustness. We find that clustering significantly increases the vulnerability of the system, which is represented by the increased value of the percolation threshold $p_c$ in interdependent networks.Comment: 6 pages, 6 figure

Financial factor influence on scaling and memory of trading volume in stock market

We study the daily trading volume volatility of 17,197 stocks in the U.S. stock markets during the period 1989--2008 and analyze the time return intervals $\tau$ between volume volatilities above a given threshold q. For different thresholds q, the probability density function P_q(\tau) scales with mean interval as P_q(\tau)=^{-1}f(\tau/) and the tails of the scaling function can be well approximated by a power-law f(x)~x^{-\gamma}. We also study the relation between the form of the distribution function P_q(\tau) and several financial factors: stock lifetime, market capitalization, volume, and trading value. We find a systematic tendency of P_q(\tau) associated with these factors, suggesting a multi-scaling feature in the volume return intervals. We analyze the conditional probability P_q(\tau|\tau_0) for $\tau$ following a certain interval \tau_0, and find that P_q(\tau|\tau_0) depends on \tau_0 such that immediately following a short/long return interval a second short/long return interval tends to occur. We also find indications that there is a long-term correlation in the daily volume volatility. We compare our results to those found earlier for price volatility.Comment: 17 pages, 6 figure

Diffusion and spectral dimension on Eden tree

We calculate the eigenspectrum of random walks on the Eden tree in two and three dimensions. From this, we calculate the spectral dimension $d_s$ and the walk dimension $d_w$ and test the scaling relation $d_s = 2d_f/d_w$ ($=2d/d_w$ for an Eden tree). Finite-size induced crossovers are observed, whereby the system crosses over from a short-time regime where this relation is violated (particularly in two dimensions) to a long-time regime where the behavior appears to be complicated and dependent on dimension even qualitatively.Comment: 11 pages, Plain TeX with J-Phys.sty style, HLRZ 93/9

Absence of kinetic effects in reaction-diffusion processes in scale-free networks

We show that the chemical reactions of the model systems of A+A->0 and A+B->0 when performed on scale-free networks exhibit drastically different behavior as compared to the same reactions in normal spaces. The exponents characterizing the density evolution as a function of time are considerably higher than 1, implying that both reactions occur at a much faster rate. This is due to the fact that the discerning effects of the generation of a depletion zone (A+A) and the segregation of the reactants (A+B) do not occur at all as in normal spaces. Instead we observe the formation of clusters of A (A+A reaction) and of mixed A and B (A+B reaction) around the hubs of the network. Only at the limit of very sparse networks is the usual behavior recovered.Comment: 4 pages, 4 figures, to be published in Physical Review Letter

Diffusion and Trapping on a one-dimensional lattice

The properties of a particle diffusing on a one-dimensional lattice where at each site a random barrier and a random trap act simultaneously on the particle are investigated by numerical and analytical techniques. The combined effect of disorder and traps yields a decreasing survival probability with broad distribution (log-normal). Exact enumerations, effective-medium approximation and spectral analysis are employed. This one-dimensional model shows rather rich behaviours which were previously believed to exist only in higher dimensionality. The possibility of a trapping-dominated super universal class is suggested.Comment: 20 pages, Revtex 3.0, 13 figures in compressed format using uufiles command, to appear in Phys. Rev. E, for an hard copy or problems e-mail to: [email protected]