Fiber Bundle model with Highly Disordered Breaking Thresholds

We present a study of the fiber bundle model using equal load sharing dynamics where the breaking thresholds of the fibers are drawn randomly from a power law distribution of the form $p(b)sim b^{-1}$ in the range $10^{-beta}$ to $10^{beta}$. Tuning the value of $beta$ continuously over a wide range, the critical behavior of the fiber bundle has been studied both analytically as well as numerically. Our results are: (i) The critical load $sigma_c(beta,N)$ for the bundle of size $N$ approaches its asymptotic value $sigma_c(beta)$ as $sigma_c(beta,N) = sigma_c(beta)+AN^{-1/ u(beta)}$ where $sigma_c(beta)$ has been obtained analytically as $sigma_c(beta) = 10^beta/(2beta eln10)$ for $beta geq beta_u = 1/(2ln10)$, and for $beta<beta_u$ the weakest fiber failure leads to the catastrophic breakdown of the entire fiber bundle, similar to brittle materials, leading to $sigma_c(beta) = 10^{-beta}$; (ii) the fraction of broken fibers right before the complete breakdown of the bundle has the form $1-1/(2beta ln10)$; (iii) the distribution $D(Delta)$ of the avalanches of size $Delta$ follows a power law $D(Delta)sim Delta^{-xi}$ with $xi = 5/2$ for $Delta gg Delta_c(beta)$ and $xi = 3/2$ for $Delta ll Delta_c(beta)$, where the crossover avalanche size $Delta_c(beta) = 2/(1-e10^{-2beta})^2$.

Scaling forms for Relaxation Times of the Fiber Bundle model

Using extensive numerical analysis of the Fiber Bundle Model with Equal Load Sharing dynamics we studied the finite-size scaling forms of the relaxation times against the deviations of applied load per fiber from the critical point. Our most crucial result is we have not found any $ln (N)$ dependence of the average relaxation time $langle T(sigma,N) rangle$ in the precritical state. The other results are: (i) The critical load $sigma_c(N)$ for the bundle of size $N$ approaches its asymptotic value $sigma_c(infty)$ as $sigma_c(N) = sigma_c(infty) + AN^{-1/ u}$. (ii) Right at the critical point the average relaxation time $langle T(sigma_c(N),N) rangle$ scales with the bundle size $N$ as: $langle T(sigma_c(N),N) rangle sim N^{eta}$ and this behavior remains valid within a small window of size $|Delta sigma| sim N^{-zeta}$ around the critical point. (iii) When $1/N < |Delta sigma| < 100N^{-zeta}$ the finite-size scaling takes the form: $langle T(sigma,N) rangle / N^{eta} sim {cal G}[{sigma_c(N)-sigma}N^{zeta}]$ so that in the limit of $N to infty$ one has $langle T(sigma) rangle sim (sigma - sigma_c)^{-tau}$. The high precision of our numerical estimates led us to verify that $ u = 3/2$, conjecture that $eta = 1/3$, $zeta = 2/3$ and therefore $tau = 1/2$.

The International Trade Network

Bilateral trade relationships in the international level between pairs of countries in the world give rise to the notion of the International Trade Network (ITN). This network has attracted the attention of network researchers as it serves as an excellent example of the weighted networks, the link weight being defined as a measure of the volume of trade between two countries. In this paper we analyzed the international trade data for 53 years and studied in detail the variations of different network related quantities associated with the ITN. Our observation is that the ITN has also a scale invariant structure like many other real-world networks.