We introduce and study a simple idealized model to describe breaking processes by analysing the current-carrying properties of a random network consisting of insulators and fuses. By increasing the value of the external voltage applied across the network, a sequence of fuses will « burn out » and change irreversible into insultating bonds. This process terminates when a conducting path no longer exists in the network. We monitor several basic quantities during this breaking process, such as the conductivity of the network and the value of the voltage needed to break the hottest fuse. Two new exponents describing the behaviour of these quantities near the percolation threshold are reported

We develop a new approach for studying the random resistor network by focusing on the distribution of voltage drops across each bond We introduce a simple model which provides a useful description of the percolating backbone, and which shows that the voltage distribution is log normal, with an infinite set of exponents required to describe the voltage moments. This latter prediction is verified by simulations of a resistor network on the square lattice.

Networks of living neurons exhibit an avalanche mode of activity, experimentally found in organotypic cultures. Here we present a model that is based on self-organized criticality and takes into account brain plasticity, which is able to reproduce the spectrum of electroencephalograms (EEG). The model consists of an electrical network with threshold firing and activity-dependent synapse strengths. The system exhibits an avalanche activity in a power-law distribution. The analysis of the power spectra of the electrical signal reproduces very robustly the power-law behavior with the exponent 0.8, experimentally measured in EEG spectra. The same value of the exponent is found on small-world lattices and for leaky neurons, indicating that universality holds for a wide class of brain models.

We investigate the breakdown properties of a random resistor-fuse network in which each network element behaves as a linear resistor if the voltage drop is less than a threshold value, but then ‘‘burns out’’ and changes irreversibly to an insulator for larger voltages. We consider a fully occupied network in which each resistor has the same resistance (in the linear regime), and with the threshold voltage drop uniformly distributed over the range $v_{\mathrm{-}}$=1-w/2 to $v_{+}$=1+w/2 (0<w≤2). The breakdown properties of this model depend crucially on w, and also on L, the linear dimension of the network. For sufficiently small w, ‘‘brittle’’ fracture occurs, in which catastrophic breaking is triggered by the failure of a vanishingly small fraction of bonds in the network. In this regime, the average voltage drop per unit length required to break the network, 〈$v_b$〉, varies as $v_{\mathrm{-}}$+O(1/$L^2$), and L→∞, and the distribution of breakdown voltages decays exponentially in $v_b$. By probabilistic arguments, we also establish the existence of a transition between this brittle regime and a ‘‘ductile’’ regime at a critical value of w=$w_c$(L), which approaches 2, as L→∞. This suggests that the fuse network fails by brittle fracture in the thermodynamic limit, except in the extreme case where the distribution of bond strengths includes the value zero. The ductile regime, w>$w_c$(L), is characterized by crack growth which is driven by increases in the external potential, before the network reaches the breaking point. For this case, numerical simulations indicate that the average breaking potential decreases as 1/(lnL${)}^y$, with y≤0.8, and that the distribution of breakdown voltages has a double experimental form. Numerical simulations are also performed to provide a geometrical description of the details of the breaking process as a function of w.