Bond percolation on a class of clustered random networks

Analytical results are derived for the bond percolation threshold and the size of the giant connected component in a class of random networks with nonzero clustering. The network's degree distribution and clustering spectrum may be prescribed and theoretical results match well with numerical simulations on both synthetic and real-world networks

Bond percolation on a class of clustered random networks

Analytical results are derived for the bond percolation threshold and the size of the giant connected component in a class of random networks with nonzero clustering. The network's degree distribution and clustering spectrum may be prescribed and theoretical results match well with numerical simulations on both synthetic and real-world networks

High-accuracy approximation of binary-state dynamics on networks

Binary-state dynamics (such as the susceptible-infected-susceptible (SIS) model of disease spread, or Glauber spin dynamics) on random networks are accurately approximated using master equations. Standard mean-field and pairwise theories are shown to result from seeking approximate solutions of the master equations. Applications to the calculation of SIS epidemic thresholds and critical points of nonequilibrium spin models are also demonstrated

Mean size of avalanches on directed random networks with arbitrary degree distributions

The mean size of unordered binary avalanches on infinite directed random networks may be determined using the damage propagation function introduced by [B. Samuelsson and J. E. S. Socolar, Phys. Rev. E 74, 036113 (2006)]. The derivation of Samuelsson and Socolar explicitly assumes a Poisson distribution of out-degrees. It is shown here that the damage propagation function method may be used whenever the in-degree and out-degree of network nodes are independently distributed; in particular, it is not necessary that the out-degree distribution be Poisson. The general case of correlated in- and out-degrees is discussed and numerical simulations (on large finite networks) are compared with the theoretical predictions (for infinite networks)

Mean size of avalanches on directed random networks with arbitrary degree distributions

The mean size of unordered binary avalanches on infinite directed random networks may be determined using the damage propagation function introduced by [B. Samuelsson and J. E. S. Socolar, Phys. Rev. E 74, 036113 (2006)]. The derivation of Samuelsson and Socolar explicitly assumes a Poisson distribution of out-degrees. It is shown here that the damage propagation function method may be used whenever the in-degree and out-degree of network nodes are independently distributed; in particular, it is not necessary that the out-degree distribution be Poisson. The general case of correlated in- and out-degrees is discussed and numerical simulations (on large finite networks) are compared with the theoretical predictions (for infinite networks)

High-accuracy approximation of binary-state dynamics on networks

Binary-state dynamics (such as the susceptible-infected-susceptible (SIS) model of disease spread, or Glauber spin dynamics) on random networks are accurately approximated using master equations. Standard mean-field and pairwise theories are shown to result from seeking approximate solutions of the master equations. Applications to the calculation of SIS epidemic thresholds and critical points of nonequilibrium spin models are also demonstrated

Transport in randomly-fluctuating spatially-periodic potentials

The motion of overdamped particles in a one-dimensional spatially-periodic potential is considered. The potential is also randomly-fluctuating in time, due to multiplicative colored noise terms, and has a deterministic tilt. Numerical simulations show two distinct parameter regimes, corresponding to free-running near-deterministic particles, and particles which are trapped in local minima of the potential with intermittent escape flights. Perturbation and asymptotic methods are developed to understand the drift velocity and diffusion coefficient in each parameter regime

Transport in randomly-fluctuating spatially-periodic potentials

The motion of overdamped particles in a one-dimensional spatially-periodic potential is considered. The potential is also randomly-fluctuating in time, due to multiplicative colored noise terms, and has a deterministic tilt. Numerical simulations show two distinct parameter regimes, corresponding to free-running near-deterministic particles, and particles which are trapped in local minima of the potential with intermittent escape flights. Perturbation and asymptotic methods are developed to understand the drift velocity and diffusion coefficient in each parameter regime

Memory-cognizant generalization to Simon’s random-copying neutral model

Simon’s classical random-copying model, introduced in 1955, has garnered much attention for its ability, in spite of an apparent simplicity, to produce characteristics similar to those observed across the spectrum of complex systems. Through a discrete-time mechanism in which items are added to a sequence based upon rich-gets-richer dynamics, Simon demonstrated that the resulting size distributions of such sequences exhibit power-law tails. The simplicity of this model arises from the approach by which copying occurs uniformly over all previous elements in the sequence. Here we propose a generalization of this model which moves away from this uniform assumption, instead incorporating memory effects that allow the copying event to occur via an arbitrary age-dependent kernel. Through this approach, we first demonstrate the potential to determine further information regarding the structure of sequences from the classical model before illustrating, via analytical study and numeric simulation, the flexibility offered by the arbitrary choice of memory. Furthermore, we demonstrate how previously proposed memory-dependent models can be further studied as specific cases of the proposed framework

Analytical results for bond percolation and k-core sizes on clustered networks

An analytical approach to calculating bond percolation thresholds, sizes of k-cores, and sizes of giant connected components on structured random networks with nonzero clustering is presented. The networks are generated using a generalization of Trapman's [P. Trapman, Theor. Popul. Biol. 71, 160 (2007)] model of cliques embedded in treelike random graphs. The resulting networks have arbitrary degree distributions and tunable degree-dependent clustering. The effect of clustering on the bond percolation thresholds for networks of this type is examined and contrasted with some recent results in the literature. For very high levels of clustering the percolation threshold in these generalized Trapman networks is increased above the value it takes in a randomly wired (unclustered) network of the same degree distribution. In assortative scale-free networks, where the variance of the degree distribution is infinite, this clustering effect can lead to a nonzero percolation (epidemic) threshold