No abstract

We study the number of distinct sites visited by N random walkers after t steps Siv(t) under the condition that all the walkers are initially at the origin. We derive asymptotic expressions for the mean number of distinct sites (Siv(t)) in one, two, and three dimensions. We find that (Siv(t)) passes through several growth regimes; at short times (Siv(t))~t " (regime I), for t» & t & t'"we find that (Siv(t))~(t ln[N Si(t)/t ])" (regime II), and for t & t'", (Siv(t))~NSi(t) (regime III). The crossover times are t " ln N for all dimensions, and t'"~o o, exp N, and N for one, two, and three dimensions, respectively. We show that in regimes II and III (Siv(t)) satisfies a scaling relation of the form (Siv(t)) t f(x), with x:-N(Si(t))/t . We also obtain asymptotic results for the complete probability distribution of Siv(t) for the one-dimensional case in the limit of large N and t,.PACS number(s): 05.40. +j

Cascading failure is a potentially devastating process that spreads on real-world complex networks and can impact the integrity of wide-ranging infrastructures, natural systems and societal cohesiveness. One of the essential features that create complex network vulnerability to failure propagation is the dependency among their components, exposing entire systems to significant risks from destabilizing hazards such as human attacks, natural disasters or internal breakdowns. Developing realistic models for cascading failures as well as strategies to halt and mitigate the failure propagation can point to new approaches to restoring and strengthening real-world networks. In this review, we summarize recent progress on models developed based on physics and complex network science to understand the mechanisms, dynamics and overall impact of cascading failures. We present models for cascading failures in single networks and interdependent networks and explain how different dynamic propagation mechanisms can lead to an abrupt collapse and a rich dynamic behaviour. Finally, we close the review with novel emerging strategies for containing cascades of failures and discuss open questions that remain to be addressed.

We study a system in which diffusing particles (species A) are injected into a reactive (i-dimensional substrate (species B) at rate A, with the rule that A+B-> C(inert). The amount of species C, C(t), and the number of surviving A particles, A(t), are calculated for substrate dimensions d= 1, 2, and 3. We find the surprising results A{t) ~ t^^^ for d = ?> and C{t) ^ vTIrTt for d = 1. We confirm our predictions by performing Monte Carlo simulations for d -1, 2, and 3 and experiments for the reaction l2(gas) + 2Ag (solid) 2AgI (solid) for d = 2.