Percolation in Directed Scale-Free Networks

Many complex networks in nature have directed links, a property that affects the network's navigability and large-scale topology. Here we study the percolation properties of such directed scale-free networks with correlated in- and out-degree distributions. We derive a phase diagram that indicates the existence of three regimes, determined by the values of the degree exponents. In the first regime we regain the known directed percolation mean field exponents. In contrast, the second and third regimes are characterized by anomalous exponents, which we calculate analytically. In the third regime the network is resilient to random dilution, i.e., the percolation threshold is p_c->1.Comment: Latex, 5 pages, 2 fig

Cross-over behaviour in a communication network

We address the problem of message transfer in a communication network. The network consists of nodes and links, with the nodes lying on a two dimensional lattice. Each node has connections with its nearest neighbours, whereas some special nodes, which are designated as hubs, have connections to all the sites within a certain area of influence. The degree distribution for this network is bimodal in nature and has finite variance. The distribution of travel times between two sites situated at a fixed distance on this lattice shows fat fractal behaviour as a function of hub-density. If extra assortative connections are now introduced between the hubs so that each hub is connected to two or three other hubs, the distribution crosses over to power-law behaviour. Cross-over behaviour is also seen if end-to-end short cuts are introduced between hubs whose areas of influence overlap, but this is much milder in nature. In yet another information transmission process, namely, the spread of infection on the network with assortative connections, we again observed cross-over behaviour of another type, viz. from one power-law to another for the threshold values of disease transmission probability. Our results are relevant for the understanding of the role of network topology in information spread processes.Comment: 12 figure

Renormalization group approach to an Abelian sandpile model on planar lattices

One important step in the renormalization group (RG) approach to a lattice sandpile model is the exact enumeration of all possible toppling processes of sandpile dynamics inside a cell for RG transformations. Here we propose a computer algorithm to carry out such exact enumeration for cells of planar lattices in RG approach to Bak-Tang-Wiesenfeld sandpile model [Phys. Rev. Lett. {\bf 59}, 381 (1987)] and consider both the reduced-high RG equations proposed by Pietronero, Vespignani, and Zapperi (PVZ) [Phys. Rev. Lett. {\bf 72}, 1690 (1994)] and the real-height RG equations proposed by Ivashkevich [Phys. Rev. Lett. {\bf 76}, 3368 (1996)]. Using this algorithm we are able to carry out RG transformations more quickly with large cell size, e.g. $3 \times 3$ cell for the square (sq) lattice in PVZ RG equations, which is the largest cell size at the present, and find some mistakes in a previous paper [Phys. Rev. E {\bf 51}, 1711 (1995)]. For sq and plane triangular (pt) lattices, we obtain the only attractive fixed point for each lattice and calculate the avalanche exponent $\tau$ and the dynamical exponent $z$. Our results suggest that the increase of the cell size in the PVZ RG transformation does not lead to more accurate results. The implication of such result is discussed.Comment: 29 pages, 6 figure

Press perturbations and indirect effects in real food webs

8 pages, 2 figures,3 tablesThe prediction of the effects of disturbances in natural systems is limited by the general lack of knowledge on the strength of species interactions, i.e., the effect of one species on the population growth rate of another, and by the uncertainty of the effects that may be manifested via indirect pathways within the food web. Here we explored the consequences of changes in species populations for the remaining species within nine exceptionally well-characterized empirical food webs, for which, unlike the vast majority of other published webs, feeding links have been fully quantified. Using the inverse of the Jacobian matrix, we found that perturbations to species with few connections have larger net effects (considering both direct and indirect pathways between two species) on the rest of the food web than do disturbances to species that are highly connected. For 40% of predator–prey links, predators had positive net effects on prey populations, due to the predominance of indirect interactions. Our results highlight the fundamental, but often counterintuitive, role of indirect effects for the maintenance of food web complexity and biodiversity.This work has been supported by ESF InterAct Network, NERC Fellowship NE/C002105/1, and a Ramon y Cajal Fellowship (to J. M. Montoya).Peer reviewe