Percolation, the formation of a macroscopic connected component, is a key feature in the description of complex networks. The dynamical properties of a variety of systems can be understood in terms of percolation, including the robustness of power grids and information networks, the spreading of epidemics and forest fires, and the stability of gene regulatory networks. Recent studies have shown that if network edges are added "competitively" in undirected networks, the onset of percolation is abrupt or "explosive." The unusual qualitative features of this phase transition have been the subject of much recent attention. Here we generalize this previously studied network growth process from undirected networks to directed networks and use finite-size scaling theory to find several scaling exponents. We find that this process is also characterized by a very rapid growth in the giant component, but that this growth is not as sudden as in undirected networks.

Boolean networks, widely used to model gene regulation, exhibit a phase transition between regimes in which small perturbations either die out or grow exponentially. We show and numerically verify that this phase transition in the dynamics can be mapped onto a static percolation problem which predicts the long-time average Hamming distance between perturbed and unperturbed orbits.

This paper explores the effectiveness of network attack when the attacker has imperfect information about the network. For Erdős-Rényi networks, we observe that dynamical importance and betweenness centrality-based attacks are surprisingly robust to the presence of a moderate amount of imperfect information and are more effective compared with simpler degree-based attacks even at moderate levels of network information error. In contrast, for scale-free networks the effectiveness of attack is much less degraded by a moderate level of information error. Furthermore, in the Erdős-Rényi case the effectiveness of network attack is much more degraded by missing links as compared with the same number of false links. [6,7], etc.). Interventions that seek to degrade [8][9][10][11][12][13] or protect [13][14][15][16] network connectivity are thus of great interest. In particular, strategies for network attack by node or link removal have been intensively studied. Key issues have been the dependence of the attack effectiveness upon network topology and the strategy for selecting nodes or links for removal. We note, however, that, while such previous studies have predominantly presumed the attacker to have perfect knowledge of the network to be attacked, this is very often not the case. Specifically, networks inferred from measurements typically have false links and miss true links. One might suppose that these errors could very much lower the effectiveness of attack strategies. The purpose of this paper is to address this important issue for the case of node removal attacks of undirected networks (directed networks are treated in the online supplementary material [17]).One example of a network attack problem is an attempt to stop the spread of a disease with a limited number of vaccinations: the people who receive the vaccinations are chosen on the basis of their position in the social network [8][9][10][11][12][13]. Another example is that of deriving gene therapies for cancer. Here the goal is to select those genes whose disabling would most inhibit cancer cell survival and proliferation [18][5]. Yet another example is the study of the resilience of the Internet to intentional attack [8,12]. The typical attack strategy is to calculate some centrality measure of each node, and to then attack (disable, vaccinate, or remove) those nodes with the highest values of this measure. However, an attacker with imperfect network information will determine values of these centrality measures with some error, and using these would be expected to degrade the effectiveness of his attack. Imperfect network information is ubiquitous in applications and can arise in various ways. Examples of link errors can be found in online social networks, where a friendship may be indicated despite the two subjects having never personally met, or inversely, if no online friendship exists between two faceto-face friends. In the previously cited example of cancer gene therapy, genes are selected for disabling based upon an estimated gene interaction net...

Networks in nature are often formed within a spatial domain in a dynamical manner, gaining links and nodes as they develop over time. Motivated by the growth and development of neuronal networks, we propose a class of spatially-based growing network models and investigate the resulting statistical network properties as a function of the dimension and topology of the space in which the networks are embedded. In particular, we consider two models in which nodes are placed one by one in random locations in space, with each such placement followed by configuration relaxation toward uniform node density, and connection of the new node with spatially nearby nodes. We find that such growth processes naturally result in networks with small-world features, including a short characteristic path length and nonzero clustering. We find no qualitative differences in these properties for two different topologies, and we suggest that results for these properties may not depend strongly on the topology of the embedding space. The results do depend strongly on dimension, and higher-dimensional spaces result in shorter path lengths but less clustering.

We study the stability of orbits in large Boolean networks with given complex topology. We impose no restrictions on the form of the update rules, which may be correlated with local topological properties of the network. While recent past work has addressed the separate effects of nontrivial network topology and certain special classes of update rules on stability, only crude results exist about how these effects interact. We present a widely applicable solution to this problem. Numerical experiments confirm our theory and show that local correlations between topology and update rules can have profound effects on the qualitative behavior of these systems.pacs: 89.75.-k (complex systems), 05.45.-a (nonlinear dynamical systems), 64.60.aq (phase transitions in networks).