Percolation Critical Exponents in Scale-Free Networks

We study the behavior of scale-free networks, having connectivity distribution P(k) k^-a, close to the percolation threshold. We show that for networks with 3<a<4, known to undergo a transition at a finite threshold of dilution, the critical exponents are different than the expected mean-field values of regular percolation in infinite dimensions. Networks with 2<a<3 possess only a percolative phase. Nevertheless, we show that in this case percolation critical exponents are well defined, near the limit of extreme dilution (where all sites are removed), and that also then the exponents bear a strong a-dependence. The regular mean-field values are recovered only for a>4.Comment: Latex, 4 page

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

Pseudofractal Scale-free Web

We find that scale-free random networks are excellently modeled by a deterministic graph. This graph has a discrete degree distribution (degree is the number of connections of a vertex) which is characterized by a power-law with exponent $\gamma=1+\ln3/\ln2$. Properties of this simple structure are surprisingly close to those of growing random scale-free networks with $\gamma$ in the most interesting region, between 2 and 3. We succeed to find exactly and numerically with high precision all main characteristics of the graph. In particular, we obtain the exact shortest-path-length distribution. For the large network ($\ln N \gg 1$) the distribution tends to a Gaussian of width $\sim \sqrt{\ln N}$ centered at $\bar{\ell} \sim \ln N$. We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent $2+\gamma$.Comment: 5 pages, 3 figure

Highly clustered scale-free networks

We propose a model for growing networks based on a finite memory of the nodes. The model shows stylized features of real-world networks: power law distribution of degree, linear preferential attachment of new links and a negative correlation between the age of a node and its link attachment rate. Notably, the degree distribution is conserved even though only the most recently grown part of the network is considered. This feature is relevant because real-world networks truncated in the same way exhibit a power-law distribution in the degree. As the network grows, the clustering reaches an asymptotic value larger than for regular lattices of the same average connectivity. These high-clustering scale-free networks indicate that memory effects could be crucial for a correct description of the dynamics of growing networks.Comment: 6 pages, 4 figure

Generic scale of the "scale-free" growing networks

We show that the connectivity distributions $P(k,t)$ of scale-free growing networks ($t$ is the network size) have the generic scale -- the cut-off at $k_{cut} \sim t^\beta$. The scaling exponent $\beta$ is related to the exponent $\gamma$ of the connectivity distribution, $\beta=1/(\gamma-1)$. We propose the simplest model of scale-free growing networks and obtain the exact form of its connectivity distribution for any size of the network. We demonstrate that the trace of the initial conditions -- a hump at $k_h \sim k_{cut} \sim t^\beta$ -- may be found for any network size. We also show that there exists a natural boundary for the observation of the scale-free networks and explain why so few scale-free networks are observed in Nature.Comment: 4 pages revtex, 3 figure

Correlations in Scale-Free Networks: Tomography and Percolation

We discuss three related models of scale-free networks with the same degree distribution but different correlation properties. Starting from the Barabasi-Albert construction based on growth and preferential attachment we discuss two other networks emerging when randomizing it with respect to links or nodes. We point out that the Barabasi-Albert model displays dissortative behavior with respect to the nodes' degrees, while the node-randomized network shows assortative mixing. These kinds of correlations are visualized by discussig the shell structure of the networks around their arbitrary node. In spite of different correlation behavior, all three constructions exhibit similar percolation properties.Comment: 6 pages, 2 figures; added reference

Comparison of Failures and Attacks on Random and Scale-Free Networks

It appeared recently that some statistical properties of complex networks like the Internet, the World Wide Web or Peer-to-Peer systems have an important influence on their resilience to failures and attacks. In particular, scale-free networks (i.e. networks with power-law degree distribution) seem much more robust than random networks in case of failures, while they are more sensitive to attacks. In this paper we deepen the study of the differences in the behavior of these two kinds of networks when facing failures or attacks. We moderate the general affirmation that scale-free networks are much more sensitive than random networks to attacks by showing that the number of links to remove in both cases is similar, and by showing that a slightly modified scenario for failures gives results similar to the ones for attacks. We also propose and analyze an efficient attack strategy against links

Singularities in ternary mixtures of k-core percolation

Heterogeneous k-core percolation is an extension of a percolation model which has interesting applications to the resilience of networks under random damage. In this model, the notion of node robustness is local, instead of global as in uniform k-core percolation. One of the advantages of k-core percolation models is the validity of an analytical mathematical framework for a large class of network topologies. We study ternary mixtures of node types in random networks and show the presence of a new type of critical phenomenon. This scenario may have useful applications in the stability of large scale infrastructures and the description of glass-forming systems.Comment: To appear in Complex Networks, Studies in Computational Intelligence, Proceedings of CompleNet 201