Percolation in hierarchical scale-free nets (original) (raw)

Effective dimensions and percolation in hierarchically structured scale-free networks

Physical review. E, Statistical, nonlinear, and soft matter physics, 2003

We introduce appropriate definitions of dimensions in order to characterize the fractal properties of complex networks. We compute these dimensions in a hierarchically structured network of particular interest. In spite of the nontrivial character of this network that displays scale-free connectivity among other features, it turns out to be approximately one dimensional. The dimensional characterization is in agreement with the results on statistics of site percolation and other dynamical processes implemented on such a network.

Percolation in directed scale-free networks

Physical Review E, 2002

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 pc → 1. 02.50.Cw, 05.40.a, 05.50.+q, 64.60.Ak

Numerical Evaluation of the Upper Critical Dimension of Percolation in Scale-Free Networks

We propose numerical methods to evaluate the upper critical dimension d c of random percolation clusters in Erdős-Rényi networks and in scale-free networks with degree distribution P͑k͒ϳk − , where k is the degree of a node and is the broadness of the degree distribution. Our results support the theoretical prediction, d c =2͑ −1͒ / ͑ −3͒ for scale-free networks with 3 ϽϽ4 and d c = 6 for Erdős-Rényi networks and scale-free networks with Ͼ4. When the removal of nodes is not random but targeted on removing the highest degree nodes we obtain d c = 6 for all Ͼ2. Our method also yields a better numerical evaluation of the critical percolation threshold p c for scale-free networks. Our results suggest that the finite size effects increases when approaches 3 from above.

Explosive Percolation in Scale-Free Networks

Physical Review Letters, 2009

We study scale-free networks constructed via a cooperative Achlioptas growth process. Links between nodes are introduced in order to produce a scale-free graph with given exponent λ for the degree distribution, but the choice of each new link depends on the mass of the clusters that this link will merge. Networks constructed via this biased procedure show a percolation transition which strongly differs from the one observed in standard percolation, where links are introduced just randomly. The different growth process leads to a phase transition with a non-vanishing percolation threshold already for λ > λc ∼ 2.2. More interestingly, the transition is continuous when λ ≤ 3 but becomes discontinuous when λ > 3. This may have important consequences both for the structure of networks and for the dynamics of processes taking place on them.

Percolation Transitions in Scale-Free Networks under the Achlioptas Process

Physical Review Letters, 2009

It has been recently shown that the percolation transition is discontinuous in Erdős-Rényi networks and square lattices in two dimensions under the Achlioptas Process (AP). Here, we show that when the structure is highly heterogeneous as in scale-free networks, a discontinuous transition does not always occur: a continuous transition is also possible depending on the degree distribution of the scale-free network. This originates from the competition between the AP that discourages the formation of a giant component and the existence of hubs that encourages it. We also estimate the value of the characteristic degree exponent that separates the two transition types.

Percolation of Partially Interdependent Scale-free Networks

We study the percolation behavior of two interdependent scale-free (SF) networks under random failure of 1-p fraction of nodes. Our results are based on numerical solutions of analytical expressions and simulations. We find that as the coupling strength between the two networks q reduces from 1 (fully coupled) to 0 (no coupling), there exist two critical coupling strengths q 1 and q 2 , which separate three different regions with different behavior of the giant component as a function of p. (i) For q q 1 , an abrupt collapse transition occurs at p = p c . (ii) For q 2 < q < q 1 , the giant component has a hybrid transition combined of both, abrupt decrease at a certain p = p jump c followed by a smooth decrease to zero for p < p jump c as p decreases to zero. (iii) For q q 2 , the giant component has a continuous second-order transition (at p = p c ). We find that (a) for λ 3, q 1 ≡ 1; and for λ > 3, q 1 decreases with increasing λ. Here, λ is the scaling exponent of the degree distribution, P (k) ∝ k −λ . (b) In the hybrid transition, at the q 2 < q < q 1 region, the mutual giant component P ∞ jumps discontinuously at p = p jump c to a very small but nonzero value, and when reducing p, P ∞ continuously approaches to 0 at p c = 0 for λ < 3 and at p c > 0 for λ > 3. Thus, the known theoretical p c = 0 for a single network with λ 3 is expected to be valid also for strictly partial interdependent networks.

Percolation on infinitely ramified fractal networks

Physics Letters A, 2018

• Percolation thresholds and critical exponents are determined for four Sierpinski carpets. • Critical exponents are dependent on three dimension numbers of the fractal. • Hyperscaling relations are governed by the Hausdorff dimension. • Percolation threshold is controlled by the topological Hausdorff dimension. • Sierpinski carpets and square lattice belong to the same universality class.

Agglomerative percolation in two dimensions

EPL (Europhysics Letters), 2012

We study a process termed agglomerative percolation (AP) in two dimensions. Instead of adding sites or bonds at random, in AP randomly chosen clusters are linked to all their neighbors. As a result the growth process involves a diverging length scale near a critical point. Picking target clusters with probability proportional to their mass leads to a runaway compact cluster. Choosing all clusters equally leads to a continuous transition in a new universality class for the square lattice, while the transition on the triangular lattice has the same critical exponents as ordinary percolation. PACS numbers: 64.60.ah, 68.43.Jk, 89.75.Da Percolation is a pervasive concept in statistical physics and an important branch of mathematics [1]. It typifies the emergence of long range connectivity in many systems such as the flow of liquids through porous media [2], transport in disordered media [3], spread of disease in populations [4], resilience of networks to attack [5], formation of gels [6] and even of social groups [7]

Percolation in self-similar networks

2010

We provide a simple proof that graphs in a general class of self-similar networks have zero percolation threshold. The considered self-similar networks include random scale-free graphs with given expected node degrees and zero clustering, scale-free graphs with finite clustering and metric structure, growing scale-free networks, and many real networks. The proof and the derivation of the giant component size do not require the assumption that networks are treelike. Our results rely only on the observation that self-similar networks possess a hierarchy of nested subgraphs whose average degree grows with their depth in the hierarchy. We conjecture that this property is pivotal for percolation in networks.

Inducing effect on the percolation transition in complex networks

Nature Communications, 2013

Percolation theory concerns the emergence of connected clusters that percolate through a networked system. Previous studies ignored the effect that a node outside the percolating cluster may actively induce its inside neighbours to exit the percolating cluster. Here we study this inducing effect on the classical site percolation and K-core percolation, showing that the inducing effect always causes a discontinuous percolation transition. We precisely predict the percolation threshold and core size for uncorrelated random networks with arbitrary degree distributions. For low-dimensional lattices the percolation threshold fluctuates considerably over realizations, yet we can still predict the core size once the percolation occurs. The core sizes of real-world networks can also be well predicted using degree distribution as the only input. Our work therefore provides a theoretical framework for quantitatively understanding discontinuous breakdown phenomena in various complex systems.