Degree Distribution Of Scale-free Networks Research Articles

Scalable Triangle Discovery Algorithm for Large Scale-Free Network with Limited Internal Memory

In complex network, the links among the nodes fabricate thousands of polygons, where triangles are the most foundamental ones. As more links are added to the network, more polygons are split to be triangles. Discovering the triangles can help understand the current status of the network. Due to the limitations of internal memory, discovering all the triangles in a large-scale real-world network, e.g., online social network (OSN), is challenging and has intrigued researchers for years. Currently, parallel computation frameworks, e.g., MapReduce, have been employed to address this issue. The last reducer is required to load all the neighbors of each node into internal memory, which causes “the curse of the last reducer.” The scale-free characteristic in real-world networks furthur exacerbate the applicability of the parallel triangle discovery algorithms. The current efforts on this issue mainly focus on dividing a large network into several small subnetworks and later employ internal memory algorithms. In this paper, we address this issue from a different perspective by employing the power-law degree distribution in scale-free networks. We theoretically and empirically prove that one node in scale-free network has a limited number of neighbors with a higher degree. Employing this finding, an novel parallel triangle discovery algorithm called degree partition (DePart) is designed. Since only the portion of neighbors with higher degrees are loaded into internal memory in the reducers, DePart solves “the curse of the last reducer” problem and works efficiently in clusters with limited internal memory. We develop DePart using MapReduce framework and examine the performance of DePart on five large real-world networks. The empirical testing results demonstrate that DePart has a favorable tradeoff between internal memory and total disk space and outperforms the current state-of-the-art MapReduce-based triangle discovery algorithms.

Mitigation Strategy for the Cascading Failure of Complex Networks Based on Node Capacity Control Function

A single node failure capacity control function taking into account attack strength, attack times, control node load intensity and the degree of attacked node is proposed in this paper to mitigate the cascading failure of complex networks under random attack. An optimal probability allocation mechanism of redundant resources is established by targeting the load of each neighbor node. Then, the node failure capacity control function and allocation mechanism are used to define the phase transition critical factor and robustness indicator that used the attack strength, control node load intensity and the degree of attacked node as the parameters. Based on the above analysis, the phase transition critical factor model of degree distribution of scale-free network and random network is derived, and the dynamic change law between the parameters and phase transition critical state as well as robust performance of classical network and real network is analyzed. The theoretical and experimental results show that in the controllable region, the smaller the degree of attacked node, the greater the control node load intensity and the more difficult the phase transition critical state to be achieved, and the better the effect of mitigating cascading failure. Besides, the robustness of the network with cascading failure is mutually affected by the control node load capacity, the degree of attacked node and phase transition critical factor within a certain range, which thus embarks on a new perspective to mitigate the failure.

The different cooperative behaviors on a kind of scale-free networks with identical degree sequence

The power-law degree distribution of scale-free networks plays an important role in the bloom of cooperation in the evolutionary games performed on them. In this paper we apply prisoner’s dilemma and public goods game on a family of scale-free networks with the same degree sequence, and show that power-law behavior alone does not determine the cooperative behavior in scale-free networks. Instead, we present that the direct connections among large-degree nodes have a crucial influence on the evolution of cooperation in the scale-free network family.

Phase transition in the biconnectivity of scale-free networks

In information-transport and biological systems, sometimes there is more than one pathway between two nodes, so that there is a backup in case one pathway becomes defective. The size of such biconnected nodes can be an important measure of the robustness of a system. The giant biconnected components of diverse real-world networks suggest the importance of scale-free topology in biconnectivity. Thus, here, we consider the critical behavior of the largest biconnected component (BC) as links are added and form a random scale-free network. The critical exponents β((BC)) and β((SC)) associated with the order parameter of the percolation transition of the biconnected component and the single-connected component (SC), respectively, are compared. We obtain a ratio β((BC))/β((SC))=λ-1 for 2<λ<3 and 2 for λ>3, where λ is the exponent of the degree distribution in scale-free networks. We also determine the finite-size scaling behavior of the order parameter analytically and numerically.

Possible thermodynamic structure underlying the laws of Zipf and Benford

We show that the laws of Zipf and Benford, obeyed by scores of numerical data generated by many and diverse kinds of natural phenomena and human activity are related to the focal expression of a generalized thermodynamic structure. This structure is obtained from a deformed type of statistical mechanics that arises when configurational phase space is incompletely visited in a severe way. Specifically, the restriction is that the accessible fraction of this space has fractal properties. The focal expression is an (incomplete) Legendre transform between two entropy (or Massieu) potentials that when particularized to first digits leads to a previously existing generalization of Benford's law. The inverse functional of this expression leads to Zipf's law; but it naturally includes the bends or tails observed in real data for small and large rank. Remarkably, we find that the entire problem is analogous to the transition to chaos via intermittency exhibited by low-dimensional nonlinear maps. Our results also explain the generic form of the degree distribution of scale-free networks.

Connection between scale-free networks and nonextensive statistical mechanics

The degree distribution of the so-called scale-free networks exhibits, quite often, the form p(k) ∝1/(k0+k)γ (with γ>0 and k0>0), in the limit of large networks. It happens that this form precisely coincides with the q-exponential p(k) ∝exp q(-k/κ) (q ≥1 and κ>0), with γ=1/(q-1) and k0=κ/(q-1). It optimises the nonadditive entropy \(S_q= \frac{k_B}{q-1}\left\{ 1-\sum_{k=1}^\infty [p(k)]^q \right\}\) with mathematically the same constraints that yield the stationary (or quasi-stationary) distribution in nonextensive statistical mechanics. In other words, the most ubiquitous form of the degree distribution of scale-free networks is a realisation of the hypothesis involved within the q-generalisation of Boltzmann-Gibbs statistical mechanics. In addition to this, we show that growth is not a necessary condition for having scale-free networks, in contrast with a widely spread belief.

Self-similarity, small-world, scale-free scaling, disassortativity, and robustness in hierarchical lattices

In this paper, firstly, we study analytically the topological features of a family of hierarchical lattices (HLs) from the view point of complex networks. We derive some basic properties of HLs controlled by a parameter $q$. Our results show that scale-free networks are not always small-world, and support the conjecture that self-similar scale-free networks are not assortative. Secondly, we define a deterministic family of graphs called small-world hierarchical lattices (SWHLs). Our construction preserves the structure of hierarchical lattices, while the small-world phenomenon arises. Finally, the dynamical processes of intentional attacks and collective synchronization are studied and the comparisons between HLs and Barab{\'asi}-Albert (BA) networks as well as SWHLs are shown. We show that degree distribution of scale-free networks does not suffice to characterize their synchronizability, and that networks with smaller average path length are not always easier to synchronize.

Optimal paths in complex networks with correlated weights: The worldwide airport network

We study complex networks with weights w(ij) associated with each link connecting node i and j. The weights are chosen to be correlated with the network topology in the form found in two real world examples: (a) the worldwide airport network and (b) the E. Coli metabolic network. Here w(ij) approximately equals x(ij)(k(i)k(j))alpha, where k(i) and k(j) are the degrees of nodes i and j , x(ij) is a random number, and alpha represents the strength of the correlations. The case alpha >0 represents correlation between weights and degree, while alpha< 0 represents anticorrelation and the case alpha=0 reduces to the case of no correlations. We study the scaling of the lengths of the optimal paths, l(opt), with the system size N in strong disorder for scale-free networks for different alpha. We find two different universality classes for l(opt), in strong disorder depending on alpha: (i) if alpha >0 , then for lambda >2 the scaling law l(opt) approximately equals N(1/3), where lambda is the power-law exponent of the degree distribution of scale-free networks, and (ii) if alpha< or =0 , then l(opt) approximately equals N((nu)(opt)) with nu(opt) identical to its value for the uncorrelated case alpha=0. We calculate the robustness of correlated scale-free networks with different alpha and find the networks with alpha< 0 to be the most robust networks when compared to the other values of alpha. We propose an analytical method to study percolation phenomena on networks with this kind of correlation, and our numerical results suggest that for scale-free networks with alpha< 0 , the percolation threshold p(c) is finite for lambda >3, which belongs to the same universality class as alpha=0 . We compare our simulation results with the real worldwide airport network, and we find good agreement.

Network of sexual contacts and sexually transmitted HIV infection in Burkina Faso

Two thirds of the people who have been infected by human immunodeficiency virus (HIV) in the world live in Sub-Saharan African countries. The results of a study measuring the degree distribution of the network of sexual contacts in Burkina Faso are described. Such a network is responsible for the spread of sexually transmitted diseases, and in particular of HIV. It has been found that the number of different sexual partners reported by males is a power law distribution with an exponent gamma = 2.9 (0.1). This is consistent with the degree distribution of scale-free networks. On the other hand, the females can be divided into two groups: the prostitutes with an average of 400 different partners per year, and females with a stable partner, having a rapidly decreasing degree distribution. Such a result may have important implications on the control of sexually transmitted diseases and in particular of HIV. Since scale-free networks have no epidemic threshold, a campaign based on prevention and anti-viral treatment of few highly connected nodes can be more successful than any policy based on enlarged but random distribution of the available anti-viral treatments.