Scale-free Model Research Articles

EXPLORING THE VULNERABILITY OF FRACTAL COMPLEX NETWORKS THROUGH CONNECTION PATTERN AND FRACTAL DIMENSION

The structure of network has a significant impact on the stability of the network. It is useful to reveal the effect of fractal structure on the vulnerability of complex network since it is a ubiquitous feature in many real-world networks. There have been many studies on the stability of the small world and scale-free models, but little has been down on the quantitative research on fractal models. In this paper, the vulnerability was studied from two perspectives: the connection pattern between hubs and the fractal dimensions of the networks. First, statistics expression of inter-connections between any two hubs was defined and used to represent the connection pattern of the whole network. Our experimental results show that statistic values of inter-connections were obvious differences for each kind of complex model, and the more inter-connections, the more stable the network was. Secondly, the fractal dimension was considered to be a key factor related to vulnerability. Here we found the quantitative power function relationship between vulnerability and fractal dimension and gave the explicit mathematical formula. The results are helpful to build stable artificial network models through the analysis and comparison of the real brain network.

Degree correlations in scale-free random graph models

AbstractWe study the average nearest-neighbour degree a(k) of vertices with degree k. In many real-world networks with power-law degree distribution, a(k) falls off with k, a property ascribed to the constraint that any two vertices are connected by at most one edge. We show that a(k) indeed decays with k in three simple random graph models with power-law degrees: the erased configuration model, the rank-1 inhomogeneous random graph, and the hyperbolic random graph. We find that in the large-network limit for all three null models, a(k) starts to decay beyond $n^{(\tau-2)/(\tau-1)}$ and then settles on a power law $a(k)\sim k^{\tau-3}$, with $\tau$ the degree exponent.

Quasi-periodic events on structured earthquake models**Project supported by the National Natural Science Foundation of China (Grant Nos. 11575072 and 11675096), the Fundamental Research Funds for the Central Universities, China (Grant No. GK201702001), and the FPALAB-SNNU, China (Grant No. 16QNGG007).

There has been much interest in studying quasi-periodic events on earthquake models. Here we investigate quasi-periodic events in the avalanche time series on structured earthquake models by the analysis of the autocorrelation function and the fast Fourier transform. For random spatial earthquake models, quasi-periodic events are robust and we obtain a simple rule for a period that is proportional to the choice of unit time and the dissipation of the system. Moreover, computer simulations validate this rule for two-dimensional lattice models and cycle graphs, but our simulation results also show that small-world models, scale-free models, and random rule graphs do not have periodic phenomena. Although the periodicity of avalanche does not depend on the criticality of the system or the average degree of the system or the size of the system, there is evidence that it depends on the time series of the average force of the system.

An Accurate Physical Model for Halo Concentrations

Abstract The relation between halo mass, M, and concentration, c, is a critical component in our understanding of the structure of dark matter halos. While numerous models for this relation have been proposed, almost none of them attempt to derive the evolution of the relation analytically. We build on previous efforts to model the c–M relation as a function of physical parameters such as the peak height, ν, and the effective power spectrum slope, , which capture the dependence of c on halo mass, redshift, and cosmology. We present three major improvements over previous models. First, we derive an analytical expression for the c–M relation that is valid under the assumption of pseudo-evolution, i.e., assuming that the density profiles of halos are static in physical coordinates while the definition of their boundary evolves. We find that this ansatz is highly successful in describing the evolution of the low-mass end of the c–M relation. Second, we employ a new physical variable, the effective exponent of linear growth, , to parameterize deviations from an Einstein–de Sitter expansion history. Third, we combine an updated definition of with the additional dependence on and propose a phenomenological extension of our analytical framework to include all halo masses. This semianalytical model matches simulated concentrations in both scale-free models and ΛCDM to 5% accuracy with very few exceptions and differs significantly from all previously proposed models. We present a publicly available code to compute the predictions of our model in the python toolkit Colossus, including updated parameters for the model of Diemer and Kravtsov.

Tight Fluctuations of Weight-Distances in Random Graphs with Infinite-Variance Degrees

We prove results for first-passage percolation on the configuration model with degrees having asymptotic finite mean, infinite variance and i.i.d. edge-weights with strictly positive support of the form Y=a+X, where a is a positive constant and the excess edge-weight X is a non-negative random variable with zero as the infimum of its support. We prove that the weight of the optimal path between two uniformly chosen vertices has tight fluctuations around the asymptotic mean of the graph-distance if and only if the following condition holds: the random variable X is such that the age-dependent branching process describing first-passage percolation exploration in the same graph with edge-weights from distribution X has a positive probability to reach infinitely many individuals in a finite time. This shows that almost-shortest paths in the graph-distance proliferate, in the sense that there even exist paths having tight total excess edge-weight for appropriate edge-weight distributions. Our proof makes use of a degree-dependent percolation model that we believe is interesting in its own right, as well as tightness results for distances in scale-free configuration models that we prove to hold under rather weak conditions on the degrees.

Impact of core-periphery structure on cascading failures in interdependent scale-free networks

Core-periphery structure is a typical meso-scale structure in networks. Previous studies on core-periphery structure mainly focus on the improvement of detection methods, while the research on the impact of core-periphery structure on cascading failures in interdependent networks is still missing. Therefore, we investigate the cascading failures of interdependent scale-free networks with different core-periphery structures and coupling preferences in the paper. First, we introduce an evaluation index to calculate the goodness of core-periphery structure. Second, we propose a new scale-free network evolution model, which can generate tunable core-periphery structures, and its degree distribution is analyzed mathematically. Finally, based on a degree-load-based cascading failure model, we mainly investigate the impact of goodness of core-periphery structure on cascading failures in both symmetrical and asymmetrical interdependent networks. Through numerical simulations, we find that with the same average degree, the networks with weak core-periphery structure will be more robust, while the initial load on node will influence the improvement of robustness. In addition, we also find that the inter-similarity coupling performs better than random coupling. These findings may be helpful for building resilient interdependent networks.

Temporal Analysis of the Diffusion of Knowledge in Networks of Software Maintenance and Development Project Team

Different approaches have been established for applications of social and complex networks involving biological systems, passing through collaborative systems in knowledge networks and organizational systems. In this latter application, we highlight the studies focused on the diffusion of information and knowledge in networks. However, most of the time, the propagation of information in these networks and the resulting process of creation and diffusion of knowledge, have been studied from static perspectives. Additionally, the very concept of diffusion inevitably implies the inclusion of the temporal dimension, due to that it is an essentially dynamic process. Although static analysis provides an important perspective in structural terms, the behavioral view that reflects the evolution of the relationships of the members of these networks over time is best described by temporal networks. Thus, it is possible to analyze both the information flow and the structural changes that occur over time, which influences the dynamics of the creation and diffusion of knowledge. This article describes the computational modeling used to elucidate the creation and diffusion of knowledge in temporal networks formed to execute software maintenance and construction projects, for the period between 2007 and 2013, in the SERVIÇO FEDERAL DE PROCESSAMENTO DE DADOS (FEDERAL DATA PROCESSING SERVICE-SERPRO)—a public organization that provides information and communication technology services. The methodological approach adopted for the study was based on techniques for analyzing social and complex networks and on the complementary extensions that address temporal modeling of these networks. We present an exploratory longitudinal study that enabled a dynamic and structural analysis of the knowledge networks formed by members of software maintenance and development project teams between 2007 and 2013. The study enabled identification of knowledge categories throughout this period, in addition to the determination that the networks have a structure with small-world and scale-free models. Finally, we concluded that, in general, the topologies of the networks studies had characteristics for facilitating the flow of knowledge within the organization.

On Modeling Shortest Path Length Distribution in Scale-Free Network Topologies

Complex and interconnected systems belonging to biological, social, economic, and technology application fields are generally described through scale-free topology models. In this context, it is essential to characterize the distribution of shortest paths in order to obtain precious insights on the network behavior. Unfortunately, the few contributions available in the current scientific literature require a case-by-case tuning of model parameters. To bridge this gap, novel Gaussian-based models are proposed hereby, whose parameters can be immediately tuned based on the number of nodes ( $N$ ) composing the network, only. In this way, given $N$ , it becomes possible to predict the distribution of shortest paths without retuning the model for each scenario of interest. The outcomes of the proposed models have been successfully validated and compared with respect to state-of-the-art approaches in a wide set of network topologies. To provide a further insight, the conceived Gaussian-based models have been also evaluated for real Internet topologies, learned from reference data sets. Obtained results highlight that the proposed models are able to reach a good tradeoff between the level of accuracy and complexity, even for real network configurations.

Scale-Free Modeling of Coupled Evolution of Discrete Dislocation Bands and Multivariant Martensitic Microstructure.

A scale-free model for the coupled evolution of discrete dislocation bands and multivariant martensitic microstructure is developed. In contrast to previous phase field models, which are limited to nanoscale specimens, this model allows for treating the nucleation and evolution of martensite at evolving dislocation pileups, twin tips, and shear bands in a sample of an arbitrary size. The model is applied for finite element simulations of plastic strain-induced phase transformations (PTs) in a polycrystalline sample under compression and shear. The solution explains the one to two orders of magnitude reduction in PT pressure by plastic shear, the existence of incompletely transformed stationary state, and optimal shear strain for the strain-induced synthesis of high pressure phases.

One energy-efficient random-walk topology evolution method for underground wireless sensor networks

Aimed at the limited energy supply and imperfect topological tolerance for underground wireless sensor networks, one energy-efficient random-walk scale-free topology model is proposed in this article, and a power network topology structure with adjustable rate index gets generated. At first, the network is divided into several clusters, and the cluster heads are selected with the use of random-walk strategy. During the growth of the network, with the introduction of preferred connection for scale-free network, together with considering both the node’s residual energy and the distance among nodes, nodes with larger residual energy present higher connectivity probability, so that the energy balance of the network gets realized. Simulation results show that the communication among the cluster heads selected by the proposed random-walk scale-free topology model presents not only the power-law characteristics of scale-free networks but also has better stability, higher fault tolerance, and it can still balance the energy consumption for nodes and the network and therefore can prolong the lifetime of the network.

CosmicGrowth Simulations—Cosmological simulations for structure growth studies

I present a large set of high resolution simulations, called CosmicGrowth Simulations, which were generated with either 8.6 billion or 29 billion particles. As the nominal cosmological model that can match nearly all observations on cosmological scales, I have adopted a flat Cold Dark Matter (CDM) model with a cosmological constant $\Lambda$ (${\Lambda}$CDM). The model parameters have been taken either from the latest result of the WMAP satellite (WMAP ${\Lambda}$CDM) or from the first year's result of the Planck satellite (Planck ${\Lambda}$CDM). Six simulations are produced in the ${\Lambda}$CDM models with two in the Planck model and the others in the WMAP model. In order for studying the nonlinear evolution of the clustering, four simulations were also produced with $8.6$ billion particles for the scale-free models of an initial power spectrum $P(k)\propto k^n$ with $n=0$,$-1$,$-1.5$ or $-2.0$. Furthermore, two radical CDM models (XCDM) are simulated with 8.6 billion particles each. Since the XCDM have some of the model parameters distinct from those of the ${\Lambda}$CDM models, they must be unable to match the observations, but are very useful for studying how the clustering properties depend on the model parameters. The Friends-of-Friends (FoF) halos were identified for each snapshot and subhalos were produced by the Hierarchical Branch Tracing (HBT) algorithm. These simulations form a powerful database to study the growth and evolution of the cosmic structures both in theories and in observations.

Scale-Free Percolation in Continuum Space

The study of real-life network modeling has become very popular in recent years. An attractive model is the scale-free percolation model on the lattice $${\mathbb Z}^d$$, $$d\ge 1$$, because it fulfills several stylized facts observed in large real-life networks. We adopt this model to continuum space which leads to a heterogeneous random-connection model on $${\mathbb R}^d$$: Particles are generated by a homogeneous marked Poisson point process on $${\mathbb R}^d$$, and the probability of an edge between two particles is determined by their marks and their distance. In this model we study several properties such as the degree distributions, percolation properties and graph distances.

A Network Approach to Link Visibility and Urban Activity Location

Performance of a range of urban amenities is influenced by their accessibility to pedestrians. Success in attracting pedestrians to a particular location depends on how they project visuospatial information. In this paper, we propose an original method for analysing the visuospatial integration of particular locations within a street network. As a case study we analyse the distribution of one type of urban amenities - food and drink public facilities. We represent them in a form of visibility graph as objects of navigational decisions within the street network. To explore how urban facilities, streets and pedestrian visual cognition are interrelated, we create and compare three cases: a street network visibility graph and two visibility graphs of amenities. The first graph is based on the existing, “natural” distribution, while the second is an “artificial”, fabricated version of the environment, where urban locations are redistributed evenly across the case study. We study the graphs’ global network properties by the use of small-world, and scale-free models. Our results demonstrate that views available for an urban traveller in the existing, “natural” setting had a particular structure. It is built of numerous weakly connected locations coexisting with a small number of hubs with an exceptionally large number of visual connections. Such organisation of urban visibility shows that visuospatial network shares morphological similarities with other natural networks, suggesting that common organizational principles underlie network structure.

A recursive method for calculating the total number of spanning trees and its applications in self-similar small-world scale-free network models

The problem of determining and calculating the number of spanning trees of any finite graph (model) is a great challenge, and has been studied in various fields, such as discrete applied mathematics, theoretical computer science, physics, chemistry and the like. In this paper, firstly, thank to lots of real-life systems and artificial networks built by all kinds of functions and combinations among some simpler and smaller elements (components), we discuss some helpful network-operation, including link-operation and merge-operation, to design more realistic and complicated network models. Secondly, we present a method for computing the total number of spanning trees. As an accessible example, we apply this method to space of trees and cycles respectively, and our results suggest that it is indeed a better one for such models. In order to reflect more widely practical applications and potentially theoretical significance, we study the enumerating method in some existing scale-free network models. On the other hand, we set up a class of new models displaying scale-free feature, that is to say, following P(k) ~ k−γ, where γ is the degree exponent. Based on detailed calculation, the degree exponent γ of our deterministic scale-free models satisfies γ > 3. In the rest of our discussions, we not only calculate analytically the solutions of average path length, which indicates our models have small-world property being prevailing in amounts of complex systems, but also derive the number of spanning trees by means of the recursive method described in this paper, which clarifies our method is convenient to research these models.

Scale-Free Registrations in 3D: 7 Degrees of Freedom with Fourier Mellin SOFT Transforms

Fourier Mellin SOFT (FMS) as a novel method for global registration of 3D data is presented. It determines the seven degrees of freedom (7-DoF) transformation, i.e., the 6-DoF rigid motion parameters plus 1-DoF scale, between two scans, i.e., two noisy, only partially overlapping views on objects or scenes. It is based on a sequence of the 3D Fourier transform, the Mellin transform and the SO(3) Fourier transform. This combination represents a non-trivial complete 3D extension of the well known Fourier-Mellin registration for 2D images. It is accordingly based on decoupling rotation and scale from translation. First, rotation—which is the main challenge for the extension to 3D data - is tackled with a SO(3) Fourier Transform (SOFT) based on Spherical Harmonics. In a second step, scale is determined via a 3D Mellin transform. Finally, translation is calculated by Phase-Matching. Experiments are presented with simulated data sets for ground truth comparisons and with real world data including object recognition and localization in Magnetic Resonance Tomography (MRT) data, registration of 2.5D RGBD scans from a Microsoft Kinect with a scale-free 3D model generated by Multi-View Vision, and 3D mapping by registration of a sequence of consecutive scans from a low-cost actuated Laser Range Finder. The results show that the method is fast and that it can robustly handle partial overlap, interfering structures, and noise. It is also shown that the method is a very interesting option for 6-DoF registration, i.e., when scale is known.

Identification of Vital Nodes in Complex Network via Belief Propagation and Node Reinsertion

Vital nodes play a pivotal part of network structure and dynamics, where finding the minimal size of a set of vital nodes belongs to an NP-hard problem and cannot be solved by a polynomial algorithm. Recent studies of vital nodes identification mostly rely on the structural information, such as collective influence and degree. However, the performance of local-based methods varies for different structure, while the complexities of global-based methods are generally high for most situations. Here, we map the problem into an optimization issue based on global information of network structure and propose a belief propagation and node reinsertion (BPR) method with almost linear time complexity, where finding the minimum feeding back vertex set is a key. Compared with several state-of-the-art heuristic methods, the BPR method has advantages of high accuracy and practicability of vital nodes identification and low computational complexity. Under two attack schemes: static and dynamical, extensive experiments of Erdős–Renyi and scale-free models and real-world networks of the power grid and traffic network convincingly demonstrate that the BPR method remarkably outperforms other methods in vital nodes identification. This helps to reassess the operational risk of a network and improve robustness ranging from network design schemes, protection strategies to failure mitigation.

Accumulating Social Capital in an Online Urban Network: The Effects of User Behaviors

Social capital is often accumulated not only in ego-networks, but in larger communities. However, these communities, although seemingly visible through social media, are still under-researched. In this work, we examine аn online friendship network spanning over an entire middle-size city amounting to 194,601 users of VK social network site. We find that the share of user’s in-city friends contributes to his/her brokerage and information influence abilities. More importantly, the number of user’s online groups - that is, user’s access to diverse or disconnected communities - is positively related with his/her bridging capital, and negatively with the bonding capital. Finally, our research questions the validity of some of the existing social capital measures. Highlights • All-city network represents a mixture ofsmall-world and scale-free graph models • Longer SNS use gives a cumulative advantage for making additional friendship ties • Participation in more SNS groups increases a user's online bridging social capital • The number of likes on a user’s wall is positively associated with online bridging • The share of local friends among all user’s VK friends increases online bridging

Efficiency of quarantine and self-protection processes in epidemic spreading control on scale-free networks.

One of the most effective mechanisms to contain the spread of an infectious disease through a population is the implementation of quarantine policies. However, its efficiency is affected by different aspects, for example, the structure of the underlining social network where highly connected individuals are more likely to become infected; therefore, the speed of the transmission of the decease is directly determined by the degree distribution of the network. Another aspect that influences the effectiveness of the quarantine is the self-protection processes of the individuals in the population, that is, they try to avoid contact with potentially infected individuals. In this paper, we investigate the efficiency of quarantine and self-protection processes in preventing the spreading of infectious diseases over complex networks with a power-law degree distribution [ P(k)∼k-ν] for different ν values. We propose two alternative scale-free models that result in power-law degree distributions above and below the exponent ν = 3 associated with the conventional Barabási-Albert model. Our results show that the exponent ν determines the effectiveness of these policies in controlling the spreading process. More precisely, we show that for the ν exponent below three, the quarantine mechanism loses effectiveness. However, the efficiency is improved if the quarantine is jointly implemented with a self-protection process driving the number of infected individuals significantly lower.

Spatial utilization predicts animal social contact networks are not scale-free.

While heterogeneity in social behaviour has been described in many human contexts it is often assumed to be less common in the animal kingdom even though scale-free networks are observed. This homogeneity raises the question of whether the patterns of behaviour necessary to account for scale-free social contact networks, where the degree distribution follows a power law, i.e. a few individuals are very highly connected but most have only a few connections, occur in animals, or whether other mechanisms are needed to produce realistic contact network architectures. We develop a space-utilization model for individual animal behaviour to predict the individuals' social contact network. Using basic properties of the χ2 distribution we present a simple analytical result that allows the model to give a range of predictions with minimal computational effort. The model results are tested on data collected in New Zealand for the social contact networks of the wild brushtail possum (Trichosurus vulpecula). Our model provides a better prediction of network architecture than other simple models, including a scale-free model.

Enhanced storage capacity with errors in scale-free Hopfield neural networks: An analytical study.

The Hopfield model is a pioneering neural network model with associative memory retrieval. The analytical solution of the model in mean field limit revealed that memories can be retrieved without any error up to a finite storage capacity of O(N), where N is the system size. Beyond the threshold, they are completely lost. Since the introduction of the Hopfield model, the theory of neural networks has been further developed toward realistic neural networks using analog neurons, spiking neurons, etc. Nevertheless, those advances are based on fully connected networks, which are inconsistent with recent experimental discovery that the number of connections of each neuron seems to be heterogeneous, following a heavy-tailed distribution. Motivated by this observation, we consider the Hopfield model on scale-free networks and obtain a different pattern of associative memory retrieval from that obtained on the fully connected network: the storage capacity becomes tremendously enhanced but with some error in the memory retrieval, which appears as the heterogeneity of the connections is increased. Moreover, the error rates are also obtained on several real neural networks and are indeed similar to that on scale-free model networks.