Scale-free Model Research Articles

COMPLEX NETWORK SIMULATION OF FOREST NETWORK SPATIAL PATTERN IN PEARL RIVER DELTA

Abstract. Forest network-construction uses for the method and model with the scale-free features of complex network theory based on random graph theory and dynamic network nodes which show a power-law distribution phenomenon. The model is suitable for ecological disturbance by larger ecological landscape Pearl River Delta consistent recovery. Remote sensing and GIS spatial data are available through the latest forest patches. A standard scale-free network node distribution model calculates the area of forest network’s power-law distribution parameter value size; The recent existing forest polygons which are defined as nodes can compute the network nodes decaying index value of the network’s degree distribution. The parameters of forest network are picked up then make a spatial transition to GIS real world models. Hence the connection is automatically generated by minimizing the ecological corridor by the least cost rule between the near nodes. Based on scale-free network node distribution requirements, select the number compared with less, a huge point of aggregation as a future forest planning network’s main node, and put them with the existing node sequence comparison. By this theory, the forest ecological projects in the past avoid being fragmented, scattered disorderly phenomena. The previous regular forest networks can be reduced the required forest planting costs by this method. For ecological restoration of tropical and subtropical in south China areas, it will provide an effective method for the forest entering city project guidance and demonstration with other ecological networks (water, climate network, etc.) for networking a standard and base datum.

Vertex Entropy As a Critical Node Measure in Network Monitoring

Understanding which node failures in a network have more impact is an important problem. Current understanding, motivated by the scale free models of network growth, places emphasis on the degree of the node. This is not a satisfactory measure; the number of connections a node has does not capture how redundantly it is connected into the whole network. Conversely, the structural entropy of a graph captures the resilience of a network well, but is expensive to compute, and, being a global measure, does not attribute any specific value to a given node. This lack of locality prevents the use of global measures as a way of identifying critical nodes. In this paper, we introduce local vertex measures of entropy which do not suffer from such drawbacks. In our theoretical analysis, we establish the possibility that our local vertex measures approximate global entropy, with the advantage of locality and ease of computation. We establish properties that vertex entropy must have in order to be useful for identifying critical nodes. We have access to a proprietary event, topology, and incident dataset from a large commercial network. Using this dataset, we demonstrate a strong correlation between vertex entropy and incident generation over events.

A quality-of-service-aware dynamic evolution model for space–ground integrated network

Space–ground integrated network, a strategic, driving, and irreplaceable infrastructure, guarantees the development of economic and national security. However, its natures of limited resources, frequent handovers, and intermittently connected links significantly reduce the quality of service. To address this issue, a quality-of-service-aware dynamic evolution model is proposed based on complex network theory. On one hand, a quality-of-service-aware strategy is adopted in the model. During evolution phases of growth and handovers, links are established or deleted according to the quality-of-service-aware preferential attachment following the rule of better quality of service getting richer and worse quality of service getting poor or to die. On the other hand, dynamic handover of nodes and intermittent connection of links are taken into account and introduced into the model. Meanwhile, node heterogeneity is analyzed and heterogeneous nodes are endowed with discriminate interactions. Theoretical analysis and simulations are utilized to explore the degree distribution and its characteristics. Results reveal that this model is a scale-free model with drift power-law distribution, fat-tail and small-world effect, and drift character of degree distribution results from dynamic handover. Furthermore, this model exerts well fault tolerance and attack resistance compared to signal-strength-based strategy. In addition, node heterogeneity and quality-of-service-aware strategy improve the attack resistance and overall quality of service of space–ground integrated network.

Stable clustering and the resolution of dissipationless cosmological N-body simulations

The determination of the resolution of cosmological N-body simulations, i.e., the range of scales in which quantities measured in them represent accurately the continuum limit, is an important open question. We address it here using scale-free models, for which self-similarity provides a powerful tool to control resolution. Such models also provide a robust testing ground for the so-called stable clustering approximation, which gives simple predictions for them. Studying large N-body simulations of such models with different force smoothing, we find that these two issues are in fact very closely related: our conclusion is that the accuracy of two point statistics in the non-linear regime starts to degrade strongly around the scale at which their behaviour deviates from that predicted by the stable clustering hypothesis. Physically the association of the two scales is in fact simple to understand: stable clustering fails to be a good approximation when there are strong interactions of structures (in particular merging) and it is precisely such non-linear processes which are sensitive to fluctuations at the smaller scales affected by discretisation. Resolution may be further degraded if the short distance gravitational smoothing scale is larger than the scale to which stable clustering can propagate. We examine in detail the very different conclusions of studies by Smith et al. (2003) and Widrow et al. (2009} and find that the strong deviations from stable clustering reported by these works are the results of over-optimistic assumptions about scales resolved accurately by the measured power spectra, and the reliance on Fourier space analysis. We emphasise the much poorer resolution obtained with the power spectrum compared to the two point correlation function..

Cascading failures of interdependent networks with different k-core structures

As one of the most common mesoscale structures in real-life networks, k-core hierarchical structure has attracted a lot of attention. Recent research about k-core always focuses on detecting influential nodes determining failure or epidemic propagation. However, few studies have attempted to understand how k-core structural properties can affect dynamic characteristics of network. In this paper, the influences of depth and coupling preferences of k-core on the cascading failures of interdependent scale-free networks are investigated. First, k-core structures of some real-life networks are analyzed, and a scale-free network evolution model with rich and successive k-core layers is proposed. Then, based on a load-based cascading model, the influence of the depth of k-core is investigated with a new evaluation index. In the end, two coupling preferences are analyzed, i.e. random coupling (RC) and assortative coupling (AC). Results show that the lower the depth is, the more robust the interdependent networks will be, and we find AC and RC perform dissimilarly when the capacity varies. Furthermore, all the effects will be affected by the initial load.

On scale-free extensions of massive (bi-)gravity

We discuss a scale-free model of bigravity, in which the mass parameter of the standard bigravity potential is promoted to a dynamical scalar field. This modification retains the ghost-free bigravity structure, in particular it remains free of the Boulware-Deser ghost. We investigate the theory’s interaction structure, focusing on its consistent scaling limits and strong coupling scales. Furthermore we explore the model’s quadratic action, both around generic background configurations and paying special attention to cosmological backgrounds and to the associated background evolution. Finally we consider the possibility of realising a phase of late-time acceleration as well as a quasi-de Sitter inflationary stage at early times, when the promoted “mass scalar” becomes the inflaton.

The research on scale-free characteristics of hypernetworks

The scale-free model is a pioneering study on complex networks. In the real world, most networks diaplay the scale-free characteristics. The formation mechanism of the scale-free networks in complex ones is based on two generic mechanisms: growth and degree preferential attachment. In this paper, it systematically studied the scale-free hypernetwork formation mechanism on the basis of the hypergraph theory. As a result, it also finds that the scale-free hypernetwork has something to with hyperedge growth and hyperdegree preferential attachment. These two factors are indispensible. This result can be used as a reference for the cause of the scale-free hypernetwork and the judgment of the scale-free hypernetwork in reality.

Gossip spread in social network Models

Gossip almost inevitably arises in real social networks. In this article we investigate the relationship between the number of friends of a person and limits on how far gossip about that person can spread in the network. How far gossip travels in a network depends on two sets of factors: (a) factors determining gossip transmission from one person to the next and (b) factors determining network topology. For a simple model where gossip is spread among people who know the victim it is known that a standard scale-free network model produces a non-monotonic relationship between number of friends and expected relative spread of gossip, a pattern that is also observed in real networks (Lind et al., 2007). Here, we study gossip spread in two social network models (Toivonen et al., 2006; Vázquez, 2003) by exploring the parameter space of both models and fitting them to a real Facebook data set. Both models can produce the non-monotonic relationship of real networks more accurately than a standard scale-free model while also exhibiting more realistic variability in gossip spread. Of the two models, the one given in Vázquez (2003) best captures both the expected values and variability of gossip spread.

Toward Energy-Efficient and Robust Large-Scale WSNs: A Scale-Free Network Approach

Due to the limited battery power of sensor nodes and harsh deployment environment, it is of fundamental importance and a great challenge to achieve high energy efficiency and strong robustness in large-scale wireless sensor networks (LS-WSNs). To this end, we propose two self-organizing schemes for LS-WSNs. The first scheme is the energy-aware common neighbor scheme, which considers the neighborhood overlap in link establishment. The second scheme is energy-aware low potential-degree common neighbor (ELDCN) scheme, which considers both neighborhood overlap in topology formation and the potential degrees of common neighbors. Both schemes generate clustering-based and scale-free-inspired LS-WSNs, which are energy-efficient and robust. However, the ELDCN scheme shows higher energy efficiency and stronger robustness to node failures, because it avoids establishing links to hub-nodes with high potential connectivity. Analytical and simulation results demonstrate that our proposed schemes outperform the existing scale-free evolution models in terms of energy efficiency and robustness.

The complex hierarchical topology of EEG functional connectivity

BackgroundUnderstanding the complex hierarchical topology of functional brain networks is a key aspect of functional connectivity research. Such topics are obscured by the widespread use of sparse binary network models which are fundamentally different to the complete weighted networks derived from functional connectivity. New methodsWe introduce two techniques to probe the hierarchical complexity of topologies. Firstly, a new metric to measure hierarchical complexity; secondly, a Weighted Complex Hierarchy (WCH) model. To thoroughly evaluate our techniques, we generalise sparse binary network archetypes to weighted forms and explore the main topological features of brain networks – integration, regularity and modularity – using curves over density. ResultsBy controlling the parameters of our model, the highest complexity is found to arise between a random topology and a strict ‘class-based’ topology. Further, the model has equivalent complexity to EEG phase-lag networks at peak performance. Comparison to existing methodsHierarchical complexity attains greater magnitude and range of differences between different networks than the previous commonly used complexity metric and our WCH model offers a much broader range of network topology than the standard scale-free and small-world models at a full range of densities. ConclusionsOur metric and model provide a rigorous characterisation of hierarchical complexity. Importantly, our framework shows a scale of complexity arising between ‘all nodes are equal’ topologies at one extreme and ‘strict class-based’ topologies at the other.

Multifractal cross-correlation effects in two-variable time series of complex network vertex observables.

We investigate the scaling of the cross-correlations calculated for two-variable time series containing vertex properties in the context of complex networks. Time series of such observables are obtained by means of stationary, unbiased random walks. We consider three vertex properties that provide, respectively, short-, medium-, and long-range information regarding the topological role of vertices in a given network. In order to reveal the relation between these quantities, we applied the multifractal cross-correlation analysis technique, which provides information about the nonlinear effects in coupling of time series. We show that the considered network models are characterized by unique multifractal properties of the cross-correlation. In particular, it is possible to distinguish between Erdös-Rényi, Barabási-Albert, and Watts-Strogatz networks on the basis of fractal cross-correlation. Moreover, the analysis of protein contact networks reveals characteristics shared with both scale-free and small-world models.

Attack Vulnerability of Network Controllability.

Controllability of complex networks has attracted much attention, and understanding the robustness of network controllability against potential attacks and failures is of practical significance. In this paper, we systematically investigate the attack vulnerability of network controllability for the canonical model networks as well as the real-world networks subject to attacks on nodes and edges. The attack strategies are selected based on degree and betweenness centralities calculated for either the initial network or the current network during the removal, among which random failure is as a comparison. It is found that the node-based strategies are often more harmful to the network controllability than the edge-based ones, and so are the recalculated strategies than their counterparts. The Barabási-Albert scale-free model, which has a highly biased structure, proves to be the most vulnerable of the tested model networks. In contrast, the Erdős-Rényi random model, which lacks structural bias, exhibits much better robustness to both node-based and edge-based attacks. We also survey the control robustness of 25 real-world networks, and the numerical results show that most real networks are control robust to random node failures, which has not been observed in the model networks. And the recalculated betweenness-based strategy is the most efficient way to harm the controllability of real-world networks. Besides, we find that the edge degree is not a good quantity to measure the importance of an edge in terms of network controllability.

Scaling of weighted spectral distribution in deterministic scale-free networks

Scale-free networks are abundant in the real world. In this paper, we investigate the scaling properties of the weighted spectral distribution in several deterministic and stochastic models of evolving scale-free networks. First, we construct a new deterministic scale-free model whose node degrees have a unified format. Using graph structure features, we derive a precise formula for the spectral metric in this model. This formula verifies that the spectral metric grows sublinearly as network size (i.e., the number of nodes) grows. Additionally, the mathematical reasoning of the precise formula theoretically provides detailed explanations for this scaling property. Finally, we validate the scaling properties of the spectral metric using some stochastic models. The experimental results show that this scaling property can be retained regardless of local world, node deleting and assortativity adjustment.

A scale-free topology model with fault-tolerance and intrusion-tolerance in wireless sensor networks

The scale-free topology is robust when confronted with random faults, but it is fragile when confronted with selective remove attacks. In this paper, we propose a new scale-free topology model which has both fault-tolerance against random faults and intrusion-tolerance against selective remove attacks at the same time. Then the mathematical expression of the topological degree distribution is derived. Through analyzing the effect of topological degree distribution on these properties of topological fault-tolerance and topological intrusion-tolerance, the optimal scale-free topology which can keep the fault-tolerance and maximize intrusion-tolerance is obtained. We performed extensive experiments on the proposed model and compared it with other existing models. The simulation results show that the new scale-free topology model can keep the character that the scale-free topology has a stronger robustness to random faults. And it also can reduce their fragility for selective remove attacks and further prolong its lifetime.

Coalitionally stable pricing schemes for inter-domain forwarding

In this work, we model and analyze the problem of stable and efficient pricing for inter-domain traffic routing in the future Internet. We consider a general network topology with multiple sources and sinks of traffic, organized into separate domains managed by Internet Service Providers (ISPs) solely interested in maximizing their own profit. In this framework, we prove that there exists a pricing scheme that attains network-wide efficiency and is yet coalitionally stable, where the coalitions correspond to the ISPs that are acting in self-interest. This implies that this pricing scheme not only maximizes the overall utility of the resulting traffic flows, but is also such that ISPs cannot expect to improve their profit through deviation from it, even if multiple ISPs deviate at the same time. Through simulations on scale-free preferential attachment network topology models as well as actual inter-domain topologies obtained from the CAIDA database, we evaluate the convergence of best-response based simple price updates, and show that they quickly attain near-optimal network utility in these network topologies.

Investigating the relationship between [formula omitted]-core and [formula omitted]-core network decompositions

Network decomposition methods, such as the much used k-core analysis, are able to identify globally central regions of networks. The decomposition approaches are hierarchical and identify nested sets of nodes with increasing centrality properties. While most studies have been concerned with unweighted networks, i.e. k-core analysis, recent works have introduced network decomposition methods that apply to weighted networks. Here, we investigate the relationship between k-core decomposition for unweighted networks and s-core decomposition for weighted networks by systematically employing a link-weight scheme that gradually discretizes the link weights. We applied this approach to the Erdős–Rényi model and the scale-free configuration model for five different weight distributions, and two empirical networks, the US air traffic network and a Facebook network. We find that (1) both uniformly random and positively correlated link-weight distributions give rise to highly stable s-core decompositions with respect to discretization levels. (2) For negatively correlated link-weight distributions, the resulting s-core decomposition has no similarity to the k-cores. Since several combinations of network topology and link-weight distributions give rise to a core-structure that is highly similar to the full s-core for a large range of link-discretization levels, it is possible to significantly speed up the numerical s-core analysis for these situations.

Complex Network Theory and Its Application Research on P2P Networks

Abstract Advances in complex networks of Peer-to-Peer (P2P) networks were reviewed and summarized. The paper outlines some important topological properties such as degree, average path length and clustering coefficient at first, and then three kinds of most important network mechanism models are introduced, including random graph model, small world model and scale-free model. A simple description about research status for P2P networks based on complex networks is made from three aspects: positive research, network mechanism model, network broadcast and control. Some developing prospects of complex networks of P2P are pointed out finally. Complex network provides new ideas and methods to deal with many complex problems including P2P networks.

Power-law ansatz in complex systems: Excessive loss of information.

The ubiquity of power-law relations in empirical data displays physicists' love of simple laws and uncovering common causes among seemingly unrelated phenomena. However, many reported power laws lack statistical support and mechanistic backings, not to mention discrepancies with real data are often explained away as corrections due to finite size or other variables. We propose a simple experiment and rigorous statistical procedures to look into these issues. Making use of the fact that the occurrence rate and pulse intensity of crumple sound obey a power law with an exponent that varies with material, we simulate a complex system with two driving mechanisms by crumpling two different sheets together. The probability function of the crumple sound is found to transit from two power-law terms to a bona fide power law as compaction increases. In addition to showing the vicinity of these two distributions in the phase space, this observation nicely demonstrates the effect of interactions to bring about a subtle change in macroscopic behavior and more information may be retrieved if the data are subject to sorting. Our analyses are based on the Akaike information criterion that is a direct measurement of information loss and emphasizes the need to strike a balance between model simplicity and goodness of fit. As a show of force, the Akaike information criterion also found the Gutenberg-Richter law for earthquakes and the scale-free model for a brain functional network, a two-dimensional sandpile, and solar flare intensity to suffer an excessive loss of information. They resemble more the crumpled-together ball at low compactions in that there appear to be two driving mechanisms that take turns occurring.

A new method optimizing the subgraph centrality of large networks

Since many realistic networks such as wireless sensor/ad hoc networks usually do not agree very well with the basic network models such as small-word and scale-free models, we often need to obtain some expected structural features such as a small average path length and a regular degree distribution while optimizing the connectivity of these networks. Although a minor addition of links for optimizing network connectivity is not likely to change the structural properties of a network, it is necessary to investigate the impact of link addition on network properties as the number of the added links increases. However, to the best of our knowledge, the study of that problem has not been found so far. Furthermore, two closely related questions to that problem, i.e., how to measure and how to improve network connectivity, have not been studied carefully enough yet. To address the three problems above, the authors derive a better measure of network connectivity for large networks and a new strategy that can increase/decrease network connectivity the most, and propose a spectral density algorithm optimizing the connectivity of large networks, which is able to indicate the impact on the structural properties of a network while increasing/decreasing its connectivity, providing us a guided optimization of network connectivity. In other words, our algorithm can optimize not only the connectivity of a large network but also its structural features. Meanwhile, our new findings about spectral density are also concluded in this paper. In addition, we may also apply this algorithm to solve all eigenvalues of an N×N matrix, with a low complexity of O(N2) at most.

Generalized scale-free homophilic network

We propose a simple network growth process where the preferential attachment contains two essential parameters: homophily, namely, the tendency of sites to link with similar ones, and the number of attaching neighbors. It jointly generalizes the Barabási–Albert model and the scale-free homophilic model with a control parameter which tunes the importance of the homophily on preferential attachment process. Our results support a detailed discussion about different kinds of correlation, in special a fitness correlation introduced in this paper, and comparisons between BA model, scale-free homophilic model, and our present model considering its topological properties: degree distribution, time dependence of the connectivity and clustering coefficient.