Scale-free Topology Research Articles

Motif formation and industry specific topologies in the Japanese business firm network

Motifs and roles are basic quantities for the characterization of interactions among 3-node subsets in complex networks. In this work, we investigate how the distribution of 3-node motifs can be influenced by modifying the rules of an evolving network model while keeping the statistics of simpler network characteristics, such as the link density and the degree distribution, invariant. We exemplify this problem for the special case of the Japanese Business Firm Network, where a well-studied and relatively simple yet realistic evolving network model is available, and compare the resulting motif distribution in the real-world and simulated networks. To better approximate the motif distribution of the real-world network in the model, we introduce both subgraph dependent and global additional rules. We find that a specific rule that allows only for the merging process between nodes with similar link directionality patterns reduces the observed excess of densely connected motifs with bidirectional links. Our study improves the mechanistic understanding of motif formation in evolving network models to better describe the characteristic features of real-world networks with a scale-free topology.

Large fluctuations in anti-coordination games on scale-free graphs

We study the influence of the complex topology of scale-free graphs on the dynamics of anti-coordination games (e.g. snowdrift games). These reference models are characterized by the coexistence (evolutionary stable mixed strategy) of two competing species, say ‘cooperators’ and ‘defectors’, and, in finite systems, by metastability and large-fluctuation-driven fixation. In this work, we use extensive computer simulations and an effective diffusion approximation (in the weak selection limit) to determine under which circumstances, depending on the individual-based update rules, the topology drastically affects the long-time behavior of anti-coordination games. In particular, we compute the variance of the number of cooperators in the metastable state and the mean fixation time when the dynamics is implemented according to the voter model (death-first/birth-second process) and the link dynamics (birth/death or death/birth at random). For the voter update rule, we show that the scale-free topology effectively renormalizes the population size and as a result the statistics of observables depend on the network’s degree distribution. In contrast, such a renormalization does not occur with the link dynamics update rule and we recover the same behavior as on complete graphs.

Domination number and minimum dominating sets in pseudofractal scale-free web and Sierpiński graph

The minimum dominating set (MDS) problem is a fundamental subject of theoretical computer science, and has found vast applications in different areas, including sensor networks, protein interaction networks, and structural controllability. However, the determination of the size of a MDS and the number of all MDSs in a general network is NP-hard, and it thus makes sense to seek particular graphs for which the MDS problem can be solved analytically. In this paper, we study the MDS problem in the pseudofractal scale-free web and the Sierpiński graph, which have the same number of vertices and edges. For both networks, we determine explicitly the domination number, as well as the number of distinct MDSs. We show that the pseudofractal scale-free web has a unique MDS, and its domination number is only half of that for the Sierpiński graph, which has many MDSs. We argue that the scale-free topology is responsible for the difference of the size and number of MDSs between the two studied graphs, which in turn indicates that power-law degree distribution plays an important role in the MDS problem and its applications in scale-free networks.

Unraveling the contact patterns and network structure of pig shipments in the United States and its association with porcine reproductive and respiratory syndrome virus (PRRSV) outbreaks

The analysis of the pork value chain is becoming key to understanding the risk of infectious disease dissemination in the swine industry. In this study, we used social network analysis to characterize the swine shipment network structure and properties in a typical multisite swine production system in the US. We also aimed to evaluate the association between network properties and porcine respiratory and reproductive syndrome virus (PRRSV) transmission between production sites. We analyzed the 109,868 swine shipments transporting over 93 million swine between more than 500 production sites from 2012 to 2014. A total of 248 PRRSV positive occurrences were reported from 79 production sites during those 3 years. The temporal dynamics of swine shipments was evaluated by computing network properties in one-month and three-month networks. The association of PRRS occurrence in sow farms with centrality properties from one-month and three-month networks was assessed by using the multilevel logistic regression. All monthly networks showed a scale-free network topology with positive degree assortativity. The regression model revealed that out-degree centrality had a negative association with PRRS occurrence in sow farms in both one-month and three-month networks [OR=0.79 (95% CI, 0.63–0.99) in one-month network and 0.56 (95% CI, 0.36, 0.88) in three-month network] and in-closeness centrality model was positively associated with PRRS occurrence in sow farms in the three-month network [OR=2.45 (95% CI, 1.14–5.26)]. We also describe how the occurrence of porcine epidemic diarrheac (PED) outbreaks severely affected the network structure as well as the PRRS occurrence reports and its association with centrality measures in sow farms. The structure of the swine shipment network and the connectivity between production sites influenced on the PRRSV transmission. The use of network topology and characteristics combining with spatial analysis based on fine scale geographical location of production sites will be useful to inform the design of more cost-efficient, risk-based surveillance and control measures for PRRSV as well as other diseases in the US swine industry.

The effect of distributed time-delays on the synchronization of neuronal networks

Here we investigate the synchronization of networks of FitzHugh-Nagumo neurons coupled in scale-free, small-world and random topologies, in the presence of distributed time delays in the coupling of neurons. We explore how the synchronization transition is affected when the time delays in the interactions between pairs of interacting neurons are non-uniform. We find that the presence of distributed time-delays does not change the behavior of the synchronization transition significantly, vis-a-vis networks with constant time-delay, where the value of the constant time-delay is the mean of the distributed delays. We also notice that a normal distribution of delays gives rise to a transition at marginally lower coupling strengths, vis-a-vis uniformly distributed delays. These trends hold across classes of networks and for varying standard deviations of the delay distribution, indicating the generality of these results. So we conclude that distributed delays, which may be typically expected in real-world situations, do not have a notable effect on synchronization. This allows results obtained with constant delays to remain relevant even in the case of randomly distributed delays.

Double Push Strategy of Knowledge for Product Design Based on Complex Network Theory

Reasonable application of design knowledge can help improve the efficiency and quality of product design. Based on complex network theory, this study proposes a double push strategy of knowledge for product design. The proposal introduces the concept of attribute similarity and triangular fuzzy number and uses the theory and method of complex network to build the knowledge network model for product design that contains creative knowledge subnetwork and engineering knowledge subnetwork. This paper is to understand the structure and dynamics of the knowledge network model and to identify and predict knowledge nodes and knowledge groups strongly related to design intent in view of the scale-free network topology analysis theory. We develop a double push strategy of product design knowledge to implement the effective auxiliary function for product design process. Finally, a design case of antalgic pump is presented to demonstrate the practicability and validity of the strategy.

Robustness of First- and Second-Order Consensus Algorithms for a Noisy Scale-Free Small-World Koch Network

In this brief, we study first- and second-order consensus algorithms for the scale-free small-world Koch network, where vertices are subject to white noise. We focus on three cases of consensus schemes: 1) first-order leaderless algorithm; 2) first-order algorithm with a single leader; and 3) second-order leaderless algorithm. We are concerned with the coherence of the Koch network in the $H_{2}$ norm, which captures the level of agreement of vertices in face of stochastic disturbances. Based on the particular network construction, we derive explicit expressions of the coherence for all the three consensus algorithms, as well as their dependence on the network size. Particularly, for the first-order leader-follower model, we show that coherence relies on the shortest-path distance between the leader and the largest-degree vertices, as well as the degree of the leader. The asymptotic behaviors for coherence of the three consensus algorithms in Koch network behave differently from those associated with other networks lacking scale-free small-world features, indicating significant influences of the scale-free small-world topology on the performance of the consensus algorithms in noisy environments.

Analysis of Large-Scale Service Network Tolerance to Cascading Failure

The future Internet will be populated with a massive number of cooperating services due to the rapid growth of publicly available services and the adoption of service-oriented computing (SOC) into the Internet of Things. The adoption of SOC enables combining the functionalities of smart devices as combining services by means of service composition. These cooperating services form a large-scale service network where the nodes and the links represent services and the dependency between services, respectively. The dependency between services potentially causes cascading failure, where the failure of a service propagates to its dependent services. Due to the lack of research in this type of cascading failure, we analyzed cascading failure in service networks for different topology and different degree of service interdependency. We found that the number of cascading failure is somewhat linear to the average number of required services, and decays exponentially over the average number of alternate services. The latter suggests that cascading failure tolerance can be significantly improved by adding few alternate services to each required service if the average number of alternate services is currently low. In addition, we also found that scale-free topology provides better tolerance, subsequently followed by exponential and random topology.

On Application of Least-delay Variation Problem in Ethernet Networks Using SDN Concept

The goal of this paper is to present an application idea of SDN in Smart Grids, particularly, in the area of L2 multicast as defined by IEC 61850-9-2. Authors propose an Integer Linear Formulation (ILP) dealing with a Least-Delay-Variation multicast forwarding problem that has a potential to utilize Ethernet networks in a new way. The proposed ILP formulation is numerically evaluated on random graph topologies and results are compared to a shortest path tree approach that is traditionally a product of Spanning Tree Protocols. Results confirm the correctness of the ILP formulation and illustrate dependency of a solution quality on the selected graph models, especially, in a case of scale-free topologies.

Fixed-points in random Boolean networks: The impact of parallelism in the Barabási–Albert scale-free topology case

Fixed points are fundamental states in any dynamical system. In the case of gene regulatory networks (GRNs) they correspond to stable genes profiles associated to the various cell types. We use Kauffman's approach to model GRNs with random Boolean networks (RBNs). In this paper we explore how the topology affects the distribution of the number of fixed points in randomly generated networks. We also study the size of the basins of attraction of these fixed points if we assume the α-asynchronous dynamics (where every node is updated independently with probability 0≤α≤1). It is well-known that asynchrony avoids the cyclic attractors into which parallel dynamics tends to fall. We observe the remarkable property that, in all our simulations, if for a given RBN with Barabási–Albert topology and α-asynchronous dynamics an initial configuration reaches a fixed point, then every configuration also reaches a fixed point. By contrast, in the parallel regime, the percentage of initial configurations reaching a fixed point (for the same networks) is dramatically smaller. We contrast the results of the simulations on Barabási–Albert networks with the classical Erdös–Rényi model of random networks. Everything indicates that Barabási–Albert networks are extremely robust. Finally, we study the mean and maximum time/work needed to reach a fixed point when starting from randomly chosen initial configurations.

Reporting a network’s most-central actor with a confidence level

This article introduces a confidence level (CL) statistic to accompany the identification of the most central actor in relational, social network data. CL is the likelihood that the most-central actor assertion is correct in light of imperfect network data. The CL value is derived from a frequency-based probability according to perturbed samples of feature-equivalent network data. Analysts often focus attention towards the most central, highest valued, top actor [or node] according to one of four traditional measures: degree, betweenness, closeness or eigenvector centrality. However, given that collected social network data often has missing relational links, the correctness of the top-actor claim becomes uncertain. This paper describes and illustrates a practical approach for estimating and applying a CL to the top-actor identification task. We provide a simple example of the technique used to derive a posterior probability, then apply the same approach to larger, more pragmatic random network by using the results of an extensive virtual experiment involving uniform random and scale-free topologies. This article has implications in organizational practice and theory; it is simple and lays groundwork for developing more intricate estimates of reliability for other network measures.

A topological insight into restricted Boltzmann machines

Restricted Boltzmann Machines (RBMs) and models derived from them have been successfully used as basic building blocks in deep artificial neural networks for automatic features extraction, unsupervised weights initialization, but also as density estimators. Thus, their generative and discriminative capabilities, but also their computational time are instrumental to a wide range of applications. Our main contribution is to look at RBMs from a topological perspective, bringing insights from network science. Firstly, here we show that RBMs and Gaussian RBMs (GRBMs) are bipartite graphs which naturally have a small-world topology. Secondly, we demonstrate both on synthetic and real-world datasets that by constraining RBMs and GRBMs to a scale-free topology (while still considering local neighborhoods and data distribution), we reduce the number of weights that need to be computed by a few orders of magnitude, at virtually no loss in generative performance. Thirdly, we show that, for a fixed number of weights, our proposed sparse models (which by design have a higher number of hidden neurons) achieve better generative capabilities than standard fully connected RBMs and GRBMs (which by design have a smaller number of hidden neurons), at no additional computational costs.

Investigation of similarity and diversity threshold networks generated from diversity-oriented and focused chemical libraries

Topological properties of chemical library networks, such as the average clustering coefficient, average path length, and existence of hubs, can serve as indicators to describe the inherent complexities of chemical libraries. We have used Diversity-Oriented Synthesis (DOS) and Focussed Libraries to investigate the appearance of scale-free properties and absence of small-world behavior in chemical libraries. DOS aims to elicit structural complexity in small compounds with respect to skeleton, functional groups, appendages and stereochemistry. Complexity here indicates incorporation of $$\hbox {sp}^{3}$$ carbons, hydrogen bond acceptors and donors in the molecule. Biological studies have shown how structural complexity enhances the interaction of molecules with complex biological macromolecules. In contrast, Focussed Libraries concentrate on specific scaffolds against a specific biological target. We have quantified the diversity in several DOS and Focussed Libraries based on properties of similarity and dissimilarity threshold networks formed from them. Similarity and dissimilarity networks were generated from diverse chemical libraries at various Tanimoto similarity coefficients ( $$\hbox {t}_{\mathrm{c}})$$ using FP2 and MACCS fingerprints. The dissimilarity networks at very low $$\hbox {t}_{\mathrm{c}}$$ threshold led to the absence of small-world behaviors, as evidenced by low average clustering coefficient and high average path length in comparison to Erdos–Renyi networks. Dissimilarity networks exhibit scale free topology as evidenced by a power law degree distribution. The similarity networks at high $$\hbox {t}_{\mathrm{c}}$$ threshold have shown high clustering coefficients and low average path lengths, without the appearance of hubs. Combining dissimilarity and similarity threshold graphs revealed assortative and dissortative behaviors in the DOS libraries, leading to the conclusion that the vertices of the dissimilarity communities are more likely to share similarity edges, but it is quite unlikely for the vertices in a similarity community to share dissimilarity edges. We propose a simple and convenient diversity quantification tool, QuaLDI (Quantitative Library Diversity Index) to quantify the diversity in DOS and Focussed libraries. We anticipate that these topological properties can be used as descriptors to quantify the diversity in chemical libraries before proceeding for synthesis.

Towards limited scale-free topology with dynamic peer participation

Growth models have been proposed for constructing the scale-free overlay topology to improve the performance of unstructured peer-to-peer (P2P) networks. However, previous growth models are able to maintain the limited scale-free topology when nodes only join but do not leave the network; the case of nodes leaving the network while preserving a precise scaling parameter is not included in the solution. Thus, the full dynamic of node participation, inherent in P2P networks, is not considered in these models. In order to handle both nodes joining and leaving the network, we propose a robust growth model E-SRA, which is capable of producing the perfect limited scale-free overlay topology with user-defined scaling parameter and hard cut-off. Scalability of our approach is ensured since no global information is required to add or remove a node. E-SRA is also tolerant to individual node failure caused by errors or attacks. Simulations have shown that E-SRA outperforms other growth models by producing topologies with high adherence to the desired scale-free property. Search algorithms, including flooding and normalized flooding, achieve higher efficiency over the topologies produced by E-SRA.

Noise Response Data Reveal Novel Controllability Gramian for Nonlinear Network Dynamics.

Control of nonlinear large-scale dynamical networks, e.g., collective behavior of agents interacting via a scale-free connection topology, is a central problem in many scientific and engineering fields. For the linear version of this problem, the so-called controllability Gramian has played an important role to quantify how effectively the dynamical states are reachable by a suitable driving input. In this paper, we first extend the notion of the controllability Gramian to nonlinear dynamics in terms of the Gibbs distribution. Next, we show that, when the networks are open to environmental noise, the newly defined Gramian is equal to the covariance matrix associated with randomly excited, but uncontrolled, dynamical state trajectories. This fact theoretically justifies a simple Monte Carlo simulation that can extract effectively controllable subdynamics in nonlinear complex networks. In addition, the result provides a novel insight into the relationship between controllability and statistical mechanics.

Exploring pig trade patterns to inform the design of risk-based disease surveillance and control strategies.

An understanding of the patterns of animal contact networks provides essential information for the design of risk-based animal disease surveillance and control strategies. This study characterises pig movements throughout England and Wales between 2009 and 2013 with a view to characterising spatial and temporal patterns, network topology and trade communities. Data were extracted from the Animal and Plant Health Agency (APHA)’s RADAR (Rapid Analysis and Detection of Animal-related Risks) database, and analysed using descriptive and network approaches. A total of 61,937,855 pigs were moved through 872,493 movements of batches in England and Wales during the 5-year study period. Results show that the network exhibited scale-free and small-world topologies, indicating the potential for diseases to quickly spread within the pig industry. The findings also provide suggestions for how risk-based surveillance strategies could be optimised in the country by taking account of highly connected holdings, geographical regions and time periods with the greatest number of movements and pigs moved, as these are likely to be at higher risk for disease introduction. This study is also the first attempt to identify trade communities in the country, information which could be used to facilitate the pig trade and maintain disease-free status across the country in the event of an outbreak.

Self-Organized Optimization of Transport on Complex Networks**Supported by the Education Foundation of Hubei Province under Grant No D20120104.

We propose a self-organized optimization mechanism to improve the transport capacity of complex gradient networks. We find that, regardless of network topology, the congestion pressure can be strongly reduced by the self-organized optimization mechanism. Furthermore, the random scale-free topology is more efficient to reduce congestion compared with the random Poisson topology under the optimization mechanism. The reason is that the optimization mechanism introduces the correlations between the gradient field and the local topology of the substrate network. Due to the correlations, the cutoff degree of the gradient network is strongly reduced and the number of the nodes exerting their maximal transport capacity consumedly increases. Our work presents evidence supporting the idea that scale-free networks can efficiently improve their transport capacity by self-organized mechanism under gradient-driven transport mode.

Time-varying causal network of the Korean financial system based on firm-specific risk premiums

The aim of this paper is to investigate the Korean financial system based on time-varying causal network. We discover many stylized facts by utilizing the firm-specific risk premiums for measuring the causality direction from a firm to firm. At first, we discover that the interconnectedness of causal network is affected by the outbreak of financial events; the co-movement of firm-specific risk premium is strengthened after each positive event, and vice versa. Secondly, we find that the major sector of the Korean financial system is the Depositories, and the financial reform in June-2011 achieves its purpose by weakening the power of risk-spillovers of Broker-Dealers. Thirdly, we identify that the causal network is a small-world network with scale-free topology where the power-law exponents of out-Degree and negative event are more significant than those of in-Degree and positive event. Lastly, we discuss that the current aspects of causal network are closely related to the long-term future scenario of the KOSPI Composite index where the direction and stability are significantly affected by the power of risk-spillovers and the power-law exponents of degree distributions, respectively.

Transport optimization on complex gradient networks

We present a strategy of the optimization of transport on complex gradient networks by introducing the correlations between the gradient field and the local topology of networks. In this strategy, the value hi of the gradient field at node i is determined by the degree kj of its own neighborsj, that is, hi=∑jkj−α(α is control parameter). We find that there exists the optimal values of α, where the structural correlated gradient field can strongly reduce jamming in both random and scale-free networks. Furthermore, the optimal scale-free networks are less congested than optimal random networks for average degree〈k〉>2. In addition, by comparing the transport capacity of network under different routing rules ranging from random diffusion to gradient-driven transport, we find that gradient-driven transport on the scale-free networks can hold the maximal transport capacity when the gradient field is optimal correlated. All these observations present the evidence supporting the idea that the scale-free topology and gradient-driven transport mode are beneficial to improve the transport capacity of networks.

Multiplex congruence network of natural numbers

Congruence theory has many applications in physical, social, biological and technological systems. Congruence arithmetic has been a fundamental tool for data security and computer algebra. However, much less attention was devoted to the topological features of congruence relations among natural numbers. Here, we explore the congruence relations in the setting of a multiplex network and unveil some unique and outstanding properties of the multiplex congruence network. Analytical results show that every layer therein is a sparse and heterogeneous subnetwork with a scale-free topology. Counterintuitively, every layer has an extremely strong controllability in spite of its scale-free structure that is usually difficult to control. Another amazing feature is that the controllability is robust against targeted attacks to critical nodes but vulnerable to random failures, which also differs from ordinary scale-free networks. The multi-chain structure with a small number of chain roots arising from each layer accounts for the strong controllability and the abnormal feature. The multiplex congruence network offers a graphical solution to the simultaneous congruences problem, which may have implication in cryptography based on simultaneous congruences. Our work also gains insight into the design of networks integrating advantages of both heterogeneous and homogeneous networks without inheriting their limitations.