Scale-free Small-world Networks Research Articles

Greedy routing optimisation in hyperbolic networks

Finding the optimal embedding of networks into low-dimensional hyperbolic spaces is a challenge that received considerable interest in recent years, with several different approaches proposed in the literature. In general, these methods take advantage of the exponentially growing volume of the hyperbolic space as a function of the radius from the origin, allowing a (roughly) uniform spatial distribution of the nodes even for scale-free small-world networks, where the connection probability between pairs decays with hyperbolic distance. One of the motivations behind hyperbolic embedding is that optimal placement of the nodes in a hyperbolic space is widely thought to enable efficient navigation on top of the network. According to that, one of the measures that can be used to quantify the quality of different embeddings is given by the fraction of successful greedy paths following a simple navigation protocol based on the hyperbolic coordinates. In the present work, we develop an optimisation scheme for this score in the native disk representation of the hyperbolic space. This optimisation algorithm can be either used as an embedding method alone, or it can be applied to improve this score for embeddings obtained from other methods. According to our tests on synthetic and real networks, the proposed optimisation can considerably enhance the success rate of greedy paths in several cases, improving the given embedding from the point of view of navigability.

Nearest-neighbor directed random hyperbolic graphs.

Undirected hyperbolic graph models have been extensively used as models of scale-free small-world networks with high clustering coefficient. Here we presented a simple directed hyperbolic model where nodes randomly distributed on a hyperbolic disk are connected to a fixed number m of their nearest spatial neighbors. We introduce also a canonical version of this network (which we call "network with varied connection radius"), where maximal length of outgoing bond is space dependent and is determined by fixing the average out-degree to m. We study local bond length, in-degree, and reciprocity in these networks as a function of spacial coordinates of the nodes and show that the network has a distinct core-periphery structure. We show that for small densities of nodes the overall in-degree has a truncated power-law distribution. We demonstrate that reciprocity of the network can be regulated by adjusting an additional temperature-like parameter without changing other global properties of the network.

Optimal Scale-Free Small-World Graphs with Minimum Scaling of Cover Time

The cover time of random walks on a graph has found wide practical applications in different fields of computer science, such as crawling and searching on the World Wide Web and query processing in sensor networks, with the application effects dependent on the behavior of the cover time: the smaller the cover time, the better the application performance. It was proved that over all graphs withNnodes, complete graphs have the minimum cover timeNlogN. However, complete graphs cannot mimic real-world networks with small average degree and scale-free small-world properties, for which the cover time has not been examined carefully, and its behavior is still not well understood. In this article, we first experimentally evaluate the cover time for various real-world networks with scale-free small-world properties, which scales asNlogN. To better understand the behavior of the cover time for real-world networks, we then study the cover time of three scale-free small-world model networks by using the connection between cover time and resistance diameter. For all the three networks, their cover time also behaves asNlogN. This work indicates that sparse networks with scale-free and small-world topology are favorable architectures with optimal scaling of cover time. Our results deepen understanding the behavior of cover time in real-world networks with scale-free small-world structure, and have potential implications in the design of efficient algorithms related to cover time.

Quantitative method for resilience assessment framework of airport network during COVID-19

The resilience and vulnerability of airport networks are significant challenges during the COVID-19 global pandemic. Previous studies considered node failure of networks under natural disasters and extreme weather. Herein, we propose a complex network methodology combined with data-driven to assess the resilience of airport networks toward global-scale disturbance using the Chinese airport network (CAN) and the European airport network (EAN) as a case study. The assessment framework includes vulnerability and resilience analyses from the network- and node-level perspectives. Subsequently, we apply the framework to analyze the airport networks in China and Europe. Specifically, real air traffic data for 232 airports in China and 82 airports in Europe are selected to form the CAN and EAN, respectively. The complex network analysis reveals that the CAN and the EAN are scale-free small-world networks, that are resilient to random attacks. However, the connectivity and vulnerability of the CAN are inferior to those of the EAN. In addition, we select the passenger throughput from the top-50 airports in China and Europe to perform a comparative analysis. By comparing the resilience evaluation of individual airports, we discovered that the factors of resilience assessment of an airport network for global disturbance considers the network metrics and the effect of government policy in actual operations. Additionally, this study also proves that a country’s emergency response-ability towards the COVID-19 has a significantly affectes the recovery of its airport network.

Nonlinear time series prediction algorithm based on AD-SSNET for artificial intelligence–powered Internet of Things

Time series have broad usage in the wireless Internet of Things. This article proposes a nonlinear time series prediction algorithm based on the Small-World Scale-Free Network after the AIC-Optimized Subtractive Clustering Algorithm (AIC-DSCA-SSNET, AD-SSNET) to predict the nonlinear and unstable time series, which improves the prediction accuracy. The AD-SSNET is introduced as a reservoir based on the echo state network to improve the predictive capability of nonlinear time series, and combined with artificial intelligence method to construct the prediction model training samples. First, the optimal clustering scheme of randomly distributed neurons in the network is adaptively obtained by the AIC-DSCA, then the AD-SSNET is constructed according to the intra-cluster priority connection algorithm. Finally, the reservoir synaptic matrix is calculated according to the synaptic information. Experimental results show that the proposed nonlinear time series prediction algorithm extends the feasible range of spectral radii of the reservoir, improves the prediction accuracy of nonlinear time series, and has great significance to time series analysis in the era of wireless Internet of Things.

Identification of key genes in osteoarthritis using bioinformatics, principal component analysis and meta-analysis.

The present study aimed to identify key genes involved in osteoarthritis (OA). Based on a bioinformatics analysis of five gene expression profiling datasets (GSE55457, GSE55235, GSE82107, GSE12021 and GSE1919), differentially expressed genes (DEGs) in OA were identified. Subsequently, a protein-protein interaction (PPI) network was constructed and its topological structure was analyzed. In addition, key genes in OA were identified following a principal component analysis (PCA) based on the DEGs in the PPI network. Finally, the functions and pathways enriched by these key genes were also analyzed. The PPI network consisted of 241 nodes and 576 interactives, including a total of 171 upregulated DEGs [e.g., aspartylglucosaminidase (AGA), CD58 and CD86] and a total of 70 downregulated DEGs (e.g., acetyl-CoA carboxylase β and dihydropyrimidine dehydrogenase). The PPI network complied with an attribute of scale-free small-world network. After PCA, 47 key genes were identified, including β-1,4-galactosyltransferase-1 (B4GALT1), AGA, CD58, CD86, ezrin, and eukaryotic translation initiation factor 4 γ 1 (EIF4G1). Subsequently, the 47 key genes were identified to be enriched in 13 Gene Ontology (GO) terms and 2 Kyoto Encyclopedia of Genes and Genomes pathways, with the GO terms involving B4GALT1 including positive regulation of developmental processes, protein amino acid terminal glycosylation and protein amino acid terminal N-glycosylation. In addition, B4GALT1 and EIF4G1 were confirmed to be downregulated in OA samples compared with healthy controls, but only EIF4G1 was determined to be significantly downregulated in OA samples, as determined via a meta-analysis of the 5 abovementioned datasets. In conclusion, B4GALT1 and EIF4G1 were indicated to have significant roles in OA, and B4GALT1 may be involved in positive regulation of developmental processes, protein amino acid terminal glycosylation and protein amino acid terminal N-glycosylation. The present study may enhance the current understanding of the molecular mechanisms of OA and provide novel therapeutic targets.

Delayed random walk on deterministic weighted scale-free small-world network with a deep trap

How to effectively control the trapping process in complex systems is of great importance in the study of trapping problem. Recently, the approach of delayed random walk has been introduced into several deterministic network models to steer trapping process. However, exploring delayed random walk on pseudo-fractal web with the co-evolution of topology and weight has remained out of reach. In this paper, we employ delayed random walk to guide trapping process on a salient deterministic weighted scale-free small-world network with the co-evolution of topology and weight. In greater detail, we first place a deep trap at one of initial nodes of the network. Then, a tunable parameter [Formula: see text] is introduced to modulate the transition probability of random walk and dominate the trapping process. Subsequently, trapping efficiency is used as readout of trapping process and average trapping time is employed to measure trapping efficiency. Finally, the closed form solution of average trapping time (ATT) is deduced analytically, which agrees with corresponding numerical solution. The analytical solution of ATT shows that the delayed parameter [Formula: see text] only modifies the prefactor of ATT, and keeps the leading scaling unchanged. In other words, ATT grows sublinearly with network size, whatever values [Formula: see text] takes. In summary, the work may serves as one piece of clues for modulating trapping process toward desired efficiency on more general deterministic networks.

Low-Mean Hitting Time for Random Walks on Heterogeneous Networks

Mean hitting time for random walks on a network, defined as the average of hitting times over all possible pairs of nodes, has found a large variety of applications in many areas. In this paper, we first prove that among all $N$ -node networks, the complete graph has the minimum mean hitting time $N-1$ , which scales linearly with network size. We then study a random walk mobility model with location heterogeneity, modeled by scale-free networks, which are ubiquitous in realistic systems such as P2P networks. For this purpose, we consider random walks on two sparse scale-free small-world networks. By using the connection between random walks and electrical networks, we derive explicit formulas about mean hitting time for both networks, the dominant scaling of which exhibits the same behavior as that of complete graphs. This paper demonstrates that heterogeneous sparse networks can have low-mean hitting time with a behavior similar to that of dense complete graphs, which is instrumental in designing networks, where search is fast between any pair of nodes.

Top Museums on Instagram

Pictures speak louder than words. In this fast-moving world where people hardly have time to read anything, photo-sharing sites become more and more popular. Instagram is being used by millions of people and has created a “sharing ecosystem” that also encourages curation, expression, and produces feedback. Museums are moving quickly to integrate Instagram into their marketing strategies, provide information, engage with audience and connect to other museums Instagram accounts. Taking into consideration that people may not see museum accounts in the same way that the other museum accounts do, the article first describes accounts' performance of the top, most visited museums worldwide and next investigates their interconnection. The analysis uses techniques from social network analysis, including visualization algorithms and calculations of well-established metrics. The research reveals the most important modes of the network by calculating the appropriate centrality metrics and shows that the network formed by the museum Instagram accounts is a scale–free small world network.

Complex network analysis of climate change in the Tarim River Basin, Northwest China

The complex network theory provides an approach for understanding the complexity of climate change from a new perspective. In this study, we used the coarse graining process to convert the data series of daily mean temperature and daily precipitation from 1961 to 2011 into symbol sequences consisting of five characteristic symbols ( i.e. , R , r , e , d and D ), and created the temperature fluctuation network (TFN) and precipitation fluctuation network (PFN) to discover the complex network characteristics of climate change in the Tarim River Basin of Northwest China. The results show that TFN and PEN both present characteristics of scale-free network and small-world network with short average path length and high clustering coefficient. The nodes with high degree in TFN are RRR , dRR and ReR while the nodes with high degree in PFN are rre , rrr , eee and err , which indicates that climate change modes represented by these nodes have large probability of occurrence. Symbol R and r are mostly included in the important nodes of TFN and PFN, which indicate that the fluctuating variation in temperature and precipitation in the Tarim River Basin mainly are rising over the past 50 years. The nodes RRR , DDD , ReR , RRd , DDd and Ree are the hub nodes in TFN, which undertake 19.71% betweenness centrality of the network. The nodes rre , rrr , eee and err are the hub nodes in PFN, which undertake 13.64% betweenness centrality of the network.

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.

Scaling of weighted spectral distribution in weighted small-world networks**Project supported by the National Natural Science Foundation of China (Grant Nos. 61402485, 61573262, and 61303061).

Many real-world systems can be modeled by weighted small-world networks with high clustering coefficients. Recent studies for rigorously analyzing the weighted spectral distribution (WSD) have focused on unweighted networks with low clustering coefficients. In this paper, we rigorously analyze the WSD in a deterministic weighted scale-free small-world network model and find that the WSD grows sublinearly with increasing network order (i.e., the number of nodes) and provides a sensitive discrimination for each input of this model. This study demonstrates that the scaling feature of the WSD exists in the weighted network model which has high and order-independent clustering coefficients and reasonable power-law exponents.

Evolution of Scale-Free Wireless Sensor Networks with Feature of Small-World Networks

Scale-free network and small-world network are the most impacting discoveries in the complex networks theories and have already been successfully proved to be highly effective in improving topology structures of wireless sensor networks. However, currently both theories are not jointly applied to have further improvements in the generation of WSN topologies. Therefore, this paper proposes a cluster-structured evolution model of WSNs considering the characteristics of both networks. With introduction of energy sensitivity and maximum limitation of degrees that a cluster head could have, the performance of our model can be ensured. In order to give an overall assessment of lifting effects of shortcuts, four placement schemes of shortcuts are analyzed. The characteristics of small-world network and scale-free network of our model are proved via theoretical derivation and simulations. Besides, we find that, by introducing shortcuts into scale-free wireless sensor network, the performance of the network can be improved concerning energy-saving and invulnerability, and we discover that the schemes constructing shortcuts between cluster heads and the sink node have better promoted effects than the scheme building shortcuts between pairs of cluster heads, and the schemes based on the preferential principle are superior to the schemes based on the random principle.

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.

Deterministic hierarchical networks

It has been shown that many networks associated with complex systems are small-world (they have both a large local clustering coefficient and a small diameter) and also scale-free (the degrees are distributed according to a power law). Moreover, these networks are very often hierarchical, as they describe the modularity of the systems that are modeled. Most of the studies for complex networks are based on stochastic methods. However, a deterministic method, with an exact determination of the main relevant parameters of the networks, has proven useful. Indeed, this approach complements and enhances the probabilistic and simulation techniques and, therefore, it provides a better understanding of the modeled systems. In this paper we find the radius, diameter, clustering coefficient and degree distribution of a generic family of deterministic hierarchical small-world scale-free networks that has been considered for modeling real-life complex systems.

Monomer–dimer problem on some networks

Abstract Zhang et al. (2012) obtained the exact formula for the number of all possible monomer–dimer arrangements and the asymptotic growth constant on a scale-free small-world network. In this note, we generalize this result and obtain the exact solution on the monomer–dimer model on many networks. Particularly, we prove that these networks have the same asymptotic growth constant of the number of monomer–dimer arrangements.

Volatilities analysis of first-passage time and first-return time on a small-world scale-free network

In this paper, we study random walks on a small-world scale-free network, also called as pseudofractal scale-free web (PSFW), and analyze the volatilities of first passage time (FPT) and first return time (FRT) by using the variance and reduced moment as measures. Note that the FRT and FPT are deeply affected by the starting or target site. We do not intend to enumerate all the possible cases and analyze them. We only study the volatilities of FRT for a given hub (i.e. node with highest degree) and the volatilities of the global FPT (GFPT) to a given hub, which is the average of the FPTs for arriving at a given hub from any possible starting site selected randomly according to the equilibrium distribution of the Markov chain. Firstly, we calculate exactly the probability generating function of the GFPT and FRT based on the self-similar structure of the PSFW. Then, we calculate the probability distribution, mean, variance and reduced moment of the GFPT and FRT by using the generating functions as a tool. Results show that: the reduced moment of FRT grows with increasing network order N and tends to infinity while ; but for the reduced moments of GFPT, it is almost a constant(1.1605) for large N. Therefore, on the PSFW of large size, the FRT has huge fluctuations and the estimate provided by MFRT is unreliable, whereas the fluctuations of the GFPT are much smaller and the estimate provided by its mean is more reliable. The method we propose can also be used to analyze the volatilities of FPT and FRT on other networks with self-similar structure, such as (u, v) flowers and recursive scale-free trees.

Neural-Based Routing Method for Alleviating Congestion in Complex Networks

In this paper, we propose a new packet routing strategy that incorporates memory information for reducing congestion in communication networks. First, we study the conventional routing strategy which selects the paths for transmitting packets to destinations using the distance information and the dynamical information such as the number of accumulating packets at adjacent nodes. Then, we evaluate the effectiveness of this routing strategy for the scale-free networks. From results of numerical simulations, we conclude that this routing strategy is not effective when the density of the packets increases due to the impermeability of the communication network. To avoid this undesirable problem, we incorporate memory information to the routing strategy. By using memory information effectively, packets are spread into the communication networks, achieving a higher performance than conventional routing strategies for various network topologies, such as scale-free networks, small-world networks, and scale-free networks with community

Social ties, homophily and extraversion--introversion to generate complex networks

Many interconnected systems and particularly social interactions can be modeled as networks. These networks often exhibit common properties such as high clustering coefficient, low average path lengths and degree distributions following power-law. Networks having these properties are called small world-scale free networks or simply complex networks. Recent interest in complex networks has catalysed the development of algorithmic models to artificially generate these networks. Often these algorithms introduce network properties in the model regardless of their social interpretation resulting in networks which are statistically similar but structurally different from real world networks. In this paper, we focus on social networks and apply concepts of social ties, homophily and extraversion-introversion to develop a model for social networks with small world and scale free properties. We claim that the proposed model produces networks which are structurally similar to real world social networks.

Eigenvalues for the transition matrix of a small-world scale-free network: Explicit expressions and applications.

The eigenvalues of the transition matrix for random walks on a network play a significant role in the structural and dynamical aspects of the network. Nevertheless, it is still not well understood how the eigenvalues behave in small-world and scale-free networks, which describe a large variety of real systems. In this paper, we study the eigenvalues for the transition matrix of a network that is simultaneously scale-free, small-world, and clustered. We derive explicit simple expressions for all eigenvalues and their multiplicities, with the spectral density exhibiting a power-law form. We then apply the obtained eigenvalues to determine the mixing time and random target access time for random walks, both of which exhibit unusual behaviors compared with those for other networks, signaling discernible effects of topologies on spectral features. Finally, we use the eigenvalues to count spanning trees in the network.