Small-world Model Research Articles

Structural comparisons of networks and model-based detection of small-worldness

In this paper, we consider the problem of assessing the “level of small-worldness” of a graph and of detecting small-worldness features in real networks. After discussing the limitations of classical approaches, based on the computation of network indicators, we propose a new procedure, which involves the comparison of network structures at different “observation scales”. This allows small-world features to be caught, even if “hidden” deeply into the network structure. Applications of the procedure to both simulated and real data show the effectiveness of the proposal, also in distinguishing between different small-world models and in detecting emerging small-worldness in dynamical networks.

Altered Rich-Club and Frequency-Dependent Subnetwork Organization in Mild Traumatic Brain Injury: A MEG Resting-State Study.

Functional brain connectivity networks exhibit “small-world” characteristics and some of these networks follow a “rich-club” organization, whereby a few nodes of high connectivity (hubs) tend to connect more densely among themselves than to nodes of lower connectivity. The Current study followed an “attack strategy” to compare the rich-club and small-world network organization models using Magnetoencephalographic (MEG) recordings from mild traumatic brain injury (mTBI) patients and neurologically healthy controls to identify the topology that describes the underlying intrinsic brain network organization. We hypothesized that the reduction in global efficiency caused by an attack targeting a model's hubs would reveal the “true” underlying topological organization. Connectivity networks were estimated using mutual information as the basis for cross-frequency coupling. Our results revealed a prominent rich-club network organization for both groups. In particular, mTBI patients demonstrated hyper-synchronization among rich-club hubs compared to controls in the δ band and the δ-γ1, θ-γ1, and β-γ2 frequency pairs. Moreover, rich-club hubs in mTBI patients were overrepresented in right frontal brain areas, from θ to γ1 frequencies, and underrepresented in left occipital regions in the δ-β, δ-γ1, θ-β, and β-γ2 frequency pairs. These findings indicate that the rich-club organization of resting-state MEG, considering its role in information integration and its vulnerability to various disorders like mTBI, may have a significant predictive value in the development of reliable biomarkers to help the validation of the recovery from mTBI. Furthermore, the proposed approach might be used as a validation tool to assess patient recovery.

On Detection and Structural Reconstruction of Small-World Random Networks

In this paper, we study detection and fast reconstruction of the celebrated Watts-Strogatz (WS) small-world random graph model \citep{watts1998collective} which aims to describe real-world complex networks that exhibit both high clustering and short average length properties. The WS model with neighborhood size $k$ and rewiring probability probability $\beta$ can be viewed as a continuous interpolation between a deterministic ring lattice graph and the Erd\H{o}s-R\'{e}nyi random graph. We study both the computational and statistical aspects of detecting the deterministic ring lattice structure (or local geographical links, strong ties) in the presence of random connections (or long range links, weak ties), and for its recovery. The phase diagram in terms of $(k,\beta)$ is partitioned into several regions according to the difficulty of the problem. We propose distinct methods for the various regions.

Collective behavior of large-scale neural networks with GPU acceleration.

In this paper, the collective behaviors of a small-world neuronal network motivated by the anatomy of a mammalian cortex based on both Izhikevich model and Rulkov model are studied. The Izhikevich model can not only reproduce the rich behaviors of biological neurons but also has only two equations and one nonlinear term. Rulkov model is in the form of difference equations that generate a sequence of membrane potential samples in discrete moments of time to improve computational efficiency. These two models are suitable for the construction of large scale neural networks. By varying some key parameters, such as the connection probability and the number of nearest neighbor of each node, the coupled neurons will exhibit types of temporal and spatial characteristics. It is demonstrated that the implementation of GPU can achieve more and more acceleration than CPU with the increasing of neuron number and iterations. These two small-world network models and GPU acceleration give us a new opportunity to reproduce the real biological network containing a large number of neurons.

The common transcriptional subnetworks of the grape berry skin in the late stages of ripening

BackgroundWine grapes are important economically in many countries around the world. Defining the optimum time for grape harvest is a major challenge to the grower and winemaker. Berry skins are an important source of flavor, color and other quality traits in the ripening stage. Senescent-like processes such as chloroplast disorganization and cell death characterize the late ripening stage.ResultsTo better understand the molecular and physiological processes involved in the late stages of berry ripening, RNA-seq analysis of the skins of seven wine grape cultivars (Cabernet Franc, Cabernet Sauvignon, Merlot, Pinot Noir, Chardonnay, Sauvignon Blanc and Semillon) was performed. RNA-seq analysis identified approximately 2000 common differentially expressed genes for all seven cultivars across four different berry sugar levels (20 to 26 °Brix). Network analyses, both a posteriori (standard) and a priori (gene co-expression network analysis), were used to elucidate transcriptional subnetworks and hub genes associated with traits in the berry skins of the late stages of berry ripening. These independent approaches revealed genes involved in photosynthesis, catabolism, and nucleotide metabolism. The transcript abundance of most photosynthetic genes declined with increasing sugar levels in the berries. The transcript abundance of other processes increased such as nucleic acid metabolism, chromosome organization and lipid catabolism. Weighted gene co-expression network analysis (WGCNA) identified 64 gene modules that were organized into 12 subnetworks of three modules or more and six higher order gene subnetworks. Some gene subnetworks were highly correlated with sugar levels and some subnetworks were highly enriched in the chloroplast and nucleus. The petal R package was utilized independently to construct a true small-world and scale-free complex gene co-expression network model. A subnetwork of 216 genes with the highest connectivity was elucidated, consistent with the module results from WGCNA. Hub genes in these subnetworks were identified including numerous members of the core circadian clock, RNA splicing, proteolysis and chromosome organization. An integrated model was constructed linking light sensing with alternative splicing, chromosome remodeling and the circadian clock.ConclusionsA common set of differentially expressed genes and gene subnetworks from seven different cultivars were examined in the skin of the late stages of grapevine berry ripening. A densely connected gene subnetwork was elucidated involving a complex interaction of berry senescent processes (autophagy), catabolism, the circadian clock, RNA splicing, proteolysis and epigenetic regulation. Hypotheses were induced from these data sets involving sugar accumulation, light, autophagy, epigenetic regulation, and fruit development. This work provides a better understanding of berry development and the transcriptional processes involved in the late stages of ripening.

Spatial connections in regional climate model rainfall outputs at different temporal scales: Application of network theory

Understanding the spatial and temporal variability of rainfall has always been a great challenge, and the impacts of climate change further complicate this issue. The present study employs the concepts of complex networks to study the spatial connections in rainfall, with emphasis on climate change and rainfall scaling. Rainfall outputs (during 1961–1990) from a regional climate model (i.e. Weather Research and Forecasting (WRF) model that downscaled the European Centre for Medium-range Weather Forecasts, ECMWF ERA-40 reanalyses) over Southeast Asia are studied, and data corresponding to eight different temporal scales (6-hr, 12-hr, daily, 2-day, 4-day, weekly, biweekly, and monthly) are analyzed. Two network-based methods are applied to examine the connections in rainfall: clustering coefficient (a measure of the network’s local density) and degree distribution (a measure of the network’s spread). The influence of rainfall correlation threshold (T) on spatial connections is also investigated by considering seven different threshold levels (ranging from 0.5 to 0.8). The results indicate that: (1) rainfall networks corresponding to much coarser temporal scales exhibit properties similar to that of small-world networks, regardless of the threshold; (2) rainfall networks corresponding to much finer temporal scales may be classified as either small-world networks or scale-free networks, depending upon the threshold; and (3) rainfall spatial connections exhibit a transition phase at intermediate temporal scales, especially at high thresholds. These results suggest that the most appropriate model for studying spatial connections may often be different at different temporal scales, and that a combination of small-world and scale-free network models might be more appropriate for rainfall upscaling/downscaling across all scales, in the strict sense of scale-invariance. The results also suggest that spatial connections in the studied rainfall networks in Southeast Asia are weak, especially when more stringent conditions are imposed (i.e. when T is very high), except at the monthly scale.

A small-world network model for the simulation of fire spread onboard naval vessels

This work is devoted to the development of a small-world network model to predict real-time fire spread onboard naval vessels. This model takes into account short-range and long-range connections between neighboring and remote network compartments. Fire ignition and flashover, as well as fire transmissions through the walls and ventilation ducts are simulated using time-dependent normal probability density functions. Mean durations of fire transmission through the walls and ducts are determined by a three-zone model and a one-dimensional CFD code, respectively. Specific experiments are conducted in a steel room, representative of a naval vessel compartment, in order to validate the zone model. Then a proof of concept is developed by applying the network model to a full-scale vessel mockup composed of 113 compartments on 7 decks. A statistical study is conducted to produce fire risk maps, classifying the vessel compartments according to their propensity to burn.

Improving Collection Browsing: Small World Networking and Gray Code Ordering

ABSTRACTDocuments in digital and paper libraries may be arranged, based on their topics, in order to facilitate browsing. It may seem intuitively obvious that ordering documents by their subject should improve browsing performance; the results presented in this article suggest that ordering library materials by their Gray code values and through using links consistent with the small world model of document relationships is consistent with improving browsing performance. Below, library circulation data, including ordering with Library of Congress Classification numbers and Library of Congress Subject Headings, are used to provide information useful in generating user-centered document arrangements, as well as user-independent arrangements. Documents may be linearly arranged so they can be placed in a line by topic, such as on a library shelf, or in a list on a computer display. Crossover links, jumps between a document and another document to which it is not adjacent, can be used in library databases to allow additional paths that one might take when browsing. The improvement that is obtained with different combinations of document orderings and different crossovers is examined and applications suggested.

Hybrid control of delay induced hopf bifurcation of dynamical small-world network.

In this paper, we focus on the Hopf bifurcation control of a small-world network model with time-delay. With emphasis on the relationship between the Hopf bifurcation and the time-delay, we investigate the effect of time-delay by choosing it as the bifurcation parameter. By using tools from control and bifurcation theory, it is proved that there exists a critical value of time-delay for the stability of the model. When the time-delay passes through the critical value, the model loses its stability and a Hopf bifurcation occurs. To enhance the stability of the model, we propose an improved hybrid control strategy in which state feedback and parameter perturbation are used. Through linear stability analysis, we show that by adjusting the control parameter properly, the onset of Hopf bifurcation of the controlled model can be delayed or eliminated without changing the equilibrium point of the model. Finally, numerical simulations are given to verify the theoretical analysis.

The role of the interaction network in the emergence of diversity of behavior.

How can systems in which individuals’ inner workings are very similar to each other, as neural networks or ant colonies, produce so many qualitatively different behaviors, giving rise to roles and specialization? In this work, we bring new perspectives to this question by focusing on the underlying network that defines how individuals in these systems interact. We applied a genetic algorithm to optimize rules and connections of cellular automata in order to solve the density classification task, a classical problem used to study emergent behaviors in decentralized computational systems. The networks used were all generated by the introduction of shortcuts in an originally regular topology, following the small-world model. Even though all cells follow the exact same rules, we observed the existence of different classes of cells’ behaviors in the best cellular automata found—most cells were responsible for memory and others for integration of information. Through the analysis of structural measures and patterns of connections (motifs) in successful cellular automata, we observed that the distribution of shortcuts between distant regions and the speed in which a cell can gather information from different parts of the system seem to be the main factors for the specialization we observed, demonstrating how heterogeneity in a network can create heterogeneity of behavior.

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.

An assessment method for aviation network optimization based on time-varying small world model

The optimization of aviation networks is of great significance for optimizing the allocation of resources, improving transport efficiency, and enhancing the competitiveness among airline companies. There have been a lot of researches which combine the theory of complex network and the actual situations to analyze the air transportation system. The present work provides a certain theoretical basis for the plan of airline schedule. Firstly, we regard an airport as a node, flight frequency as a link weight, and build a heterogeneous network. Through empirical analysis, we find that the aviation network has small-world and scale-free properties. In addition, considering that the instant network consists of current flights changing over time, time-varying is another important characteristic of aviation network. Also, a spatiotemporal correspondence between the flight frequency and route geometric distance is demonstrated to be τij~rij-C. Secondly, by Monte Carlo simulation, we know that the time-ordered topologies influence the optimal navigation structure and make it different from those from traditional static models. Specially, we can obtain a unique restriction between C and optimal structural exponent α, which unveils a new optimization principle in route design and schedule arrangement. Applying these features to the cost-minimized optimization model, a method to evaluate the optimization of network is proposed, by which we can directly predict the overall optimal distribution of flight distances and corresponding flight frequencies only based on the information about the passenger flow assignment. Thirdly, China aviation network data from 2001 to 2010 are used for empirical study. It is found that the predictions consist with the actual data. Compared with traditional optimization methods, it can simplify the computational complexity, and therefore it takes full advantage of the structural convenience and provides a new perspective for the overall scheduling of air transportation system. In this case, companies are able to estimate route adjustments easily to see whether they are reasonable and analyze the development trend of network to provide suggestions for future optimization.

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.

The most common friend first immunization**Project supported by the National Natural Science Foundation of China (Grant No. 61263019), the Program for International Science and Technology Cooperation Projects of Gansu Province, China (Grant No. 144WCGA166), and the Program for Longyuan Young Innovation Talents and the Doctoral Foundation of Lanzhou University of Technology, China.

In this paper, a standard susceptible-infected-recovered-susceptible(SIRS) epidemic model based on the Watts–Strogatz (WS) small-world network model and the Barabsi–Albert (BA) scale-free network model is established, and a new immunization scheme — “the most common friend first immunization” is proposed, in which the most common friend’s node is described as being the first immune on the second layer protection of complex networks. The propagation situations of three different immunization schemes — random immunization, high-risk immunization, and the most common friend first immunization are studied. At the same time, the dynamic behaviors are also studied on the WS small-world and the BA scale-free network. Moreover, the analytic and simulated results indicate that the immune effect of the most common friend first immunization is better than random immunization, but slightly worse than high-risk immunization. However, high-risk immunization still has some limitations. For example, it is difficult to accurately define who a direct neighbor in the life is. Compared with the traditional immunization strategies having some shortcomings, the most common friend first immunization is effective, and it is nicely consistent with the actual situation.

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.

An embedded self-adapting network service framework for networked manufacturing system

To improve the self-adapting ability and real-time performance of client/server based networked manufacturing system (NMS), this paper introduces the universal plug and play (UPnP), an intelligent network middleware, into networked manufacturing area, and proposes an embedded self-adapting network framework and related service methods. Referring to small world model and scale-free principles, a complex network model oriented to digital manufacturing is set up. Based on the model, an improved entropy vector projection algorithm is proposed to evaluate the network complexity and reveal the evolution regulars. Then, the self-adapting services for NMS are performed by UPnP service-calling and inter-process communication methods. Finally, the case studies and industrial field experiments verify the effectiveness of the proposed service framework.

A study on small-world characteristics of Chinaäó»s high-speed railway networks

The small-world model provides a useful perspective and method to study the topological structure and intrinsic characteristics of high-speed rail networks (HRNs). In this paper, the P-space method is used to examine global and local HRNs in China, meanwhile the adjacency matrix is developed, then the social network analysis and visualization tool UCINET is used to calculate the spatial and attribute data of HRNs at national and local levels in China. The small-world characteristics of whole HRNs are discussed, three networks which have different properties are determined, and a comparative analysis of the small-world effect is detected. Then, the relationship between the construction of high-speed rail and regional development of China is analysed. The results show that: 1) China's HRNs have small average path length ( L ) and large clustering coefficient (C ), representing a typical small-world network; 2) Local HRNs have a certain correlation with economic development. The reasons for the difference of HRNs with respect to characteristics among regions are eventually discussed.

Using Graph and Vertex Entropy to Compare Empirical Graphs with Theoretical Graph Models

Over the years, several theoretical graph generation models have been proposed. Among the most prominent are: the Erdős–Renyi random graph model, Watts–Strogatz small world model, Albert–Barabási preferential attachment model, Price citation model, and many more. Often, researchers working with real-world data are interested in understanding the generative phenomena underlying their empirical graphs. They want to know which of the theoretical graph generation models would most probably generate a particular empirical graph. In other words, they expect some similarity assessment between the empirical graph and graphs artificially created from theoretical graph generation models. Usually, in order to assess the similarity of two graphs, centrality measure distributions are compared. For a theoretical graph model this means comparing the empirical graph to a single realization of a theoretical graph model, where the realization is generated from the given model using an arbitrary set of parameters. The similarity between centrality measure distributions can be measured using standard statistical tests, e.g., the Kolmogorov–Smirnov test of distances between cumulative distributions. However, this approach is both error-prone and leads to incorrect conclusions, as we show in our experiments. Therefore, we propose a new method for graph comparison and type classification by comparing the entropies of centrality measure distributions (degree centrality, betweenness centrality, closeness centrality). We demonstrate that our approach can help assign the empirical graph to the most similar theoretical model using a simple unsupervised learning method.

A small-world network model of facial emotion recognition.

Various models have been proposed to increase understanding of the cognitive basis of facial emotions. Despite those efforts, interactions between facial emotions have received minimal attention. If collective behaviours relating to each facial emotion in the comprehensive cognitive system could be assumed, specific facial emotion relationship patterns might emerge. In this study, we demonstrate that the frameworks of complex networks can effectively capture those patterns. We generate 81 facial emotion images (6 prototypes and 75 morphs) and then ask participants to rate degrees of similarity in 3240 facial emotion pairs in a paired comparison task. A facial emotion network constructed on the basis of similarity clearly forms a small-world network, which features an extremely short average network distance and close connectivity. Further, even if two facial emotions have opposing valences, they are connected within only two steps. In addition, we show that intermediary morphs are crucial for maintaining full network integration, whereas prototypes are not at all important. These results suggest the existence of collective behaviours in the cognitive systems of facial emotions and also describe why people can efficiently recognize facial emotions in terms of information transmission and propagation. For comparison, we construct three simulated networks--one based on the categorical model, one based on the dimensional model, and one random network. The results reveal that small-world connectivity in facial emotion networks is apparently different from those networks, suggesting that a small-world network is the most suitable model for capturing the cognitive basis of facial emotions.

A new ranking algorithm and an evolutionary model based on comprehensive weighted clique degree for complex network

• A new algorithm for import nodes ranking based on clique degree is proposed. • The algorithm reflects the importance rank well and has low time complexity. • A new evolutionary model of complex network based on CWCDR algorithm is proposed. • Model has good degree distribution, average shortest path, clustering coefficient. This paper proposes a new ranking algorithm based on comprehensive weighted clique degree (CWCDR) for ranking importance of nodes in complex network. Simulation results show that CWCDR algorithm can overcome the limitation of degree ranking algorithm and find important nodes in complex networks more precisely and effectively. To the shortage of small-world model and BA model, this paper proposes an evolutionary model of complex network based on CWCDR algorithms, named CWCDR model. Simulation results show that the CWCDR model accords with power-law distribution. Compared with BA model, this model has better average shortest path length and clustering coefficient. Therefore, the CWCDR model is more consistent with the real network.