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Journal articles on the topic 'Small World Network model'

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Author: Grafiati

Published: 23 September 2022

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1

SHANKER, O., and TAD HOGG. "EPIDEMIOLOGY MODEL ON SHORTCUT AND SMALL WORLD NETWORKS." Modern Physics Letters B 23, no. 10 (April 20, 2009): 1249–62. http://dx.doi.org/10.1142/s0217984909019387.

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We show that the behavior of an epidemiology model depends sensitively on the shortcut density in the shortcut network. This is consistent with an earlier work on other processes on the shortcut network. We analytically study the reason for the sensitivity. The shortcut network is similar to the small world network, and it has the advantage that the model dependence on the shortcut density can be analytically studied. The model would be relevant to the spread of diseases in human, animal, plant or other populations, to the spread of viruses in computer networks, or to the spread of social contagion in social networks. It would also be relevant in understanding the variations in the load on routers connecting different computer networks, as the network topology gets extended by the addition of new links, and in analyzing the placement of certain special sensors in a sensor network laid out in a non-random way with some shortcut links.

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2

Lee, Kang-won, Hee-kwan Uhm, and Hye-jin Choe. "Tunable Network Generation Model for Small-World and Scale-Free Network." Journal of Korean Institute of Communications and Information Sciences 42, no. 7 (July 31, 2017): 1392–401. http://dx.doi.org/10.7840/kics.2017.42.7.1392.

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3

Shang, Jun, Hao Qiang Liu, Qiang Liu, and Zi Qi Liu. "Design of the Small World Model by NS2." Applied Mechanics and Materials 496-500 (January 2014): 2338–41. http://dx.doi.org/10.4028/www.scientific.net/amm.496-500.2338.

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WSN is the network which is used mostly in the world nowadays, and it has the characteristics that lower cost and better functions than other kinds of the network, and the WSN network is built by the ordinary nodes and the super nodes.Theoretical study of the complex network is widely involved in the fields of computer networks, and the applied research becomes more and more important in the future. It has caused many academic attention about how to apply the complex network theory among the specific application in recent years. In the complex network theory, there has been a number of important research results about the use of the small-world network, scale-free network in the field of transportation.

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4

ZHU, LING-ZAN, BEI-BEI YIN, LEI ZHAO, and KAI-YUAN CAI. "SCALE-FREE NETWORKS CAN BE LINEAR-WORLD." International Journal of Modern Physics B 25, no. 32 (December 30, 2011): 4593–603. http://dx.doi.org/10.1142/s0217979211059206.

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It was generally believed that scale-free networks would be small-world. In this paper, two models, named Model A and Model B, are proposed to show that certain scale-free networks can be linear-world instead of small-world. By linear-world, it means that the average path length L of the network grows linearly with the total number of nodes N, i.e., L~N. Model A generates a deterministic scale-free network with high assortativity and numerical simulations demonstrate that the network is linear-world when it satisfies degree exponent λ>1. Model B constructs a partially deterministic scale-free network, which is connected by identical small scale-free networks following certain rules. Analytical arguments and numerical simulations both yield L~N which suggests that it is also linear-world. It is further discussed in this paper that the network generated by Model Bcould be either correlated or uncorrelated. This suggests that, inconsistent with the results in related works, uncorrelated scale-free networks can also be linear-world.

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5

CAI, TIAN-YI, and ZHEN-YA LI. "ISING MODEL ON A SMALL WORLD NETWORK." International Journal of Modern Physics B 18, no. 17n19 (July 30, 2004): 2575–78. http://dx.doi.org/10.1142/s0217979204025695.

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The properties of 2D Ising model on a small world network are investigated. It is found that Curie temperature increases with the increase of small world links. The relations between the Curie temperature and the concentration of small world links are found. The possibility of using Ising model to describe real network is discussed.

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6

Jia, Huanshen, Guona Hu, and Haixing Zhao. "Topological Properties of a 3-Regular Small World Network." Discrete Dynamics in Nature and Society 2014 (2014): 1–4. http://dx.doi.org/10.1155/2014/160740.

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Complex networks have seen much interest from all research fields and have found many potential applications in a variety of areas including natural, social, biological, and engineering technology. The deterministic models for complex networks play an indispensable role in the field of network model. The construction of a network model in a deterministic way not only has important theoretical significance, but also has potential application value. In this paper, we present a class of 3-regular network model with small world phenomenon. We determine its relevant topological characteristics, such as diameter and clustering coefficient. We also give a calculation method of number of spanning trees in the 3-regular network and derive the number and entropy of spanning trees, respectively.

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7

WU, ZHENGPING, and RENMING WANG. "DISTANCE PREFERENCES SMALL-WORLD COMMUNICATION TOPOLOGY FOR AGENT NETWORK." International Journal of Modern Physics B 24, no. 11 (April 30, 2010): 1489–99. http://dx.doi.org/10.1142/s0217979210054361.

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In multi-agent system (MAS), the communication topology of agent network plays a very important role in its collaboration. Small-world networks are the networks with high local clustering and small average path length, and the communication networks of MAS can be analyzed within the frame of small-world topology. Yet the real multiagent communication networks are abundant and the classical WS small-world model is not suitable for all cases. In this paper, two new small-world network models are presented. One is based on random graph substrate and local nodes preference reconnection and the other is based on regular graph substrate and long-range nodes preference reconnection. The characteristic of the network parameter such as the clustering coefficients, average path length, and eigenvalue λ2 and λn of the Laplacian matrix for these two models and WS model is studied. The consensus problem that based on these three models is also studied. An example is given and the conclusions are made in the end.

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8

Zhang, Tiankui, Jinlong Cao, Yue Chen, Laurie Cuthbert, and Maged Elkashlan. "A Small World Network Model for Energy Efficient Wireless Networks." IEEE Communications Letters 17, no. 10 (October 2013): 1928–31. http://dx.doi.org/10.1109/lcomm.2013.081313.131394.

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9

Schilling, Melissa. "A "Small-World" Network Model of Cognitive Insight." Creativity Research Journal 17, no. 2 (July 1, 2005): 131–54. http://dx.doi.org/10.1207/s15326934crj1702&3_2.

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10

Schilling, Melissa A. "A "Small-World" Network Model of Cognitive Insight." Creativity Research Journal 17, no. 2-3 (July 2005): 131–54. http://dx.doi.org/10.1080/10400419.2005.9651475.

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11

Li, Min, Rui Qiu Ou, and Yan Ling Song. "Principal-Agent Activities on Small World Network." Applied Mechanics and Materials 727-728 (January 2015): 969–75. http://dx.doi.org/10.4028/www.scientific.net/amm.727-728.969.

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As a medium forcommunication and coalition, social network plays an important role in economicactivities. Empirical researches indicate that most of social networks performsignificant small-world characteristics, so this paper assumes the socialnetwork among merchants is a small-world network, and then studies the agencyactivities of merchants by means of simulation based on a game model. We findthat if the average degree of the network increases, the wage of agents willdecrease and social efficiency will increase, but the probability of emergenceof third-part institutions such as courts, which is the basis of modern marketeconomy, will decrease.

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12

KOMURO, Nobuyoshi, Sho MOTEGI, Kosuke SANADA, Jing MA, Zhetao LI, Tingrui PEI, Young-June CHOI, and Hiroo SEKIYA. "Small-World-Network Model Based Routing Method for Wireless Sensor Networks." IEICE Transactions on Communications E99.B, no. 11 (2016): 2315–22. http://dx.doi.org/10.1587/transcom.2016nep0016.

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13

WAGNER, C., and R. STOOP. "NEOCORTEX'S SMALL WORLD OF FRACTAL COUPLING." International Journal of Bifurcation and Chaos 17, no. 10 (October 2007): 3409–14. http://dx.doi.org/10.1142/s0218127407019135.

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Biological neocortical neurons are arranged in a columnar clustered architecture. Using a mathematical model in which the clustering properties can be monitored by means of a connectivity probability function, we investigate the information propagation in the associated networks, by means of simulations and a semi-analytical approach. Our analysis demonstrates that for systems with n-nearest neighbor coupling, the information propagation increases linearly in the neighbor order n. For fractal coupling, shown to give rise to small-world network characteristics, in contrast, an enhanced dependence is found, that, in our model of the neocortex, quickly saturates at a high level, indicating the superiority of this network type.

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14

Santos, José I., David J. Poza, José M. Galán, and Adolfo López-Paredes. "Evolution of Equity Norms in Small-World Networks." Discrete Dynamics in Nature and Society 2012 (2012): 1–18. http://dx.doi.org/10.1155/2012/482481.

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The topology of interactions has been proved very influential in the results of models based on learning and evolutionary game theory. This paper is aimed at investigating the effect of structures ranging from regular ring lattices to random networks, including small-world networks, in a model focused on property distribution norms. The model considers a fixed and finite population of agents who play the Nash bargaining game repeatedly. Our results show that regular networks promote the emergence of the equity norm, while less-structured networks make possible the appearance of fractious regimes. Additionally, our analysis reveals that the speed of adoption can also be affected by the network structure.

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15

Christensen, Alexander P., and Hudson Golino. "Factor or Network Model? Predictions From Neural Networks." Journal of Behavioral Data Science 1, no. 1 (May 2021): 85–126. http://dx.doi.org/10.35566/jbds/v1n1/p5.

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The nature of associations between variables is important for constructing theory about psychological phenomena. In the last decade, this topic has received renewed interest with the introduction of psychometric network models. In psychology, network models are often contrasted with latent variable (e.g., factor) models. Recent research has shown that differences between the two tend to be more substantive than statistical. One recently developed algorithm called the Loadings Comparison Test (LCT) was developed to predict whether data were generated from a factor or small-world network model. A significant limitation of the current LCT implementation is that it's based on heuristics that were derived from descriptive statistics. In the present study, we used artificial neural networks to replace these heuristics and develop a more robust and generalizable algorithm. We performed a Monte Carlo simulation study that compared neural networks to the original LCT algorithm as well as logistic regression models that were trained on the same data. We found that the neural networks performed as well as or better than both methods for predicting whether data were generated from a factor, small-world network, or random network model. Although the neural networks were trained on small-world networks, we show that they can reliably predict the data-generating model of random networks, demonstrating generalizability beyond the trained data. We echo the call for more formal theories about the relations between variables and discuss the role of the LCT in this process.

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16

Xiao, Wen Jun, Shi Zhong Jiang, and Guan Rong Chen. "A Small-World Model of Scale-Free Networks: Features and Verifications." Applied Mechanics and Materials 50-51 (February 2011): 166–70. http://dx.doi.org/10.4028/www.scientific.net/amm.50-51.166.

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It is now well known that many large-sized complex networks obey a scale-free power-law vertex-degree distribution. Here, we show that when the vertex degrees of a large-sized network follow a scale-free power-law distribution with exponent  2, the number of degree-1 vertices, if nonzero, is of order N and the average degree is of order lower than log N, where N is the size of the network. Furthermore, we show that the number of degree-1 vertices is divisible by the least common multiple of , , . . ., , and l is less than log N, where l = < is the vertex-degree sequence of the network. The method we developed here relies only on a static condition, which can be easily verified, and we have verified it by a large number of real complex networks.

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17

Morales, Pablo, and Jorge Finke. "Small-World Networks of corruption." Revista CIFE: Lecturas de Economía Social 17, no. 26 (July 11, 2016): 19. http://dx.doi.org/10.15332/s0124-3551.2015.0026.01.

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<p>Collective behavior forms and spreads through social contact. This thesis introduces a framework for understanding how the structure of social ties may impact the evolution of bribery. We represent relationships as highly clustered networks with small characteristic path lengths (i.e., small-world models having “local” and “long-range” contacts). Based on a principal-agent-client formulation, our model focuses on the effects of clustering on an equilibrium of persistent bribery. Collective outcomes depend on decision-making mechanisms that rely on sensitivity functions, which capture the level of influence between local contacts. Moreover, we represent the evolution of the network as a system of differential equations and identify its region of parameters for which the equilibrium of persistent bribery is stable. Our results show that an increase in clustering tends to decrease the levels of bribery. A more sensitive response to the behavior of neighbors, on the other hand, tends to increase bribery, but only up to a certain point. Beyond this threshold, the expected level of bribery remains constant, despite variations in the structural properties of the network.</p>

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18

NEAL, ZACHARY P. "How small is it? Comparing indices of small worldliness." Network Science 5, no. 1 (March 2017): 30–44. http://dx.doi.org/10.1017/nws.2017.5.

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AbstractMany studies have attempted to determine whether an observed network exhibits a so-called “small-world structure.” Such determinations have often relied on a conceptual definition of small worldliness proposed by Watts and Strogatz in their seminal 1998 paper, but recently several quantitative indices of network small worldliness have emerged. This paper reviews and compares three such indices—the small-world quotient (Q), a small-world metric (ω), and the small-world index(SWI)—in the canonical Watts–Strogatz re-wiring model and in four real-world networks. These analyses suggest that researchers should avoid Q, and identify considerations that should guide the choice between ω and SWI.

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19

Newman, M. E. J., and D. J. Watts. "Renormalization group analysis of the small-world network model." Physics Letters A 263, no. 4-6 (December 1999): 341–46. http://dx.doi.org/10.1016/s0375-9601(99)00757-4.

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20

Takehara, Takuma, Fumio Ochiai, and Naoto Suzuki. "A small-world network model of facial emotion recognition." Quarterly Journal of Experimental Psychology 69, no. 8 (August 2016): 1508–29. http://dx.doi.org/10.1080/17470218.2015.1086393.

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21

Newman, M. E. J., and D. J. Watts. "Scaling and percolation in the small-world network model." Physical Review E 60, no. 6 (December 1, 1999): 7332–42. http://dx.doi.org/10.1103/physreve.60.7332.

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22

Stone, Thomas E., and Susan R. McKay. "Majority-vote model on a dynamic small-world network." Physica A: Statistical Mechanics and its Applications 419 (February 2015): 437–43. http://dx.doi.org/10.1016/j.physa.2014.10.032.

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23

Takehara, Takuma, Fumio Ochiai, and Naoto Suzuki. "A Small-world Network Model of Facial Emotion Recognition." Proceedings of the Annual Convention of the Japanese Psychological Association 79 (September 22, 2015): 2AM—079–2AM—079. http://dx.doi.org/10.4992/pacjpa.79.0_2am-079.

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24

Newman, M. E. J., C. Moore, and D. J. Watts. "Mean-Field Solution of the Small-World Network Model." Physical Review Letters 84, no. 14 (April 3, 2000): 3201–4. http://dx.doi.org/10.1103/physrevlett.84.3201.

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25

Elettreby, M. F. "Multiobjective Bak–Sneppen model on a small-world network." Chaos, Solitons & Fractals 26, no. 3 (November 2005): 1009–17. http://dx.doi.org/10.1016/j.chaos.2005.01.048.

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26

Adou, J. K., Y. Billaud, D. A. Brou, J. P. Clerc, J. L. Consalvi, A. Fuentes, A. Kaiss, et al. "Simulating wildfire patterns using a small-world network model." Ecological Modelling 221, no. 11 (June 2010): 1463–71. http://dx.doi.org/10.1016/j.ecolmodel.2010.02.015.

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27

Amin, Farhan, and Gyu Sang Choi. "Advanced Service Search Model for Higher Network Navigation Using Small World Networks." IEEE Access 9 (2021): 70584–95. http://dx.doi.org/10.1109/access.2021.3077655.

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28

LUZ, EDINA M. S., and F. W. S. LIMA. "MAJORITY-VOTE ON DIRECTED SMALL-WORLD NETWORKS." International Journal of Modern Physics C 18, no. 08 (August 2007): 1251–61. http://dx.doi.org/10.1142/s0129183107011297.

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On directed small-world networks the majority-vote model with noise is now studied through Monte Carlo simulations. In this model, the order-disorder phase transition of the order parameter is well defined. We calculate the value of the critical noise parameter qc for several values of rewiring probability p of the directed small-world network. The critical exponents β/ν, γ/ν and 1/ν were calculated for several values of p.

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29

Chen, Li Qiang. "The Wireless Sensor Network Routing Algorithm Based on the Small World Model." Applied Mechanics and Materials 325-326 (June 2013): 1045–48. http://dx.doi.org/10.4028/www.scientific.net/amm.325-326.1045.

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Small-world networks have the higher clustering coefficient and shorter average path length. According to the design requirements of topology and routing algorithm of the WSN, we apply small world theory into the WSN, and propose the routing algorithm based on Newman Watts small-world network model. This algorithm judge the cluster number whether same to decide the communication type. Data is transmitted to super node firstly, and then the packets are sent by the shortest transmission paths which get from the super node ring. Experiments show that the routing algorithm improves the network throughput and network transmission efficiency, the common node energy consumption become small, so the service life of wireless sensor network is prolonged.

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30

Ni, Yan, Yinghua Wang, Tao Yu, and Xiaoli Li. "Analysis of Epileptic Seizures with Complex Network." Computational and Mathematical Methods in Medicine 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/283146.

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Epilepsy is a disease of abnormal neural activities involving large area of brain networks. Until now the nature of functional brain network associated with epilepsy is still unclear. Recent researches indicate that the small world or scale-free attributes and the occurrence of highly clustered connection patterns could represent a general organizational principle in the human brain functional network. In this paper, we seek to find whether the small world or scale-free property of brain network is correlated with epilepsy seizure formation. A mass neural model was adopted to generate multiple channel EEG recordings based on regular, small world, random, and scale-free network models. Whether the connection patterns of cortical networks are directly associated with the epileptic seizures was investigated. The results showed that small world and scale-free cortical networks are highly correlated with the occurrence of epileptic seizures. In particular, the property of small world network is more significant during the epileptic seizures.

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31

Li, Wei. "Freemium of Digital Products with Small-World Network." Mathematical Problems in Engineering 2020 (April 27, 2020): 1–7. http://dx.doi.org/10.1155/2020/6367051.

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“Freemium” is a popular business model adopted by the vendors of digital products, and it has aroused extensive attention in the academia. The existing research studies commonly explore the business model from the perspective of network effect, but lack the attention to the small-world features of network effect. In order to explore the effect of the small-world network, the current work presents a two-period optimization model of monopolist. The optimization model is incorporating with the Freemium model and the small-world feature of consumer base. The optimization model is solved analytically, and the comparative static results show that if the integrating network effect caused by the strong and weak relationship group is sufficiently high (or the small-world feature of the user group is prominent), the user group network exerts a positive effect; if the integrating network effect is not sufficiently high (or the small-world feature of the user group is not prominent), the user group network exerts a negative effect; especially, if the integrating network effect is low or moderate, the premium product is supposed to be free for the consumers. The conclusions enrich the understanding on the operation of digital products firms in the academia and industry.

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32

CARTWRIGHT, EDWARD J. "CONTAGION AND THE SPEED OF ADJUSTMENT IN SMALL WORLDS." International Game Theory Review 09, no. 04 (December 2007): 689–704. http://dx.doi.org/10.1142/s0219198907001667.

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We model a simple dynamic process in which myopic agents are matched amongst each other to play a coordination game. The network of player interaction is varied between a regular lattice and a random network allowing us to model contagion in small world networks. Weighting times for an equilibrium shift from the risk dominated to risk dominant equilibrium are shown to be smallest in small world networks.

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33

Hu, Xiao-Bing, Ming Wang, and Mark S. Leeson. "Ripple-Spreading Network Model Optimization by Genetic Algorithm." Mathematical Problems in Engineering 2013 (2013): 1–15. http://dx.doi.org/10.1155/2013/176206.

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Small-world and scale-free properties are widely acknowledged in many real-world complex network systems, and many network models have been developed to capture these network properties. The ripple-spreading network model (RSNM) is a newly reported complex network model, which is inspired by the natural ripple-spreading phenomenon on clam water surface. The RSNM exhibits good potential for describing both spatial and temporal features in the development of many real-world networks where the influence of a few local events spreads out through nodes and then largely determines the final network topology. However, the relationships between ripple-spreading related parameters (RSRPs) of RSNM and small-world and scale-free topologies are not as obvious or straightforward as in many other network models. This paper attempts to apply genetic algorithm (GA) to tune the values of RSRPs, so that the RSNM may generate these two most important network topologies. The study demonstrates that, once RSRPs are properly tuned by GA, the RSNM is capable of generating both network topologies and therefore has a great flexibility to study many real-world complex network systems.

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34

Sohn, Insoo. "Small-World and Scale-Free Network Models for IoT Systems." Mobile Information Systems 2017 (2017): 1–9. http://dx.doi.org/10.1155/2017/6752048.

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It is expected that Internet of Things (IoT) revolution will enable new solutions and business for consumers and entrepreneurs by connecting billions of physical world devices with varying capabilities. However, for successful realization of IoT, challenges such as heterogeneous connectivity, ubiquitous coverage, reduced network and device complexity, enhanced power savings, and enhanced resource management have to be solved. All these challenges are heavily impacted by the IoT network topology supported by massive number of connected devices. Small-world networks and scale-free networks are important complex network models with massive number of nodes and have been actively used to study the network topology of brain networks, social networks, and wireless networks. These models, also, have been applied to IoT networks to enhance synchronization, error tolerance, and more. However, due to interdisciplinary nature of the network science, with heavy emphasis on graph theory, it is not easy to study the various tools provided by complex network models. Therefore, in this paper, we attempt to introduce basic concepts of graph theory, including small-world networks and scale-free networks, and provide system models that can be easily implemented to be used as a powerful tool in solving various research problems related to IoT.

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35

Wu, Yu, Yuan Yao, and Li Wang. "A Novel Emergence Model of Public Opinion Based on Small-World Network." Key Engineering Materials 474-476 (April 2011): 2263–68. http://dx.doi.org/10.4028/www.scientific.net/kem.474-476.2263.

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From the view of complex networks and emergent computation, a new emergence model of public opinion is built. It is based on small-world model, and takes Internet users as agents. Then the system parameters and realistic interactions in this model are set. Simulation results show that our model can demonstrate the whole evolution process of formed or unformed public opinion. The formation evolution of public opinion is in accordance with the real network of public opinion. We can get all kinds of public opinion forms via setting different model parameters. By comparing with the existing network model, there is an obvious advantage for the interaction rules and forms in our model, and it is realistic and reasonable. As a new model for the complex system, it can be used as one of the objects for studying the network behaviors and emergent computation.

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36

Jeong, Wonhee, Hoseung Jang, and Unjong Yu. "Highly clustered complex networks in the configuration model: Random regular small-world network." EPL (Europhysics Letters) 128, no. 1 (November 26, 2019): 16001. http://dx.doi.org/10.1209/0295-5075/128/16001.

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37

WEN, GUANGHUI, and ZHISHENG DUAN. "DYNAMICS BEHAVIORS OF WEIGHTED LOCAL-WORLD EVOLVING NETWORKS WITH EXTENDED LINKS." International Journal of Modern Physics C 20, no. 11 (November 2009): 1719–35. http://dx.doi.org/10.1142/s0129183109014692.

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In this paper, we present a local-world evolving model to characterize weighted networks. By introducing the extended links to mimic the weak interactions between the nodes in different local-worlds, the model yields scale-free behavior as well as the small-world property, as confirmed in many real networks. With the increase of the local information, the generated network undergoes a transition from assortative to disassortative, meanwhile the small-world property is preserved. It indicates that the small-world property is a universal characteristic in our model. The numerical simulation results are in good agreement with the analytical expressions.

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38

Zhang, Zhongzhi, Yibin Sheng, and Qiang Jiang. "Monomer–dimer model on a scale-free small-world network." Physica A: Statistical Mechanics and its Applications 391, no. 3 (February 2012): 828–33. http://dx.doi.org/10.1016/j.physa.2011.08.007.

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39

Addario-Berry, Louigi, and Tao Lei. "The Mixing Time of the Newman-Watts Small-World Model." Advances in Applied Probability 47, no. 01 (March 2015): 37–56. http://dx.doi.org/10.1017/s0001867800007692.

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‘Small worlds’ are large systems in which any given node has only a few connections to other points, but possessing the property that all pairs of points are connected by a short path, typically logarithmic in the number of nodes. The use of random walks for sampling a uniform element from a large state space is by now a classical technique; to prove that such a technique works for a given network, a bound on the mixing time is required. However, little detailed information is known about the behaviour of random walks on small-world networks, though many predictions can be found in the physics literature. The principal contribution of this paper is to show that for a famous small-world random graph model known as the Newman-Watts small-world model, the mixing time is of order log2 n. This confirms a prediction of Richard Durrett [5, page 22], who proved a lower bound of order log2 n and an upper bound of order log3 n.

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40

Addario-Berry, Louigi, and Tao Lei. "The Mixing Time of the Newman-Watts Small-World Model." Advances in Applied Probability 47, no. 1 (March 2015): 37–56. http://dx.doi.org/10.1239/aap/1427814580.

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‘Small worlds’ are large systems in which any given node has only a few connections to other points, but possessing the property that all pairs of points are connected by a short path, typically logarithmic in the number of nodes. The use of random walks for sampling a uniform element from a large state space is by now a classical technique; to prove that such a technique works for a given network, a bound on the mixing time is required. However, little detailed information is known about the behaviour of random walks on small-world networks, though many predictions can be found in the physics literature. The principal contribution of this paper is to show that for a famous small-world random graph model known as the Newman-Watts small-world model, the mixing time is of order log2n. This confirms a prediction of Richard Durrett [5, page 22], who proved a lower bound of order log2n and an upper bound of order log3n.

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Draief, M., and A. Ganesh. "Efficient routeing in Poisson small-world networks." Journal of Applied Probability 43, no. 03 (September 2006): 678–86. http://dx.doi.org/10.1017/s0021900200002023.

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In a recent paper, Kleinberg (2000) considered a small-world network model consisting of a d-dimensional lattice augmented with shortcuts. The probability of a shortcut being present between two points decays as a power, r -α, of the distance, r, between them. Kleinberg showed that greedy routeing is efficient if α = d and that there is no efficient decentralised routeing algorithm if α ≠ d. The results were extended to a continuum model by Franceschetti and Meester (2003). In our work, we extend the result to more realistic models constructed from a Poisson point process wherein each point is connected to all its neighbours within some fixed radius, and possesses random shortcuts to more distant nodes as described above.

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Draief, M., and A. Ganesh. "Efficient routeing in Poisson small-world networks." Journal of Applied Probability 43, no. 3 (September 2006): 678–86. http://dx.doi.org/10.1239/jap/1158784938.

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In a recent paper, Kleinberg (2000) considered a small-world network model consisting of a d-dimensional lattice augmented with shortcuts. The probability of a shortcut being present between two points decays as a power, r-α, of the distance, r, between them. Kleinberg showed that greedy routeing is efficient if α = d and that there is no efficient decentralised routeing algorithm if α ≠ d. The results were extended to a continuum model by Franceschetti and Meester (2003). In our work, we extend the result to more realistic models constructed from a Poisson point process wherein each point is connected to all its neighbours within some fixed radius, and possesses random shortcuts to more distant nodes as described above.

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Liu, Ming, and Yihong Xiao. "Modeling and Analysis of Epidemic Diffusion within Small-World Network." Journal of Applied Mathematics 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/841531.

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To depict the rule of epidemic diffusion, two different models, the Susceptible-Exposure-Infected-Recovered-Susceptible (SEIRS) model and the Susceptible-Exposure-Infected-Quarantine-Recovered-Susceptible (SEIQRS) model, are proposed and analyzed within small-world network in this paper. Firstly, the epidemic diffusion models are constructed with mean-filed theory, and condition for the occurrence of disease diffusion is explored. Then, the existence and global stability of the disease-free equilibrium and the endemic equilibrium for these two complex epidemic systems are proved by differential equations knowledge and Routh-Hurwiz theory. At last, a numerical example which includes key parameters analysis and critical topic discussion is presented to test how well the proposed two models may be applied in practice. These works may provide some guidelines for decision makers when coping with epidemic diffusion controlling problems.

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Li, Xiao Hu, Feng Xu, Jin Hua Zhang, and Su Nan Wang. "A New Small-World Neural Network with its Performance on Fault Tolerance." Advanced Materials Research 629 (December 2012): 719–24. http://dx.doi.org/10.4028/www.scientific.net/amr.629.719.

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Many artificial neural networks are the simple simulation of brain neural network’s architecture and function. However, how to rebuild new artificial neural network which architecture is similar to biological neural networks is worth studying. In this study, a new multilayer feedforward small-world neural network is presented using the results form research on complex network. Firstly, a new multilayer feedforward small-world neural network which relies on the rewiring probability heavily is built up on the basis of the construction ideology of Watts-Strogatz networks model and community structure. Secondly, fault tolerance is employed in investigating the performances of new small-world neural network. When the network with connection fault or neuron damage is used to test the fault tolerance performance under different rewiring probability, simulation results show that the fault tolerance capability of small-world neural network outmatches that of the same scale regular network when the fault probability is more than 40%, while random network has the best fault tolerance capability.

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Duan, Ying, Xiuwen Fu, Wenfeng Li, Yu Zhang, and Giancarlo Fortino. "Evolution of Scale-Free Wireless Sensor Networks with Feature of Small-World Networks." Complexity 2017 (2017): 1–15. http://dx.doi.org/10.1155/2017/2516742.

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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.

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Zhao, Yi, Jianwen Feng, and Jingyi Wang. "Synchronizability of Small-World Networks Generated from a Two-Dimensional Kleinberg Model." Mathematical Problems in Engineering 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/451960.

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This paper investigates the synchronizability of small-world networks generated from a two-dimensional Kleinberg model, which is more general than NW small-world network. The three parameters of the Kleinberg model, namely, the distance of neighbors, the number of edge-adding, and the edge-adding probability, are analyzed for their impacts on its synchronizability and average path length. It can be deduced that the synchronizability becomes stronger as the edge-adding probability increases, and the increasing edge-adding probability could make the average path length of the Kleinberg small-world network go smaller. Moreover, larger distance among neighbors and more edges to be added could play positive roles in enhancing the synchronizability of the Kleinberg model. The lorentz oscillators are employed to verify the conclusions numerically.

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Song, Shuang, Xiangdong Chen, and Gupeng Zhang. "Structure of Small World Innovation Network and Learning Performance." Mathematical Problems in Engineering 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/860216.

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This paper examines the differences of learning performance of 5 MNCs (multinational corporations) that filed the largest number of patents in China. We establish the innovation network with the patent coauthorship data by these 5 MNCs and classify the networks by the tail of distribution curve of connections. To make a comparison of the learning performance of these 5 MNCs with differing network structures, we develop an organization learning model by regarding the reality as havingmdimensions, which denotes the heterogeneous knowledge about the reality. We further setninnovative individuals that are mutually interactive and own unique knowledge about the reality. A longer (shorter) distance between the knowledge of the individual and the reality denotes a lower (higher) knowledge level of that individual. Individuals interact with and learn from each other within the small-world network. By making 1,000 numerical simulations and averaging the simulated results, we find that the differing structure of the small-world network leads to the differences of learning performance between these 5 MNCs. The network monopolization negatively impacts and network connectivity positively impacts learning performance. Policy implications in the conclusion section suggest that to improve firm learning performance, it is necessary to establish a flat and connective network.

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Gong, Yuhui, and Qian Yu. "Evolution of Conformity Dynamics in Complex Social Networks." Symmetry 11, no. 3 (February 28, 2019): 299. http://dx.doi.org/10.3390/sym11030299.

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Conformity is a common phenomenon among people in social networks. In this paper, we focus on customers’ conformity behaviors in a symmetry market where customers are located in a social network. We establish a conformity model and analyze it in ring network, random network, small-world network, and scale-free network. Our simulations shown that topology structure, network size, and initial market share have significant effects on the evolution of customers’ conformity behaviors. The market will likely converge to a monopoly state in small-world networks but will form a duopoly market in scale networks. As the size of the network increases, there is a greater possibility of forming a dominant group of preferences in small-world network, and the market will converge to the monopoly of the product which has the initial selector in the market. Also, network density will become gradually significant in small-world networks.

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Whigham, P. A., G. Dick, and M. Parry. "Network rewiring dynamics with convergence towards a star network." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, no. 2194 (October 2016): 20160236. http://dx.doi.org/10.1098/rspa.2016.0236.

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Network rewiring as a method for producing a range of structures was first introduced in 1998 by Watts & Strogatz ( Nature 393 , 440–442. ( doi:10.1038/30918 )). This approach allowed a transition from regular through small-world to a random network. The subsequent interest in scale-free networks motivated a number of methods for developing rewiring approaches that converged to scale-free networks. This paper presents a rewiring algorithm (RtoS) for undirected, non-degenerate, fixed size networks that transitions from regular, through small-world and scale-free to star-like networks. Applications of the approach to models for the spread of infectious disease and fixation time for a simple genetics model are used to demonstrate the efficacy and application of the approach.

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YANG, XUHUA, BO WANG, WANLIANG WANG, and YOUXIAN SUN. "A NOVEL SMALL-WORLD NETWORK MODEL: KEEPING CONNECTIVITY WITHOUT ADDING EDGES." International Journal of Modern Physics B 22, no. 29 (November 20, 2008): 5229–34. http://dx.doi.org/10.1142/s0217979208049273.

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Considering the problems of potentially generating a disconnected network in the WS small-world network model [Watts and Strogatz, Nature393, 440 (1998)] and of adding edges in the NW small-world network model [Newman and Watts, Phys. Lett. A263, 341 (1999)], we propose a novel small-world network model. First, generate a regular ring lattice of N vertices. Second, randomly rewire each edge of the lattice with probability p. During the random rewiring procedure, keep the edges between the two nearest neighbor vertices, namely, always keep a connected ring. This model need not add edges and can maintain connectivity of the network at all times in the random rewiring procedure. Simulation results show that the novel model has the typical small-world properties which are small characteristic path length and high clustering coefficient. For large N, the model is approximately equal to the WS model. For large N and small p, the model is approximately equal to the WS model or the NW model.

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