Abstract:
Quantitative descriptions of network structure in big data can provide fundamental insights into the function of interconnected complex systems. Small-world structure, commonly diagnosed by high local clustering yet short average path length between any two nodes, directly enables information flow in coupled systems, a key function that can differ across conditions or between groups. However, current techniques to quantify small-world structure are dependent on nuisance variables such as density and agnostic to critical variables such as the strengths of connections between nodes, thereby hampering accurate and comparable assessments of small-world structure in different networks. Here, we address both limitations with a novel metric called the Small-World Propensity (SWP). In its binary instantiation, the SWP provides an unbiased assessment of small-world structure in networks of varying densities. We extend this concept to the case of weighted networks by developing (i) a standardized procedure for generating weighted small-world networks, (ii) a weighted extension of the SWP, and (iii) a stringent and generalizable method for mapping real-world data onto the theoretical model. In applying these techniques to real world brain networks, we uncover the surprising fact that the canonical example of a biological small-world network, the C. elegans neuronal network, has strikingly low SWP in comparison to other examined brain networks. These metrics, models, and maps form a coherent toolbox for the assessment of architectural properties in real-world networks and their statistical comparison across conditions.

Abstract:
In all empirical-network studies, the observed properties of economic networks are informative only if compared with a well-defined null model that can quantitatively predict the behavior of such properties in constrained graphs. However, predictions of the available null-model methods can be derived analytically only under assumptions (e.g., sparseness of the network) that are unrealistic for most economic networks like the World Trade Web (WTW). In this paper we study the evolution of the WTW using a recently-proposed family of null network models. The method allows to analytically obtain the expected value of any network statistic across the ensemble of networks that preserve on average some local properties, and are otherwise fully random. We compare expected and observed properties of the WTW in the period 1950-2000, when either the expected number of trade partners or total country trade is kept fixed and equal to observed quantities. We show that, in the binary WTW, node-degree sequences are sufficient to explain higher-order network properties such as disassortativity and clustering-degree correlation, especially in the last part of the sample. Conversely, in the weighted WTW, the observed sequence of total country imports and exports are not sufficient to predict higher-order patterns of the WTW. We discuss some important implications of these findings for international-trade models.

Abstract:
We introduce the concept of efficiency of a network, measuring how efficiently it exchanges information. By using this simple measure small-world networks are seen as systems that are both globally and locally efficient. This allows to give a clear physical meaning to the concept of small-world, and also to perform a precise quantitative a nalysis of both weighted and unweighted networks. We study neural networks and man-made communication and transportation systems and we show that the underlying general principle of their construction is in fact a small-world principle of high efficiency.

Abstract:
We propose a deterministic weighted scale-free small-world model for considering pseudofractal web with the coevolution of topology and weight. In the model, we have the degree distribution exponent $\gamma$ restricted to a range between 2 and 3, simultaneously tunable with two parameters. At the same time, we provide a relatively complete view of topological structure and weight dynamics characteristics of the networks: weight and strength distribution; degree correlations; average clustering coefficient and degree-cluster correlations; as well as the diameter. We show that our model is particularly effective at mimicing weighted scale-free small-world networks with a high and relatively stable clustering coefficient, which rapidly decline with the network size in most previous models.

Abstract:
Small-world networks (SWN) are found to be closer to the real social systems than both regular and random lattices. Then, a model for the evolution of economic systems is generalized to SWN. The Sznajd model for the two-state opinion formation problem is applied to SWN. Then a simple definition of leaders is included. These models explain some socio-economic aspects.

Abstract:
Small-world networks are the focus of recent interest because they appear to circumvent many of the limitations of either random networks or regular lattices as frameworks for the study of interaction networks of complex systems. Here, we report an empirical study of the statistical properties of a variety of diverse real-world networks. We present evidence of the occurrence of three classes of small-world networks: (a) scale-free networks, characterized by a vertex connectivity distribution that decays as a power law; (b) broad-scale networks, characterized by a connectivity distribution that has a power-law regime followed by a sharp cut-off; (c) single-scale networks, characterized by a connectivity distribution with a fast decaying tail. Moreover, we note for the classes of broad-scale and single-scale networks that there are constraints limiting the addition of new links. Our results suggest that the nature of such constraints may be the controlling factor for the emergence of different classes of networks.

Abstract:
We report numerical evidence that an epidemic-like model, which can be interpreted as the propagation of a rumor, exhibits critical behavior at a finite randomness of the underlying small-world network. The transition occurs between a regime where the rumor "dies" in a small neighborhood of its origin, and a regime where it spreads over a finite fraction of the whole population. Critical exponents are evaluated, and the dependence of the critical randomness with the network connectivity is studied. The behavior of this system as a function of the small-network randomness bears noticeable similarities with an epidemiological model reported recently [M. Kuperman and G. Abramson, Phys. Rev. Lett. 86, 2909 (2001)], in spite of substantial differences in the respective dynamical rules.

Abstract:
Connections in complex networks are inherently fluctuating over time and exhibit more dimensionality than analysis based on standard static graph measures can capture. Here, we introduce the concepts of temporal paths and distance in time-varying graphs. We define as temporal small world a time-varying graph in which the links are highly clustered in time, yet the nodes are at small average temporal distances. We explore the small-world behavior in synthetic time-varying networks of mobile agents, and in real social and biological time-varying systems.

Abstract:
We investigate the role of clustering on the critical behavior of the contact process (CP) on small-world networks using the Watts-Strogatz (WS) network model with an edge rewiring probability p. The critical point is well predicted by a homogeneous cluster-approximation for the limit of vanishing clustering (p close to 1). The critical exponents and dimensionless moment ratios of the CP are in agreement with those predicted by the mean-field theory for any p > 0. This independence on the network clustering shows that the small-world property is a sufficient condition for the mean-field theory to correctly predict the universality of the model. Moreover, we compare the CP dynamics on WS networks with rewiring probability p = 1 and random regular networks and show that the weak heterogeneity of the WS network slightly changes the critical point but does not alter other critical quantities of the model.

Abstract:
Since some realistic networks are influenced not only by increment behavior but also by tunable clustering mechanism with new nodes to be added to networks, it is interesting to characterize the model for those actual networks. In this paper, a weighted local-world model, which incorporates increment behavior and tunable clustering mechanism, is proposed and its properties are investigated, such as degree distribution and clustering coefficient. Numerical simulations are fit to the model characters and also display good right skewed scale-free properties. Furthermore, the correlation of vertices in our model is studied which shows the assortative property. Epidemic spreading process by weighted transmission rate on the model shows that the tunable clustering behavior has a great impact on the epidemic dynamic. Keywords: Weighted network, increment behavior, tun- able cluster, epidemic spreading.