Small-world Model Research Articles

The navigability of strong ties: Small worlds, tie strength, and network topology

We examine data on and models of small world properties and parameters of social networks. Our focus, on tie-strength, multilevel networks and searchability in strong-tie social networks, allows us to extend some of the questions and findings of recent research and the fit of small world models to sociological and anthropological data on human communities. We offer a navigability of strong ties hypothesis about network topologies tested with data from kinship systems, and potentially applicable to corporate cultures and business networks.

Using the small-world model to improve freenet performance

Efficient data retrieval in an unstructured peer-to-peer system like Freenet is a challenging problem. In this paper, we study the impact of workload on the performance of Freenet. We find that there is a steep reduction in the hit ratio of document requests with increasing load in Freenet. We show that a slight modification of Freenet's routing table cache replacement scheme (from LRU to a replacement scheme that enforces clustering in the key space) can significantly improve performance. Our modification is based on intuition from the small-world models and theoretical results by Kleinberg; our replacement scheme forces the routing tables to resemble neighbor relationships in a small-world acquaintance graph--clustering with light randomness. Our simulations show that this new scheme improves the request hit ratio dramatically while keeping the small average hops per successful request comparable to LRU. A simple, highly idealized model of Freenet under clustering with light randomness proves that the expected message delivery time in Freenet is O(log n) if the routing tables satisfy the small-world model and have the size θ(log2n).

Étude statistique et dynamique de la propagation d'épidémies dans un réseau de petit monde

We study numerically in this work the statistical and dynamical properties of the clusters in a one dimensional small world model. The parameters chosen correspond to a realistic network of children of school age where a disease like measles can propagate. Extensive results on the statistical behavior of the clusters around the percolation threshold, as well as the evoltion with time, are discussed. To cite this article: N. Zekri, J.P. Clerc, C. R. Physique 3 (2002) 741–747.

Topological properties of citation and metabolic networks.

Topological properties of "scale-free" networks are investigated by determining their spectral dimensions d(S), which reflect a diffusion process in the corresponding graphs. Data bases for citation networks and metabolic networks together with simulation results from the growing network model [A.-L. Barabasi and R. Albert, Science 286, 509 (1999)] are probed. For completeness and comparisons lattice, random and small-world models are also investigated. We find that d(S) is around 3 for citation and metabolic networks, which is significantly different from the growing network model, for which d(S) is approximately 7.5. This signals a substantial difference in network topology despite the observed similarities in vertex-order distributions. In addition, the diffusion analysis indicates that the citation networks are treelike in structure, whereas the metabolic networks contain many loops.

Small‐world networks in geophysics

Many geophysical processes can be modelled as interconnected networks. The small‐world network model has recently attracted much attention in physics and applied sciences. In this paper, we modify and apply the small‐world network theory to model geophysical processes such as diffusion and transport in disordered porous rocks. We develop an analytical approach as well as numerical simulations to characterize the properties of small‐world networks in geophysics using system saturation time and fractal dimensions.

Fractal analysis of wave functions at the localization-delocalization transition in a disordered quantum small-world-network model

Localization-delocalization transition in disordered quantum small-world network model is confirmed by using the numerical multifractal analysis of the wave functions at the critical point. A linear relation between the numerator and the denominator of the parametric representation for the singularity spectrum of the wave functions near the transition point is obtained, serving as a criterion of the critical behavior. The singularity spectrum $f(\ensuremath{\alpha})$ at the critical point is size independent. The variation of the fractal intensity ${\ensuremath{\alpha}}_{q}$ at $q=0$ and $q=1$ in changing the rewiring probability p reveals the phase transition as shown in the level statistics on the same model.

Chaos in small-world networks.

A nonlinear small-world network model has been presented to investigate the effect of nonlinear interaction and time delay on the dynamic properties of small-world networks. Both numerical simulations and analytical analysis for networks with time delay and nonlinear interaction show chaotic features in the system response when nonlinear interaction is strong enough or the length scale is large enough. In addition, the small-world system may behave very differently on different scales. Time-delay parameter also has a very strong effect on properties such as the critical length and response time of small-world networks.

Mean-field solution of the small-world network model.

The small-world network model is a simple model of the structure of social networks, which possesses characteristics of both regular lattices and random graphs. The model consists of a one-dimensional lattice with a low density of shortcuts added between randomly selected pairs of points. These shortcuts greatly reduce the typical path length between any two points on the lattice. We present a mean-field solution for the average path length and for the distribution of path lengths in the model. This solution is exact in the limit of large system size and either a large or small number of shortcuts.

On the properties of small-world network models

We study the small-world networks recently introduced by Watts and Strogatz [Nature 393, 440 (1998)], using analytical as well as numerical tools. We characterize the geometrical properties resulting from the coexistence of a local structure and random long-range connections, and we examine their evolution with size and disorder strength. We show that any finite value of the disorder is able to trigger a “small-world” behaviour as soon as the initial lattice is big enough, and study the crossover between a regular lattice and a “small-world” one. These results are corroborated by the investigation of an Ising model defined on the network, showing for every finite disorder fraction a crossover from a high-temperature region dominated by the underlying one-dimensional structure to a mean-field like low-temperature region. In particular there exists a finite-temperature ferromagnetic phase transition as soon as the disorder strength is finite. [0.5cm]

Renormalization group analysis of the small-world network model

We study the small-world network model, which mimics the transition between regular-lattice and random-lattice behavior in social networks of increasing size. We contend that the model displays a critical point with a divergent characteristic length as the degree of randomness tends to zero. We propose a real-space renormalization group transformation for the model and demonstrate that the transformation is exact in the limit of large system size. We use this result to calculate the exact value of the single critical exponent for the system, and to derive the scaling form for the average number of `degrees of separation' between two nodes on the network as a function of the three independent variables. We confirm our results by extensive numerical simulation.

Scaling and percolation in the small-world network model.

In this paper we study the small-world network model of Watts and Strogatz, which mimics some aspects of the structure of networks of social interactions. We argue that there is one nontrivial length-scale in the model, analogous to the correlation length in other systems, which is well-defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero, giving a normal critical point in this limit. This length-scale governs the crossover from large- to small-world behavior in the model, as well as the number of vertices in a neighborhood of given radius on the network. We derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex-vertex distance on the network, and, using series expansion and Padé approximants, find an approximate analytic form for the scaling function. We calculate the effective dimension of small-world graphs and show that this dimension varies as a function of the length-scale on which it is measured, in a manner reminiscent of multifractals. We also study the problem of site percolation on small-world networks as a simple model of disease propagation, and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms (in epidemiological terms, the point at which an epidemic occurs). The typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph. All our analytic results are confirmed by extensive numerical simulations of the model.

Bounded rationality and search over small-world models

The ideal Bayesian agent reasons from a global probability model. Real agents must use simplified models which they know to be adequate only in restricted circumtances. Very little formal theory has been developed to help fallibly rational agents manage the process of constructing and revising small world models. The goal of this paper is to present a theoretical framework for analyzing model management approaches and to illustrate the approach by analyzing a particular class of model management strategies. For a probability forecasting problem, a search process over small-world models is analyzed as an approximation to a larger-world model which the agent cannot explicitly enumerate or compute. Conditions are given under which the sequence of small-world forecasts converges to those produced by the larger-world model.

Metareasoning and the problem of small worlds

Practical decision theoretic reasoning requires the construction of local, problem-specific models in which attention is confined to a restricted universe of propositions. Such a restricted universe is called a small world. Managing the construction and revision of small world models can itself be viewed as a meta-level decision problem. This paper presents a theoretical framework for understanding and analyzing many of the issues associated with the management of small world models. >

ARMOD: A Small Numerical Macroeconomic World Model with Non-Clearing Markets

The purpose of this paper is to demonstrate the usefulness of a non-Walrasian numerical modelling approach to current macroeconomic problems of excessive unemployment and trade imbalances in the OECD area. The model, which consists of five world regions, generates a sequence of short-run equilibria. The model captures international transmission mechanisms of business-cycle impulses and of the effects of monetary and fiscal policy, assuming flexible interest and exchange rates, full capital mobility and adaptive expectations. A scenario with successful international policy coordination illustrates how the model works.

Comment on E. Steigum, "ARMOD: A Small Numerical Macroeconomic World Model with Non-Clearing Markets"

Comment on E. Steigum, "ARMOD: A Small Numerical Macroeconomic World Model with Non-Clearing Markets"

Redes Sociais na Internet: Considerações Iniciais

O presente artigo discute a insuficiência dos modelos de redes igualitárias, pequenos mundos e redes sem escalas para dar conta do problema das redes sociais estabelecidas através da comunicação mediada por computador