Small-world Model Research Articles

Super-spreaders and the rate of transmission of the SARS virus

We describe a stochastic small-world network model of transmission of the SARS virus. Unlike the standard Susceptible-Infected-Removed models of disease transmission, our model exhibits both geographically localised outbreaks and “super-spreaders”. Moreover, the combination of localised and long range links allows for more accurate modelling of partial isolation and various public health policies. From this model, we derive an expression for the probability of a widespread outbreak and a condition to ensure that the epidemic is controlled. Moreover, multiple simulations are used to make predictions of the likelihood of various eventual scenarios for fixed initial conditions. The main conclusions of this study are: (i) “super-spreaders” may occur even if the infectiousness of all infected individuals is constant; (ii) consistent with previous reports, extended exposure time beyond 3–5 days (i.e. significant nosocomial transmission) was the key factor in the severity of the SARS outbreak in Hong Kong; and, (iii) the spread of SARS can be effectively controlled by either limiting long range links (imposing a partial quarantine) or enforcing rapid hospitalisation and isolation of symptomatic individuals.

Greedy pathlengths and small world graphs

We use matrix analysis to study a cycle plus random, uniform shortcuts—the classic small world model. For such graphs, in addition to the usual edge and vertex information there is an underlying metric that determines distance between vertices. The metric induces a natural greedy algorithm for navigating between vertices and we use this to define a pathlength. This pathlength definition, which is implicit in [J. Kleinberg, The small-world phenomenon: an algorithmic perspective, in: Proceedings of the 32nd ACM Symposium on Theory of Computing, 2000] is entirely appropriate in many message passing contexts. Using a Markov chain formulation, we set up a linear system to determine the expected greedy pathlengths and then use techniques from numerical analysis to find a continuum limit. This gives an asymptotically correct expression for the expected greedy pathlength in the limit of large network size: both the leading term and a sharp estimate of the remainder are produced. The results quantify how the greedy pathlength drops as the number of shortcuts is increased. Further, they allow us to measure the amount by which the greedy pathlength, which is based on local information, exceeds the traditional pathlength, which requires knowledge of the whole network. The analysis allows for either O(1) shortcuts per node or O(1) shortcuts per network. In both cases we find that the greedy algorithm fails to exploit fully the existence of short paths.

An Analysis of the Transition Zone Between the Various Scaling Regimes in the Small-World Model

We analyse the so-called small-world network model (originally devised by Strogatz and Watts), treating it, among other things, as a case study of non-linear coupled difference or differential equations. We derive a system of evolution equations containing more of the previously neglected (possibly relevant) non-linear terms. As an exact solution of this entangled system of equations is out of question we develop a (as we think, promising) method of enclosing the ``exact'' solutions for the expected quantities by upper and lower bounds, which represent solutions of a slightly simpler system of differential equation. Furthermore we discuss the relation between difference and differential equations and scrutinize the limits of the spreading idea for random graphs. We then show that there exists in fact a ``broad'' (with respect to scaling exponents) crossover zone, smoothly interpolating between linear and logarithmic scaling of the diameter or average distance. We are able to corroborate earlier findings in certain regions of phase or parameter space (as e.g. the finite size scaling ansatz) but find also deviations for other choices of the parameters. Our analysis is supplemented by a variety of numerical calculations, which, among other things, quantify the effect of various approximations being made. With the help of our analytical results we manage to calculate another important network characteristic, the (fractal) dimension, and provide numerical values for the case of the small-world network. Catchwords: Small-World Networks, Non-linear Difference Equations

Discrete small world networks

Small world models are networks consisting of many local links and fewer long range `shortcuts', used to model networks with a high degree of local clustering but relatively small diameter. Here, we concern ourselves with the distribution of typical inter-point network distances. We establish approximations to the distribution of the graph distance in a discrete ring network with extra random links, and compare the results to those for simpler models, in which the extra links have zero length and the ring is continuous.

Interactions in economic models: Statistical mechanics and networks

During the last decade, the interaction based models have received increased attention in economics, mainly with the recognition that modeling aggregate patterns of behavior requires viewing individuals in their social environments continuously in interaction with each other. The existing literature suggests that statistical mechanics tools can be useful to model interactions among economic agents. In addition to statistical mechanics, the network approach has also gained popularity, as is evident in the rising attention attributed to small world models and scale free network topologies. These developments point to the fact that interdisciplinary research in economics, mainly using the tools of physics has accelerated to a great extent. In this paper, we review the analogies made between molecules and economic agents in statistical mechanics models, as they are utilized in economics literature. We perform a simulation study by using the small world model and statistical mechanics, to demonstrate the influence of network structure in the spreading of organizational forms, in particular the research collaboration decisions of firms.

Random quantum Ising chains with competing interactions

In this paper we discuss the criticality of a quantum Ising spin chain with competing random ferromagnetic and antiferromagnetic couplings. Quantum fluctuations are introduced via random local transverse fields. First we consider the chain with couplings between first and second neighbors only and then generalize the study to a quantum analog of the Viana-Bray model, defined on a small world random lattice. We use the Dasgupta-Ma decimation technique, both analytically and numerically, and focus on the scaling of the lattice topology, whose determination is necessary to define any infinite disorder transition beyond the chain. In the first case, at the transition the model renormalizes towards the chain, with the infinite disorder fixed point described by Fisher. This corresponds to the irrelevance of the competition induced by the second neighbors couplings. As opposed to this case, this infinite disorder transition is found to be unstable towards the introduction of an arbitrary small density of long range couplings in the small world models.

A Novel Small-World Model: Using Social Mirror Identities for Epidemic Simulations

The authors propose a small-world network model that combines cellular automata with the social mirror identities of daily-contact networks for purposes of performing epidemiological simulations. The social mirror identity concept was established to integrate human long-distance movement and daily visits to fixed locations. After showing that the model is capable of displaying such small-world effects as low degree of separation and relatively high degree of clustering on a societal level, the authors offer proof of its ability to display R0 properties—considered central to all epidemiological studies. To test their model, they simulated the 2003 severe acute respiratory syndrome (SARS) outbreak.

Matrix analysis of a Markov chain small-world model

Recently D. Higham showed that the small-world phenomenon arising in a ring network of N nodes can be modelled by a Markov chain which depends on a parameter ϵ of the form ϵ = K/ N α , where K ⩾ 0 and α > 1. The parameter can be viewed as a transitional factor interpolating between the completely local and completely global configurations of the network. Higham analyzed the Markov chain model by a combination of matrix perturbation theory and finite difference approximations to an underlying boundary value problem. Using such tools he obtained asymptotic results for the limiting case when N is sufficiently large. Furthermore, for large N, Higham verified the small-world phenomenon on the network in the case when α = 3. Motivated by Higham’s work, we show that the Markov chain of the small-world model can be investigated more completely by direct matrix-theoretic methods which produces exact results for all N and for all ϵ. Our results therefore allow a fuller examination of the behavior of the small-world phenomenon in the Markov chain model for small to moderate values of N and for all α > 1.

The impacts of network topology on disease spread

Individuals in a population susceptible to a disease may be represented as vertices in a network, with the edges that connect vertices representing social and/or spatial contact between individuals. Networks, which explicitly included six different patterns of connection between vertices, were created. Both scale-free networks and random graphs showed a different response in path level to increasing levels of clustering than regular lattices. Clustering promoted short path lengths in all network types, but randomly assembled networks displayed a logarithmic relationship between degree and path length; whereas this response was linear in regular lattices. In all cases, small-world models, generated by rewiring the connections of a regular lattice, displayed properties, which spanned the gap between random and regular networks. Simulation of a disease in these networks showed a strong response to connectance pattern, even when the number of edges and vertices were approximately equal. Epidemic spread was fastest, and reached the largest size, in scale-free networks, then in random graphs. Regular lattices were the slowest to be infected, and rewired lattices were intermediate between these two extremes. Scale-free networks displayed the capacity to produce an epidemic even at a likelihood of infection, which was too low to produce an epidemic for the other network types. The interaction between the statistical properties of the network and the results of epidemic spread provides a useful tool for assessing the risk of disease spread in more realistic networks.

ON SPREADING DYNAMICS IN DISCRETE SMALL-WORLD NETWORKS

The dynamical behaviors of some general spreading phenomena in discrete small-world networks are investigated. A new discrete small-world spreading model is proposed, where typical spreading dynamics, including period-doubling bifurcations and chaos, are analyzed on the small-world probability and the nonlinear interaction gain constant.

A Study of Information Effect of Human Network : Focused on the Concept of Small World Model

A Study of Information Effect of Human Network : Focused on the Concept of Small World Model

Analyzing and enhancing the resilience of structured peer-to-peer systems

In this paper, we propose an approach to analyze the resilience of structured peer-to-peer (P2P) systems under failures. The approach is Markov-chain based, and can be applied to systems with relatively stable size and uniform distribution of nodes. We apply our approach to several well-known structured P2P systems. We find that different system features (types of neighbors of nodes) in P2P systems have different impacts on their resilience against failures. Following this observation, we propose to add some extra neighbor(s) to CAN using small-world model principles to form a so-called CAN-SW system. We then apply the proposed approach to analyze its resilience. We find that the performance is improved significantly, particularly, in terms of the average path length.

L-percolations of complex networks

Given a complex network, its L-paths correspond to sequences of L+1 distinct nodes connected through L distinct edges. The L-conditional expansion of a complex network can be obtained by connecting all its pairs of nodes which are linked through at least one L-path, and the respective conditional L-expansion of the original network is defined as the intersection between the original network and its L-expansion. Such expansions are verified to act as filters enhancing the network connectivity, consequently contributing to the identification of communities in small-world models. It is shown in this paper for L=2 and 3, in both analytical and experimental fashions, that an evolving complex network with a fixed number of nodes undergoes successive phase transitions--the so-called L-percolations, giving rise to Eulerian giant clusters. It is also shown that the critical values of such percolations are a function of the network size and that the network percolates for L=3 before L=2 .

Stability and bifurcation of disease spreading in complex networks

A general nonlinear model of disease spreading is proposed, describing the effect of the new link-adding probability p in the topological transition of the N-W small-world network model. The new nonlinear model covers both limiting cases of regular lattices and random networks, and presents a more flexible internal nonlinear interaction than a previous model. Hopf bifurcation is proved to exist during disease spreading in all typical cases of regular lattices, small-world networks, and random networks described by this model. It is shown that probability p not only determines the topological transition of the N-W small-world network model, but also dominates the stability of the local equilibria and bifurcating periodic solutions, and moreover can be further applied to stabilize a periodic spreading behaviour onto a stable equilibrium over the network.

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(log 2 n).

Chaotic Behavior of Small-World Communication Netwoks with Imperfect Transmission

Internet can be considered as a Complex System where the temporal evolution of the transmission time between two fixed machines in the network it is known to be a chaotic time series of a dynamical system with positive Lyapunov exponents. This effect is due to the highly nonlinear behavior present in modern communication networks. Here a nonlinear small-world model of a communication network with time delay and package dropping due to imperfect transmision is analyzed both analytically and numerically. Both time-delay and package dropping have an effect on transmission properties such as the response time.

Temporal series analysis approach to spectra of complex networks.

The spacing of nearest levels of the spectrum of a complex network can be regarded as a time series. Joint use of the multifractal detrended fluctuation approach (MF-DFA) and diffusion entropy (DE) is employed to extract characteristics from this time series. For the Watts-Strogatz small-world model, there exists a critical point at rewiring probability P(r) =0.32. For a network generated in the range 0< P(r) <0.32, the correlation exponent is in the range of 1.0-1.64. Above this critical point, all the networks behave similar to that at p(r) =1. For the Erdos-Renyi model, the time series behaves like fractional Brownian motion noise at p(ER) =1/N. For the growing random network (GRN) model, the values of the long-range correlation exponent are in the range of 0.74-0.83. For most of the GRN networks the probability distribution function of a constructed time series obeys a Gaussian form. In the joint use of MF-DFA and DE, the shuffling procedure in DE is essential to obtain a reliable result.

VORONOI AND FRACTAL COMPLEX NETWORKS AND THEIR CHARACTERIZATION

Real world networks are often characterized by spatial constraints such as the relative position and adjacency of nodes. The present work describes how Voronoi tessellations of the space where the network is embedded provide not only a natural means for relating such networks with metric spaces, but also a natural means for obtaining fractal complex networks. A series of comprehensive measurements closely related to spatial aspects of these networks is proposed, which includes the effective length, adjacency, as well as the fractal dimension of the network in terms of the spatial scales defined by successive shortest paths starting from a specific node. The potential of such features is illustrated with respect to the random, small-world, scale-free and fractal network models.

Stability of a neural network model with small-world connections.

Small-world networks are highly clustered networks with small distances among the nodes. There are many biological neural networks that present this kind of connection. There are no special weightings in the connections of most existing small-world network models. However, this kind of simply connected model cannot characterize biological neural networks, in which there are different weights in synaptic connections. In this paper, we present a neural network model with weighted small-world connections and further investigate the stability of this model.

Local stability and Hopf bifurcation in small-world delayed networks

The notion of small-world networks, recently introduced by Watts and Strogatz, has attracted increasing interest in studying the interesting properties of complex networks. Notice that, a signal or influence travelling on a small-world network often is associated with time-delay features, which are very common in biological and physical networks. Also, the interactions within nodes in a small-world network are often nonlinear. In this paper, we consider a small-world networks model with nonlinear interactions and time delays, which was recently considered by Yang. By choosing the nonlinear interaction strength as a bifurcation parameter, we prove that Hopf bifurcation occurs. We determine the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation by applying the normal form theory and the center manifold theorem. Finally, we show a numerical example to verify the theoretical analysis.