Small-world networks: links with long-tailed distributions

State-of-art review of information diffusion models and their impact on social network vulnerabilities

With the development of information society and network technology, people increasingly depend on information found on the Internet. At the same time, the models of information diffusion on the Internet are changing as well. However, these models experience the problem due to the fast development of network technologies. There is no thorough research in regards to the latest models and their applications and advantages. As a result, it is essential to have a comprehensive study of information diffusion models. The primary goal of this research is to provide a comparative study on the existing models such as the Ising model, Sznajd model, SIR model, SICR model, Game theory and social networking services models. We discuss several of their applications with the existing limitations and further categorizations. Vulnerabilities and privacy challenges of information diffusion models are extensively explored. Furthermore, categorization including strengths and weaknesses are discussed. Finally, limitations and recommendations are suggested with diverse solutions for the improvement of the information diffusion models and envisioned future research directions.

Network architectures supporting learnability

Human learners acquire complex interconnected networks of relational knowledge. The capacity for such learning naturally depends on two factors: the architecture (or informational structure) of the knowledge network itself and the architecture of the computational unit-the brain-that encodes and processes the information. That is, learning is reliant on integrated network architectures at two levels: the epistemic and the computational, or the conceptual and the neural. Motivated by a wish to understand conventional human knowledge, here, we discuss emerging work assessing network constraints on the learnability of relational knowledge, and theories from statistical physics that instantiate the principles of thermodynamics and information theory to offer an explanatory model for such constraints. We then highlight similarities between those constraints on the learnability of relational networks, at one level, and the physical constraints on the development of interconnected patterns in neural systems, at another level, both leading to hierarchically modular networks. To support our discussion of these similarities, we employ an operational distinction between the modeller (e.g. the human brain), the model (e.g. a single human's knowledge) and the modelled (e.g. the information present in our experiences). We then turn to a philosophical discussion of whether and how we can extend our observations to a claim regarding explanation and mechanism for knowledge acquisition. What relation between hierarchical networks, at the conceptual and neural levels, best facilitate learning? Are the architectures of optimally learnable networks a topological reflection of the architectures of comparably developed neural networks? Finally, we contribute to a unified approach to hierarchies and levels in biological networks by proposing several epistemological norms for analysing the computational brain and social epistemes, and for developing pedagogical principles conducive to curious thought. This article is part of the theme issue 'Unifying the essential concepts of biological networks: biological insights and philosophical foundations'.

Spatial Transmission Models: A Taxonomy and Framework

Within risk analysis and, more broadly, the decision behind the choice of which modeling technique to use to study the spread of disease, epidemics, fires, technology, rumors, or, more generally, spatial dynamics, is not well documented. While individual models are well defined and the modeling techniques are well understood by practitioners, there is little deliberate choice made as to the type of model to be used, with modelers using techniques that are well accepted in the field, sometimes with little thought as to whether alternative modeling techniques could or should be used. In this article, we divide modeling techniques for spatial transmission into four main categories: population-level models, where a macro-level estimate of the infected population is required; cellular models, where the transmission takes place between connected domains, but is restricted to a fixed topology of neighboring cells; network models, where host-to-host transmission routes are modeled, either as planar spatial graphs or where shortcuts can take place as in social networks; and, finally, agent-based models that model the local transmission between agents, either as host-to-host geographical contacts, or by modeling the movement of the disease vector, with dynamic movement of hosts and vectors possible, on a Euclidian space or a more complex space deformed by the existence of information about the topology of the landscape. We summarize these techniques by introducing a taxonomy classifying these modeling approaches. Finally, we present a framework for choosing the most appropriate spatial modeling method, highlighting the links between seemingly disparate methodologies, bearing in mind that the choice of technique rests with the subject expert.

Geometric assortative growth model for small-world networks

It has been shown that both humanly constructed and natural networks are often characterized by small-world phenomenon and assortative mixing. In this paper, we propose a geometrically growing model for small-world networks. The model displays both tunable small-world phenomenon and tunable assortativity. We obtain analytical solutions of relevant topological properties such as order, size, degree distribution, degree correlation, clustering, transitivity, and diameter. It is also worth noting that the model can be viewed as a generalization for an iterative construction of Farey graphs.

Competition between quenched disorder and long-range connections: a numerical study of diffusion

The problem of random walk is considered in one dimension in the simultaneous presence of a quenched random force field and long-range connections, the probability of which decays with the distance algebraically as p(l)≃βl(-s). The dynamics are studied mainly by a numerical strong disorder renormalization group method. According to the results, for s>2 the long-range connections are irrelevant, and the mean-square displacement increases as <x(2)(t)>∼(lnt)(2/ψ) with the barrier exponent ψ=1/2, which is known in one-dimensional random environments. For s<2, instead, the quenched disorder is found to be irrelevant, and the dynamical exponent is z=1 like in a homogeneous environment. At the critical point, s=2, the interplay between quenched disorder and long-range connections results in activated scaling, however, with a nontrivial barrier exponent ψ(β), which decays continuously with β but is independent of the form of the quenched disorder. Upper and lower bounds on ψ(β) are established, and numerical estimates are given for various values of β. Besides random walks, accurate numerical estimates of the graph dimension and the resistance exponent are given for various values of β at s=2.

Effect of the nature of randomness on quenching dynamics of the Ising model on complex networks

Randomness is known to affect the dynamical behavior of many systems to a large extent. In this paper we investigate how the nature of randomness affects the dynamics in a zero-temperature quench of the Ising model on two types of random networks. In both networks, which are embedded in a one-dimensional space, the first-neighbor connections exist and the average degree is 4 per node. In random model A the second-neighbor connections are rewired with a probability p, while in random model B additional connections between neighbors at a Euclidean distance l(l > 1) are introduced with a probability P(l) proportionally l(-α). We find that for both models, the dynamics leads to freezing such that the system gets locked in a disordered state. The point at which the disorder of the nonequilibrium steady state is maximum is located. The behavior of dynamical quantities such as residual energy, order parameter, and persistence are discussed and compared. Overall, the behavior of physical quantities are similar, although subtle differences are observed due to the difference in the nature of randomness.

Interarrival times of message propagation on directed networks

One of the challenges in fighting cybercrime is to understand the dynamics of message propagation on botnets, networks of infected computers used to send viruses, unsolicited commercial emails (SPAM) or denial of service attacks. We map this problem to the propagation of multiple random walkers on directed networks and we evaluate the interarrival time distribution between successive walkers arriving at a target. We show that the temporal organization of this process, which models information propagation on unstructured peer to peer networks, has the same features as SPAM reaching a single user. We study the behavior of the message interarrival time distribution on three different network topologies using two different rules for sending messages. In all networks the propagation is not a pure Poisson process. It shows universal features on Poissonian networks and a more complex behavior on scale free networks. Results open the possibility to indirectly learn about the process of sending messages on networks with unknown topologies, by studying interarrival times at any node of the network.

Dynamic model of time-dependent complex networks

The characterization of the "most connected" nodes in static or slowly evolving complex networks has helped in understanding and predicting the behavior of social, biological, and technological networked systems, including their robustness against failures, vulnerability to deliberate attacks, and diffusion properties. However, recent empirical research of large dynamic networks (characterized by irregular connections that evolve rapidly) has demonstrated that there is little continuity in degree centrality of nodes over time, even when their degree distributions follow a power law. This unexpected dynamic centrality suggests that the connections in these systems are not driven by preferential attachment or other known mechanisms. We present an approach to explain real-world dynamic networks and qualitatively reproduce these dynamic centrality phenomena. This approach is based on a dynamic preferential attachment mechanism, which exhibits a sharp transition from a base pure random walk scheme.

Scaling behavior of the contact process in networks with long-range connections

We present simulation results for the contact process on regular cubic networks that are composed of a one-dimensional lattice and a set of long edges with unbounded length. Networks with different sets of long edges are considered that are characterized by different shortest-path dimensions and random-walk dimensions. We provide numerical evidence that an absorbing phase transition occurs at some finite value of the infection rate and the corresponding dynamical critical exponents depend on the underlying network. Furthermore, the time-dependent quantities exhibit log-periodic oscillations in agreement with the discrete scale invariance of the networks. In case of spreading from an initial active seed, the critical exponents are found to depend on the location of the initial seed and break the hyperscaling law of the directed percolation universality class due to the inhomogeneity of the networks. However, if the cluster-spreading quantities are averaged over initial sites, the hyperscaling law is restored.

Stripe domain coarsening in geographical small-world networks on a Euclidean lattice

We study phase ordering dynamics of spatially periodic striped patterns on the small-world network that is derived from a two-dimensional regular lattice with distance-dependent random connections. It is demonstrated numerically that addition of spatial disorder in the form of shortcuts makes the growth of domains much slower or even frozen at late times.

Superdiffusion in a class of networks with marginal long-range connections

A class of cubic networks composed of a regular one-dimensional lattice and a set of long-range links is introduced. Networks parametrized by a positive integer k are constructed by starting from a one-dimensional lattice and iteratively connecting each site of degree 2 with a kth neighboring site of degree 2. Specifying the way pairs of sites to be connected are selected, various random and regular networks are defined, all of which have a power-law edge-length distribution of the form P_{>}(l) approximately l;{-s} with the marginal exponent s=1 . In all these networks, lengths of shortest paths grow as a power of the distance and random walk is superdiffusive. Applying a renormalization group method, the corresponding shortest-path dimensions and random-walk dimensions are calculated exactly for k=1 networks and for k=2 regular networks; in other cases, they are estimated by numerical methods. Although, s=1 holds for all representatives of this class, the above quantities are found to depend on the details of the structure of networks controlled by k and other parameters.

Random-walk access times on partially disordered complex networks: an effective medium theory

An analytic effective medium theory is constructed to study the mean access times for random walks on hybrid disordered structures formed by embedding complex networks into regular lattices, considering transition rates F that are different for steps across lattice bonds from the rates f across network shortcuts. The theory is developed for structures with arbitrary shortcut distributions and applied to a class of partially disordered traversal enhanced networks in which shortcuts of fixed length are distributed randomly with finite probability. Numerical simulations are found to be in excellent agreement with predictions of the effective medium theory on all aspects addressed by the latter. Access times for random walks on these partially disordered structures are compared to those on small-world networks, which on average appear to provide the most effective means of decreasing access times uniformly across the network.

Quantum transport on small-world networks: a continuous-time quantum walk approach

We consider the quantum mechanical transport of (coherent) excitons on small-world networks (SWNs). The SWNs are built from a one-dimensional ring of N nodes by randomly introducing B additional bonds between them. The exciton dynamics is modeled by continuous-time quantum walks, and we evaluate numerically the ensemble-averaged transition probability to reach any node of the network from the initially excited one. For sufficiently large B we find that the quantum mechanical transport through the SWNs is, first, very fast, given that the limiting value of the transition probability is reached very quickly, and second, that the transport does not lead to equipartition, given that on average the exciton is most likely to be found at the initial node.

Spatially embedded random networks

Many real-world networks analyzed in modern network theory have a natural spatial element; e.g., the Internet, social networks, neural networks, etc. Yet, aside from a comparatively small number of somewhat specialized and domain-specific studies, the spatial element is mostly ignored and, in particular, its relation to network structure disregarded. In this paper we introduce a model framework to analyze the mediation of network structure by spatial embedding; specifically, we model connectivity as dependent on the distance between network nodes. Our spatially embedded random networks construction is not primarily intended as an accurate model of any specific class of real-world networks, but rather to gain intuition for the effects of spatial embedding on network structure; nevertheless we are able to demonstrate, in a quite general setting, some constraints of spatial embedding on connectivity such as the effects of spatial symmetry, conditions for scale free degree distributions and the existence of small-world spatial networks. We also derive some standard structural statistics for spatially embedded networks and illustrate the application of our model framework with concrete examples.

Phase transition of a one-dimensional Ising model with distance-dependent connections

The critical behavior of the Ising model on a one-dimensional network, which has long-range connections at distances l>1 with the probability theta(l) approximately l(-m), is studied by using Monte Carlo simulations. Through analyzing the Ising model on networks with different m values, this paper discusses the impact of the global correlation, which decays with the increase of m, on the phase transition of the Ising model. Adding the analysis of the finite-size scaling of the order parameter [M], it is observed that in the whole range of 0<m<2, a finite-temperature transition exists, and the critical exponents show consistence with mean-field values, which indicates a mean-field nature of the phase transition.

Nature-inspired interconnects for self-assembled large-scale network-on-chip designs

Future nanoscale electronics built up from an Avogadro number of components need efficient, highly scalable, and robust means of communication in order to be competitive with traditional silicon approaches. In recent years, the networks-on-chip (NoC) paradigm emerged as a promising solution to interconnect challenges in silicon-based electronics. Current NoC architectures are either highly regular or fully customized, both of which represent implausible assumptions for emerging bottom-up self-assembled molecular electronics that are generally assumed to have a high degree of irregularity and imperfection. Here, we pragmatically and experimentally investigate important design tradeoffs and properties of an irregular, abstract, yet physically plausible three-dimensional (3D) small-world interconnect fabric that is inspired by modern network-on-chip paradigms. We vary the framework's key parameters, such as the connectivity, number of switch nodes, and distribution of long- versus short-range connections, and measure the network's relevant communication characteristics. We further explore the robustness against link failures and the ability and efficiency to solve a simple toy problem, the synchronization task. The results confirm that (1) computation in irregular assemblies is a promising and disruptive computing paradigm for self-assembled nanoscale electronics and (2) that 3D small-world interconnect fabrics with a power-law decaying distribution of shortcut lengths are physically plausible and have major advantages over local two-dimensional and 3D regular topologies.

Phase transitions in an Ising model on a Euclidean network

A one-dimensional network on which there are long-range bonds at lattice distances l>1 with the probability P(l) proportional to l(-delta) has been taken under consideration. We investigate the critical behavior of the Ising model on such a network where spins interact with these extra neighbors apart from their nearest neighbors for 0<or=delta<2. It is observed that there is a finite temperature phase transition in the entire range. For 0<or=delta<1, finite-size scaling behavior of various quantities are consistent with mean-field exponents while for 1<or=delta<or=2, the exponents depend on delta. The results are discussed in the context of earlier observations on the topology of the underlying network.

Synchronization landscapes in small-world-connected computer networks

Motivated by a synchronization problem in distributed computing we studied a simple growth model on regular and small-world networks, embedded in one and two dimensions. We find that the synchronization landscape (corresponding to the progress of the individual processors) exhibits Kardar-Parisi-Zhang-like kinetic roughening on regular networks with short-range communication links. Although the processors, on average, progress at a nonzero rate, their spread (the width of the synchronization landscape) diverges with the number of nodes (desynchronized state) hindering efficient data management. When random communication links are added on top of the one and two-dimensional regular networks (resulting in a small-world network), large fluctuations in the synchronization landscape are suppressed and the width approaches a finite value in the large system-size limit (synchronized state). In the resulting synchronization scheme, the processors make close-to-uniform progress with a nonzero rate without global intervention. We obtain our results by "simulating the simulations," based on the exact algorithmic rules, supported by coarse-grained arguments.

Efficiency of quantum and classical transport on graphs

We propose a measure to quantify the efficiency of classical and quantum mechanical transport processes on graphs. The measure only depends on the density of states (DOS), which contains all the necessary information about the graph. For some given (continuous) DOS, the measure shows a power law behavior, where the exponent for the quantum transport is twice the exponent of its classical counterpart. For small-world networks, however, the measure shows rather a stretched exponential law but still the quantum transport outperforms the classical one. Some finite tree graphs have a few highly degenerate eigenvalues, such that, on the other hand, on them the classical transport may be more efficient than the quantum one.

Two-dimensional small-world networks: navigation with local information

A navigation process is studied on a variant of the Watts-Strogatz small-world network model embedded on a square lattice. With probability , each vertex sends out a long-range link, and the probability of the other end of this link falling on a vertex at lattice distance away decays as r(-a). Vertices on the network have knowledge of only their nearest neighbors. In a navigation process, messages are forwarded to a designated target. For alpha < 3 and alpha not equal to 2, a scaling relation is found between the average actual path length and , where is the average length of the additional long range links. Given pL > 1, a dynamic small world effect is observed, and the behavior of the scaling function at large enough is obtained. At alpha = 2 and 3, this kind of scaling breaks down, and different functions of the average actual path length are obtained. For alpha > 3, the average actual path length is nearly linear with network size.

Physical realizability of small-world networks

Supplementing a lattice with long-range connections effectively models small-world networks characterized by a high local and global interconnectedness observed in systems ranging from society to the brain. If the links have a wiring cost associated with their length l, the corresponding distribution q(l) plays a crucial role. Uniform length distributions have received the most attention despite indications that q(l) approximately l(-alpha) exists-e.g., for integrated circuits, the Internet, and cortical networks. While length distributions of this type were previously examined in the context of navigability, we here discuss for such systems the emergence and physical realizability of small-world topology. Our simple argument allows us to understand under which condition and at what expense a small world results.

Elastic properties of small-world spring networks

We construct small-world spring networks based on a one-dimensional chain and study its static and quasistatic behavior with respect to external forces. Regular bonds and shortcuts are assigned linear springs of constant k and k', respectively. In our models, shortcuts can only stand extensions less than deltac beyond which they are removed from the network. First we consider the simple cases of a hierarchical small-world network and a complete network. In the main part of this paper we study random small-world networks (RSWN) in which each pair of nodes is connected by a shortcut with probability p. We obtain a scaling relation for the effective stiffness of RSWN when k=k'. In this case the extension distribution of shortcuts is scale free with the exponent -2. There is a strong positive correlation between the extension of shortcuts and their betweenness. We find that the chemical end-to-end distance (CEED) could change either abruptly or continuously with respect to the external force. In the former case, the critical force is determined by the average number of shortcuts emanating from a node. In the latter case, the distribution of changes in CEED obeys power laws of the exponent -alpha with alpha < or = 3/2.

Traversal times for random walks on small-world networks

We study the mean traversal time for a class of random walks on Newman-Watts small-world networks, in which steps around the edge of the network occur with a transition rate that is different from the rate for steps across small-world connections. When f>>F, the mean time to traverse the network exhibits a transition associated with percolation of the random graph (i.e., small-world) part of the network, and a collapse of the data onto a universal curve. This transition was not observed in earlier studies in which equal transition rates were assumed for all allowed steps. We develop a simple self-consistent effective-medium theory and show that it gives a quantitatively correct description of the traversal time in all parameter regimes except the immediate neighborhood of the transition, as is characteristic of most effective-medium theories.

Diffusion processes on power-law small-world networks

We consider diffusion processes on power-law small-world networks in different dimensions. In one dimension, we find a rich phase diagram, with different transient and recurrent phases, including a critical line with continuously varying exponents. The results were obtained using self-consistent perturbation theory and can also be understood in terms of a scaling theory, which provides a general framework for understanding processes on small-world networks with different distributions of long-range links.

Maximal planar networks with large clustering coefficient and power-law degree distribution

In this article, we propose a simple rule that generates scale-free networks with very large clustering coefficient and very small average distance. These networks are called random Apollonian networks (RANs) as they can be considered as a variation of Apollonian networks. We obtain the analytic results of power-law exponent gamma=3 and clustering coefficient C= (46/3)-36 ln 3/2 approximately 0.74, which agree with the simulation results very well. We prove that the increasing tendency of average distance of RANs is a little slower than the logarithm of the number of nodes in RANs. Since most real-life networks are both scale-free and small-world networks, RANs may perform well in mimicking the reality. The RANs possess hierarchical structure as C(k) approximately k(-1) that are in accord with the observations of many real-life networks. In addition, we prove that RANs are maximal planar networks, which are of particular practicability for layout of printed circuits and so on. The percolation and epidemic spreading process are also studied and the comparisons between RANs and Barabási-Albert (BA) as well as Newman-Watts (NW) networks are shown. We find that, when the network order N (the total number of nodes) is relatively small (as N approximately 10(4)), the performance of RANs under intentional attack is not sensitive to N , while that of BA networks is much affected by N. And the diseases spread slower in RANs than BA networks in the early stage of the susceptible-infected process, indicating that the large clustering coefficient may slow the spreading velocity, especially in the outbreaks.

Self-avoiding walks on scale-free networks

Several kinds of walks on complex networks are currently used to analyze search and navigation in different systems. Many analytical and computational results are known for random walks on such networks. Self-avoiding walks (SAW's) are expected to be more suitable than unrestricted random walks to explore various kinds of real-life networks. Here we study long-range properties of random SAW's on scale-free networks, characterized by a degree distribution P(k) approximately k(-gamma). In the limit of large networks (system size N-->infinity), the average number sn of SAW's starting from a generic site increases as mu(n) , with mu = k2/k-1 . For finite N, sn is reduced due to the presence of loops in the network, which causes the emergence of attrition of the paths. For kinetic growth walks, the average maximum length L increases as a power of the system size: L approximately Nalpha, with an exponent alpha increasing as the parameter gamma is raised. We discuss the dependence of alpha on the minimum allowed degree in the network. A similar power-law dependence is found for the mean self-intersection length of nonreversal random walks. Simulation results support our approximate analytical calculations.

Navigation in a small world with local information

It is commonly known that there exist short paths between vertices in a network showing the small-world effect. Yet vertices, for example, the individuals living in society, usually are not able to find the shortest paths, due to the very serious limit of information. To study this issue theoretically, here the navigation process of launching messages toward designated targets is investigated on a variant of the one-dimensional small-world network (SWN). In the network structure considered, the probability of a shortcut falling between a pair of nodes is proportional to r(-alpha) , where r is the lattice distance between the nodes. When alpha=0 , it reduces to the SWN model with random shortcuts. The system shows the dynamic small-world effect, which is different from the well-studied static SW effect. We study the effective network diameter, the path length as a function of the lattice distance, and the dynamics. They are controlled by multiple parameters, and we use data collapse to show that the parameters are correlated. The central finding is that, in the one-dimensional network studied, the dynamic SW effect exists for 0</=alpha</=2 . For each given value of alpha in this region, the point where the dynamic SW effect arises is M L' approximately 1 , where M is the number of useful shortcuts and L' is their average reduced (effective) length.

Dynamical scaling behavior of percolation clusters in scale-free networks

In this work we investigate the spectra of Laplacian matrices that determine many dynamic properties of scale-free networks below and at the percolation threshold. We use a replica formalism to develop analytically, based on an integral equation, a systematic way to determine the ensemble averaged eigenvalue spectrum for a general type of treelike networks. Close to the percolation threshold we find characteristic scaling functions for the density of states rho(lambda) of scale-free networks. rho(lambda) shows characteristic power laws rho (lambda) approximately lambda (alpha(1) ) or rho (lambda) approximately lambda (d(2) ) for small lambda, where alpha(1) holds below and alpha(2) at the percolation threshold. In the range where the spectra are accessible from a numerical diagonalization procedure the two methods lead to very similar results.

Particle-cluster aggregation on a small-world network

To describe the aggregation behaviors on substrates with long-range jump paths, a model of particle-cluster aggregation on a two-dimensional small-world network is presented. This model is characterized by two parameters: the clustering exponent alpha and the long-range connection rate phi. The results show that there exists an asymptotic fractal dimension D(max)(f) that depends upon alpha. With decrement of alpha, D(max)(f) varies from 1.7 to 2.0, which corresponds to a crossover from diffusion-limited-aggregation-like to dense growth. The change of the aggregation pattern results from the long-range connection in the network, which reduces the effect of screening during the aggregation. When the system size is not large enough, the effective fractal dimension D(f) depends upon phi because of the finite-size effect. With primitive analysis, we obtain the expression of the effective fractal dimension D(f) with the network parameters alpha and phi.

Self-avoiding walks and connective constants in small-world networks

Long-distance characteristics of small-world networks have been studied by means of self-avoiding walks (SAW's). We consider networks generated by rewiring links in one- and two-dimensional regular lattices. The number of SAW's u(n) was obtained from numerical simulations as a function of the number of steps n on the considered networks. The so-called connective constant, mu=lim(n-->infinity)u(n)/u(n-1), which characterizes the long-distance behavior of the walks, increases continuously with disorder strength (or rewiring probability p). For small p, one has a linear relation mu=mu(0)+ap, mu(0) and a being constants dependent on the underlying lattice. Close to p=1 one finds the behavior expected for random graphs. An analytical approach is given to account for the results derived from numerical simulations. Both methods yield results agreeing with each other for small p, and differ for p close to 1, because of the different connectivity distributions resulting in both cases.

Small-world properties of the Indian railway network

Structural properties of the Indian railway network is studied in the light of recent investigations of the scaling properties of different complex networks. Stations are considered as "nodes" and an arbitrary pair of stations is said to be connected by a "link" when at least one train stops at both stations. Rigorous analysis of the existing data shows that the Indian railway network displays small-world properties. We define and estimate several other quantities associated with this network.

Introducing small-world network effects to critical dynamics

We analytically investigate the kinetic Gaussian model and the one-dimensional kinetic Ising model of two typical small-world networks (SWN), the adding type and the rewiring type. The general approaches and some basic equations are systematically formulated. The rigorous investigation of the Glauber-type kinetic Gaussian model shows the mean-field-like global influence on the dynamic evolution of the individual spins. Accordingly a simplified method is presented and tested, which is believed to be a good choice for the mean-field transition widely (in fact, without exception so far) observed for SWN. It yields the evolving equation of the Kawasaki-type Gaussian model. In the one-dimensional Ising model, the p dependence of the critical point is analytically obtained and the nonexistence of such a threshold p(c), for a finite-temperature transition, is confirmed. The static critical exponents gamma and beta are in accordance with the results of the recent Monte Carlo simulations, and also with the mean-field critical behavior of the system. We also prove that the SWN effect does not change the dynamic critical exponent z=2 for this model. The observed influence of the long-range randomness on the critical point indicates two obviously different hidden mechanisms.

Quasistatic scale-free networks

A network is formed using the N sites of a one-dimensional lattice in the shape of a ring as nodes and each node with the initial degree k(in)=2. N links are then introduced to this network, each link starts from a distinct node, the other end being connected to any other node with degree k randomly selected with an attachment probability proportional to k(alpha). Tuning the control parameter alpha, we observe a transition where the average degree of the largest node <k(m)(alpha,N)> changes its variation from N0 to N at a specific transition point of alpha(c). The network is scale free, i.e., the nodal degree distribution has a power law decay for alpha> or = alpha(c).

Modulated scale-free network in Euclidean space

A random network is grown by introducing at unit rate randomly selected nodes on the Euclidean space. A node is randomly connected to its ith predecessor of degree k(i) with a directed link of length l using a probability proportional to k(i)l(alpha). Our numerical study indicates that the network is scale free for all values of alpha>alpha(c) and the degree distribution decays stretched exponentially for the other values of alpha. The link length distribution follows a power law: D(l) approximately l(delta), where delta is calculated exactly for the whole range of values of alpha.

Simple models of small-world networks with directed links

We investigate the effect of directed short- and long-range connections in a simple model of a small-world network. Our model is one in which we can determine many quantities of interest by an exact analytical method. We calculate the function V(T), defined as the number of sites affected up to time T when a naive spreading process starts in the network. As opposed to shortcuts, the presence of unfavorable bonds has a negative effect on this quantity. Hence, the spreading process may not be able to affect all of the network. We define and calculate a quantity identified as the average size of the accessible world in our model. The interplay of shortcuts and unfavorable bonds on the small world properties is studied.

Phase transitions in a network with a range-dependent connection probability

We consider a one-dimensional network in which the nodes at Euclidean distance l can have long range connections with a probability P(l) approximately l(-delta) in addition to nearest neighbor connections. This system has been shown to exhibit small-world behavior for delta<2, above which its behavior is like a regular lattice. From the study of the clustering coefficients, we show that there is a transition to a random network at delta=1. The finite size scaling analysis of the clustering coefficients obtained from numerical simulations indicates that a continuous phase transition occurs at this point. Using these results, we find that the two transitions occurring in this network can be detected in any dimension by the behavior of a single quantity, the average bond length. The phase transitions in all dimensions are nontrivial in nature.

Shortest paths on systems with power-law distributed long-range connections

We discuss shortest-path lengths l(r) on periodic rings of size L supplemented with an average of pL randomly located long-range links whose lengths are distributed according to P(l) approximately l(-mu). Using rescaling arguments and numerical simulation on systems of up to 10(7) sites, we show that a characteristic length xi exists such that l(r) approximately r for r<xi but l(r) approximately r(theta(s)(mu)) for r>>xi. For small p we find that the shortest-path length satisfies the scaling relation l(r,mu,p)/xi=f(mu,r/xi). Three regions with different asymptotic behaviors are found, respectively: (a) mu>2 where theta(s)=1, (b) 1<mu<2 where 0<theta(s)(mu)<1/2, and (c) mu<1 where l(r) behaves logarithmically, i.e., theta(s)=0. The characteristic length xi is of the form xi approximately p(-nu) with nu=1/(2-mu) in region (b), but depends on L as well in region (c). A directed model of shortest paths is solved and compared with numerical results.

Correlation effects in a simple model of a small-world network

We analyze the effect of correlations in a simple model of a small-world network by obtaining exact analytical expressions for the distribution of shortest paths in the network. We enter correlations into a simple model with a distinguished site, by taking the random connections to this site from an Ising distribution. Our method shows how the transfer-matrix technique can be used in the new context of small-world networks.

Trapping of random walks on small-world networks

We investigate the trapping of random walkers on small-world networks (SWN's), irregular graphs. We derive bounds for the survival probability Phi(SWN)(n) and display its analysis through cumulant expansions. Computer simulations are performed for large SWNs. We show that in the limit of infinite sizes, trapping on SWNs is equivalent to trapping on a certain class of random trees, which are grown during the random walk.

Target problem on small-world networks

In this work we focus on reactions on small-world networks (SWN's), disordered graphs of much recent interest. We study the target problem, since it allows an exact solution on regular lattices. On SWN's we find that the decay of the targets (for which we extend the formalism to disordered lattices) is again related to S(n), the mean number of distinct sites visited in n steps, although the S(n) vs n dependence changes here drastically in going from regular linear chains to their SWN.