Dynamic domain networks

A model for a dynamic network consisting of changing local interactions is presented in this work. While the network maintains solely local connections, certain properties known only to Small World Networks may be extracted due to the dynamic nature of the model. At each time step the individuals are grouped into clusters creating neighborhoods or domains of fully connected agents. The boundaries of these domains change in time, corresponding to a situation where the links between individuals are dynamic only throughout the history of the network. A question that we pose is whether our model, which maintains a local structure such that diffusion calculations are possible, might lead to analytic or conceptual advances for the much more complicated case of diffusion on a static disordered network that exhibits the same macroscopic properties as our dynamic ordered network. To answer this, we compare certain properties which characterize the dynamic domain network to those of a Small World Network, and then analyze the diffusion coefficients for three possible domain mutations. We close with a comparison and confirmation of previous epidemiological work carried out on networks.     

Modeling network growth with assortative mixing

We propose a model of an underlying mechanism responsible for the formation of assortative mixing in networks between “similar” nodes or vertices based on generic vertex properties. Existing models focus on a particular type of assortative mixing, such as mixing by vertex degree, or present methods of generating a network with certain properties, rather than modeling a mechanism driving assortative mixing during network growth. The motivation is to model assortative mixing by non-topological vertex properties, and the influence of these non-topological properties on network topology. The model is studied in detail for discrete and hierarchical vertex properties, and we use simulations to study the topology of resulting networks. We show that assortative mixing by generic properties directly drives the formation of community structure beyond a threshold assortativity of r ∼0.5, which in turn influences other topological properties. This direct relationship is demonstrated by introducing a new measure to characterise the correlation between assortative mixing and community structure in a network. Additionally, we introduce a novel type of assortative mixing in systems with hierarchical vertex properties, from which a hierarchical community structure is found to result. Electronic supplementary material Supplementary Online Material     

Agreement dynamics of finite-memory language games on networks

We propose a Finite-Memory Naming Game (FMNG) model with respect to the bounded rationality of agents or finite resources for information storage in communication systems. We study its dynamics on several kinds of complex networks, including random networks, small-world networks and scale-free networks. We focus on the dynamics of the FMNG affected by the memory restriction as well as the topological properties of the networks. Interestingly, we found that the most important quantity, the convergence time of reaching the consensus, shows some non-monotonic behaviors by varying the average degrees of the networks with the existence of the fastest convergence at some specific average degrees. We also investigate other main quantities, such as the success rate in negotiation, the total number of words in the system and the correlations between agents of full memory and the total number of words, which clearly explain the nontrivial behaviors of the convergence. We provide some analytical results which help better understand the dynamics of the FMNG. We finally report a robust scaling property of the convergence time, which is regardless of the network structure and the memory restriction.     

An evolutionary model of social networks

Social networks in communities, markets, and societies self-organise through the interactions of many individuals. In this paper we use a well-known mechanism of social interactions — the balance of sentiment in triadic relations — to describe the development of social networks. Our model contrasts with many existing network models, in that people not only establish but also break up relations whilst the network evolves. The procedure generates several interesting network features such as a variety of degree distributions and degree correlations. The resulting network converges under certain conditions to a steady critical state where temporal disruptions in triangles follow a power-law distribution.     

Evolving pseudofractal networks

We present a family of scale-free network model consisting of cliques, which is established by a simple recursive algorithm. We investigate the networks both analytically and numerically. The obtained analytical solutions show that the networks follow a power-law degree distribution, with degree exponent continuously tuned between 2 and 3. The exact expression of clustering coefficient is also provided for the networks. Furthermore, the investigation of the average path length reveals that the networks possess small-world feature. Interestingly, we find that a special case of our model can be mapped into the Yule process.     

Dynamics of congestion transition triggered by multiple walkers on complex networks

The congestion transition triggered by multiple walkers walking along the shortest path on complex networks is numerically investigated. These networks are composed of nodes that have a finite capacity in analogy to the buffer memory of a computer. It is found that a transition from free-flow phase to congestion phase occurs at a critical walker density f_c, which varies for complex networks with different topological structures. The dynamic pictures of congestion for networks with different topological structures show that congestion on scale-free networks is a percolation process of congestion clusters, while the dynamics of congestion transition on non-scale-free networks is mainly a process of nucleation.     

Can a few fanatics influence the opinion of a large segment of a society?

Models that provide insight into how extreme positions regarding any social phenomenon may spread in a society or at the global scale are of great current interest. A realistic model must account for the fact that globalization, internet, and other means of mass communications have given rise to scale-free networks of interactions between people. We propose a novel model which takes into account the nature of the interactions network, and provides some key insights into this phenomenon. These include, (1) the existence of a fundamental difference between a hierarchical network whereby people are influenced by those that are higher in the hierarchy but not by those below them, and a symmetrical network where person-on-person influence works mutually, and (2) that a few “fanatics” can influence a large fraction of the population either temporarily (in the hierarchical networks) or permanently (in symmetrical networks). Even if the “fanatics” disappear, the population may still remain susceptible to the positions originally advocated by them. The model is, however, general and applicable to any phenomenon for which there is a degree of enthusiasm or susceptibility to in the population.     

Growing distributed networks with arbitrary degree distributions

We consider distributed networks, such as peer-to-peer networks, whose structure can be manipulated by adjusting the rules by which vertices enter and leave the network. We focus in particular on degree distributions and show that, with some mild constraints, it is possible by a suitable choice of rules to arrange for the network to have any degree distribution we desire. We also describe a mechanism based on biased random walks by which appropriate rules could be implemented in practice. As an example application, we describe and simulate the construction of a peer-to-peer network optimized to minimize search times and bandwidth requirements.     

Concentric circular arrays of Josephson junctions as dressed models of granular superconductors

In this paper networks that optimize a combined measure of local and global synchronizability are evolved. It is shown that for low coupling improvements in the local synchronizability dominate network evolution. This leads to an expressed grouping of elements with similar native frequency into cliques, allowing for an early onset of synchronization, but rendering full synchronization hard to achieve. In contrast, for large coupling the network evolution is governed by improvements towards full synchronization, preventing any expressed community structure. Such networks exhibit strong coupling between dissimilar oscillators. Albeit a rapid transition to full synchronization is achieved, the onset of synchronization is delayed in comparison to the first type of networks. The paper illustrates that an early onset of synchronization (which relates to clustering) and global synchronization are conflicting demands on network topology.     

Cumulative Laws, Team Assembling Mechanisms Determining Network Structure

A number of researching works have shed light on the field of complex networks recently. We investigate a wide range of real-world networks and find several interesting phenomena. Firstly, almost all of these networks evolve by overlapping new small graphs on former networks. Secondly, not only the degree sequence of the mature network follows a power-law distribution, but also the distribution of the cumulative occurrence times during the growing process are revealed to have a heavy tail. Existing network evolving models do not provide interpretation to these phenomena. We suggest a model based on the team assembling mechanism, which is extracted from the growing processes of real-world networks and requires simple parameters, and produces networks exhibiting these properties observed in the present study and in previous works.     

Preservation of network degree distributions from non-uniform failures

There has been a considerable amount of interest in recent years on the robustness of networks to failures. Many previous studies have concentrated on the effects of node and edge removals on the connectivity structure of a static network; the networks are considered to be static in the sense that no compensatory measures are allowed for recovery of the original structure. Real world networks such as the world wide web, however, are not static and experience a considerable amount of turnover, where nodes and edges are both added and deleted. Considering degree-based node removals, we examine the possibility of preserving networks from these types of disruptions. We recover the original degree distribution by allowing the network to react to the attack by introducing new nodes and attaching their edges via specially tailored schemes. We focus particularly on the case of non-uniform failures, a subject that has received little attention in the context of evolving networks. Using a combination of analytical techniques and numerical simulations, we demonstrate how to preserve the exact degree distribution of the studied networks from various forms of attack.     

Analyzing and modeling real-world phenomena with complex networks: a survey of applications

The success of new scientific areas can be assessed by their potential in contributing to new theoretical approaches and in applications to real-world problems. Complex networks have fared extremely well in both of these aspects, with their sound theoretical basis being developed over the years and with a variety of applications. In this survey, we analyze the applications of complex networks to real-world problems and data, with emphasis in representation, analysis and modeling. A diversity of phenomena are surveyed, which may be classified into no less than 11 areas, providing a clear indication of the impact of the field of complex networks.     

A Law of Gravitation for Complex Networks

Inspiring Newton＇s law of universal gravitation and empirical studies, we propose a concept of virtual network mass and network gravitational force in complex networks. Then a network gravitational model for complex networks is presented. In the model, each node in the network is described with its position, edges （links） and virtual network mass. The proposed model is examined by experiments to show its potential applications.     

Inevitable self-similar topology of binary trees and their diverse hierarchical density

Self-similar topology, which can be characterized as power law size distribution, has been found in diverse tree networks ranging from river networks to taxonomic trees. In this study, we find that the statistical self-similar topology is an inevitable consequence of any full binary tree organization. We show this by coding a binary tree as a unique bifurcation string. This coding scheme allows us to investigate trees over the realm from deterministic to entirely random trees. To obtain partial random trees, partial random perturbation is added to the deterministic trees by an operator similar to that used in genetic algorithms. Our analysis shows that the hierarchical density of binary trees is more diverse than has been described in earlier studies. We find that the connectivity structure of river networks is far from strict self-similar trees. On the other hand, organization of some social networks is close to deterministic supercritical trees.     

Accelerating networks: Effects of preferential connections

Networks are commonly observed structures in complex systems with interacting and interdependent parts that self-organize. For nonlinearly growing networks, when the total number of connections increases faster than the total number of nodes, the network is said to accelerate. We propose a systematic model for the dynamics of growing networks represented by distribution kinetics equations. We define the nodal-linkage distribution, construct a population dynamics equation based on the association-dissociation process, and perform the moment calculations to describe the dynamics of such networks. For nondirectional networks with finite numbers of nodes and connections, the moments are the total number of nodes, the total number of connections, and the degree (the average number of connections per node), represented by the average moment. Size independent rate coefficients yield an exponential network describing the network without preferential attachment, and size dependent rate coefficients produce a power law network with preferential attachment. The model quantitatively describes accelerating network growth data for a supercomputer (Earth Simulator), for regulatory gene networks, and for the Internet.     

Network evolution by different rewiring schemes

Many real world networks, such as social networks, are characterized by rearrangements of the links between nodes (rewiring). Indeed, very few natural networks are static in time, and it is therefore important to study the properties of networks in which rewiring occurs. In this paper, two different rewiring schemes are formulated and compared using a general ordinary differential equation (ODE) model. The equilibrium distributions are analytically derived. It is found that by uniformly choosing a node and a link connected to it, rewiring from different ends of the link yields different equilibrium degree distributions. Rewiring from the neighbor generally produces more high degree nodes. The equilibrium distributions of the ODE model are compared with simulation results of the corresponding stochastic process for rewiring. Conditions are discussed under which our ODE provides a good approximation for the mean of the corresponding stochastic process.     

On the rich-club effect in dense and weighted networks

For many complex networks present in nature only a single instance, usually of large size, is available. Any measurement made on this single instance cannot be repeated on different realizations. In order to detect significant patterns in a real-world network it is therefore crucial to compare the measured results with a null model counterpart. Here we focus on dense and weighted networks, proposing a suitable null model and studying the behaviour of the degree correlations as measured by the rich-club coefficient. Our method solves an existing problem with the randomization of dense unweighted graphs, and at the same time represents a generalization of the rich-club coefficient to weighted networks which is complementary to other recently proposed ones.     

Rank-based model for weighted network with hierarchical organization and disassortative mixing

In this paper, we study a rank-based model for weighted network. The evolution rule of the network is based on the ranking of node strength, which couples the topological growth and the weight dynamics. Analytically and by simulations, we demonstrate that the generated networks recover the scale-free distributions of degree and strength in the whole region of the growth dynamics parameter (α>0). Moreover, this network evolution mechanism can also produce scale-free property of weight, which adds deeper comprehension of the networks growth in the presence of incomplete information. We also characterize the clustering and correlation properties of this class of networks. It is showed that at α=1 a structural phase transition occurs, and for α>1 the generated network simultaneously exhibits hierarchical organization and disassortative degree correlation, which is consistent with a wide range of biological networks.     

Jamming transition in air transportation networks

In this work we present a model of an air transportation traffic system from the complex network modelling viewpoint. In the network, every node corresponds to a given airport, and two nodes are connected by means of flight routes. Each node is weighted according to its load capacity, and links are weighted according to the Euclidean distance that separates each pair of nodes. Local rules describing the behaviour of individual nodes in terms of the surrounding flow have been also modelled, and a random network topology has been chosen in a baseline approach. Numerical simulations describing the diffusion of a given number of agents (aircraft) in this network show the onset of a jamming transition that distinguishes an efficient regime with null amount of airport queues and high diffusivity (free phase) and a regime where bottlenecks suddenly take place, leading to a poor aircraft diffusion (congested phase). Fluctuations are maximal around the congestion threshold, suggesting that the transition is critical. We then proceed by exploring the robustness of our results in neutral random topologies by embedding the model in heterogeneous networks. Specifically, we make use of the European air transportation network formed by 858 airports and 11 170 flight routes connecting them, which we show to be scale-free. The jamming transition is also observed in this case. These results and methodologies may introduce relevant decision-making procedures in order to optimize the air transportation traffic.     

Decline of minorities in stubborn societies

Opinion compromise models can give insight into how groups of individuals may either come to form consensus or clusters of opinion groups, corresponding to parties. We consider models where randomly selected individuals interact pairwise. If the opinions of the interacting agents are not within a certain confidence threshold, the agents retain their own point of view. Otherwise, they constructively dialogue and smooth their opinions. Persuasible agents are inclined to compromise with interacting individuals. Stubborn individuals slightly modify their opinion during the interaction. Collective states for persuasible societies include extremist minorities, which instead decline in stubborn societies. We derive a mean field approximation for the compromise model in stubborn populations. Bifurcation and clustering analysis of this model compares favorably with Monte Carlo analysis found in the literature.