Nonlinear Preferential Attachment Research Articles

Intra-community link formation and modularity in ultracold growing hyperbolic networks

Hyperbolic network models offer a straightforward yet powerful method for understanding the small-world, scale-free, highly clustered, and modular characteristics typical of complex systems that are often called as real-world networks. These models involve randomly positioning nodes in a hyperbolic space and connecting them based on a probability that diminishes with distance. In this study, we examine the community structure within networks created by the Popularity Similarity Optimization model, a fundamental hyperbolic model, when the temperature parameter (responsible for tuning the clustering coefficient) is set to the limiting value of zero. We focus on link formation within communities and show their close relation with non-linear preferential attachment processes. Based on this, we derive analytical expressions that significantly improve previous estimates of the expected modularity for partitions formed by equally sized angular sectors in the 2d hyperbolic space. Our formulas can now predict average modularity, confirmed by numerical simulations, with high accuracy over a broader range of model parameters and community sizes relative to the whole network. These results advance our understanding of module formation in hyperbolic networks, highlighting the surprising emergence of communities despite the lack of explicit community formation steps in the model definition.

Measuring the variability of local characteristics in complex networks: Empirical and analytical analysis.

We examine the dynamics for the average degree of a node's neighbors in complex networks. It is a Markov stochastic process, and at each moment of time, this quantity takes on its values in accordance with some probability distribution. We are interested in some characteristics of this distribution: its expectation and its variance, as well as its coefficient of variation. First, we look at several real communities to understand how these values change over time in social networks. The empirical analysis of the behavior of these quantities for real networks shows that the coefficient of variation remains at high level as the network grows. This means that the standard deviation and the mean degree of the neighbors are comparable. Then, we examine the evolution of these three quantities over time for networks obtained as simulations of one of the well-known varieties of the Barabási-Albert model, the growth model with nonlinear preferential attachment (NPA) and a fixed number of attached links at each iteration. We analytically show that the coefficient of variation for the average degree of a node's neighbors tends to zero in such networks (albeit very slowly). Thus, we establish that the behavior of the average degree of neighbors in Barabási-Albert networks differs from its behavior in real networks. In this regard, we propose a model based on the NPA mechanism with the rule of random number of edges added at each iteration in which the dynamics of the average degree of neighbors is comparable to its dynamics in real networks.

An Investigation into the Trend Stationarity of Local Characteristics in Media and Social Networks

We studied the evolution of complex social networks over time. The elements of the networks are users, and the connections correspond to the interactions between them. At a particular moment in time, each node of a complex network has such characteristics as its degree, as well as the total degree of its neighbors. Obviously, in the process of network growth, these characteristics are constantly changing due to the fact that new edges are attached to this node or its neighbors. In this paper, we study the dynamics of these characteristics over time for networks generated on the basis of a nonlinear preferential attachment mechanism, and we find both the asymptotics of their expected values and the characteristics of their spread around the mean. In addition, we analyze the behavior of these local characteristics for three real social networks. The applicability of the findings to actual problems in the study of social media in the digital humanities is discussed.

A detailed characterization of complex networks using Information Theory

Understanding the structure and the dynamics of networks is of paramount importance for many scientific fields that rely on network science. Complex network theory provides a variety of features that help in the evaluation of network behavior. However, such analysis can be confusing and misleading as there are many intrinsic properties for each network metric. Alternatively, Information Theory methods have gained the spotlight because of their ability to create a quantitative and robust characterization of such networks. In this work, we use two Information Theory quantifiers, namely Network Entropy and Network Fisher Information Measure, to analyzing those networks. Our approach detects non-trivial characteristics of complex networks such as the transition present in the Watts-Strogatz model from k-ring to random graphs; the phase transition from a disconnected to an almost surely connected network when we increase the linking probability of Erdős-Rényi model; distinct phases of scale-free networks when considering a non-linear preferential attachment, fitness, and aging features alongside the configuration model with a pure power-law degree distribution. Finally, we analyze the numerical results for real networks, contrasting our findings with traditional complex network methods. In conclusion, we present an efficient method that ignites the debate on network characterization.

The impact of node arrival process and stochastic edge growth on scale-free distribution in complex networks

ABSTRACT A variety of proposals has been introduced to dynamically complete scale-free (SF) networks that are utilized to generate Barabási-Albert (BA) graph models. The analytical method for analysis of real networks, which is based on uniform distributions proposed in the BA model, does not seem so perfect. Consequently, some scholars refined and extended the BA model among which non-linear preferential attachment (NLPA), dynamical edge rewiring, fitness models and other growth models presented in the literature. In a network, features like how individuals enter the network, the probability distribution of edge growth, and the probability distribution of nodes’ age have considerable significance. A key question that neglects in the BA model is that ‘Can LPA phenomenon basically generate SF graphs apart from the arrival process of nodes and the probability distribution of nodes’ age?’. In this paper, we demonstrate the resulted graphs belong to the class of small-world or random networks contrary to the employ of LPA in the proposed model. Hence, what neglect in the BA model prevents us from attaining a clear interpretation of all properties of real-world networks. This study attempts to uncover these features by proposing an extended BA model.

The study of the duration of transient processes in growing random graphs with a non-linear preferential attachment rule

A study of the duration of transients processes in random graphs with preferred attachment was carried out, during which it was found out that for some random graphs transient processes can last very long. For some known growing graphs the absence of stationary distribution of the vertices attachment degree has been found out for the first time. This makes the calculation methods and the techniques for analyzing transient processes up-to-date, because real networks modelled by this graphs, do not have stationary characteristics either. Real networks having stationary characteristics, can for a long time, even if they are too large, stay in the transient mode which does not let estimate these networks by stationary solutions. While modeling these networks, the methods for analyzing the transient modes are also urgent as they provide for controlling the errors of stationary solutions application. The paper gives the example of the stationary solution error control for a random graph, widely used in modeling of growing networks.

Growing graphs with addition of communities

Paper proposes a model of large networks based on a random preferential attachment graph with addition of complete subgraphs (cliques). The proposed model refers to models of random graphs following the nonlinear preferential attachment rule, and takes into account the possibility of «adding» entire communities of nodes to the network. In the derivation of the relations that determine the vertex degree distribution, the technique of finite-difference equations describing stationary states of a graph is used. The obtained results are tested empirically (by generating large graphs), special cases correspond to known mathematical relations.

The investigation of social networks based on multi-component random graphs

The methods of non-homogeneous random graphs calibration are developed for social networks simulation. The graphs are calibrated by the degree distributions of the vertices and the edges. The mathematical foundation of the methods is formed by the theory of random graphs with the nonlinear preferential attachment rule and the theory of Erdôs-Rényi random graphs. In fact, well-calibrated network graph models and computer experiments with these models would help developers (owners) of the networks to predict their development correctly and to choose effective strategies for controlling network projects.

Non-uniform Evolving Hypergraphs and Weighted Evolving Hypergraphs.

Firstly, this paper proposes a non-uniform evolving hypergraph model with nonlinear preferential attachment and an attractiveness. This model allows nodes to arrive in batches according to a Poisson process and to form hyperedges with existing batches of nodes. Both the number of arriving nodes and that of chosen existing nodes are random variables so that the size of each hyperedge is non-uniform. This paper establishes the characteristic equation of hyperdegrees, calculates changes in the hyperdegree of each node, and obtains the stationary average hyperdegree distribution of the model by employing the Poisson process theory and the characteristic equation. Secondly, this paper constructs a model for weighted evolving hypergraphs that couples the establishment of new hyperedges, nodes and the dynamical evolution of the weights. Furthermore, what is obtained are respectively the stationary average hyperdegree and hyperstrength distributions by using the hyperdegree distribution of the established unweighted model above so that the weighted evolving hypergraph exhibits a scale-free behavior for both hyperdegree and hyperstrength distributions.

Growing network: Models following nonlinear preferential attachment rule

We investigate the preferential attachment graphs proceeding from the following two assumptions. The first one: the probability that a new vertex connects to a vertex i is proportional to an arbitrary nonnegative function f of a vertex degree k. The second assumption: a new vertex can have a random number of edges. We derive formulas for any f to determine the vertex degree distribution {Qk} in generated graphs. The inverse problem is solved: we have obtained formulas, that allow from a given distribution {Qk} to determine f (the problem of a model calibration). The formulas allowing for any f to calculate the joint distribution of vertex degrees at the ends of randomly selected edge are also obtained. Some other results are presented in the paper.

Nonlinear growth and condensation in multiplex networks.

Different types of interactions coexist and coevolve to shape the structure and function of a multiplex network. We propose here a general class of growth models in which the various layers of a multiplex network coevolve through a set of nonlinear preferential attachment rules. We show, both numerically and analytically, that by tuning the level of nonlinearity these models allow us to reproduce either homogeneous or heterogeneous degree distributions, together with positive or negative degree correlations across layers. In particular, we derive the condition for the appearance of a condensed state in which one node in each layer attracts an extensive fraction of all the edges.

Emergence of scaling in non-uniform hypernetworksdoes the rich get richer lead to a power-law distribution?

In this paper, we propose a hypernetwork model with a nonlinear preferential attachment, and study the evolving mechanism and topological properties of the hypernetwork. We analyze the model by using a Poisson process theory and a continuous technique, and give a characteristic equation of hyperdegrees. We obtain the stationary average hyperdegree distribution of the hypernetwork by the characteristic equation. The analytical result shows that the hypernetwork has a phenomenon of the rich get richer, and it accords well with the simulation. It is shown in this paper that the hyperdegree distribution of the dynamic model exhibits a stretched exponential distribution with the increase of the hypernetwork size. It proves that the rich get richer does not necessarily induce a power-law distribution.

Entropy rate of nonequilibrium growing networks

New entropy measures have been recently introduced for the quantification of the complexity of networks. Most of these entropy measures apply to static networks or to dynamical processes defined on static complex networks. In this paper we define the entropy rate of growing network models. This entropy rate quantifies how many labeled networks are typically generated by the growing network models. We analytically evaluate the difference between the entropy rate of growing tree network models and the entropy of tree networks that have the same asymptotic degree distribution. We find that the growing networks with linear preferential attachment generated by dynamical models are exponentially less than the static networks with the same degree distribution for a large variety of relevant growing network models. We study the entropy rate for growing network models showing structural phase transitions including models with nonlinear preferential attachment. Finally, we bring numerical evidence that the entropy rate above and below the structural phase transitions follows a different scaling with the network size.

Local preferential attachment model for hierarchical networks

In real-life networks, incomers may only connect to a few others in a local area for their limited information, and individuals in a local area are likely to have close relations. Accordingly, we propose a local preferential attachment model. Here, a local-area-network stands for a node and all its neighbors, and the new nodes perform nonlinear preferential attachment, π ( k i ) ∝ k i α , in local areas. The stable degree distribution and clustering-degree correlations are analytically obtained. With the increasing of α , the clustering coefficient increases, while assortativity decreases from positive to negative. In addition, by adjusting the parameter α , the model can generate different kinds of degree distribution, from exponential to power-law. The hierarchical organization, independent of α , is the most significant character of this model.

MODELLING COLLABORATION NETWORKS BASED ON NONLINEAR PREFERENTIAL ATTACHMENT

In this paper, we propose an alternative model for collaboration networks based on nonlinear preferential attachment. Depending on a single free parameter "preferential exponent", this model interpolates between networks with a scale-free and an exponential degree distribution. The degree distribution in the present networks can be roughly classified into four patterns, all of which are observed in empirical data. And this model exhibits small-world effect, which means the corresponding networks are of very short average distance and highly large clustering coefficient. More interesting, we find a peak distribution of act-size from empirical data which has not been emphasized before. Our model can produce the peak act-size distribution naturally that agrees with the empirical data well.

Nonlocal evolution of weighted scale-free networks

We introduce the notion of globally updating evolution for a class of weighted networks, in which the weight of a link is characterized by the amount of data packet transport flowing through it. By noting that the packet transport over the network is determined nonlocally, this approach can explain the generic nonlinear scaling between the strength and the degree of a node. We demonstrate by a simple model that the strength-driven evolution scheme recently introduced can be generalized to a nonlinear preferential attachment rule, generating the power-law behaviors in degree and in strength simultaneously.

Accurately modeling the internet topology

Based on measurements of the internet topology data, we found that there are two mechanisms which are necessary for the correct modeling of the internet topology at the autonomous systems (AS) level: the interactive growth of new nodes and new internal links, and a nonlinear preferential attachment, where the preference probability is described by a positive-feedback mechanism. Based on the above mechanisms, we introduce the positive-feedback preference (PFP) model which accurately reproduces many topological properties of the AS-level internet, including degree distribution, rich-club connectivity, the maximum degree, shortest path length, short cycles, disassortative mixing, and betweenness centrality. The PFP model is a phenomenological model which provides an insight into the evolutionary dynamics of real complex networks.