Cyclic Preferential Attachment in Complex Networks

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.

Scale-free networks, as one major class of complex networks, are characterized by a specific structural feature of power-law degree distributions. Examples of scale-free real-world networks include World Wide Web (WWW), email networks, citation and co-author scientific networks, social networks (Facebook, Twitter), networks of Internet routers, and protein-protein interaction networks. In this paper we model and evaluate the reliability of scale-free complex networks using a binary decision diagram (BDD)-based method. Two-terminal reliability is considered, which is defined as the probability that a specified pair of nodes can communication through at least one fault-tree path in the network. This reliability metric can measure, for example, how reliable the information can be shared between two parties or in general how reliable the information can be disseminated within a social network. The Barabasi-Albert model is used for generating the sample scale-free networks studied in this paper. The reliability performance of scale-free networks is compared to that of random networks under the Erdos-Renyi model. Degree distributions and clustering coefficients (average fraction of pairs of neighbors of a node which are also neighbors of each other) of sample scale-free and random networks are illustrated and compared. Their effects on the performance of the BDD-based method for the reliability analysis of both scale-free and random networks are also discussed.

Whether real-world complex networks are scale free or not has long been controversial. Recently, in Broido and Clauset [A. D. Broido, A. Clauset, Nat. Commun. 10, 1017 (2019)], it was claimed that the degree distributions of real-world networks are rarely power law under statistical tests. Here, we attempt to address this issue by defining a fundamental property possessed by each link, the degree-degree distance, the distribution of which also shows signs of being power law by our empirical study. Surprisingly, although full-range statistical tests show that degree distributions are not often power law in real-world networks, we find that in more than half of the cases the degree-degree distance distributions can still be described by power laws. To explain these findings, we introduce a bidirectional preferential selection model where the link configuration is a randomly weighted, two-way selection process. The model does not always produce solid power-law distributions but predicts that the degree-degree distance distribution exhibits stronger power-law behavior than the degree distribution of a finite-size network, especially when the network is dense. We test the strength of our model and its predictive power by examining how real-world networks evolve into an overly dense stage and how the corresponding distributions change. We propose that being scale free is a property of a complex network that should be determined by its underlying mechanism (e.g., preferential attachment) rather than by apparent distribution statistics of finite size. We thus conclude that the degree-degree distance distribution better represents the scale-free property of a complex network.

Robustness and Optimization of Complex Networks: Reconstructability, Algorithms and Modeling

Scale-free networks beyond power-law degree distribution

The study of complex networks has garnered significant attention in recent years due to their ability to visualize intricate systems. However, understanding the fundamental structures underlying their formation dynamics has remained elusive due to the enigmatic interplay between local and global properties. A hallmark of real-world complex networks is the presence of power-law distributions, which account for many of their intriguing characteristics. Preferential attachment has emerged as a common mechanism for generating scale-free networks. This research introduces a novel strategy for creating scale-free networks by capturing the spatiotemporal properties of an Elementary Cellular Automata trajectory. The process has been validated through the analysis of information-theoretic measures such as correlation information and entropy rate. Our findings demonstrate how the system’s sub-linear movement towards entropy reduction and increased correlations leads to state space localization, manifesting as a power-law distribution in the resulting network. The proposed method implicitly offers various forms of preferential attachment (sub-linear, linear, and super-linear) based on the sampling time pattern, circumventing the need for the first-mover advantage to generate hubs and requiring no global system information. We evaluate the resulting network using well-established criteria.

Network Growth Models such as Preferential Attachment and Duplication/Divergence are popular generative models with which to study complex networks in biology, sociology, and computer science. However, analyzing them within the framework of model selection and statistical inference is often complicated and computationally difficult, particularly when comparing models that are not directly related or nested. In practice, ad hoc methods are often used with uncertain results. If possible, the use of standard likelihood-based statistical model selection techniques is desirable. With this in mind, we develop an Adaptive Importance Sampling algorithm for estimating likelihoods of Network Growth Models. We introduce the use of the classic Plackett-Luce model of rankings as a family of importance distributions. Updates to importance distributions are performed iteratively via the Cross-Entropy Method with an additional correction for degeneracy/over-fitting inspired by the Minimum Description Length principle. This correction can be applied to other estimation problems using the Cross-Entropy method for integration/approximate counting, and it provides an interpretation of Adaptive Importance Sampling as iterative model selection. Empirical results for the Preferential Attachment model are given, along with a comparison to an alternative established technique, Annealed Importance Sampling.

The out-degree distribution is one of the most reported topological properties to characterize real complex networks. This property describes the probability that a node in the network has a particular number of outgoing links. It has been found that in many real complex networks the out-degree has a behavior similar to a power-law distribution, therefore some network growth models have been proposed to approximate this behavior. This paper introduces a new growth model that allows to produce out-degree distributions that decay as a power-law with an exponent in the range from 1 to ∞.

Scale-free networks by super-linear preferential attachment rule

We propose a fluctuation analysis to quantify spatial correlations in complex networks. The approach considers the sequences of degrees along shortest paths in the networks and quantifies the fluctuations in analogy to time series. In this work, the Barabasi-Albert (BA) model, the Cayley tree at the percolation transition, a fractal network model, and examples of real-world networks are studied. While the fluctuation functions for the BA model show exponential decay, in the case of the Cayley tree and the fractal network model the fluctuation functions display a power law behavior. The fractal network model comprises long-range anticorrelations. The results suggest that the fluctuation exponent provides complementary information to the fractal dimension.

Hidden symmetries in real and theoretical networks

Agent-based modeling (ABM) approach is used to reassess the Barabasi-Albert (BA) model, the classical algorithm used to describe the emergent mechanism of scale-free networks. This approach allows for the incorporation of agent heterogeneity which is rarely considered in BA model and its extended models. The authors argue that, in social networks, people's intention to connect is not only affected by popularity, but also strongly affected by the extent of similarity. The authors propose that in forming social networks, agents are constantly balancing between instrumental and intrinsic preferences. The proposed model allows for varying the weighting of instrumental and intrinsic preferences on the agents attachment choices. The authors also find that changing preferences of individuals can lead to significant deviations from power-law degree distribution. Given the importance of intrinsic consideration in social networking, the findings emerged from this study is conducive to future studies of social networks.

In many real-world networks, the rates of node and link addition are time dependent. This observation motivates the definition of accelerating networks. There has been relatively little investigation of accelerating networks and previous efforts at analyzing their degree distributions have employed mean-field techniques. By contrast, we show that it is possible to apply a master-equation approach to such network development. We provide full time-dependent expressions for the evolution of the degree distributions for the canonical situations of random and preferential attachment in networks undergoing constant acceleration. These results are in excellent agreement with results obtained from simulations. We note that a growing nonequilibrium network undergoing constant acceleration with random attachment is equivalent to a classical random graph, bridging the gap between nonequilibrium and classical equilibrium networks.

The scale-free (SF) property is a major concept in complex networks, and it is based on the definition that an SF network has a degree distribution that follows a power-law (PL) pattern. This paper highlights that not all networks with a PL degree distribution arise through a Barabási-Albert (BA) preferential attachment growth process, a fact that, although evident from the literature, is often overlooked by many researchers. For this purpose, it is demonstrated, with simulations, that established measures of network topology do not suffice to distinguish between BA networks and other (random-like and lattice-like) SF networks with the same degree distribution. Additionally, it is examined whether an existing self-similarity metric proposed for the definition of the SF property is also capable of distinguishing different SF topologies with the same degree distribution. To contribute to this discrimination, this paper introduces a spectral metric, which is shown to be more capable of distinguishing between different SF topologies with the same degree distribution, in comparison with the existing metrics.

Preferential survival in models of complex ad hoc networks