We propose a model of network growth in which the network is co-evolving together with the dynamics of a quantum mechanical system, namely a quantum walk taking place over the network. The model naturally generalizes the Barabási-Albert model of preferential attachment and has a rich set of tunable parameters, such as the initial conditions of the dynamics or the interaction of the system with its environment. We show that the model produces networks with two-modal power-law degree distributions, super-hubs, finite clustering coefficient, small-world behaviour and non-trivial degree-degree correlations.
Preferential attachment is commonly invoked to explain the emergence of those heavy-tailed degree distributions characteristic of growing network representations of diverse real-world phenomena. Experimentally confirming this hypothesis in real-world growing networks is an important frontier in network science research. Conventional preferential attachment estimation methods require that a growing network be observed across at least two snapshots in time. Numerous publicly available growing network datasets are, however, only available as single snapshots, leaving the applied network scientist…
We analyze the growth models for complex networks including preferential attachment (A.-L. Barabasi and R. Albert, Science 286, 509 (1999)) and fitness model (Caldarelli et al., Phys. Rev. Lett. 89, 258702 (2002)) and demonstrate that, under very general conditions, these two models yield the same dynamic equation of network growth, $\frac{dK}{dt}=A(t)(K+K_{0})$, where $A(t)$ is the aging constant, $K$ is the node's degree, and $K_{0}$ is the initial attractivity. Basing on this result, we show that the fitness model provides an underlying microscopic basis for the preferential attachment mech…
This paper expands the degree-based consideration of the preferential attachment growth process and applies five different connectivity criteria (node degree, clustering coefficient, betweenness centrality, closeness centrality, and eigenvector centrality) to define the development of new links in the networks. Based on statistical inference, the analysis shows that all the available control attributes are capable generating SF networks, that the proposed generalized preferential attachment growth process produces networks of statistically different topologies, under different control-attribut…
The Mean Field Approximation (MFA), or continuum method, is often used in courses on Networks to derive the degree distribution of preferential attachment networks. This method is simple and the outcome is close to the correct answer. However, this paper shows that the method is flawed in several aspects, leading to unresolvable contradictions. More importantly, the MFA is not explicitly derived from a mathematical model. An analysis of the implied model shows that it makes an approximation which is far from the truth and another one which can not be motivated in general. The success of the MF…
Preferential attachment drives the evolution of many complex networks. Its analytical studies mostly consider the simplest case of a network that grows uniformly in time despite the accelerating growth of many real networks. Motivated by the observation that the average degree growth of nodes is time-invariant in empirical network data, we study the degree dynamics in the relevant class of network models where preferential attachment is combined with heterogeneous node fitness and aging. We propose a novel analytical framework based on the time-invariance of the studied systems and show that i…
In this article we presented a brief study of the main network models with growth and preferential attachment. Such models are interesting because they present several characteristics of real systems. We started with the classical model proposed by Barabasi and Albert: nodes are added to the network connecting preferably to other nodes that are more connected. We also presented models that consider more representative elements from social perspectives, such as the homophily between the vertices or the fitness that each node has to build connections. Furthermore, we showed a version of these mo…
In this paper we define a family of preferential attachment models for random graphs with fitness in the following way: independently for each node, at each time step a random fitness is drawn according to the position of a moving average process with positive increments. We will define two regimes in which our graph reproduces some features of two well known preferential attachment models: the Bianconi-Barabási and the Barabási-Albert models. We will discuss a few conjectures on these models, including the convergence of the degree sequence and the appearance of Bose-Einstein condensation in …
We study the following preferential attachment variant of the classical Erdos-Renyi random graph process. Starting with an empty graph on n vertices, new edges are added one-by-one, and each time an edge is chosen with probability roughly proportional to the product of the current degrees of its endpoints (note that the vertex set is fixed). We determine the asymptotic size of the giant component in the supercritical phase, confirming a conjecture of Pittel from 2010. Our proof uses a simple method: we condition on the vertex degrees (of a multigraph variant), and use known results for the con…
Bitcoin and Ethereum transactions present one of the largest real-world complex networks that are publicly available for study, including a detailed picture of their time evolution. As such, they have received a considerable amount of attention from the network science community, beside analysis from an economic or cryptography perspective. Among these studies, in an analysis on the early instance of the Bitcoin network, we have shown the clear presence of the preferential attachment, or "rich-get-richer" phenomenon. Now, we revisit this question, using a recent version of the Bitcoin network …
Network Inequality through Preferential Attachment, Triadic Closure, and Homophily
Complexity Science Hub（复杂性科学中心）；Department of Network and Data Science, Central European University（中欧大学网络与数据科学系）；Graz University of Technology（格拉茨工业大学）；Grupo de Sistemas Complejos, ETS de Arquitectura de Madrid, Universidad Politécnica de Madrid（马德里理工大学马德里建筑高等技术学院复杂系统小组）；Faculty of Engineering, Free University of Bozen-Bolzano（博尔扎诺-博岑自由大学工程学院）；Center for Computational and Stochastic Mathematics, Instituto Superior Tecnico（里斯本高等理工学院计算与随机数学中心）
Reciprocity characterizes the information exchange between users in a network, and some empirical studies have revealed that social networks have a high proportion of reciprocal edges. Classical directed preferential attachment (PA) models, though generating scale-free networks, may give networks with low reciprocity. This points out one potential problem of fitting a classical PA model to a given network dataset with high reciprocity, and indicates alternative models need to be considered. We give one possible modification of the classical PA model by including another parameter which control…
Reciprocity in social networks helps understand information exchange between two individuals, and indicates interaction patterns between pairs of users. A recent study indicates the reciprocity coefficient of a classical directed preferential attachment (PA) model does not match empirical evidence. In this paper, we extend the classical 3-scenario directed PA model by adding an additional parameter that controls the probability of creating a reciprocal edge. Our proposed model also allows edge creation between two existing nodes, making it a more realistic choice for fitting to real datasets. …
We propose a statistical method to estimate simultaneously the non-parametric transitivity and preferential attachment functions in a growing network, in contrast to conventional methods that either estimate each function in isolation or assume some functional form for them. Our model is shown to be a good fit to two real-world co-authorship networks and be able to bring to light intriguing details of the preferential attachment and transitivity phenomena that would be unavailable under traditional methods. We also introduce a method to quantify the amount of contributions of those phenomena i…
Numerous works have been proposed to generate random graphs preserving the same properties as real-life large scale networks. However, many real networks are better represented by hypergraphs. Few models for generating random hypergraphs exist and no general model allows to both preserve a power-law degree distribution and a high modularity indicating the presence of communities. We present a dynamic preferential attachment hypergraph model which features partition into communities. We prove that its degree distribution follows a power-law and we give theoretical lower bounds for its modularit…
Many networks generated by nature have two generic properties: they are formed in the process of {preferential attachment} and they are scale-free. Considering these features, by interfering with mechanism of the {preferential attachment}, we propose a generalisation of the Barabási--Albert model---the 'Fractional Preferential Attachment' (FPA) scale-free network model---that generates networks with time-independent degree distributions $p(k)\sim k^{-γ}$ with degree exponent $2<γ\leq3$ (where $γ=3$ corresponds to the typical value of the BA model). In the FPA model, the element controlling the…
Empirical studies show that online social networks have not only in- and out-degree distributions with Pareto-like tails but also a high proportion of reciprocal edges. A classical directed preferential attachment (PA) model generates in- and out-degree distribution with power-law tails, but theoretical properties of the reciprocity feature in this model have not yet been studied. We derive the asymptotic results on the number of reciprocal edges between two fixed nodes, as well as the proportion of reciprocal edges in the entire PA network. We see that with certain choices of parameters, the …
In this paper, we propose that relations between high order moments of data distributions, for example between the skewness (S) and kurtosis (K), allow to point to theoretical models with understandable structural parameters. The illustrative data concerns two cases: (i) the distribution of income taxes and (ii) that of inhabitants, after aggregation over each city in each province of Italy in 2011. Moreover, from the rank-size relationship, for either S or K, in both cases, it is shown that one obtains the parameters of the underlying (hypothetical) modeling distribution: in the present cases…
Many real-world systems are profitably described as complex networks that grow over time. Preferential attachment and node fitness are two simple growth mechanisms that not only explain certain structural properties commonly observed in real-world systems, but are also tied to a number of applications in modeling and inference. While there are statistical packages for estimating various parametric forms of the preferential attachment function, there is no such package implementing non-parametric estimation procedures. The non-parametric approach to the estimation of the preferential attachment…
In this paper, we study the clustering properties of the Spatial Preferential Attachment (SPA) model introduced by Aiello et al. in 2009. This model naturally combines geometry and preferential attachment using the notion of spheres of influence. It was previously shown in several research papers that graphs generated by the SPA model are similar to real-world networks in many aspects. For example, the vertex degree distribution was shown to follow a power law. In the current paper, we study the behaviour of C(d), which is the average local clustering coefficient for the vertices of degree d. …
Preferential attachment, homophily and, their consequences such as the glass ceiling effect have been well-studied in the context of undirected networks. However, the lack of an intuitive, theoretically tractable model of a directed, bi-populated~(i.e.,~containing two groups) network with variable levels of preferential attachment, homophily and growth dynamics~(e.g.,~the rate at which new nodes join, whether the new nodes mostly follow existing nodes or the existing nodes follow them, etc.) has largely prevented such consequences from being explored in the context of directed networks, where …
