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 modularity. We compare its characteristics with a real-life co-authorship network and show that our model achieves good performances. We believe that our hypergraph model will be an interesting tool that may be used in many research domains in order to reflect better real-life phenomena.
The random graph model has recently been extended to a random preferential attachment graph model, in order to enable the study of general asymptotic properties in network types that are better represented by the preferential attachment evolution model than by the ordinary (uniform) evolution lodel. Analogously, this paper extends the random {\em hypergraph} model to a random {\em preferential attachment hypergraph} model. We then analyze the degree distribution of random preferential attachment hypergraphs and show that they possess heavy tail degree distribution properties similar to those o…
We analyze whether preferential attachment in scientific coauthorship networks is different for authors with different forms of centrality. Using a complete database for the scientific specialty of research about "steel structures," we show that betweenness centrality of an existing node is a significantly better predictor of preferential attachment by new entrants than degree or closeness centrality. During the growth of a network, preferential attachment shifts from (local) degree centrality to betweenness centrality as a global measure. An interpretation is that supervisors of PhD projects …
Many networks exhibit scale free behavior where their degree distribution obeys a power law for large vertex degrees. Models constructed to explain this phenomena have relied on preferential attachment where the networks grow by the addition of both vertices and edges, and the edges attach themselves to a vertex with a probability proportional to its degree. Simulations hint, though not conclusively, that both growth and preferential attachment are necessary for scale free behavior. We derive analytic expressions for degree distributions for networks that grow by the addition of edges to a fix…
In principle, the rules of links formation of a network model can be considered as a kind of link prediction algorithm. By revisiting the preferential attachment mechanism for generating a scale-free network, here we propose a class of preferential attachment indices which are different from the previous one. Traditionally, the preferential attachment index is defined by the product of the related nodes degrees, while the new indices will define the similarity score of a pair of nodes by either the maximum in the two nodes degrees or the summarization of their degrees. Extensive experiments ar…
Relevance and importance are the main factors when humans build network connections. We propose an evolutionary network model based on preferential attachment(PA) considering these factors. We analyze and compute several important features of the network class generated by this algorithm including scale free degree distribution, high clustering coefficient, small world property and core-periphery structure. We then compare this model with other network models and empirical data such as inter-city road transportation and air traffic networks.
Preferential attachment is a popular model of growing networks. We consider a generalized model with random node removal, and a combination of preferential and random attachment. Using a high-degree expansion of the master equation, we identify a topological phase transition depending on the rate of node removal and the relative strength of preferential vs. random attachment, where the degree distribution goes from a power law to one with an exponential tail.
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…
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…
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…
We perform an empirical study of the preferential attachment phenomenon in temporal networks and show that on the Web, networks follow a nonlinear preferential attachment model in which the exponent depends on the type of network considered. The classical preferential attachment model for networks by Barabási and Albert (1999) assumes a linear relationship between the number of neighbors of a node in a network and the probability of attachment. Although this assumption is widely made in Web Science and related fields, the underlying linearity is rarely measured. To fill this gap, this paper pe…
We investigate a class of network growth rules that are based on a redirection algorithm wherein new nodes are added to a network by linking to a randomly chosen target node with some probability 1-r or linking to the parent node of the target node with probability r. For fixed 0<r<1, the redirection algorithm is equivalent to linear preferential attachment. We show that when r is a decaying function of the degree of the parent of the initial target, the redirection algorithm produces sublinear preferential attachment network growth. We also argue that no local redirection algorithm can produc…
In this paper, we characterise the notion of preferential attachment in networks as action at a distance, and argue that it can only be an emergent phenomenon -- the actual mechanism by which networks grow always being the closing of triangles. After a review of the concepts of triangle closing and preferential attachment, we present our argument, as well as a simplified model in which preferential attachment can be derived mathematically from triangle closing. Additionally, we perform experiments on synthetic graphs to demonstrate the emergence of preferential attachment in graph growth model…
We propose a simple preferential attachment model of growing network using the complementary probability of Barabási-Albert (BA) model, i.e., $Π(k_i) \propto 1-\frac{k_i}{\sum_j k_j}$. In this network, new nodes are preferentially attached to not well connected nodes. Numerical simulations, in perfect agreement with the master equation solution, give an exponential degree distribution. This suggests that the power law degree distribution is a consequence of preferential attachment probability together with "rich get richer" phenomena. We also calculate the average degree of a target node at ti…
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 …
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 introduce a new model of competition on growing networks. This extends the preferential attachment model, with the key property that node choices evolve simultaneously with the network. When a new node joins the network, it chooses neighbours by preferential attachment, and selects its type based on the number of initial neighbours of each type. The model is analysed in detail, and in particular, we determine the possible proportions of the various types in the limit of large networks. An important qualitative feature we find is that, in contrast to many current theoretical models, often se…
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…
The spatial preferential attachment (SPA) is a model for complex networks. In the SPA model, nodes are embedded in a metric space, and each node has a sphere of influence whose size increases if the node gains an in-link, and otherwise decreases with time. In this paper, we study the behaviour of the SPA model when the distribution of the nodes is non-uniform. Specifically, the space is divided into dense and sparse regions, where it is assumed that the dense regions correspond to coherent communities. We prove precise theoretical results regarding the degree of a node, the number of common ne…
In this paper, we provide a rigorous analysis of preferential attachment with fitness, a random graph model introduced by Bianconi and Barabasi. Depending on the shape of the fitness distribution, we observe three distinct phases: a first-mover-advantage phase, a fit-get-richer phase and an innovation-pays-off phase.
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…
We consider a preferential attachment process in which a multigraph is built one node at a time. The number of edges added at stage $t$, emanating from the new node, is given by some prescribed function $f(t)$, generalising a model considered by Kleinberg and Kleinberg in 2005 where $f$ was presumed constant. We show that if $f(t)$ is asymptotically bounded above and below by linear functions in $t$, then with probability $1$ the infinite limit of the process will be isomorphic to the \emph{Rado multigraph}. This structure is the natural multigraph analogue of the Rado graph, which we introduc…
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）
Preferential attachment models have been widely studied in complex networks, because they can explain the formation of many networks like social networks, citation networks, power grids, and biological networks, to name a few. Motivated by the application of key predistribution in wireless sensor networks (WSN), we initiate the study of preferential attachment with degree bound. Our paper has two important contributions to two different areas. The first is a contribution in the study of complex networks. We propose preferential attachment model with degree bound for the first time. In the norm…
All real networks are different, but many have some structural properties in common. There seems to be no consensus on what the most common properties are, but scale-free degree distributions, strong clustering, and community structure are frequently mentioned without question. Surprisingly, there exists no simple generative mechanism explaining all the three properties at once in growing networks. Here we show how latent network geometry coupled with preferential attachment of nodes to this geometry fills this gap. We call this mechanism geometric preferential attachment (GPA), and validate i…
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…
This paper presents an analysis of a temporal network that describes the social connections of a large-scale (~ 30,000) sample of online social network users, inhabitants of a fixed city. We tested how the main network formation determinants - transitivity, preferential attachment, and social selection - contribute to network evolution. Among other things, we found that opinion social selection does affect tie appearing whereas its impact on tie removing is rather unclear. We report that transitivity displayed the strongest effect on network dynamics. Surprisingly, a closer look revealed an in…
For fitness preferential attachment random networks, we define the empirical degree and pair measure, which counts the number of vertices of a given degree and the number of edges with given fits, and the sample path empirical degree distribution. For the empirical degree and pair distribution for the fitness preferential attachment random networks, we find a large deviation upper bound. From this result we obtain a weak law of large numbers for the empirical degree and pair distribution, and the basic information theorem or an asymptotic equipartition property for fitness preferential attachm…
Prediction and control of network dynamics are grand-challenge problems in network science. The lack of understanding of fundamental laws driving the dynamics of networks is among the reasons why many practical problems of great significance remain unsolved for decades. Here we study the dynamics of networks evolving according to preferential attachment, known to approximate well the large-scale growth dynamics of a variety of real networks. We show that this dynamics is Hamiltonian, thus casting the study of complex networks dynamics to the powerful canonical formalism, in which the time evol…
Motivated by the complexity of network data, we propose a directed hybrid random network that mixes preferential attachment (PA) rules with uniform attachment (UA) rules. When a new edge is created, with probability $p\in [0,1]$, it follows the PA rule. Otherwise, this new edge is added between two uniformly chosen nodes. Such mixture makes the in- and out-degrees of a fixed node grow at a slower rate, compared to the pure PA case, thus leading to lighter distributional tails. Useful inference methods for the proposed hybrid model are then provided and applied to both synthetic and real datase…
In the Yule-Simon process, selection of words follows the preferential attachment mechanism, resulting in the power-law growth in the cumulative number of individual word occurrences. This is derived using mean-field approximation, assuming a continuum limit of both the time and number of word occurrences. However, time and word occurrences are inherently discrete in the process, and it is natural to assume that the cumulative number of word occurrences has a certain fluctuation around the average behavior predicted by the mean-field approximation. We derive the exact and approximate forms of …
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…
Many real-world networks exhibit degree-assortativity, with nodes of similar degree more likely to link to one another. Particularly in social networks, the contribution to the total assortativity varies with degree, featuring a distinctive peak slightly past the average degree. The way traditional models imprint assortativity on top of pre-defined topologies is via degree-preserving link permutations, which however destroy the particular graph's hierarchical traits of clustering. Here, we propose the first generative model which creates heterogeneous networks with scale-free-like properties a…
We generalize the scale-free network model of Barabàsi and Albert [Science 286, 509 (1999)] by proposing a class of stochastic models for scale-free interdependent networks in which interdependent nodes are not randomly connected but rather are connected via preferential attachment (PA). Each network grows through the continuous addition of new nodes, and new nodes in each network attach preferentially and simultaneously to (a) well-connected nodes within the same network and (b) well-connected nodes in other networks. We present analytic solutions for the power-law exponents as functions of t…
This paper provides time-dependent expressions for the expected degree distribution of a given network that is subject to growth, as a function of time. We consider both uniform attachment, where incoming nodes form links to existing nodes selected uniformly at random, and preferential attachment, when probabilities are assigned proportional to the degrees of the existing nodes. We consider the cases of single and multiple links being formed by each newly-introduced node. The initial conditions are arbitrary, that is, the solution depends on the degree distribution of the initial graph which i…
We obtain the degree distribution for a class of growing network models on flat and curved spaces. These models evolve by preferential attachment weighted by a function of the distance between nodes. The degree distribution of these models is similar to the one of the fitness model of Bianconi and Barabasi, with a fitness distribution dependent on the metric and the density of nodes. We show that curvature singularities in these spaces can give rise to asymptotic Bose-Einstein condensation, but transient condensation can be observed also in smooth hyperbolic spaces with strong curvature. We pr…
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…
The classical preferential attachment model is sensitive to the choice of the initial configuration of the network. As the number of initial nodes and their degree grow, so does the time needed for an equilibrium degree distribution to be established. We study this phenomenon, provide estimates of the equilibration time, and characterize the degree distribution cutoff observed at finite times. When the initial network is dense and exceeds a certain small size, there is no equilibration and a suitable statistical test can always discern the produced degree distribution from the equilibrium one.…
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…
We investigate centrality and root-inference properties in a class of growing random graphs known as sublinear preferential attachment trees. We show that a continuous time branching processes called the Crump-Mode-Jagers (CMJ) branching process is well-suited to analyze such random trees, and prove that almost surely, a unique terminal tree centroid emerges, having the property that it becomes more central than any other fixed vertex in the limit of the random growth process. Our result generalizes and extends previous work establishing persistent centrality in uniform and linear preferential…
We study the influence of the seed graph in the preferential attachment model, focusing on the case of trees. We first show that the seed has no effect from a weak local limit point of view. On the other hand, we conjecture that different seeds lead to different distributions of limiting trees from a total variation point of view. We take a first step in proving this conjecture by showing that seeds with different degree profiles lead to different limiting distributions for the (appropriately normalized) maximum degree, implying that such seeds lead to different (in total variation) limiting t…
We study information aggregation in networks where agents make binary decisions (labeled incorrect or correct). Agents initially form independent private beliefs about the better decision, which is correct with probability $1/2+δ$. The dynamics we consider are asynchronous (each round, a single agent updates their announced decision) and non-Bayesian (agents simply copy the majority announcements among their neighbors, tie-breaking in favor of their private signal). Our main result proves that when the network is a tree formed according to the preferential attachment model \cite{BarabasiA99}, …
We study preferential attachment mechanisms in random graphs that are parameterized by (i) a constant bias affecting the degree-biased distribution on the vertex set and (ii) the distribution of times at which new vertices are created by the model. The class of random graphs so defined admits a representation theorem reminiscent of residual allocation, or "stick-breaking" schemes. We characterize how the vertex arrival times affect the asymptotic degree distribution, and relate the latter to neutral-to-the-left processes. Our random graphs generate edges "one end at a time", which sets up a on…
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 …
Preferential attachment models are a common class of graph models which have been used to explain why power-law distributions appear in the degree sequences of real network data. One of the things they lack, however, is higher-order network clustering, including non-trivial clustering coefficients. In this paper we present a specific Triangle Generalized Preferential Attachment Model (TGPA) that, by construction, has nontrivial clustering. We further prove that this model has a power-law in both the degree distribution and eigenvalue spectra. We use this model to investigate a recent finding t…
We provide an analytic expression for the quantity described in the title. Namely, we perform a preferential attachment growth process to generate a scale-free network. At each stage we add a new node with $m$ new links. Let $k$ denote the degree of a node, and $N$ the number of nodes in the network. The degree distribution is assumed to converge to a power-law (for $k\geq m$) of the form $k^{-γ}$ and we obtain an exact implicit relationship for $γ$, $m$ and $N$. We verify this with numerical calculations over several orders of magnitude. Although this expression is exact, it provides only an …
