Preferential Attachment Networks Research Articles

A dandelion structure of eigenvector preferential attachment networks

In this paper we introduce a new type of preferential attachment network, the growth of which is based on the eigenvector centrality. In this network, the agents attach most probably to the nodes with larger eigenvector centrality which represents that the agent has stronger connections. A new network is presented, namely a dandelion network, which shares some properties of star-like structure and also a hierarchical network. We show that this network, having hub-and-spoke topology is not generally scale free, and shows essential differences with respect to the Barabási–Albert preferential attachment model. Most importantly, there is a super hub agent in the system (identified by a pronounced peak in the spectrum), and the other agents are classified in terms of the distance to this super-hub. We explore a plenty of statistical centralities like the nodes degree, the betweenness and the eigenvector centrality, along with various measures of structure like the community and hierarchical structures, and the clustering coefficient. Global measures like the shortest path statistics and the self-similarity are also examined.

Generating General Preferential Attachment Networks with R Package wdnet

Preferential attachment (PA) network models have a wide range of applications in various scientific disciplines. Efficient generation of large-scale PA networks helps uncover their structural properties and facilitate the development of associated analytical methodologies. Existing software packages only provide limited functions for this purpose with restricted configurations and efficiency. We present a generic, user-friendly implementation of weighted, directed PA network generation with R package wdnet. The core algorithm is based on an efficient binary tree approach. The package further allows adding multiple edges at a time, heterogeneous reciprocal edges, and user-specified preference functions. The engine under the hood is implemented in C++. Usages of the package are illustrated with detailed explanation. A benchmark study shows that wdnet is efficient for generating general PA networks not available in other packages. In restricted settings that can be handled by existing packages, wdnet provides comparable efficiency.

PageRank asymptotics on directed preferential attachment networks

We characterize the tail behavior of the distribution of the PageRank of a uniformly chosen vertex in a directed preferential attachment graph and show that it decays as a power law with an explicit exponent that is described in terms of the model parameters. Interestingly, this power law is heavier than the tail of the limiting in-degree distribution, which goes against the commonly accepted power law hypothesis. This deviation from the power law hypothesis points at the structural differences between the inbound neighborhoods of typical vertices in a preferential attachment graph versus those in static random graph models where the power law hypothesis has been proven to hold (e.g., directed configuration models and inhomogeneous random digraphs). In addition to characterizing the PageRank distribution of a typical vertex, we also characterize the explicit growth rate of the PageRank of the oldest vertex as the network size grows.

Measuring reciprocity in a directed preferential attachment network

AbstractEmpirical studies (e.g. Jiang et al. (2015) and Mislove et al. (2007)) 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 distributions with power-law tails, but the theoretical properties of the reciprocity feature in this model have not yet been studied. We derive 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 proportion of reciprocal edges in a directed PA network is close to 0, which differs from the empirical observation. This points out one potential problem of fitting a classical PA model to a given network dataset with high reciprocity, and indicates that alternative models need to be considered.

Preferential attachment network model with aging and initial attractiveness

In this paper, we generalize the growing network model with preferential attachment for new links to simultaneously include aging and initial attractiveness of nodes. The network evolves with the addition of a new node per unit time, and each new node has m new links that with probability Π i are connected to nodes i already present in the network. In our model, the preferential attachment probability Π i is proportional not only to k i + A, the sum of the old node i's degree k i and its initial attractiveness A, but also to the aging factor , where τ i is the age of the old node i. That is, . Based on the continuum approximation, we present a mean-field analysis that predicts the degree dynamics of the network structure. We show that depending on the aging parameter α two different network topologies can emerge. For α < 1, the network exhibits scaling behavior with a power-law degree distribution P(k) ∝ k −γ for large k where the scaling exponent γ increases with the aging parameter α and is linearly correlated with the ratio A/m. Moreover, the average degree k(t i , t) at time t for any node i that is added into the network at time t i scales as where 1/β is a linear function of A/m. For α > 1, such scaling behavior disappears and the degree distribution is exponential.

Percolation phase transition in weight-dependent random connection models

AbstractWe investigate spatial random graphs defined on the points of a Poisson process in d-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point is assigned an independent weight. Given the weight and position of the points, we form an edge between any pair of points independently with a probability depending on the two weights of the points and their distance. Preference is given to short edges and connections to vertices with large weights. We characterize the parameter regime where there is a non-trivial percolation phase transition and show that it depends not only on the power-law exponent of the degree distribution but also on a geometric model parameter. We apply this result to characterize robustness of age-based spatial preferential attachment networks.

Fragility of nonconvergence in preferential attachment graphs with three types

Preferential attachment networks are a type of random network where new nodes are connected to existing ones at random, and are more likely to connect to those that already have many connections. We investigate further a family of models introduced by Antunovi\'{c}, Mossel and R\'{a}cz where each vertex in a preferential attachment graph is assigned a type, based on the types of its neighbours. Instances of this type of process where the proportions of each type present do not converge over time seem to be rare. Previous work found that a "rock-paper-scissors" setup where each new node's type was determined by a rock-paper-scissors contest between its two neighbours does not converge. Here, two cases similar to that are considered, one which is like the above but with an arbitrarily small chance of picking a random type and one where there are four neighbours which perform a knockout tournament to determine the new type. These two new setups, despite seeming very similar to the rock-paper-scissors model, do in fact converge, perhaps surprisingly.

An agent-based system for modeling users’ acquisition and retention in startup apps

Startup companies boost the quality of everyday life in almost all dimensions, and their products and services are of relevance everywhere. One of the most important goals that startups pursue is to increase the number of their users quickly. Users are of two types, new and returning. The present study presents an agent-based model to simultaneously deal with these two types of users by placing them in a preferential attachment network to interact. In this model, new users can be added at each time step according to word-of-mouth (WOM) and marketing activities. To define the retention probability for an agent, a set of real users in the records with the same properties as the agent are looked for to check what they have done in the same situation. To validate and test the model, the agent-based system is first thoroughly verified and then applied to the real data of a startup in the game app industry. After the experiments on a real scenario, the best decisions are made about the users to focus on or incentivize, and the best combination of acquisition and retention policies is adopted. The results show that user retention on the early days of adoption is better than the acquisition of new users. In this regard, acquisition should be focused on when the retention is in an acceptable state. Furthermore, the highest increase in the number of users occurs when there is a good balance between acquisition and retention.

Competing growth processes with random growth rates and random birth times

Comparing individual contributions in a strongly interacting system of stochastic growth processes can be a very difficult problem. This is particularly the case when new growth processes are initiated depending on the state of previous ones and the growth rates of the individual processes are themselves random. We propose a novel technique to deal with such problems and show how it can be applied to a broad range of examples where it produces new insight and surprising results. The method relies on two steps: In the first step, which is highly problem dependent, the growth processes are jointly embedded into continuous time so that their evolutions after initiation become approximately independent while we retain some control over the initiation times. Once such an embedding is achieved, the second step is to apply a Poisson limit theorem that enables a comparison of the state of the processes initiated in a critical window and therefore allows an asymptotic description of the extremal process. In this paper we prove a versatile limit theorem of this type and show how this tool can be applied to obtain novel asymptotic results for a variety of interesting stochastic processes. These include (a) the maximal degree in different types of preferential attachment networks with fitnesses like the well-known Bianconi–Barabási tree and a network model of Dereich, (b) the most successful mutant in branching processes evolving by selection and mutation, and (c) the ratio between the largest and second largest cycles in a random permutation with random cycle weights, which can also be interpreted as a disordered version of Pitman’s Chinese restaurant process.

Network comparison and the within-ensemble graph distance

Quantifying the differences between networks is a challenging and ever-present problem in network science. In recent years, a multitude of diverse, ad hoc solutions to this problem have been introduced. Here, we propose that simple and well-understood ensembles of random networks—such as Erdős–Rényi graphs, random geometric graphs, Watts–Strogatz graphs, the configuration model and preferential attachment networks—are natural benchmarks for network comparison methods. Moreover, we show that the expected distance between two networks independently sampled from a generative model is a useful property that encapsulates many key features of that model. To illustrate our results, we calculate this within-ensemble graph distance and related quantities for classic network models (and several parameterizations thereof) using 20 distance measures commonly used to compare graphs. The within-ensemble graph distance provides a new framework for developers of graph distances to better understand their creations and for practitioners to better choose an appropriate tool for their particular task.

Information Diffusion in Halal Food Social Media: A Social Network Approach

Few studies have explored the distinct role played by social media in information diffusion. This study investigates the structure, interaction patterns, opinion leaders and communities’ distribution of actors involved in cross-national halal food Twitter networks. A social network analysis (SNA) is employed using a data scraping software to collect around 12,000 halal food tweets. A relatively low density and heterogeneous hub-and-spoke network emerged, with only few influencers/social mediators within small communities controlling the diffusion of information through the network. This result lends strong support to the “Matthew Effect”, which is a variation of the “rich get richer” model. Robustness checks confirmed the existence of small world preferential attachment network linking digital halal food consumers. Our findings hold important implications for different stakeholders as they underscore the relational nature of halal food networks and the importance of the strategic management of social media at the international marketing level as a critical communication tool.

Dynamics in Affinity-Weighted Preferential Attachment Networks

During the formation process of stochastic networks, nodes tend to establish edges based on selective linkage mechanisms. In general, these mechanisms involve probability distributions that underlie the selection of target nodes. In social networks, edges are often associated to relationships that are homophilic with respect to individual traits. Such traits include, for example, gender and age, and are referred to as node types. Our work considers an affinity-weighted preferential attachment model that characterizes the tendency of two group of nodes to connect to other nodes of the same type. We derive mathematical expressions for the dynamics and convergence of homophily measures at the node, group, and network level. Furthermore, we characterize the convergence of network modularity and show that the formation of community structures can be expressed as a function of network homophily.

Three-State Majority-vote Model on Scale-Free Networks and the Unitary Relation for Critical Exponents

We investigate the three-state majority-vote model for opinion dynamics on scale-free and regular networks. In this model, an individual selects an opinion equal to the opinion of the majority of its neighbors with probability 1 − q, and different to it with probability q. The parameter q is called the noise parameter of the model. We build a network of interactions where z neighbors are selected by each added site in the system, a preferential attachment network with degree distribution k−λ, where λ = 3 for a large number of nodes N. In this work, z is called the growth parameter. Using finite-size scaling analysis, we obtain that the critical exponents beta /bar{nu } and gamma /bar{nu } associated with the magnetization and the susceptibility, respectively. Using Monte Carlo simulations, we calculate the critical noise parameter qc as a function of z for the scale-free networks and obtain the phase diagram of the model. We find that the critical exponents add up to unity when using a special volumetric scaling, regardless of the dimension of the network of interactions. We verify this result by obtaining the critical noise and the critical exponents for the two and three-state majority-vote model on cubic lattice networks.

Modeling the Adoption and Diffusion of Mobile Telecommunications Technologies in Iran: A Computational Approach Based on Agent-Based Modeling and Social Network Theory

Understanding the mechanism underlying the mobile telecommunications technologies (MTTs) diffusion in a country is crucial for telecom planners to know how to accelerate their diffusion by designing appropriate scenarios. Considering the technology diffusion as a bottom-up process, this study is aimed at exploring this mechanism, drawing on insights from diffusion of innovation theory and social network theory. Accordingly, an agent-based model is proposed to investigate how MTTs are diffused in Iran over time. The results of this study show, (1) social network of Iranian society seems more similar to a Watts–Strogatz small-world network than a Barabási–Albert preferential attachment network, where the clustering coefficient is high and average path length is low, (2) compared to the compatibility parameter, the advertisement parameter not only is less influential on diffusion of a targeted MTT (i.e., 4G) but also is not necessary for it, and (3) scenarios having the least number of steps and turning points are more appropriate for continuous diffusion of 4G. The proposed study is empirically validated against real-world data ranging from 7/1/2017 to 12/31/2017. We believe it provides telecom planners insights regarding MTTs diffusion mechanism in a social complex structure and the how of scenario designing for increasing their diffusion.

Time-invariant degree growth in preferential attachment network models.

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 an analytical framework based on the time invariance of the studied systems and show that it is self-consistent only for two special network growth forms: the uniform and the exponential network growth. Conversely, the breaking of such time invariance explains the winner-takes-all effect in some model settings, revealing the connection between the Bose-Einstein condensation in the Bianconi-Barabási model and similar gelation in superlinear preferential attachment. Aging is necessary to reproduce realistic node degree growth curves and can prevent the winner-takes-all effect under weak conditions. Our results are verified by extensive numerical simulations.

Inference for the degree distributions of preferential attachment networks with zero-degree nodes

The tail of the logarithmic degree distribution of networks decays linearly with respect to the logarithmic degree is known as the power law and is ubiquitous in daily lives. A commonly used technique in modeling the power law is preferential attachment (PA), which sequentially joins each new node to the existing nodes according to the conditional probability law proportional to a linear function of their degrees. Although effective, it is tricky to apply PA to real networks because the number of nodes and that of edges have to satisfy a linear constraint. This paper enables real application of PA by making each new node as an isolated node that attaches to other nodes according to PA scheme in some later epochs. This simple and novel strategy provides an additional degree of freedom to relax the aforementioned constraint to the observed data and uses the PA scheme to compute the implied proportion of the unobserved zero-degree nodes. By using martingale convergence theory, the degree distribution of the proposed model is shown to follow the power law and its asymptotic variance is proved to be the solution of a Sylvester matrix equation, a class of equations frequently found in the control theory (see Hansen and Sargent (2008, 2014)). These results give a strongly consistent estimator for the power-law parameter and its asymptotic normality. Note that this statistical inference procedure is non-iterative and is particularly applicable for big networks such as the World Wide Web presented in Section 6. Moreover, the proposed model offers a theoretically coherent framework that can be used to study other network features, such as clustering and connectedness, as given in Cheung (2016).

Time-invariant degree growth in preferential attachment network models

Time-invariant degree growth in preferential attachment network models

Game theoretical inference of human behavior in social networks

Social networks emerge as a result of actors’ linking decisions. We propose a game-theoretical model of socio-strategic network formation on directed weighted graphs, in which every actors’ benefit is a parametric trade-off between centrality measure, brokerage opportunities, clustering coefficient, and sociological network patterns. We use two different stability definitions to infer individual behavior of homogeneous, rational agents from network structure, and to quantify the impact of cooperation. Our theoretical analysis confirms results known for specific network motifs studied previously in isolation, yet enables us to precisely quantify the trade-offs in the space of user preferences. To deal with complex networks of heterogeneous and irrational actors, we construct a statistical behavior estimation method using Nash equilibrium conditions. We provide evidence that our results are consistent with empirical, historical, and sociological observations on real-world data-sets. Furthermore, our method offers sociological and strategic interpretations of random networks models, such as preferential attachment and small-world networks.

The age-dependent random connection model

We investigate a class of growing graphs embedded into the $d$-dimensional torus where new vertices arrive according to a Poisson process in time, are randomly placed in space and connect to existing vertices with a probability depending on time, their spatial distance and their relative ages. This simple model for a scale-free network is called the age-based spatial preferential attachment network and is based on the idea of preferential attachment with spatially induced clustering. We show that the graphs converge weakly locally to a variant of the random connection model, which we call the age-dependent random connection model. This is a natural infinite graph on a Poisson point process where points are marked by a uniformly distributed age and connected with a probability depending on their spatial distance and both ages. We use the limiting structure to investigate asymptotic degree distribution, clustering coefficients and typical edge lengths in the age-based spatial preferential attachment network.

Lyapunov–based Anomaly Detection in Preferential Attachment Networks

Abstract Network models aim to explain patterns of empirical relationships based on mechanisms that operate under various principles for establishing and removing links. The principle of preferential attachment forms a basis for the well-known Barabási–Albert model, which describes a stochastic preferential attachment process where newly added nodes tend to connect to the more highly connected ones. Previous work has shown that a wide class of such models are able to recreate power law degree distributions. This paper characterizes the cumulative degree distribution of the Barabási–Albert model as an invariant set and shows that this set is not only a global attractor, but it is also stable in the sense of Lyapunov. Stability in this context means that, for all initial configurations, the cumulative degree distributions of subsequent networks remain, for all time, close to the limit distribution. We use the stability properties of the distribution to design a semi-supervised technique for the problem of anomalous event detection on networks.