Preferential Attachment Model Research Articles

Degree evolution in a general growing network

We consider the preferential attachment model introduced by Deijfen and Lindholm (2009) in which, at every discrete time step: (i) either we add a vertex and connect it to an older vertex; or (ii) we add an edge between two random vertices; or (iii) we delete one edge. We show that, when the deletion probability equals 1/3, the expected degree of any given vertex grows logarithmically, thus correcting a statement made in Lindholm and Vallier (2011). Moreover we show that, when the deletion probability is strictly less than 1/3, then the function which scales the expected degree of a given vertex, identified in Lindholm and Vallier (2011), also guarantees almost sure convergence for the degree process of a given vertex.

Generating preferential attachment graphs via a Pólya urn with expanding colors

AbstractWe introduce a novel preferential attachment model using the draw variables of a modified Pólya urn with an expanding number of colors, notably capable of modeling influential opinions (in terms of vertices of high degree) as the graph evolves. Similar to the Barabási-Albert model, the generated graph grows in size by one vertex at each time instance; in contrast however, each vertex of the graph is uniquely characterized by a color, which is represented by a ball color in the Pólya urn. More specifically at each time step, we draw a ball from the urn and return it to the urn along with a number of reinforcing balls of the same color; we also add another ball of a new color to the urn. We then construct an edge between the new vertex (corresponding to the new color) and the existing vertex whose color ball is drawn. Using color-coded vertices in conjunction with the time-varying reinforcing parameter allows for vertices added (born) later in the process to potentially attain a high degree in a way that is not captured in the Barabási-Albert model. We study the degree count of the vertices by analyzing the draw vectors of the underlying stochastic process. In particular, we establish the probability distribution of the random variable counting the number of draws of a given color which determines the degree of the vertex corresponding to that color in the graph. We further provide simulation results presenting a comparison between our model and the Barabási-Albert network.

Recent advances on mechanisms of network generation: Community, exchangeability, and scale‐free properties

AbstractThe mechanisms of network generation have undergone extensive analysis and found broad applications in various real‐world scenarios. Among the fruitful literature on network models, numerous studies seek to explore and interpret fundamental graph structure properties, including the clustering effect, exchangeability, and scale‐free properties. In this paper, we present a comprehensive review of the statistical modeling methods for the mechanisms of network generation. We specifically focus on three representative classes of models, namely the stochastic block models, the exchangeable network models, and the preferential attachment models. For each model type, our approach begins by reviewing existing methods and model setups, followed by an exploration of the core modeling principles behind them. We also summarize relevant statistical inference techniques and provide a unified understanding of theoretical analyses. Furthermore, we emphasize several challenges and open problems that could shed light on future research. We conclude this review with the identification of some possible directions for future study.This article is categorized under: Statistical and Graphical Methods of Data Analysis > Modeling Methods and Algorithms Algorithms and Computational Methods > Networks and Security

Preferential attachment with reciprocity: properties and estimation

Abstract Reciprocity in social networks is a measure of information exchange between two individuals, and indicates interaction patterns between pairs of users. A recent study finds that the reciprocity coefficient of a classical directed preferential attachment (PA) model does not match empirical evidence. Towards remedying this deficiency, we extend the classical three-scenario directed PA model by adding a parameter that controls the probability of creating a reciprocal edge. This proposed model also allows edge creation between two existing nodes, making it a realistic candidate for fitting to datasets. We provide and compare two estimation procedures for fitting the new reciprocity model and demonstrate the methods on simulated and real datasets. One estimation method requires careful analysis of the heavy tail properties of the model. The fitted models provide a good match with the empirical tail distributions of both in- and out-degrees but other mismatched diagnostics suggest that further generalization of the model is warranted.

Modeling and analysis of affiliation networks with preferential attachment and subsumption.

Preferential attachment describes a variety of graph-based models in which a network grows incrementally via the sequential addition of new nodes and edges, and where existing nodes acquire new neighbors at a rate proportional to their degree. Some networks, however, are better described as groups of nodes rather than a set of pairwise connections. These groups are called affiliations, and the corresponding networks affiliation networks. When viewed as graphs, affiliation networks do not necessarily exhibit the power law distribution of node degrees that is typically associated with preferential attachment. We propose a preferential attachment mechanism for affiliation networks that highlights the power law characteristic of these networks when presented as hypergraphs and simplicial complexes. The two representations capture affiliations in similar ways, but the latter offers an intrinsic feature of the model called subsumption, where an affiliation cannot be a subset of another. Our model of preferential attachment has interesting features, both algorithmic and analytic, including implicit preferential attachment (node sampling does not require knowledge of node degrees), a locality property where the neighbors of a newly added node are also neighbors, the emergence of a power law distribution of degrees (defined in hypergraphs and simplicial complexes rather than at a graph level), implicit deletion of affiliations (through subsumption in the case of simplicial complexes), and to some extent a control over the affiliation size distribution. By varying the parameters of the model, the generated affiliation networks can resemble different types of real-world examples, so the framework also serves as a synthetic generation algorithm for simulation and experimental studies.

Popularity Ratio Maximization: Surpassing Competitors through Influence Propagation

In this paper, we present an algorithmic study on how to surpass competitors in popularity by strategic promotions in social networks. We first propose a novel model, in which we integrate the Preferential Attachment (PA) model for popularity growth with the Independent Cascade (IC) model for influence propagation in social networks called PA-IC model. In PA-IC, a popular item and a novice item grab shares of popularity from the natural popularity growth via the PA model, while the novice item tries to gain extra popularity via influence cascade in a social network. The popularity ratio is defined as the ratio of the popularity measure between the novice item and the popular item. We formulate Popularity Ratio Maximization (PRM) as the problem of selecting seeds in multiple rounds to maximize the popularity ratio in the end. We analyze the popularity ratio and show that it is monotone but not submodular. To provide an effective solution, we devise a surrogate objective function and show that empirically it is very close to the original objective function while theoretically, it is monotone and submodular. We design two efficient algorithms, one for the overlapping influence and non-overlapping seeds (across rounds) setting and the other for the non-overlapping influence and overlapping seed setting, and further discuss how to deal with other models and problem variants. Our empirical evaluation further demonstrates that our proposed method consistently achieves the best popularity promotion compared to other methods. Our theoretical and empirical analyses shed light on the interplay between influence maximization and preferential attachment in social networks.

The Structure of Explanations in Network Formation Models: An Analysis of Barabasi-Albert Preferential Attachment Model

The Structure of Explanations in Network Formation Models: An Analysis of Barabasi-Albert Preferential Attachment Model

The developmental trajectories of L2 lexical-semantic networks

This study explored the developmental trajectories of L2 Lexical-semantic networks by comparing the structure of lexical-semantic networks of Chinese EFL learners at two levels of English: intermediate and advanced. Analyses of data from semantic fluency tasks found that with the improvement of language proficiency, the L2 lexical-semantic network becomes more connected but less centralized. Analyses of community structure revealed that the lexical-semantic network of advanced learners has denser clusters than that of intermediate learners. Examination of individual words in the L2 lexical-semantic network shows that words tend to reorganize themselves with the expanse of the network. In this process, more frequent and connected words tend to remain in the network as central words while less frequent words with fewer connections have the risk of being lost, although a large number of new words keep joining in the network. The results provide empirical evidence for the preferential attachment model of lexical-semantic network growth.

Estimating the Topology of Preferential Attachment Graphs Under Partial Observability

This work addresses the problem of learning the topology of a network from the signals emitted by the network nodes. These signals are generated over time through a linear diffusion process, where neighboring nodes exchange messages according to the underlying network topology, and aggregate them according to a certain combination matrix. We consider the demanding setting of graph learning under partial observability, where signals are available only from a limited fraction of nodes, and we want to establish whether the topology of these nodes can be estimated faithfully, despite the presence of possibly many latent nodes. Recent results examined this problem when the network topology is generated according to an Erdős-Rényi random model. However, Erdős-Rényi graphs feature homogeneity across nodes and independence across edges, while, over several real-world networks, significant heterogeneity is observed between very connected “hubs” and scarcely connected peripheral nodes, and the edge construction process entails significant dependencies across edges. Preferential attachment random graphs were conceived primarily to fill these gaps. We tackle the problem of graph learning over preferential attachment graphs by focusing on the following setting: first-order vector autoregressive models equipped with a stable Laplacian combination matrix, and a network topology drawn according to the popular Bollobás-Riordan preferential attachment model. The main result established in this work is that a combination-matrix estimator known as Granger estimator achieves graph learning under partial observability.

Scale-free growth in regional scientific capacity building explains long-term scientific dominance

The regional capability of performing front-running research and technological development has been identified as a necessary condition for future wealth creation. The quality and amount of scientific capabilities in specific fields vary dramatically across different world regions. Capabilities, knowledge, and skills are embodied by scientists working in research institutions or companies. The conditions for the emergence of a leading regional scientific environment – and the resulting early technological leadership – are poorly understood. The existence of a critical mass – the threshold above which a region can build comparably strong scientific capabilities – of scientists is often assumed. Using a unique dataset of global scientific activity and researcher mobility over several decades, we present empirical evidence in three scientific areas (semiconductor research, embryonic stem cells, and Internet research) that the process of scientific knowledge accumulation is remarkably general and applies to practically all regions. Regional knowledge accumulation data obtained from an analysis of scientists’ geolocated trajectories follow a preferential attachment mechanism characterized by a sub-linear kernel with a robust growth exponent. Scale-free growth patterns suggest that regions that move early into new technologies tend to dominate the corresponding scientific fields. We find no evidence that critical mass is required to achieve prolonged scientific dominance. We propose a simple preferential attachment model that explains the empirical data and allows us to understand deviations from the growth exponent as focused interventions to strategically attract scientists at regional level. We demonstrate this explicitly for China in the three scientific fields examined.

Distance evolutions in growing preferential attachment graphs

We study the evolution of the graph distance and weighted distance between two fixed vertices in dynamically growing random graph models. More precisely, we consider preferential attachment models with power-law exponent τ∈(2,3), sample two vertices ut, vt uniformly at random when the graph has t vertices and study the evolution of the graph distance between these two fixed vertices as the surrounding graph grows. This yields a discrete-time stochastic process in t′≥t, called the distance evolution. We show that there is a tight strip around the function 4loglog(t)−log(log(t′/t)∨1)|log(τ−2)|∨2 that the distance evolution never leaves with high probability as t tends to infinity. We extend our results to weighted distances, where every edge is equipped with an i.i.d. copy of a nonnegative random variable L.

Dynamical fitness models: evidence of universality classes for preferential attachment graphs

AbstractIn 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 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 the network when the drift of the fitness process has order comparable to the graph size.

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.

Asymptotic dependence of in- and out-degrees in a preferential attachment model with reciprocity

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 controls the probability of adding a reciprocated edge at each step. Asymptotic analyses suggest that large in- and out-degrees become fully dependent in this modified model, as a result of the additional reciprocated edges.

An ephemerally self-exciting point process

AbstractAcross a wide variety of applications, the self-exciting Hawkes process has been used to model phenomena in which the history of events influences future occurrences. However, there may be many situations in which the past events only influence the future as long as they remain active. For example, a person spreads a contagious disease only as long as they are contagious. In this paper, we define a novel generalization of the Hawkes process that we call theephemerally self-exciting process. In this new stochastic process, the excitement from one arrival lasts for a randomly drawn activity duration, hence the ephemerality. Our study includes exploration of the process itself as well as connections to well-known stochastic models such as branching processes, random walks, epidemics, preferential attachment, and Bayesian mixture models. Furthermore, we prove a batch scaling construction of general, marked Hawkes processes from a general ephemerally self-exciting model, and this novel limit theorem both provides insight into the Hawkes process and motivates the model contained herein as an attractive self-exciting process in its own right.

Biocode: A Data-Driven Procedure to Learn the Growth of Biological Networks.

Probabilistic biological network growth models have been utilized for many tasks including but not limited to capturing mechanism and dynamics of biological growth activities, null model representation, capturing anomalies, etc. Well-known examples of these probabilistic models are Kronecker model, preferential attachment model, and duplication-based model. However, we should frequently keep developing new models to better fit and explain the observed network features while new networks are being observed. Additionally, it is difficult to develop a growth model each time we study a new network. In this paper, we propose Biocode, a framework to automatically discover novel biological growth models matching user-specified graph attributes in directed and undirected biological graphs. Biocode designs a basic set of instructions which are common enough to model a number of well-known biological graph growth models. We combine such instruction-wise representation with a genetic algorithm based optimization procedure to encode models for various biological networks. We mainly evaluate the performance of Biocode in discovering models for biological collaboration networks, gene regulatory networks, and protein interaction networks which features such as assortativity, clustering coefficient, degree distribution closely match with the true ones in the corresponding real biological networks.

Clustering in preferential attachment random graphs with edge-step

AbstractWe prove concentration inequality results for geometric graph properties of an instance of the Cooper–Frieze [5] preferential attachment model withedge-steps. More precisely, we investigate a random graph model that at each time$t\in \mathbb{N}$, with probabilitypadds a new vertex to the graph (avertex-stepoccurs) or with probability$1-p$an edge connecting two existent vertices is added (anedge-stepoccurs). We prove concentration results for theglobal clustering coefficientas well as theclique number. More formally, we prove that the global clustering, with high probability, decays as$t^{-\gamma(p)}$for a positive function$\gamma$ofp, whereas the clique number of these graphs is, up to subpolynomially small factors, of order$t^{(1-p)/(2-p)}$.

Limits of sparse configuration models and beyond: Graphexes and MultiGraphexes

We investigate structural properties of large, sparse random graphs through the lens of sampling convergence (Borgs et al. (Ann. Probab. 47 (2019) 2754–2800). Sampling convergence generalizes left convergence to sparse graphs, and describes the limit in terms of a graphex. We introduce a notion of sampling convergence for sequences of multigraphs, and establish the graphex limit for the configuration model, a preferential attachment model, the generalized random graph and a bipartite variant of the configuration model. The results for the configuration model, preferential attachment model and bipartite configuration model provide necessary and sufficient conditions for these random graph models to converge. The limit for the configuration model and the preferential attachment model is an augmented version of an exchangeable random graph model introduced by Caron and Fox (J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 (2017) 1295–1366).

Dynamics of node influence in network growth models

Many classes of network growth models have been proposed in the literature for capturing real-world complex networks. Existing research primarily focuses on global characteristics of these models, e.g., degree distribution. We aim to shift the focus towards studying the network growth dynamics from the perspective of individual nodes. In this paper, we study how a metric for node influence in network growth models behaves over time as the network evolves. This metric, which we call node visibility, captures the probability of the node to form new connections. First, we conduct an investigation on three popular network growth models — preferential attachment, additive, and multiplicative fitness models; and primarily look into the “influential nodes” or “leaders” to understand how their visibility evolves over time. Subsequently, we consider a generic fitness model and observe that the multiplicative model strikes a balance between allowing influential nodes to maintain their visibility, while at the same time making it possible for new nodes to gain visibility in the network. Finally, we observe that a spatial growth model with multiplicative fitness can curtail the global reach of influential nodes, thereby allowing the emergence of a multiplicity of “local leaders” in the network.

Efficient topology control of blockchain peer to peer network based on SDN paradigm

Blockchain allows to securely store, using cryptography functions, validated transactions and other data across its peer to peer (P2P) network. This P2P network is generated and maintained by the participating peers in a distributed manner through peer discovery, neighbor selection and managing inbound/outbound connections. As these tasks require extra consumption of network bandwidth, energy, memory and CPU usage, resource-limited devices may not support inherent blockchain applications. Further, changes in the topology control functions or reconfiguration should be considered by all peers to fully benefit from the new performance, which can take time. The aim of this paper is to build a blockchain network in a manner that greatly reduce topology control overhead while guaranteeing the properties such as high flexibility, fast reconfigurability, connectivity, small diameter and clustering. For this, we propose to use the Software-Defined Networking (SDN) paradigm to manage the blockchain P2P network. This way, the topology control tasks are moved off the peers to a secure overlay layer composed of multiple servers having synchronized databases. This new layer is responsible for building and managing the topological structure of the P2P network layer based on random r-out digraphs. Next, we mathematically discuss r-out digraphs generation using binomial distribution and preferential attachment models. Then, and in order to reduce the number of connections per peer, we establish lower and minimum upper bounds on outbound and inbound connections respectively that still guarantee the P2P network feasibility and connectivity. Further, we investigate topological properties of blockchain P2P network such as connectivity, diameter and clustering. Finally, we provide extensive simulation and numerical results to verify the efficiency of our approach and illustrate the effects of centralized topology control on network performance.