Linear Preferential Attachment Research Articles

Local weak limit of preferential attachment random trees with additive fitness

AbstractWe consider linear preferential attachment trees with additive fitness, where fitness is the random initial vertex attractiveness. We show that when the fitnesses are independent and identically distributed and have positive bounded support, the local weak limit can be constructed using a sequence of mixed Poisson point processes. We also provide a rate of convergence for the total variation distance between the r-neighbourhoods of a uniformly chosen vertex in the preferential attachment tree and the root vertex of the local weak limit. The proof uses a Pólya urn representation of the model, for which we give new estimates for the beta and product beta variables in its construction. As applications, we obtain limiting results and convergence rates for the degrees of the uniformly chosen vertex and its ancestors, where the latter are the vertices that are on the path between the uniformly chosen vertex and the initial vertex.

Cross‐affiliation collaboration and power laws for research output of institutions: Evidence and theory from top three finance journals

AbstractCross‐affiliation emerges as a new and fast‐developing means to promote collaboration in financial research. We find that the average number of affiliations reported per author in the top‐three finance journals increases steadily from 1.1 to 1.3 from 1995 to 2016. Scale‐free power laws characterize the resulting highly‐skewed distributions of top finance journal publications of worldwide institutions. We propose an explanation of the scale‐invariance, based on a network model featuring nonlinear growth and linear preferential attachment. The model indicates that success‐breeds‐success engenders 87% of total publications and hence the dispersion in research output, while accelerated growth of collaboration reduces the heterogeneity.

Broadcasting‐induced colorings of preferential attachment trees

AbstractWe consider random two‐colorings of random linear preferential attachment trees, which includes recursive trees, plane‐oriented recursive trees, binary search trees, and a class of d‐ary trees. The random coloring is defined by assigning the root the color red or blue with equal probability, and all other vertices are assigned the color of their parent with probability and the other color otherwise. These colorings have been previously studied in other contexts, including Ising models and broadcasting, and can be considered as generalizations of bond percolation. With the help of Pólya urns, we prove limiting distributions, after proper rescalings, for the number of vertices, monochromatic subtrees, and leaves of each color, as well as the number of fringe subtrees with two‐colorings. Using methods from analytic combinatorics, we also provide precise descriptions of the limiting distribution after proper rescaling of the size of the root cluster; the largest monochromatic subtree containing the root.

Extremal properties of evolving networks: local dependence and heavy tails

A network evolution with predicted tail and extremal indices of PageRank and the Max-Linear Model used as node influence indices in random graphs is considered. The tail index shows a heaviness of the distribution tail. The extremal index is a measure of clustering (or local dependence) of the stochastic process. The cluster implies a set of consecutive exceedances of the process over a sufficiently high threshold. Our recent results concerning sums and maxima of non-stationary random length sequences of regularly varying random variables are extended to random graphs. Starting with a set of connected stationary seed communities as a hot spot and ranking them with regard to their tail indices, the tail and extremal indices of new nodes that are appended to the network may be determined. This procedure allows us to predict a temporal network evolution in terms of tail and extremal indices. The extremal index determines limiting distributions of a maximum of the PageRank and the Max-Linear Model of newly attached nodes. The exposition is provided by algorithms and examples. To validate our theoretical results, our simulation and real data study concerning a linear preferential attachment as a tool for network growth are provided.

Tail Index Estimation of PageRanks in Evolving Random Graphs

Random graphs are subject to the heterogeneities of the distributions of node indices and their dependence structures. Superstar nodes to which a large proportion of nodes attach in the evolving graphs are considered. In the present paper, a statistical analysis of the extremal part of random graphs is considered. We used the extreme value theory regarding sums and maxima of non-stationary random length sequences to evaluate the tail index of the PageRanks and max-linear models of superstar nodes in the evolving graphs where existing nodes or edges can be deleted or not. The evolution is provided by a linear preferential attachment. Our approach is based on the analysis of maxima and sums of the node PageRanks over communities (block maxima and block sums), which can be independent or weakly dependent random variables. By an empirical study, it was found that tail indices of the block maxima and block sums are close to the minimum tail index of representative series extracted from the communities. The tail indices are estimated by data of simulated graphs.

Root finding algorithms and persistence of Jordan centrality in growing random trees

We consider models of growing random trees {Tf(n):n≥1} with model dynamics driven by an attachment function f:Z+→R+. At each stage a new vertex enters the system and connects to a vertex v in the current tree with probability proportional to f(degree(v)). The main goal of this study is to understand the performance of root finding algorithms. A large body of work (e.g., Random Structures Algorithms 50 (2017) 158–172; IEEE Trans. Netw. Sci. Eng. 4 (2017) 1–12; Random Structures Algorithms 52 (2018) 136–157) has emerged in the last few years in using techniques based on the Jordan centrality measure (J. Reine Angew. Math. 70 (1869) 185–190) and its variants to develop root finding algorithms. Given an unlabeled unrooted tree, one computes the Jordan centrality for each vertex in the tree and for a fixed budget K outputs the optimal K vertices (as measured by Jordan centrality). Under general conditions on the attachment function f, we derive necessary and sufficient bounds on the budget K(ε) in order to recover the root with probability at least 1−ε. For canonical examples such as linear preferential attachment and uniform attachment, these general results give matching upper and lower bounds for the budget. We also prove persistence of the optimal K Jordan centers for any K, that is, the existence of an almost surely finite random time n∗ such that for n≥n∗ the identity of the K-optimal Jordan centers in {Tf(n):n≥n∗} does not change, thus describing robustness properties of this measure. Key technical ingredients in the proofs of independent interest include sufficient conditions for the existence of exponential moments for limits of (appropriately normalized) continuous time branching processes within which the models {Tf(n):n≥1} can be embedded, as well as rates of convergence results to these limits.

Broadcasting on random recursive trees

We study the broadcasting problem when the underlying tree is a random recursive tree. The root of the tree has a random bit value assigned. Every other vertex has the same bit value as its parent with probability 1−q and the opposite value with probability q, where q∈[0,1]. The broadcasting problem consists in estimating the value of the root bit upon observing the unlabeled tree, together with the bit value associated with every vertex. In a more difficult version of the problem, the unlabeled tree is observed but only the bit values of the leaves are observed. When the underlying tree is a uniform random recursive tree, in both variants of the problem we characterize the values of q for which the optimal reconstruction method has a probability of error bounded away from 1/2. We also show that the probability of error is bounded by a constant times q. Two simple reconstruction rules are analyzed in detail. One of them is the simple majority vote, the other is the bit value of the centroid of the tree. Most results are extended to linear preferential attachment trees as well.

A Hypergraph Approach for Estimating Growth Mechanisms of Complex Networks

Temporal datasets that describe complex interactions between individuals over time are increasingly common in various domains. Conventional graph representations of such datasets may lead to information loss since higher-order relationships between more than two individuals must be broken into multiple pairwise relationships in graph representations. In those cases, a hypergraph representation is preferable since it can preserve higher-order relationships by using hyperedges. However, existing hypergraph models of temporal complex networks often employ some data-independent growth mechanism, which is the linear preferential attachment in most cases. In principle, this pre-specification is undesirable since it completely ignores the data at hand. Our work proposes a new hypergraph growth model with a data-driven preferential attachment mechanism estimated from observed data. A key component of our method is a recursive formula that allows us to overcome a bottleneck in computing the normalizing factors in our model. We also treat an often-neglected selection bias in modeling the emergence of new edges with new nodes. Fitting the proposed hypergraph model to 13 real-world datasets from diverse domains, we found that all estimated preferential attachment functions deviates substantially from the linear form. This demonstrates the need of doing away with the linear preferential attachment assumption and adopting a data-driven approach. We also showed that our model outperformed conventional models in replicating the observed first-order and second-order structures in these real-world datasets.

Possible origin of scaling laws in preferential attachment growth networks

The degree distribution of network nodes exhibits scale-free features, which is a universal phenomenon in natural and social networks. It is commonly believed that growth and linear preferential attachment are the evolutionary mechanisms of scaling laws. However, the underlying cause behind these mechanisms remains unknown. In this paper, we propose the concept of the topology state of network nodes, and then define the network homomorphism ratio. Combining the total energy of the network defined in previous studies, we systematically investigate the connection between these two quantities in preferential attachment growth networks. Our results show that under the thermodynamic limit condition, the emergence of the scaling laws of the preferential attachment growth network is dominated by two principles. One principle is that the homomorphism ratio gradually tends to zero as the network grows, and the other principle is that the total energy of the system is the lowest under the condition of satisfying principle one. Interestingly, these two principles of the network are similar to the arrangement rules of electrons outside the nucleus which dominated by the Pauli exclusion principle and the principle of minimum energy.

Heterogeneous Preferential Attachment in Key Ethereum-Based Cryptoassets

In this study, we analyse the aggregated transaction networks of Ether (the native cryptocurrency in Ethereum) and the three most market-capitalised ERC-20 tokens in this platform at the time of writing: Binance, USDT, and Chainlink. We analyse a comprehensive dataset from 2015 to 2020 (encompassing 87,780,546 nodes and 856,207,725 transactions) to understand the mechanism that drives their growth. In a seminal analysis, Kondor et al. (PLoS ONE, 2014, 9: e86197) showed that during its first year, the aggregated Bitcoin transaction network grew following linear preferential attachment. For the Ethereum-based cryptoassets, we find that they present in general super-linear preferential attachment, i.e., the probability for a node to receive a new incoming link is proportional to kα, where k is the node’s degree. Specifically, we find an exponent α = 1.2 for Binance and Chainlink, for Ether α = 1.1, and for USDT α = 1.05. These results reveal that few nodes become hubs rapidly. We then analyse wealth and degree correlation between tokens since many nodes are active simultaneously in different networks. We conclude that, similarly to what happens in Bitcoin, “the rich indeed get richer” in Ethereum and related tokens as well, with wealth much more concentrated than in-degree and out-degree.

Inference on the History of a Randomly Growing Tree

Abstract The spread of infectious disease in a human community or the proliferation of fake news on social media can be modelled as a randomly growing tree-shaped graph. The history of the random growth process is often unobserved but contains important information such as the source of the infection. We consider the problem of statistical inference on aspects of the latent history using only a single snapshot of the final tree. Our approach is to apply random labels to the observed unlabelled tree and analyse the resulting distribution of the growth process, conditional on the final outcome. We show that this conditional distribution is tractable under a shape exchangeability condition, which we introduce here, and that this condition is satisfied for many popular models for randomly growing trees such as uniform attachment, linear preferential attachment and uniform attachment on a D-regular tree. For inference of the root under shape exchangeability, we propose O(n log n) time algorithms for constructing confidence sets with valid frequentist coverage as well as bounds on the expected size of the confidence sets. We also provide efficient sampling algorithms which extend our methods to a wide class of inference problems.

Growth of common friends in a preferential attachment model

The number of common friends (or connections) in a graph is a commonly used measure of proximity between two nodes. Such measures are used in link prediction algorithms and recommendation systems in large online social networks. We obtain the rate of growth of the number of common friends in a linear preferential attachment model. We apply our result to develop an estimate for the number of common friends that can significantly reduce the cost of computing these measures. We also observe a phase transition in the limiting behavior of the number of common friends; depending on the range of the parameters of the model, the growth is either power-law, or, logarithmic, or static with the size of the graph.

Metric dimension of critical Galton–Watson trees and linear preferential attachment trees

The metric dimension of a graph G is the minimal size of a subset R of vertices of G that, upon reporting their graph distance from a distinguished (source) vertex v⋆, enable unique identification of the source vertex v⋆ among all possible vertices of G. In this paper we show a Law of Large Numbers (LLN) for the metric dimension of some classes of trees: critical Galton–Watson trees conditioned to have size n, and growing general linear preferential attachment trees. The former class includes uniform random trees, the latter class includes Yule-trees (also called random recursive trees), m-ary increasing trees, binary search trees, and positive linear preferential attachment trees. In all these cases, we are able to identify the limiting constant in the LLN explicitly. Our result relies on the insight that the metric dimension can be related to subtree properties, and hence we can make use of the powerful fringe-tree literature developed by Aldous and Janson et al.

Invisible market for online personal data: An examination

Despite the widespread knowledge that corporations collect and exchange user online personal data (OPD) between themselves in a market for OPD, there have been few attempts to systematically understand the nature and structure of these markets or answer basic questions about the behavior of parties in these markets. This paper addresses these questions using records of data sharing behavior by 218 websites across eight economic sectors. Two datasets, collected 4 years apart, are analyzed using social network analysis (SNA). Findings indicate linear preferential attachment is the most likely coordinating mechanism in the OPD market. Further, this market has a much higher number of brokers (intermediary corporations that facilitate exchange between other corporations) than comparable markets. Building on these findings, implications for research and practice are presented along with future research directions.

Identifying time dependence in network growth

Identifying power-law scaling in real networks - indicative of preferential attachment - has proved controversial. Critics argue that measuring the temporal evolution of a network directly is better than measuring the degree distribution when looking for preferential attachment. However, many of the established methods do not account for any potential time-dependence in the attachment kernels of growing networks, or methods assume that node degree is the key observable determining network evolution. In this paper, we argue that these assumptions may lead to misleading conclusions about the evolution of growing networks. We illustrate this by introducing a simple adaptation of the Barab{\'a}si-Albert model, the "k2 model", where new nodes attach to nodes in the existing network in proportion to the number of nodes one or two steps from the target node. The k2 model results in time dependent degree distributions and attachment kernels, despite initially appearing to grow as linear preferential attachment, and without the need to include explicit time dependence in key network parameters (such as the average out-degree). We show that similar effects are seen in several real world networks where constant network growth rules do not describe their evolution. This implies that measurements of specific degree distributions in real networks are also likely to change over time.

Power Laws for Research Output of Institutions: Evidence from Financial Economics

Cross-affiliation emerges as a new and fast-developing means to promote collaboration in financial research. We find that the average number of affiliations reported per author in the top-three finance journals increases steadily from 1.1 to 1.3 from 1995 to 2016. Scale-free power laws characterize the resulting highly-skewed distributions of top finance journal publications of worldwide institutions. We propose an explanation of the scale-in-variance, based on a network model featuring nonlinear growth and linear preferential attachment. We show that preferential allocation of publications through success-breeds-success engenders disparity in institutions’ research output, while acceleration in the growth of collaboration reduces the dispersion.

Consistency of Hill estimators in a linear preferential attachment model

Preferential attachment is widely used to model power-law behavior of degree distributions in both directed and undirected networks. Practical analyses on the tail exponent of the power-law degree distribution use the Hill estimator as one of the key summary statistics, whose consistency is justified mostly for iid data. The major goal in this paper is to answer the question whether the Hill estimator is still consistent when applied to non-iid network data. To do this, we first derive the asymptotic behavior of the degree sequence via embedding the degree growth of a fixed node into a birth immigration process. We also need to show the convergence of the tail empirical measure, from which the consistency of Hill estimators is obtained. This step requires checking the concentration of degree counts. We give a proof for a particular linear preferential attachment model and use simulation results as an illustration in other choices of models.

Random Recursive Trees and Preferential Attachment Trees are Random Split Trees

We consider linear preferential attachment trees, and show that they can be regarded as random split trees in the sense of Devroye (1999), although with infinite potential branching. In particular, this applies to the random recursive tree and the standard preferential attachment tree. An application is given to the sum over all pairs of nodes of the common number of ancestors.

Relating Topological Determinants of Complex Networks to Their Spectral Properties: Structural and Dynamical Effects

The largest eigenvalue of a network's adjacency matrix and its associated principal eigenvector are key elements for determining the topological structure and the properties of dynamical processes mediated by it. We present a physically grounded expression relating the value of the largest eigenvalue of a given network to the largest eigenvalue of two network subgraphs, considered as isolated: The hub with its immediate neighbors and the densely connected set of nodes with maximum $K$-core index. We validate this formula showing that it predicts with good accuracy the largest eigenvalue of a large set of synthetic and real-world topologies. We also present evidence of the consequences of these findings for broad classes of dynamics taking place on the networks. As a byproduct, we reveal that the spectral properties of heterogeneous networks built according to the linear preferential attachment model are qualitatively different from those of their static counterparts.

Multivariate Normal Limit Laws for the Numbers of Fringe Subtrees in $m$-ary Search Trees and Preferential Attachment Trees

We study fringe subtrees of random $m$-ary search trees and of preferential attachment trees, by putting them in the context of generalised Pólya urns. In particular we show that for the random $m$-ary search trees with $ m\leq 26 $ and for the linear preferential attachment trees, the number of fringe subtrees that are isomorphic to an arbitrary fixed tree $ T $ converges to a normal distribution; more generally, we also prove multivariate normal distribution results for random vectors of such numbers for different fringe subtrees. Furthermore, we show that the number of protected nodes in random $m$-ary search trees for $m\leq 26$ has asymptotically a normal distribution.