Preferential Attachment Scheme Research Articles

Directed Acyclic Graph-Type Distributed Ledgers via Young-Age Preferential Attachment

Distributed ledger technologies provide a mechanism to achieve ordering among transactions that are scattered on multiple participants with no prerequisite trust relations. This mechanism is essentially based on the idea of new transactions referencing older ones in a chain structure. Recently, directed acyclic graph (DAG)-type distributed ledgers that are based on DAGs were proposed to increase the system scalability through sacrificing the total order of transactions. In this paper, we develop a mathematical model to study the process that governs the addition of new transactions to the DAG-type distributed ledger. We propose a simple model for DAG-type distributed ledgers that are obtained from a recursive young-age preferential attachment scheme (i.e., new connections are made preferably to transactions that have not been in the system for very long). We determine the asymptotic degree structure of the resulting graph and show that a forward component of linear size arises if the edge density is chosen sufficiently large in relation to the “young-age preference” that tunes how quickly old transactions become unattractive. Funding: The research of C. Mönch is supported by the Deutsche Forschungsgemeinschaft [Grant 443916008].

Directed Acyclic Graph-Type Distributed Ledgers via Young-Age Preferential Attachment

Distributed ledger technologies provide a mechanism to achieve ordering among transactions that are scattered on multiple participants with no prerequisite trust relations. This mechanism is essentially based on the idea of new transactions referencing older ones in a chain structure. Recently, directed acyclic graph (DAG)-type distributed ledgers that are based on DAGs were proposed to increase the system scalability through sacrificing the total order of transactions. In this paper, we develop a mathematical model to study the process that governs the addition of new transactions to the DAG-type distributed ledger. We propose a simple model for DAG-type distributed ledgers that are obtained from a recursive young-age preferential attachment scheme (i.e., new connections are made preferably to transactions that have not been in the system for very long). We determine the asymptotic degree structure of the resulting graph and show that a forward component of linear size arises if the edge density is chosen sufficiently large in relation to the “young-age preference” that tunes how quickly old transactions become unattractive. Funding: The research of C. Mönch is supported by the Deutsche Forschungsgemeinschaft [Grant 443916008].

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).

The configuration model for Barabasi-Albert networks

We develop and test a rewiring method (originally proposed by Newman) which allows to build random networks having pre-assigned degree distribution and two-point correlations. For the case of scale-free degree distributions, we discretize the tail of the distribution according to the general prescription by Dorogovtsev and Mendes. The application of this method to Barabasi-Albert (BA) networks is possible thanks to recent analytical results on their correlations, and allows to compare the ensemble of random networks generated in the configuration model with that of “real” networks obtained from preferential attachment. For β≥2 (β is the number of parent nodes in the preferential attachment scheme) the networks obtained with the configuration model are completely connected (giant component equal to 100%). In both generation schemes a clear disassortativity of the small degree nodes is demonstrated from the computation of the function knn. We also develop an efficient rewiring method which produces tunable variations of the assortativity coefficient r, and we use it to obtain maximally disassortative networks having the same degree distribution of BA networks with given β. Possible applications of this method concern assortative social networks.

Generative Models for Global Collaboration Relationships

When individuals interact with each other and meaningfully contribute toward a common goal, it results in a collaboration. The artifacts resulting from collaborations are best captured using a hypergraph model, whereas the relation of who has collaborated with whom is best captured via an abstract simplicial complex (SC). We propose a generative algorithm GENESCs for SCs modeling fundamental collaboration relations. The proposed network growth process favors attachment that is preferential not to an individual’s degree, i.e., how many people has he/she collaborated with, but to his/her facet degree, i.e., how many maximal groups or facets has he/she collaborated within. Based on our observation that several real-world facet size distributions have significant deviation from power law–mainly since larger facets tend to subsume smaller ones–we adopt a data-driven approach. We prove that the facet degree distribution yielded by GENESCs is power law distributed for large SCs and show that it is in agreement with real world co-authorship data. Finally, based on our intuition of collaboration formation in domains such as collaborative scientific experiments and movie production, we propose two variants of GENESCs based on clamped and hybrid preferential attachment schemes, and show that they perform well in these domains.

A scaling limit for the degree distribution in sublinear preferential attachment schemes

We consider a general class of preferential attachment schemes evolving by a reinforcement rule with respect to certain sublinear weights. In these schemes, which grow a random network, the sequence of degree distributions is an object of interest which sheds light on the evolving structures. In this article, we use a fluid limit approach to prove a functional law of large numbers for the degree structure in this class, starting from a variety of initial conditions. The method appears robust and applies in particular to ‘non-tree’ evolutions where cycles may develop in the network. A main part of the argument is to show that there is a unique nonnegative solution to an infinite system of coupled ODEs, corresponding to a rate formulation of the law of large numbers limit, through C0-semigroup/dynamical systems methods. These results also resolve a question in Chung, Handjani and Jungreis (2003). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

Modeling of technology adoption in Botswana

Purpose– The conceptualisation of technology adoption has largely been based on the Bass or some Bass-derived model – notably, the logistic model. Logistic-type models offer limited insights regarding the adoption process dynamics or the utility value of their results. The purpose of this paper is to outline an alternative technology adoption framework based on complex adaptive networks.Design/methodology/approach– An agent-based methodological approach is proposed. In it the actors, factors, goals, and adaptive learning influences driving solar energy technology adoption (SETA) process are first substantiated by empirical evidence gathered using field questionnaires and then incorporated in the simulation of a dynamic complex adaptive network of SETA. The complex adaptive network model is based on simple heuristic rules applied using a modified preferential attachment scheme within a NetLogo simulation environment.Findings– The interim results suggest an emergent network where prominent hub “driver” agents underlining the robustness of the model are statistically discernible.Research limitations/implications– The research is limited to solar photovoltaic and solar water heating technology adoption in Botswana households; however, its results are far-reaching.Practical implications– These results can be related to sustainable energy policy design. There, targeted incentive mechanisms can be formulated against the backdrop of the identified environmental factors and actors; the aim being to accelerate and cascade SETA.Social implications– The results could also be cascaded to other sectors and other non-solar technologies, thus providing a general alternative framework for enabling the widespread adoption of technologies.Originality/value– This research therefore represents a novel way of utilizing the new science of networks to accelerate SETA.

A Evolution Model of the Express Freight Network

A Evolution Model of the Express Freight Network is proposed based on the evolution theory of complex weighting network. The freight demands and space distance are considered into the preferential attachment scheme. Applying the model on the evolution of the express freight trains service network by China Railway Express, results show that the model is capable to embody the evolution laws with some indices slightly better than the real network.

Large deviations for the degree structure in preferential attachment schemes

Preferential attachment schemes, where the selection mechanism is linear and possibly time-dependent, are considered, and an infinite-dimensional large deviation principle for the sample path evolution of the empirical degree distribution is found by Dupuis-Ellis-type methods. Interestingly, the rate function, which can be evaluated, contains a term which accounts for the cost of assigning a fraction of the total degree to an "infinite" degree component, that is, when an atypical "condensation" effect occurs with respect to the degree structure. As a consequence of the large deviation results, a sample path a.s. law of large numbers for the degree distribution is deduced in terms of a coupled system of ODEs from which power law bounds for the limiting degree distribution are given. However, by analyzing the rate function, one can see that the process can deviate to a variety of atypical nonpower law distributions with finite cost, including distributions typically associated with sub and superlinear selection models.

The bounds of heavy-tailed return distributions in evolving complex networks

We consider the evolution of scale-free networks according to preferential attachment schemes and show the conditions for which the exponent characterizing the degree distribution is bounded by upper and lower values. Our framework is an agent model, presented in the context of economic networks of trades, which shows the emergence of critical behavior. Starting from a brief discussion about the main features of the evolving network of trades, we show that the logarithmic return distributions have bounded heavy-tails, and the corresponding bounding exponent values can be derived. Finally, we discuss these findings in the context of model risk.

Soil porous system as heterogeneous complex network

In this paper we present a complex network model based on a heterogeneous preferential attachment scheme as a new way to quantify the porous structure of soils and to relate them with soil texture. We consider the pores as nodes, the properties of which, such as position and size, are described by fixed states in a metric space. An affinity function is introduced to bias the attachment probabilities of links according to these properties. We perform an analytical study of the degree distributions in the model and develop a numerical analysis of the degree distributions in the model for a combination of parameters corresponding to eleven empirical soil samples with different physical properties and five different textures.

A statistical construction of power-law networks

We propose a new mechanism for generating networks with a wide variety of degree distributions. The idea is a modification of the well-studied preferential attachment scheme in which the degree of each node is used to determine its evolving connectivity. Modifications to this base protocol to include features other than connectivity have been considered in building the network. However, some of the existing models are merely formulaic and do not offer an explanation that can be interpreted naturally or intuitively, while the others require certain information that is not available in many real-world circumstances. We propose instead a protocol based only on a single statistical feature which results from the reasonable assumption that the effect of various attributes, which determine the ability of each node to attract other nodes, is multiplicative. This composite attribute or fitness is lognormally distributed and is used in forming the complex network. We show that, by varying the parameters of the lognormal distribution, we can recover both exponential and power-law degree distributions. The exponents for the power-law case are in the correct range seen in real-world networks. Further, as power-law networks with exponents in the same range are a crucial ingredient of efficient search algorithms in P2P networks, we believe our network construct may serve as a basis for new protocols that will enable P2P networks to efficiently establish a topology conducive to optimised search procedures.

Multiscaling of porous soils as heterogeneous complex networks

Abstract. In this paper we present a complex network model based on a heterogeneous preferential attachment scheme to quantify the structure of porous soils. Under this perspective pores are represented by nodes and the space for the flow of fluids between them is represented by links. Pore properties such as position and size are described by fixed states in a metric space, while an affinity function is introduced to bias the attachment probabilities of links according to these properties. We perform an analytical study of the degree distributions in the soil model and show that under reasonable conditions all the model variants yield a multiscaling behavior in the connectivity degrees, leaving a empirically testable signature of heterogeneity in the topology of pore networks. We also show that the power-law scaling in the degree distribution is a robust trait of the soil model and analyze the influence of the parameters on the scaling exponents. We perform a numerical analysis of the soil model for a combination of parameters corresponding to empirical samples with different properties, and show that the simulation results exhibit a good agreement with the analytical predictions.

Kernel-Granger causality and the analysis of dynamical networks

We propose a method of analysis of dynamical networks based on a recent measure of Granger causality between time series, based on kernel methods. The generalization of kernel-Granger causality to the multivariate case, here presented, shares the following features with the bivariate measures: (i) the nonlinearity of the regression model can be controlled by choosing the kernel function and (ii) the problem of false causalities, arising as the complexity of the model increases, is addressed by a selection strategy of the eigenvectors of a reduced Gram matrix whose range represents the additional features due to the second time series. Moreover, there is no a priori assumption that the network must be a directed acyclic graph. We apply the proposed approach to a network of chaotic maps and to a simulated genetic regulatory network: it is shown that the underlying topology of the network can be reconstructed from time series of node's dynamics, provided that a sufficient number of samples is available. Considering a linear dynamical network, built by preferential attachment scheme, we show that for limited data use of the bivariate Granger causality is a better choice than methods using L1 minimization. Finally we consider real expression data from HeLa cells, 94 genes and 48 time points. The analysis of static correlations between genes reveals two modules corresponding to well-known transcription factors; Granger analysis puts in evidence 19 causal relationships, all involving genes related to tumor development.

Effects of preference for attachment to low-degree nodes on the degree distributions of a growing directed network and a simple food-web model

We study the growth of a directed network, in which the growth is constrained by the cost of adding links to the existing nodes. We propose a preferential-attachment scheme, in which a new node attaches to an existing node i with probability II(k(i)) approximately k(-1), where k(i) is the number of outgoing links at i. We calculate the degree distribution for the outgoing links in the asymptotic regime t --> infinity, n(k) both analytically and by Monte Carlo simulations. The distribution decays like kmu(k)/Tau(k) for large k, where is a constant. We investigate the effect of this preferential-attachment scheme, by comparing the results to an equivalent growth model with a degree-independent probability of attachment, which gives an exponential outdegree distribution. Also, we relate this mechanism to simple food-web models by implementing it in the cascade model. We show that the low-degree preferential-attachment mechanism breaks the symmetry between in- and outdegree distributions in the cascade model. It also causes a faster decay in the tails of the outdegree distributions for both our network growth model and the cascade model.

EFFECTS OF AGING AND LINKS REMOVAL ON EPIDEMIC DYNAMICS IN SCALE-FREE NETWORKS

We study the combined effects of aging and links removal on epidemic dynamics in the Barabási–Albert scale-free networks. The epidemic is described by a susceptible-infected-refractory (SIR) model. The aging effect of a node introduced at time ti is described by an aging factor of the form (t-ti)-β in the probability of being connected to newly added nodes in a growing network under the preferential attachment scheme based on popularity of the existing nodes. SIR dynamics is studied in networks with a fraction 1-p of the links removed. Extensive numerical simulations reveal that there exists a threshold pc such that for p≥pc, epidemic breaks out in the network. For p<pc, only a local spread results. The dependence of pc on β is studied in detail. The function pc(β) separates the space formed by β and p into regions corresponding to local and global spreads, respectively.

Accelerated growth in outgoing links in evolving networks: deterministic versus stochastic picture.

In several real-world networks such as the Internet, World Wide Web, etc., the number of links grow in time in a nonlinear fashion. We consider growing networks in which the number of outgoing links is a nonlinear function of time but new links between older nodes are forbidden. The attachments are made using a preferential attachment scheme. In the deterministic picture, the number of outgoing links m (t) at any time t is taken as N (t)(theta) where N (t) is the number of nodes present at that time. The continuum theory predicts a power-law decay of the degree distribution: P (k) proportional to k-(1-2/ (1-theta ) ), while the degree of the node introduced at time t(i) is given by k(t(i),t)=t(theta)(i) [t/t(i) ]((1+theta)/2) when the network is evolved till time t. Numerical results show a growth in the degree distribution for small k values at any nonzero theta. In the stochastic picture, m (t) is a random variable. As long as <m (t) > is independent of time, the network shows a behavior similar to the Barabási-Albert (BA) model. Different results are obtained when <m (t) > is time dependent, e.g., when m (t) follows a distribution P (m) proportional to m(-lambda). The behavior of P (k) changes significantly as lambda is varied: for lambda>3, the network has a scale-free distribution belonging to the BA class as predicted by the mean field theory; for smaller values of lambda it shows different behavior. Characteristic features of the clustering coefficients in both models have also been discussed.

Generalizations of Polya's urn Problem

We consider generalizations of the classical Polya urn problem: Given finitely many bins each containing one ball, suppose that additional balls arrive one at a time. For each new ball, with probability p, create a new bin and place the ball in that bin; with probability 1−p, place the ball in an existing bin, such that the probability that the ball is placed in a bin is proportional to $ m^\gamma $, where m is the number of balls in that bin. For p=0, the number of bins is fixed and finite, and the behavior of the process depends on whether γ is greater than, equal to, or less than 1. We survey the known results and give new proofs for all three cases. We then consider the case p>0. When γ=1, this is equivalent to the so-called preferential attachment scheme which leads to power law distribution for bin sizes. When γ>1, we prove that a single bin dominates, i.e., as the number of balls goes to infinity, the probability that any new ball either goes into that bin or creates a new bin converges to 1. When p > 0 and γ < 1, we show that under the assumption that certain limits exist, the fraction of bins having m balls shrinks exponentially as a function of m. We then discuss further generalizations and pose several open problems.