arXiv subjects

Tiandong Wang

Publications and source records attributed to Tiandong Wang.

At least 37 records · Page 2Linked to original sources

arXiv2022

Random Networks with Heterogeneous Reciprocity

Users of social networks display diversified behavior and online habits. For instance, a user's tendency to reply to a post can depend on the user and the person posting. For convenience, we group users into aggregated behavioral patterns, focusing here on the tendency to reply to or reciprocate messages. The reciprocity feature in social networks reflects the information exchange among users. We study the properties of a preferential attachment model with heterogeneous reciprocity levels, give the growth rate of model edge counts, and prove convergence of empirical degree frequencies to a limiting distribution. This limiting distribution is not only multivariate regularly varying, but also has the property of hidden regular variation.

cs.SI↗

arXiv2022

EVIboost for the Estimation of Extreme Value Index under Heterogeneous Extremes

Modeling heterogeneity on heavy-tailed distributions under a regression framework is challenging, and classical statistical methodologies usually place conditions on the distribution models to facilitate the learning procedure. However, these conditions are likely to overlook the complex dependence structure between the heaviness of tails and the covariates. Moreover, data sparsity on tail regions also makes the inference method less stable, leading to largely biased estimates for extreme-related quantities. This paper proposes a gradient boosting algorithm to estimate a functional extreme value index with heterogeneous extremes. Our proposed algorithm is a data-driven procedure that captures complex and dynamic structures in tail distributions. We also conduct extensive simulation studies to show the prediction accuracy of the proposed algorithm. In addition, we apply our method to a real-world data set to illustrate the state-dependent and time-varying properties of heavy-tail phenomena in the financial industry.

stat.ME↗

arXiv2022

Preferential Attachment with Reciprocity: Properties and Estimation

Reciprocity in social networks helps understand information exchange between two individuals, and indicates interaction patterns between pairs of users. A recent study indicates the reciprocity coefficient of a classical directed preferential attachment (PA) model does not match empirical evidence. In this paper, we extend the classical 3-scenario directed PA model by adding an additional parameter that controls the probability of creating a reciprocal edge. Our proposed model also allows edge creation between two existing nodes, making it a more realistic choice for fitting to real datasets. In addition to analysis of the theoretical properties of this PA model with reciprocity, we provide and compare two estimation procedures for the fitting of the extended model to both simulated and real datasets. The fitted models provide a good match with the empirical tail distributions of both in- and out-degrees. Other mismatched diagnostics suggest that further generalization of the model is warranted.

physics.soc-ph↗

arXiv2022

An Efficient Algorithm for Generating Directed Networks with Predetermined Assortativity Measures

Assortativity coefficients are important metrics to analyze both directed and undirected networks. In general, it is not guaranteed that the fitted model will always agree with the assortativity coefficients in the given network, and the structure of directed networks is more complicated than the undirected ones. Therefore, we provide a remedy by proposing a degree-preserving rewiring algorithm, called DiDPR, for generating directed networks with given directed assortativity coefficients. We construct the joint edge distribution of the target network by accounting for the four directed assortativity coefficients simultaneously, provided that they are attainable, and obtain the desired network by solving a convex optimization problem.Our algorithm also helps check the attainability of the given assortativity coefficients. We assess the performance of the proposed algorithm by simulation studies with focus on two different network models, namely Erdös--Rényi and preferential attachment random networks. We then apply the algorithm to a Facebook wall post network as a real data example. The codes for implementing our algorithm are publicly available in R package wdnet.

stat.CO↗

arXiv2021

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.

physics.soc-ph↗

arXiv2021

Asymptotic Behavior of Common Connections in Sparse Random Networks

Random network models generated using sparse exchangeable graphs have provided a mechanism to study a wide variety of complex real-life networks. In particular, these models help with investigating power-law properties of degree distributions, number of edges, and other relevant network metrics which support the scale-free structure of networks. Previous work on such graphs imposes a marginal assumption of univariate regular variation (e.g., power-law tail) on the bivariate generating graphex function. In this paper, we study sparse exchangeable graphs generated by graphex functions which are multivariate regularly varying. We also focus on a different metric for our study: the distribution of the number of common vertices (connections) shared by a pair of vertices. The number being high for a fixed pair is an indicator of the original pair of vertices being connected. We find that the distribution of number of common connections are regularly varying as well, where the tail indices of regular variation are governed by the type of graphex function used. Our results are verified on simulated graphs by estimating the relevant tail index parameters.

math.ST↗

arXiv2021

PageRank centrality and algorithms for weighted, directed networks with applications to World Input-Output Tables

PageRank (PR) is a fundamental tool for assessing the relative importance of the nodes in a network. In this paper, we propose a measure, weighted PageRank (WPR), extended from the classical PR for weighted, directed networks with possible non-uniform node-specific information that is dependent or independent of network structure. A tuning parameter leveraging node degree and strength is introduced. An efficient algorithm based on R program has been developed for computing WPR in large-scale networks. We have tested the proposed WPR on widely used simulated network models, and found it outperformed other competing measures in the literature. By applying the proposed WPR to the real network data generated from World Input-Output Tables, we have seen the results that are consistent with the global economic trends, which renders it a preferred measure in the analysis.

physics.soc-ph↗

arXiv2021

Measuring Reciprocity in a Directed Preferential Attachment Network

Empirical studies 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 distribution with power-law tails, but theoretical properties of the reciprocity feature in this model have not yet been studied. We derive the 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 alternative models need to be considered.

physics.soc-ph↗

arXiv2021

Directed Hybrid Random Networks Mixing Preferential Attachment with Uniform Attachment Mechanisms

Motivated by the complexity of network data, we propose a directed hybrid random network that mixes preferential attachment (PA) rules with uniform attachment (UA) rules. When a new edge is created, with probability $p\in [0,1]$, it follows the PA rule. Otherwise, this new edge is added between two uniformly chosen nodes. Such mixture makes the in- and out-degrees of a fixed node grow at a slower rate, compared to the pure PA case, thus leading to lighter distributional tails. Useful inference methods for the proposed hybrid model are then provided and applied to both synthetic and real datasets. We see that with extra flexibility given by the parameter $p$, the hybrid random network provides a better fit to real-world scenarios, where lighter tails from in- and out-degrees are observed.

cs.SI↗

arXiv2020

Functional data analysis: An application to COVID-19 data in the United States

The COVID-19 pandemic so far has caused huge negative impacts on different areas all over the world, and the United States (US) is one of the most affected countries. In this paper, we use methods from the functional data analysis to look into the COVID-19 data in the US. We explore the modes of variation of the data through a functional principal component analysis (FPCA), and study the canonical correlation between confirmed and death cases. In addition, we run a cluster analysis at the state level so as to investigate the relation between geographical locations and the clustering structure. Lastly, we consider a functional time series model fitted to the cumulative confirmed cases in the US, and make forecasts based on the dynamic FPCA. Both point and interval forecasts are provided, and the methods for assessing the accuracy of the forecasts are also included.

stat.AP↗

arXiv2020

A Directed Preferential Attachment Model with Poisson Measurement

When modeling a directed social network, one choice is to use the traditional preferential attachment model, which generates power-law tail distributions. In a traditional directed preferential attachment, every new edge is added sequentially into the network. However, for real datasets, it is common to only have coarse timestamps available, which means several new edges are created at the same timestamp. Previous analyses on the evolution of social networks reveal that after reaching a stable phase, the growth of edge counts in a network follows a non-homogeneous Poisson process with a constant rate across the day but varying rates from day to day. Taking such empirical observations into account, we propose a modified preferential attachment model with Poisson measurement, and study its asymptotic behavior. This modified model is then fitted to real datasets, and we see it provides a better fit than the traditional one.

math.PR↗

arXiv2020

On a minimum distance procedure for threshold selection in tail analysis

Power-law distributions have been widely observed in different areas of scientific research. Practical estimation issues include how to select a threshold above which observations follow a power-law distribution and then how to estimate the power-law tail index. A minimum distance selection procedure (MDSP) is proposed in Clauset et al. (2009) and has been widely adopted in practice, especially in the analyses of social networks. However, theoretical justifications for this selection procedure remain scant. In this paper, we study the asymptotic behavior of the selected threshold and the corresponding power-law index given by the MDSP. We find that the MDSP tends to choose too high a threshold level and leads to Hill estimates with large variances and root mean squared errors for simulated data with Pareto-like tails.

math.ST↗

arXiv2019

Common Growth Patterns for Regional Social Networks: a Point Process Approach

Although recent research on social networks emphasizes microscopic dynamics such as retweets and social connectivity of an individual user, we focus on macroscopic growth dynamics of social network link formation. Rather than focusing on one particular dataset, we find invariant behavior in regional social networks that are geographically concentrated. Empirical findings suggest that the startup phase of a regional network can be modeled by a self-exciting point process. After the startup phase ends, the growth of the links can be modeled by a non-homogeneous Poisson process with constant rate across the day but varying rates from day to day, plus a nightly inactive period when local users are expected to be asleep. Conclusions are drawn based on analyzing four different datasets, three of which are regional and a non-regional one is included for contrast.

stat.AP↗

arXiv2018

Degree Growth Rates and Index Estimation in a Directed Preferential Attachment Model

Preferential attachment is widely used to model power-law behavior of degree distributions in both directed and undirected networks. In a directed preferential attachment model, despite the well-known marginal power-law degree distributions, not much investigation has been done on the joint behavior of the in- and out-degree growth. Also, statistical estimates of the marginal tail exponent of the power-law degree distribution often use the Hill estimator as one of the key summary statistics, even though no theoretical justification has been given. This paper focuses on convergence of the joint empirical measure for in- and out-degrees and proves the consistency of the Hill estimator. To do this, we first derive the asymptotic behavior of the joint degree sequences by embedding the in- and out-degrees of a fixed node into a pair of switched birth processes with immigration and then establish the convergence of the joint tail empirical measure. From these steps, the consistency of the Hill estimators is obtained.

math.PR↗

arXiv2017

Are Extreme Value Estimation Methods Useful for Network Data?

Preferential attachment is an appealing edge generating mechanism for modeling social networks. It provides both an intuitive description of network growth and an explanation for the observed power laws in degree distributions. However, there are often limitations in fitting parametric network models to data due to the complex nature of real-world networks. In this paper, we consider a semi-parametric estimation approach by looking at only the nodes with large in- or out-degrees of the network. This method examines the tail behavior of both the marginal and joint degree distributions and is based on extreme value theory. We compare it with the existing parametric approaches and demonstrate how it can provide more robust estimates of parameters associated with the network when the data are corrupted or when the model is misspecified.

stat.ME↗

arXiv2017

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.

math.PR↗

arXiv2017

Fitting the Linear Preferential Attachment Model

Preferential attachment is an appealing mechanism for modeling power-law behavior of the degree distributions in directed social networks. In this paper, we consider methods for fitting a 5-parameter linear preferential model to network data under two data scenarios. In the case where full history of the network formation is given, we derive the maximum likelihood estimator of the parameters and show that it is strongly consistent and asymptotically normal. In the case where only a single-time snapshot of the network is available, we propose an estimation method which combines method of moments with an approximation to the likelihood. The resulting estimator is also strongly consistent and performs quite well compared to the MLE estimator. We illustrate both estimation procedures through simulated data, and explore the usage of this model in a real data example.

stat.ME↗

arXiv2016

Multivariate Regular Variation of Discrete Mass Functions with Applications to Preferential Attachment Networks

Regular variation of a multivariate measure with a Lebesgue density implies the regular variation of its density provided the density satisfies some regularity conditions. Unlike the univariate case, the converse also requires regularity conditions. We extend these arguments to discrete mass functions and their associated measures using the concept that the the mass function can be embedded in a continuous density function. We give two different conditions, monotonicity and convergence on the unit sphere, both of which can make the discrete function embeddable. Our results are then applied to the preferential attachment network model, and we conclude that the joint mass function of in- and out-degree is embeddable and thus regularly varying.

math.PR↗