General results on preferential attachment and clustering coefficient

This is a review paper that covers some recent results on the behavior of the clustering coefficient in preferential attachment networks and scale-free networks in general. The paper focuses on general approaches to network science. In other words, instead of discussing different fully specified random graph models, we describe some generic results which hold for classes of models. Namely, we first discuss a generalized class of preferential attachment models which includes many classical models. It turns out that some properties can be analyzed for the whole class without specifying the model. Such properties are the degree distribution and the global and average local clustering coefficients. Finally, we discuss some surprising results on the behavior of the global clustering coefficient in scale-free networks. Here we do not assume any underlying model.     

Small subgraphs in preferential attachment networks

In this paper, we give an overview of old and new results on the global and average local clustering coefficients as well as on the distribution of small subgraphs in the Bollobás–Riordan and Buckley–Osthus models of web graphs.     

Directed hybrid random networks mixing preferential attachment with uniform attachment mechanisms

Annals of the Institute of Statistical Mathematics - Motivated by the complexity of network data, we propose a directed hybrid random network that mixes preferential attachment (PA) rules with...     

On the asymptotic normality of estimating the affine preferential attachment network models with random initial degrees

We consider the estimation of the affine parameter and power-law exponent in the preferential attachment model with random initial degrees. We derive the likelihood, and show that the maximum likelihood estimator (MLE) is asymptotically normal and efficient. We also propose a quasi-maximum-likelihood estimator (QMLE) to overcome the MLE’s dependence on the history of the initial degrees. To demonstrate the power of our idea, we present numerical simulations.     

A preferential attachment model with random initial degrees

In this paper, a random graph process {G(t)} _t≥1 is studied and its degree sequence is analyzed. Let {W _t } _t≥1 be an i.i.d. sequence. The graph process is defined so that, at each integer time t, a new vertex with W _t edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on G(t-1), the probability that a given edge of vertex t is connected to vertex i is proportional to d _i (t-1)+δ, where d _i (t-1) is the degree of vertex i at time t-1, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent τ=min{τ_W,τ_P}, where τ_W is the power-law exponent of the initial degrees {W _t } _t≥1 and τ_P the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze.     

A preferential attachment model with Poisson growth for scale-free networks

We propose a scale-free network model with a tunable power-law exponent. The Poisson growth model, as we call it, is an offshoot of the celebrated model of Barabási and Albert where a network is generated iteratively from a small seed network; at each step a node is added together with a number of incident edges preferentially attached to nodes already in the network. A key feature of our model is that the number of edges added at each step is a random variable with Poisson distribution, and, unlike the Barabási–Albert model where this quantity is fixed, it can generate any network. Our model is motivated by an application in Bayesian inference implemented as Markov chain Monte Carlo to estimate a network; for this purpose, we also give a formula for the probability of a network under our model.     

Growth of preferential attachment random graphs via continuous-time branching processes

Some growth asymptotics of a version of ‘preferential attachment’ random graphs are studied through an embedding into a continuous-time branching scheme. These results complement and extend previous work in the literature.     

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

Extremes - 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....     

Multigraph limit of the dense configuration model and the preferential attachment graph

The configuration model is the most natural model to generate a random multigraph with a given degree sequence. We use the notion of dense graph limits to characterize the special form of limit objects of convergent sequences of configuration models. We apply these results to calculate the limit object corresponding to the dense preferential attachment graph and the edge reconnecting model. Our main tools in doing so are (1) the relation between the theory of graph limits and that of partially exchangeable random arrays (2) an explicit construction of our random graphs that uses urn models.     

Multiple autoregressive models with random coefficients

This paper derives conditions for the stationarity of a class of multiple autoregressive models with random coefficients. The models considered include as special cases those previously discussed by Andel (Ann. Math. Statist.42 (1971), 755–759; Math. Operationsforsch. Statist.7 (1976), 735–741).     

Predator-prey models with almost periodic coefficients

This paper deals with the question of persistence of differential equations with almost periodic coefficients which model predator-prey systems. It is shown that persistence does occur in many models that incorporate almost periodic time dependence in the prey equation     

Convergence in competition models with small diffusion coefficients

It is well known that for reaction-diffusion 2-species Lotka-Volterra competition models with spatially independent reaction terms, global stability of an equilibrium for the reaction system implies global stability for the reaction-diffusion system. This is not in general true for spatially inhomogeneous models. We show here that for an important range of such models, for small enough diffusion coefficients, global convergence to an equilibrium holds for the reaction-diffusion system, if for each point in space the reaction system has a globally attracting hyperbolic equilibrium. This work is planned as an initial step towards understanding the connection between the asymptotics of reaction-diffusion systems with small diffusion coefficients and that of the corresponding reaction systems.     

Local solvability of PDE with constant coefficients

The fundamental result of the paper consists in the following assertion: Let D be an open set in Rⁿ,f £ L₂(D), and let be a distribution in Rⁿ with a compact support, whose Fourier transform satisfies the condition

Fitting semiparametric clustering models to dissimilarity data

The cluster analysis problem of partitioning a set of objects from dissimilarity data is here handled with the statistical model-based approach of fitting the “closest” classification matrix to the observed dissimilarities. A classification matrix represents a clustering structure expressed in terms of dissimilarities. In cluster analysis there is a lack of methodologies widely used to directly partition a set of objects from dissimilarity data. In real applications, a hierarchical clustering algorithm is applied on dissimilarities and subsequently a partition is chosen by visual inspection of the dendrogram. Alternatively, a “tandem analysis” is used by first applying a Multidimensional Scaling (MDS) algorithm and then by using a partitioning algorithm such as k-means applied on the dimensions specified by the MDS. However, neither the hierarchical clustering algorithms nor the tandem analysis is specifically defined to solve the statistical problem of fitting the closest partition to the observed dissimilarities. This lack of appropriate methodologies motivates this paper, in particular, the introduction and the study of three new object partitioning models for dissimilarity data, their estimation via least-squares and the introduction of three new fast algorithms.     

Local Shalika models and functoriality

We prove, over a p-adic local field F, that an irreducible supercuspidal representation of GL_2n (F) is a local Langlands functorial transfer from SO_2n+1(F) if and only if it has a nonzero Shalika model (Corollary 5.2, Proposition 5.4 and Theorem 5.5). Based on this, we verify (Sect. 6) in our cases a conjecture of Jacquet and Martin, a conjecture of Kim, and a conjecture of Speh in the theory of automorphic forms.     

Unifying data units and models in (co-)clustering

Advances in Data Analysis and Classification - Statisticians are already aware that any task (exploration, prediction) involving a modeling process is largely dependent on the measurement units for...     

分组数据下线性回归模型的参数估计

讨论了分组数据下线性回归模型参数的MLE的存在、唯一性.通过EM算法获得MLE的近似解.通过SEM算法获得MLE的渐近协方差阵.