Evolutionary of Online Social Networks Driven by Pareto Wealth Distribution and Bidirectional Preferential Attachment

Understanding of evolutionary mechanism of online social networks is greatly significant for the development of network science. However, present researches on evolutionary mechanism of online social networks are neither deep nor clear enough. In this study, we empirically showed the essential evolution characteristics of Renren online social network. From the perspective of Pareto wealth distribution and bidirectional preferential attachment, the origin of online social network evolution is analyzed and the evolution mechanism of online social networks is explained. Then a novel model is proposed to reproduce the essential evolution characteristics which are consistent with the ones of Renren online social network, and the evolutionary analytical solution to the model is presented. The model can also well predict the ordinary power-law degree distribution. In addition, the universal bowing phenomenon of the degree distribution in many online social networks is explained and predicted by the model. The results suggest that Pareto wealth distribution and bidirectional preferential attachment can play an important role in the evolution process of online social networks and can help us to understand the evolutionary origin of online social networks. The model has significant implications for dynamic simulation researches of social networks, especially in information diffusion through online communities and infection spreading in real societies.Comment: 19 pages, 8 figures,31 reference

Early thermalization of quark-gluon matter by elastic 3-to-3 scattering

The early thermalization is crucial to the quark-gluon plasma as a perfect liquid and results from many-body scattering. We calculate squared amplitudes for elastic parton-parton-parton scattering in perturbative QCD. Transport equations with the squared amplitudes are established and solved to obtain the thermalization time of initially produced quark-gluon matter and the initial temperature of quark-gluon plasma. We find that the thermalization times of quark matter and gluon matter are different.Comment: 5 pages, 1 figure, proceedings for Extreme QCD 201

Early Thermalization at RHIC

Triple-gluon elastic scatterings are briefly reviewed since the scatterings explain the early thermalization puzzle in Au-Au collisions at RHIC energies. A numerical solution of the transport equation with the triple-gluon elastic scatterings demonstrates gluon momentum isotropy achieved at a time of the order of 0.65 fm/c. Triple-gluon scatterings lead to a short thermalization time of gluon matter.Comment: LaTex, 8 pages and 4 figures, talk presented in the Weihai workshop on relativistic heavy ion collision

Early thermalization of quark-gluon matter initially created in high-energy nucleus-nucleus collisions

Elastic parton-parton-parton scattering is briefly reviewed and is included in transport equations of quark-gluon matter. We solve the transport equations and get thermal states from initially produced quark-gluon matter. Both gluon matter and quark matter take early thermalization, but gloun matter has a shorter thermalization time than quark matter.Comment: 7 pages, 4 figures, proceedings for the 29th Winter Workshop on Nuclear Dynamic

Origin of Temperature of Quark-Gluon Plasma in Heavy Ion Collisions

Initially produced quark-gluon matter at RHIC and LHC does not have a temperature. A quark-gluon plasma has a high temperature. From this quark-gluon matter to the quark-gluon plasma is the early thermalization or the rapid creation of temperature. Elastic three-parton scattering plays a key role in the process. The temperature originates from the two-parton scattering, the three-parton scattering, the four-parton scattering and so forth in quark-gluon matter.Comment: 6 pages, proceedings for the XXX-th International Workshop on High Energy Physic

Hall algebras associated to triangulated categories

By counting with triangles and the octohedral axiom, we find a direct way to prove the formula of To\"en in \cite{Toen2005} for a triangulated category with (left) homological-finite condition.Comment: 12 pages. Final version, to appear in Duk

Energy Optimal Interpolation in Quantum Evolution

We introduce the concept of interpolation in quantum evolution and present a general framework to find the energy optimal Hamiltonian for a quantum system evolving among a given set of middle states using variational and geometric methods. A few special cases are carefully studied. The quantum brachistochrone problem is proved as a special case.Comment: 8 pages, 0 figure

Understanding Weight Normalized Deep Neural Networks with Rectified Linear Units

This paper presents a general framework for norm-based capacity control for $L_{p,q}$ weight normalized deep neural networks. We establish the upper bound on the Rademacher complexities of this family. With an $L_{p,q}$ normalization where $q\le p^*$, and $1/p+1/p^{*}=1$, we discuss properties of a width-independent capacity control, which only depends on depth by a square root term. We further analyze the approximation properties of $L_{p,q}$ weight normalized deep neural networks. In particular, for an $L_{1,\infty}$ weight normalized network, the approximation error can be controlled by the $L_1$ norm of the output layer, and the corresponding generalization error only depends on the architecture by the square root of the depth

The Impact of Alternation

Alternating automata have been widely used to model and verify systems that handle data from finite domains, such as communication protocols or hardware. The main advantage of the alternating model of computation is that complementation is possible in linear time, thus allowing to concisely encode trace inclusion problems that occur often in verification. In this paper we consider alternating automata over infinite alphabets, whose transition rules are formulae in a combined theory of booleans and some infinite data domain, that relate past and current values of the data variables. The data theory is not fixed, but rather it is a parameter of the class. We show that union, intersection and complementation are possible in linear time in this model and, though the emptiness problem is undecidable, we provide two efficient semi-algorithms, inspired by two state-of-the-art abstraction refinement model checking methods: lazy predicate abstraction \cite{HJMS02} and the \impact~ semi-algorithm \cite{mcmillan06}. We have implemented both methods and report the results of an experimental comparison

Green's formula with \bbc^{*}-action and Caldero-Keller's formula for cluster algebras

It is known that Green's formula over finite fields gives rise to the comultiplications of Ringel-Hall algebras and quantum groups (see\cite{Green}, also see \cite{Lusztig}). In this paper, we deduce the projective version of Green's formula in a geometric way. Then following the method of Hubery in \cite{Hubery2005}, we apply this formula to proving Caldero-Keller's multiplication formula for acyclic cluster algebras of arbitrary type.Comment: 26 page