12,787 research outputs found
Binomial Difference Ideal and Toric Difference Variety
In this paper, the concepts of binomial difference ideals and toric
difference varieties are defined and their properties are proved. Two canonical
representations for Laurent binomial difference ideals are given using the
reduced Groebner basis of Z[x]-lattices and regular and coherent difference
ascending chains, respectively. Criteria for a Laurent binomial difference
ideal to be reflexive, prime, well-mixed, perfect, and toric are given in terms
of their support lattices which are Z[x]-lattices. The reflexive, well-mixed,
and perfect closures of a Laurent binomial difference ideal are shown to be
binomial. Four equivalent definitions for toric difference varieties are
presented. Finally, algorithms are given to check whether a given Laurent
binomial difference ideal I is reflexive, prime, well-mixed, perfect, or toric,
and in the negative case, to compute the reflexive, well-mixed, and perfect
closures of I. An algorithm is given to decompose a finitely generated perfect
binomial difference ideal as the intersection of reflexive prime binomial
difference ideals.Comment: 72 page
Influence of Reciprocal links in Social Networks
In this Letter, we empirically study the influence of reciprocal links, in
order to understand its role in affecting the structure and function of
directed social networks. Experimental results on two representative datesets,
Sina Weibo and Douban, demonstrate that the reciprocal links indeed play a more
important role than non-reciprocal ones in both spreading information and
maintaining the network robustness. In particular, the information spreading
process can be significantly enhanced by considering the reciprocal effect. In
addition, reciprocal links are largely responsible for the connectivity and
efficiency of directed networks. This work may shed some light on the in-depth
understanding and application of the reciprocal effect in directed online
social networks
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