105,260 research outputs found
Community Structures Are Definable in Networks: A Structural Theory of Networks
We found that neither randomness in the ER model nor the preferential
attachment in the PA model is the mechanism of community structures of
networks, that community structures are universal in real networks, that
community structures are definable in networks, that communities are
interpretable in networks, and that homophyly is the mechanism of community
structures and a structural theory of networks. We proposed the notions of
entropy- and conductance-community structures. It was shown that the two
definitions of the entropy- and conductance-community structures and the notion
of modularity proposed by physicists are all equivalent in defining community
structures of networks, that neither randomness in the ER model nor
preferential attachment in the PA model is the mechanism of community
structures of networks, and that the existence of community structures is a
universal phenomenon in real networks. This poses a fundamental question: What
are the mechanisms of community structures of real networks? To answer this
question, we proposed a homophyly model of networks. It was shown that networks
of our model satisfy a series of new topological, probabilistic and
combinatorial principles, including a fundamental principle, a community
structure principle, a degree priority principle, a widths principle, an
inclusion and infection principle, a king node principle and a predicting
principle etc. The new principles provide a firm foundation for a structural
theory of networks. Our homophyly model demonstrates that homophyly is the
underlying mechanism of community structures of networks, that nodes of the
same community share common features, that power law and small world property
are never obstacles of the existence of community structures in networks, that
community structures are {\it definable} in networks, and that (natural)
communities are {\it interpretable}
Homophyly Networks -- A Structural Theory of Networks
A grand challenge in network science is apparently the missing of a
structural theory of networks. The authors have showed that the existence of
community structures is a universal phenomenon in real networks, and that
neither randomness nor preferential attachment is a mechanism of community
structures of network \footnote{A. Li, J. Li, and Y. Pan, Community structures
are definable in networks, and universal in the real world, To appear.}. This
poses a fundamental question: What are the mechanisms of community structures
of real networks? Here we found that homophyly is the mechanism of community
structures and a structural theory of networks. We proposed a homophyly model.
It was shown that networks of our model satisfy a series of new topological,
probabilistic and combinatorial principles, including a fundamental principle,
a community structure principle, a degree priority principle, a widths
principle, an inclusion and infection principle, a king node principle, and a
predicting principle etc, leading to a structural theory of networks. Our model
demonstrates that homophyly is the underlying mechanism of community structures
of networks, that nodes of the same community share common features, that power
law and small world property are never obstacles of the existence of community
structures in networks, and that community structures are definable in
networks.Comment: arXiv admin note: substantial text overlap with arXiv:1310.803
Community Structures Are Definable in Networks, and Universal in Real World
Community detecting is one of the main approaches to understanding networks
\cite{For2010}.
However it has been a longstanding challenge to give a definition for
community structures of networks. Here we found that community structures are
definable in networks, and are universal in real world. We proposed the notions
of entropy- and conductance-community structure ratios. It was shown that the
definitions of the modularity proposed in \cite{NG2004}, and our entropy- and
conductance-community structures are equivalent in defining community
structures of networks, that randomness in the ER model \cite{ER1960} and
preferential attachment in the PA \cite{Bar1999} model are not mechanisms of
community structures of networks, and that the existence of community
structures is a universal phenomenon in real networks. Our results demonstrate
that community structure is a universal phenomenon in the real world that is
definable, solving the challenge of definition of community structures in
networks. This progress provides a foundation for a structural theory of
networks.Comment: arXiv admin note: substantial text overlap with arXiv:1310.803
Permanence and almost periodic solutions for a single-species system with impulsive effects on time scales
In this paper, we first propose a single-species system with impulsive
effects on time scales and by establishing some new comparison theorems of
impulsive dynamic equations on time scales, we obtain sufficient conditions to
guarantee the permanence of the system. Then we prove a Massera type theorem
for impulsive dynamic equations on time scales and based on this theorem, we
establish a criterion for the existence and uniformly asymptotic stability of
unique positive almost periodic solution of the system. Finally, we give an
example to show the feasibility of our main results. Our example also shows
that the continuous time system and its corresponding discrete time system have
the same dynamics. Our results of this paper are completely new.Comment: 19 page
The -curvature on a 4-dimensional Riemannian manifold with
In this paper we study the solutions of the -curvature equation on a
4-dimensional Riemannian manifold with , proving
some sufficient conditions for the existence
Two addition theorems on polynomials of prime variables
We extend a recent result of Khalfalah and Szemeredi to the polynomials of
prime variables.Comment: This is a very preliminary draft, and maybe contains some minor
mistake
Unobstructedness of deformations of Calabi-Yau varieties with a line bundle
We generalize the Tian-Todorov Theorem in the case of Calabi-Yau varieties
equipped with a line bundle.Comment: In this version, we proved the generalized Tian-Todorov Theorem in
the case of Calabi-Yau varieties equipped with a line bundle completel
Kirchhoff index, multiplicative degree-Kirchhoff index and spanning trees of the linear crossed polyomino chains
Let be a linear crossed polyomino chain with four-order complete
graphs. In this paper, explicit formulas for the Kirchhoff index, the
multiplicative degree-Kirchhoff index and the number of spanning trees of
are determined, respectively. It is interesting to find that the Kirchhoff
(resp. multiplicative degree-Kirchhoff) index of is approximately one
quarter of its Wiener (resp. Gutman) index. More generally, let
be the set of subgraphs obtained by deleting vertical
edges of , where . For any graph , its Kirchhoff index and number of spanning trees are
completely determined, respectively. Finally, we show that the Kirchhoff index
of is approximately one quarter of its Wiener index
Primes in the form
Let \beta be a real number. Then for almost all irrational \alpha>0 (in the
sense of Lebesgue measure)
\limsup_{x\to\infty}\pi_{\alpha,\beta}^*(x)(\log x)^2/x>=1, where
\pi_{\alpha,\beta}^*(x)={p<=x: both p and [\alpha p+\beta] are primes}
Submodular Hypergraphs: p-Laplacians, Cheeger Inequalities and Spectral Clustering
We introduce submodular hypergraphs, a family of hypergraphs that have
different submodular weights associated with different cuts of hyperedges.
Submodular hypergraphs arise in clustering applications in which higher-order
structures carry relevant information. For such hypergraphs, we define the
notion of p-Laplacians and derive corresponding nodal domain theorems and k-way
Cheeger inequalities. We conclude with the description of algorithms for
computing the spectra of 1- and 2-Laplacians that constitute the basis of new
spectral hypergraph clustering methods.Comment: A short version of this paper is presented in ICML 2018. This version
includes the definition of a sequence of eigenvalues for 1-Laplacia
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