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}

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

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

The QQ-curvature on a 4-dimensional Riemannian manifold (M,g)(M,g) with ∫MQdVg=8π2\int_MQdV_g=8\pi^2

In this paper we study the solutions of the $Q$-curvature equation on a 4-dimensional Riemannian manifold $(M,g)$ with $\int_MQdV_g=8\pi^2$, proving some sufficient conditions for the existence

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

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 $G_n$ be a linear crossed polyomino chain with $n$ 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 $G_n$ are determined, respectively. It is interesting to find that the Kirchhoff (resp. multiplicative degree-Kirchhoff) index of $G_n$ is approximately one quarter of its Wiener (resp. Gutman) index. More generally, let $\mathcal{G}^r_n$ be the set of subgraphs obtained by deleting $r$ vertical edges of $G_n$, where $0\leqslant r\leqslant n+1$. For any graph $G^r_n\in \mathcal{G}^r_{n}$, its Kirchhoff index and number of spanning trees are completely determined, respectively. Finally, we show that the Kirchhoff index of $G^r_n$ is approximately one quarter of its Wiener index

Primes in the form [αp+β][\alpha p+\beta]

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