Moment bounds for large autocovariance matrices under dependence

The goal of this paper is to obtain expectation bounds for the deviation of large sample autocovariance matrices from their means under weak data dependence. While the accuracy of covariance matrix estimation corresponding to independent data has been well understood, much less is known in the case of dependent data. We make a step towards filling this gap, and establish deviation bounds that depend only on the parameters controlling the "intrinsic dimension" of the data up to some logarithmic terms. Our results have immediate impacts on high dimensional time series analysis, and we apply them to high dimensional linear VAR($d$) model, vector-valued ARCH model, and a model used in Banna et al. (2016).Comment: to appear in Journal of Theoretical Probabilit

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

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}

Estimating and Forecasting the Smoking-Attributable Mortality Fraction for Both Genders Jointly in Over 60 Countries

Smoking is one of the preventable threats to human health and is a major risk factor for lung cancer, upper aero-digestive cancer, and chronic obstructive pulmonary disease. Estimating and forecasting the smoking attributable fraction (SAF) of mortality can yield insights into smoking epidemics and also provide a basis for more accurate mortality and life expectancy projection. Peto et al. (1992) proposed a method to estimate the SAF using the lung cancer mortality rate as an indicator of exposure to smoking in the population of interest. Here we use the same method to estimate the all-age SAF (ASAF) for both genders for over 60 countries. We document a strong and cross-nationally consistent pattern of the evolution of the SAF over time. We use this as the basis for a new Bayesian hierarchical model to project future male and female ASAF from over 60 countries simultaneously. This gives forecasts as well as predictive distributions that can be used to find uncertainty intervals for any quantities of interest. We assess the model using out-of-sample predictive validation, and find that it provides good forecasts and well calibrated forecast intervals

Homophyly and Randomness Resist Cascading Failure in Networks

The universal properties of power law and small world phenomenon of networks seem unavoidably obstacles for security of networking systems. Existing models never give secure networks. We found that the essence of security is the security against cascading failures of attacks and that nature solves the security by mechanisms. We proposed a model of networks by the natural mechanisms of homophyly, randomness and preferential attachment. It was shown that homophyly creates a community structure, that homophyly and randomness introduce ordering in the networks, and that homophyly creates inclusiveness and introduces rules of infections. These principles allow us to provably guarantee the security of the networks against any attacks. Our results show that security can be achieved provably by structures, that there is a tradeoff between the roles of structures and of thresholds in security engineering, and that power law and small world property are never obstacles for security of networks

Dimensions, Structures and Security of Networks

One of the main issues in modern network science is the phenomenon of cascading failures of a small number of attacks. Here we define the dimension of a network to be the maximal number of functions or features of nodes of the network. It was shown that there exist linear networks which are provably secure, where a network is linear, if it has dimension one, that the high dimensions of networks are the mechanisms of overlapping communities, that overlapping communities are obstacles for network security, and that there exists an algorithm to reduce high dimensional networks to low dimensional ones which simultaneously preserves all the network properties and significantly amplifies security of networks. Our results explore that dimension is a fundamental measure of networks, that there exist linear networks which are provably secure, that high dimensional networks are insecure, and that security of networks can be amplified by reducing dimensions.Comment: arXiv admin note: text overlap with arXiv:1310.804

Provable Security of Networks

We propose a definition of {\it security} and a definition of {\it robustness} of networks against the cascading failure models of deliberate attacks and random errors respectively, and investigate the principles of the security and robustness of networks. We propose a {\it security model} such that networks constructed by the model are provably secure against any attacks of small sizes under the cascading failure models, and simultaneously follow a power law, and have the small world property with a navigating algorithm of time complex $O(\log n)$. It is shown that for any network $G$ constructed from the security model, $G$ satisfies some remarkable topological properties, including: (i) the {\it small community phenomenon}, that is, $G$ is rich in communities of the form $X$ of size poly logarithmic in $\log n$ with conductance bounded by $O(\frac{1}{|X|^{\beta}})$ for some constant $\beta$, (ii) small diameter property, with diameter $O(\log n)$ allowing a navigation by a $O(\log n)$ time algorithm to find a path for arbitrarily given two nodes, and (iii) power law distribution, and satisfies some probabilistic and combinatorial principles, including the {\it degree priority theorem}, and {\it infection-inclusion theorem}. By using these principles, we show that a network $G$ constructed from the security model is secure for any attacks of small scales under both the uniform threshold and random threshold cascading failure models. Our security theorems show that networks constructed from the security model are provably secure against any attacks of small sizes, for which natural selections of {\it homophyly, randomness} and {\it preferential attachment} are the underlying mechanisms.Comment: arXiv admin note: text overlap with arXiv:1310.8038, arXiv:1310.804

Structure Entropy and Resistor Graphs

We propose the notion of {\it resistance of a graph} as an accompanying notion of the structure entropy to measure the force of the graph to resist cascading failure of strategic virus attacks. We show that for any connected network $G$, the resistance of $G$ is $\mathcal{R}(G)=\mathcal{H}^1(G)-\mathcal{H}^2(G)$, where $\mathcal{H}^1(G)$ and $\mathcal{H}^2(G)$ are the one- and two-dimensional structure entropy of $G$, respectively. According to this, we define the notion of {\it security index of a graph} to be the normalized resistance, that is, $\theta (G)=\frac{\mathcal{R}(G)}{\mathcal{H}^1(H)}$. We say that a connected graph is an $(n,\theta)$-{\it resistor graph}, if $G$ has $n$ vertices and has security index $\theta (G)\geq\theta$. We show that trees and grid graphs are $(n,\theta)$-resistor graphs for large constant $\theta$, that the graphs with bounded degree $d$ and $n$ vertices, are $(n,\frac{2}{d}-o(1))$-resistor graphs, and that for a graph $G$ generated by the security model \cite{LLPZ2015, LP2016}, with high probability, $G$ is an $(n,\theta)$-resistor graph, for a constant $\theta$ arbitrarily close to $1$, provided that $n$ is sufficiently large. To the opposite side, we show that expander graphs are not good resistor graphs, in the sense that, there is a global constant $\theta_0<1$ such that expander graphs cannot be $(n,\theta)$-resistor graph for any $\theta\geq\theta_0$. In particular, for the complete graph $G$, the resistance of $G$ is a constant $O(1)$, and hence the security index of $G$ is $\theta (G)=o(1)$. Finally, we show that for any simple and connected graph $G$, if $G$ is an $(n, 1-o(1))$-resistor graph, then there is a large $k$ such that the $k$-th largest eigenvalue of the Laplacian of $G$ is $o(1)$, giving rise to an algebraic characterization for the graphs that are secure against intentional virus attack

Accounting for Smoking in Forecasting Mortality and Life Expectancy

Smoking is one of the main risk factors that has affected human mortality and life expectancy over the past century. Smoking accounts for a large part of the nonlinearities in the growth of life expectancy and of the geographic and sex differences in mortality. As Bongaarts (2006) and Janssen (2018) suggested, accounting for smoking could improve the quality of mortality forecasts due to the predictable nature of the smoking epidemic. We propose a new Bayesian hierarchical model to forecast life expectancy at birth for both sexes and for 69 countries with good data on smoking-related mortality. The main idea is to convert the forecast of the non-smoking life expectancy at birth (i.e., life expectancy at birth removing the smoking effect) into life expectancy forecast through the use of the age-specific smoking attributable fraction (ASSAF). We introduce a new age-cohort model for the ASSAF and a Bayesian hierarchical model for non-smoking life expectancy at birth. The forecast performance of the proposed method is evaluated by out-of-sample validation compared with four other commonly used methods for life expectancy forecasting. Improvements in forecast accuracy and model calibration based on the new method are observed