721 research outputs found
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() 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 . It is shown that for any network constructed from
the security model, satisfies some remarkable topological properties,
including: (i) the {\it small community phenomenon}, that is, is rich in
communities of the form of size poly logarithmic in with
conductance bounded by for some constant ,
(ii) small diameter property, with diameter allowing a navigation
by a 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
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 , the resistance of is
, where
and are the one- and two-dimensional structure entropy of
, respectively. According to this, we define the notion of {\it security
index of a graph} to be the normalized resistance, that is, . We say that a connected graph is
an -{\it resistor graph}, if has vertices and has security
index . We show that trees and grid graphs are
-resistor graphs for large constant , that the graphs with
bounded degree and vertices, are -resistor
graphs, and that for a graph generated by the security model
\cite{LLPZ2015, LP2016}, with high probability, is an -resistor
graph, for a constant arbitrarily close to , provided that 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
such that expander graphs cannot be -resistor graph
for any . In particular, for the complete graph , the
resistance of is a constant , and hence the security index of is
. Finally, we show that for any simple and connected graph
, if is an -resistor graph, then there is a large such
that the -th largest eigenvalue of the Laplacian of is , 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
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