21,577 research outputs found
A survey of statistical network models
Networks are ubiquitous in science and have become a focal point for
discussion in everyday life. Formal statistical models for the analysis of
network data have emerged as a major topic of interest in diverse areas of
study, and most of these involve a form of graphical representation.
Probability models on graphs date back to 1959. Along with empirical studies in
social psychology and sociology from the 1960s, these early works generated an
active network community and a substantial literature in the 1970s. This effort
moved into the statistical literature in the late 1970s and 1980s, and the past
decade has seen a burgeoning network literature in statistical physics and
computer science. The growth of the World Wide Web and the emergence of online
networking communities such as Facebook, MySpace, and LinkedIn, and a host of
more specialized professional network communities has intensified interest in
the study of networks and network data. Our goal in this review is to provide
the reader with an entry point to this burgeoning literature. We begin with an
overview of the historical development of statistical network modeling and then
we introduce a number of examples that have been studied in the network
literature. Our subsequent discussion focuses on a number of prominent static
and dynamic network models and their interconnections. We emphasize formal
model descriptions, and pay special attention to the interpretation of
parameters and their estimation. We end with a description of some open
problems and challenges for machine learning and statistics.Comment: 96 pages, 14 figures, 333 reference
What Is a Macrostate? Subjective Observations and Objective Dynamics
We consider the question of whether thermodynamic macrostates are objective
consequences of dynamics, or subjective reflections of our ignorance of a
physical system. We argue that they are both; more specifically, that the set
of macrostates forms the unique maximal partition of phase space which 1) is
consistent with our observations (a subjective fact about our ability to
observe the system) and 2) obeys a Markov process (an objective fact about the
system's dynamics). We review the ideas of computational mechanics, an
information-theoretic method for finding optimal causal models of stochastic
processes, and argue that macrostates coincide with the ``causal states'' of
computational mechanics. Defining a set of macrostates thus consists of an
inductive process where we start with a given set of observables, and then
refine our partition of phase space until we reach a set of states which
predict their own future, i.e. which are Markovian. Macrostates arrived at in
this way are provably optimal statistical predictors of the future values of
our observables.Comment: 15 pages, no figure
Gallavotti-Cohen-Type symmetry related to cycle decompositions for Markov chains and biochemical applications
We slightly extend the fluctuation theorem obtained in \cite{LS} for sums of
generators, considering continuous-time Markov chains on a finite state space
whose underlying graph has multiple edges and no loop. This extended frame is
suited when analyzing chemical systems. As simple corollary we derive in a
different method the fluctuation theorem of D. Andrieux and P. Gaspard for the
fluxes along the chords associated to a fundamental set of oriented cycles
\cite{AG2}.
We associate to each random trajectory an oriented cycle on the graph and we
decompose it in terms of a basis of oriented cycles. We prove a fluctuation
theorem for the coefficients in this decomposition. The resulting fluctuation
theorem involves the cycle affinities, which in many real systems correspond to
the macroscopic forces. In addition, the above decomposition is useful when
analyzing the large deviations of additive functionals of the Markov chain. As
example of application, in a very general context we derive a fluctuation
relation for the mechanical and chemical currents of a molecular motor moving
along a periodic filament.Comment: 23 pages, 5 figures. Correction
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