83 research outputs found
Kramers-Wannier Duality and Random Bond Ising Model
We present a new combinatorial approach to the Ising model incorporating
arbitrary bond weights on planar graphs. In contrast to existing methodologies,
the exact free energy is expressed as the determinant of a set of ordered and
disordered operators defined on vertices and dual vertices respectively,
thereby explicitly demonstrating the Kramers-Wannier duality. The implications
of our derived formula for the random bond Ising model are further elucidated
Quantum Geometry of Expectation Values
We propose a novel framework for the quantum geometry of expectation values
over arbitrary sets of operators and establish a link between this geometry and
the eigenstates of Hamiltonian families generated by these operators. We show
that the boundary of expectation value space corresponds to the ground state,
which presents a natural bound that generalizes Heisenberg's uncertainty
principle. To demonstrate the versatility of our framework, we present several
practical applications, including providing a stronger nonlinear quantum bound
that violates the Bell inequality and an explicit construction of the density
functional. Our approach provides an alternative time-independent quantum
formulation that transforms the linear problem in a high-dimensional Hilbert
space into a nonlinear algebro-geometric problem in a low dimension, enabling
us to gain new insights into quantum systems
Theory of random packings
We review a recently proposed theory of random packings. We describe the
volume fluctuations in jammed matter through a volume function, amenable to
analytical and numerical calculations. We combine an extended statistical
mechanics approach 'a la Edwards' (where the role traditionally played by the
energy and temperature in thermal systems is substituted by the volume and
compactivity) with a constraint on mechanical stability imposed by the
isostatic condition. We show how such approaches can bring results that can be
compared to experiments and allow for an exploitation of the statistical
mechanics framework. The key result is the use of a relation between the local
Voronoi volume of the constituent grains and the number of neighbors in contact
that permits a simple combination of the two approaches to develop a theory of
random packings. We predict the density of random loose packing (RLP) and
random close packing (RCP) in close agreement with experiments and develop a
phase diagram of jammed matter that provides a unifying view of the disordered
hard sphere packing problem and further shedding light on a diverse spectrum of
data, including the RLP state. Theoretical results are well reproduced by
numerical simulations that confirm the essential role played by friction in
determining both the RLP and RCP limits. Finally we present an extended
discussion on the existence of geometrical and mechanical coordination numbers
and how to measure both quantities in experiments and computer simulations.Comment: 9 pages, 5 figures. arXiv admin note: text overlap with
arXiv:0808.219
From Micro to Macro: Uncovering and Predicting Information Cascading Process with Behavioral Dynamics
Cascades are ubiquitous in various network environments. How to predict these
cascades is highly nontrivial in several vital applications, such as viral
marketing, epidemic prevention and traffic management. Most previous works
mainly focus on predicting the final cascade sizes. As cascades are typical
dynamic processes, it is always interesting and important to predict the
cascade size at any time, or predict the time when a cascade will reach a
certain size (e.g. an threshold for outbreak). In this paper, we unify all
these tasks into a fundamental problem: cascading process prediction. That is,
given the early stage of a cascade, how to predict its cumulative cascade size
of any later time? For such a challenging problem, how to understand the micro
mechanism that drives and generates the macro phenomenons (i.e. cascading
proceese) is essential. Here we introduce behavioral dynamics as the micro
mechanism to describe the dynamic process of a node's neighbors get infected by
a cascade after this node get infected (i.e. one-hop subcascades). Through
data-driven analysis, we find out the common principles and patterns lying in
behavioral dynamics and propose a novel Networked Weibull Regression model for
behavioral dynamics modeling. After that we propose a novel method for
predicting cascading processes by effectively aggregating behavioral dynamics,
and propose a scalable solution to approximate the cascading process with a
theoretical guarantee. We extensively evaluate the proposed method on a large
scale social network dataset. The results demonstrate that the proposed method
can significantly outperform other state-of-the-art baselines in multiple tasks
including cascade size prediction, outbreak time prediction and cascading
process prediction.Comment: 10 pages, 11 figure
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