13,482 research outputs found
Analysis of heat kernel highlights the strongly modular and heat-preserving structure of proteins
In this paper, we study the structure and dynamical properties of protein
contact networks with respect to other biological networks, together with
simulated archetypal models acting as probes. We consider both classical
topological descriptors, such as the modularity and statistics of the shortest
paths, and different interpretations in terms of diffusion provided by the
discrete heat kernel, which is elaborated from the normalized graph Laplacians.
A principal component analysis shows high discrimination among the network
types, either by considering the topological and heat kernel based vector
characterizations. Furthermore, a canonical correlation analysis demonstrates
the strong agreement among those two characterizations, providing thus an
important justification in terms of interpretability for the heat kernel.
Finally, and most importantly, the focused analysis of the heat kernel provides
a way to yield insights on the fact that proteins have to satisfy specific
structural design constraints that the other considered networks do not need to
obey. Notably, the heat trace decay of an ensemble of varying-size proteins
denotes subdiffusion, a peculiar property of proteins
Alternative approach to community detection in networks
The problem of community detection is relevant in many disciplines of science
and modularity optimization is the widely accepted method for this purpose. It
has recently been shown that this approach presents a resolution limit by which
it is not possible to detect communities with sizes smaller than a threshold
which depends on the network size. Moreover, it might happen that the
communities resulting from such an approach do not satisfy the usual
qualitative definition of commune, i.e., nodes in a commune are more connected
among themselves than to nodes outside the commune. In this article we
introduce a new method for community detection in complex networks. We define
new merit factors based on the weak and strong community definitions formulated
by Radicchi et al (Proc. Nat. Acad. Sci. USA 101, 2658-2663 (2004)) and we show
that this local definitions avoid the resolution limit problem found in the
modularity optimization approach.Comment: 17 pages, 6 figure
The Shimura-Taniyama Conjecture and Conformal Field Theory
The Shimura-Taniyama conjecture states that the Mellin transform of the
Hasse-Weil L-function of any elliptic curve defined over the rational numbers
is a modular form. Recent work of Wiles, Taylor-Wiles and
Breuil-Conrad-Diamond-Taylor has provided a proof of this longstanding
conjecture. Elliptic curves provide the simplest framework for a class of
Calabi-Yau manifolds which have been conjectured to be exactly solvable. It is
shown that the Hasse-Weil modular form determined by the arithmetic structure
of the Fermat type elliptic curve is related in a natural way to a modular form
arising from the character of a conformal field theory derived from an affine
Kac-Moody algebra
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