4,354 research outputs found
A Topological Framework for the Computation of the HOMFLY Polynomial and Its Application to Proteins
Polymers can be modeled as open polygonal paths and their closure generates
knots. Knotted proteins detection is currently achieved via high-throughput
methods based on a common framework insensitive to the handedness of knots.
Here we propose a topological framework for the computation of the HOMFLY
polynomial, an handedness-sensitive invariant. Our approach couples a
multi-component reduction scheme with the polynomial computation. After
validation on tabulated knots and links the framework was applied to the entire
Protein Data Bank along with a set of selected topological checks that allowed
to discard artificially entangled structures. This led to an up-to-date table
of knotted proteins that also includes two newly detected right-handed trefoil
knots in recently deposited protein structures. The application range of our
framework is not limited to proteins and it can be extended to the topological
analysis of biological and synthetic polymers and more generally to arbitrary
polygonal paths.Comment: 20 pages, 7 figure
Hyperbolic Geometry of Complex Networks
We develop a geometric framework to study the structure and function of
complex networks. We assume that hyperbolic geometry underlies these networks,
and we show that with this assumption, heterogeneous degree distributions and
strong clustering in complex networks emerge naturally as simple reflections of
the negative curvature and metric property of the underlying hyperbolic
geometry. Conversely, we show that if a network has some metric structure, and
if the network degree distribution is heterogeneous, then the network has an
effective hyperbolic geometry underneath. We then establish a mapping between
our geometric framework and statistical mechanics of complex networks. This
mapping interprets edges in a network as non-interacting fermions whose
energies are hyperbolic distances between nodes, while the auxiliary fields
coupled to edges are linear functions of these energies or distances. The
geometric network ensemble subsumes the standard configuration model and
classical random graphs as two limiting cases with degenerate geometric
structures. Finally, we show that targeted transport processes without global
topology knowledge, made possible by our geometric framework, are maximally
efficient, according to all efficiency measures, in networks with strongest
heterogeneity and clustering, and that this efficiency is remarkably robust
with respect to even catastrophic disturbances and damages to the network
structure
A framework for proving the self-organization of dynamic systems
This paper aims at providing a rigorous definition of self- organization, one
of the most desired properties for dynamic systems (e.g., peer-to-peer systems,
sensor networks, cooperative robotics, or ad-hoc networks). We characterize
different classes of self-organization through liveness and safety properties
that both capture information re- garding the system entropy. We illustrate
these classes through study cases. The first ones are two representative P2P
overlays (CAN and Pas- try) and the others are specific implementations of
\Omega (the leader oracle) and one-shot query abstractions for dynamic
settings. Our study aims at understanding the limits and respective power of
existing self-organized protocols and lays the basis of designing robust
algorithm for dynamic systems
MAP: Medial Axis Based Geometric Routing in Sensor Networks
One of the challenging tasks in the deployment of dense wireless networks (like sensor networks) is in devising a routing scheme for node to node communication. Important consideration includes scalability, routing complexity, the length of the communication paths and the load sharing of the routes. In this paper, we show that a compact and expressive abstraction of network connectivity by the medial axis enables efficient and localized routing. We propose MAP, a Medial Axis based naming and routing Protocol that does not require locations, makes routing decisions locally, and achieves good load balancing. In its preprocessing phase, MAP constructs the medial axis of the sensor field, defined as the set of nodes with at least two closest boundary nodes. The medial axis of the network captures both the complex geometry and non-trivial topology of the sensor field. It can be represented compactly by a graph whose size is comparable with the complexity of the geometric features (e.g., the number of holes). Each node is then given a name related to its position with respect to the medial axis. The routing scheme is derived through local decisions based on the names of the source and destination nodes and guarantees delivery with reasonable and natural routes. We show by both theoretical analysis and simulations that our medial axis based geometric routing scheme is scalable, produces short routes, achieves excellent load balancing, and is very robust to variations in the network model
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