2,559 research outputs found
A New Stable Peer-to-Peer Protocol with Non-persistent Peers
Recent studies have suggested that the stability of peer-to-peer networks may
rely on persistent peers, who dwell on the network after they obtain the entire
file. In the absence of such peers, one piece becomes extremely rare in the
network, which leads to instability. Technological developments, however, are
poised to reduce the incidence of persistent peers, giving rise to a need for a
protocol that guarantees stability with non-persistent peers. We propose a
novel peer-to-peer protocol, the group suppression protocol, to ensure the
stability of peer-to-peer networks under the scenario that all the peers adopt
non-persistent behavior. Using a suitable Lyapunov potential function, the
group suppression protocol is proven to be stable when the file is broken into
two pieces, and detailed experiments demonstrate the stability of the protocol
for arbitrary number of pieces. We define and simulate a decentralized version
of this protocol for practical applications. Straightforward incorporation of
the group suppression protocol into BitTorrent while retaining most of
BitTorrent's core mechanisms is also presented. Subsequent simulations show
that under certain assumptions, BitTorrent with the official protocol cannot
escape from the missing piece syndrome, but BitTorrent with group suppression
does.Comment: There are only a couple of minor changes in this version. Simulation
tool is specified this time. Some repetitive figures are remove
Card shuffling and diophantine approximation
The ``overlapping-cycles shuffle'' mixes a deck of cards by moving either
the th card or the th card to the top of the deck, with probability
half each. We determine the spectral gap for the location of a single card,
which, as a function of and , has surprising behavior. For example,
suppose is the closest integer to for a fixed real
. Then for rational the spectral gap is
, while for poorly approximable irrational numbers ,
such as the reciprocal of the golden ratio, the spectral gap is
.Comment: Published in at http://dx.doi.org/10.1214/07-AAP484 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The Jammed Phase of the Biham-Middleton-Levine Traffic Model
Initially a car is placed with probability p at each site of the
two-dimensional integer lattice. Each car is equally likely to be East-facing
or North-facing, and different sites receive independent assignments. At odd
time steps, each North-facing car moves one unit North if there is a vacant
site for it to move into. At even time steps, East-facing cars move East in the
same way. We prove that when p is sufficiently close to 1 traffic is jammed, in
the sense that no car moves infinitely many times. The result extends to
several variant settings, including a model with cars moving at random times,
and higher dimensions.Comment: 15 pages, 5 figures; revised journal versio
Modular Graph Functions
In earlier work we studied features of non-holomorphic modular functions
associated with Feynman graphs for a conformal scalar field theory on a
two-dimensional torus with zero external momenta at all vertices. Such
functions, which we will refer to as modular graph functions, arise, for
example, in the low energy expansion of genus-one Type II superstring
amplitudes. We here introduce a class of single-valued elliptic multiple
polylogarithms, which are defined as elliptic functions associated with Feynman
graphs with vanishing external momenta at all but two vertices. These functions
depend on a coordinate, , on the elliptic curve and reduce to modular
graph functions when is set equal to . We demonstrate that these
single-valued elliptic multiple polylogarithms are linear combinations of
multiple polylogarithms, and that modular graph functions are sums of
single-valued elliptic multiple polylogarithms evaluated at the identity of the
elliptic curve, in both cases with rational coefficients. This insight suggests
the many interrelations between modular graph functions (a few of which were
established in earlier papers) may be obtained as a consequence of identities
involving multiple polylogarithms, and explains an earlier observation that the
coefficients of the Laurent polynomial at the cusp are given by rational
numbers times single-valued multiple zeta values.Comment: 42 pages, significant clarifications added in section 5, minor typos
corrected, and references added in version
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