2,123 research outputs found
Creation and Growth of Components in a Random Hypergraph Process
Denote by an -component a connected -uniform hypergraph with
edges and vertices. We prove that the expected number of
creations of -component during a random hypergraph process tends to 1 as
and tend to with the total number of vertices such that
. Under the same conditions, we also show that
the expected number of vertices that ever belong to an -component is
approximately . As an immediate
consequence, it follows that with high probability the largest -component
during the process is of size . Our results
give insight about the size of giant components inside the phase transition of
random hypergraphs.Comment: R\'{e}sum\'{e} \'{e}tend
Random trees with superexponential branching weights
We study rooted planar random trees with a probability distribution which is
proportional to a product of weight factors associated to the vertices of
the tree and depending only on their individual degrees . We focus on the
case when grows faster than exponentially with . In this case the
measures on trees of finite size converge weakly as tends to infinity
to a measure which is concentrated on a single tree with one vertex of infinite
degree. For explicit weight factors of the form with
we obtain more refined results about the approach to the infinite
volume limit.Comment: 19 page
Delayed feedback as a means of control of noise-induced motion
Time--delayed feedback is exploited for controlling noise--induced motion in
coherence resonance oscillators. Namely, under the proper choice of time delay,
one can either increase or decrease the regularity of motion. It is shown that
in an excitable system, delayed feedback can stabilize the frequency of
oscillations against variation of noise strength. Also, for fixed noise
intensity, the phenomenon of entrainment of the basic oscillation period by the
delayed feedback occurs. This allows one to steer the timescales of
noise-induced motion by changing the time delay.Comment: 4 pages, 4 figures. In the replacement file Fig. 2 and Fig. 4(b),(d)
were amended. The reason is numerical error found, that affected the
quantitative estimates of correlation time, but did not affect the main
messag
Bridging frustrated-spin-chain and spin-ladder physics: quasi-one-dimensional magnetism of BiCu2PO6
We derive and investigate the microscopic model of the quantum magnet
BiCu2PO6 using band structure calculations, magnetic susceptibility and
high-field magnetization measurements, as well as ED and DMRG techniques. The
resulting quasi-one-dimensional spin model is a two-leg AFM ladder with
frustrating next-nearest-neighbor couplings along the legs. The individual
couplings are estimated from band structure calculations and by fitting the
magnetic susceptibility with theoretical predictions, obtained using ED. The
nearest-neighbor leg coupling J1, the rung coupling J4, and one of the
next-nearest-neighbor couplings J2 amount to 120-150 K, while the second
next-nearest-neighbor coupling is J2'~J2/2. The spin ladders do not match the
structural chains, and although the next-nearest-neighbor interactions J2 and
J2' have very similar superexchange pathways, they differ substantially in
magnitude due to a tiny difference in the O-O distances and in the arrangement
of non-magnetic PO4 tetrahedra. An extensive ED study of the proposed model
provides the low-energy excitation spectrum and shows that the system is in the
strong rung coupling regime. The strong frustration by the
next-nearest-neighbor couplings leads to a triplon branch with an
incommensurate minimum. This is further corroborated by a strong-coupling
expansion up to second order in the inter-rung coupling. Based on high-field
magnetization measurements, we estimate the spin gap of 32 K and suggest the
likely presence of antisymmetric DM anisotropy and inter-ladder coupling J3. We
also provide a tentative description of the physics of BiCu2PO6 in magnetic
field, in the light of the low-energy excitation spectra and numerical
calculations based on ED and DMRG. In particular, we raise the possibility for
a rich interplay between one- and two-component Luttinger liquid phases and a
magnetization plateau at 1/2 of the saturation value
The cut metric, random graphs, and branching processes
In this paper we study the component structure of random graphs with
independence between the edges. Under mild assumptions, we determine whether
there is a giant component, and find its asymptotic size when it exists. We
assume that the sequence of matrices of edge probabilities converges to an
appropriate limit object (a kernel), but only in a very weak sense, namely in
the cut metric. Our results thus generalize previous results on the phase
transition in the already very general inhomogeneous random graph model we
introduced recently, as well as related results of Bollob\'as, Borgs, Chayes
and Riordan, all of which involve considerably stronger assumptions. We also
prove corresponding results for random hypergraphs; these generalize our
results on the phase transition in inhomogeneous random graphs with clustering.Comment: 53 pages; minor edits and references update
Diameters in preferential attachment models
In this paper, we investigate the diameter in preferential attachment (PA-)
models, thus quantifying the statement that these models are small worlds. The
models studied here are such that edges are attached to older vertices
proportional to the degree plus a constant, i.e., we consider affine PA-models.
There is a substantial amount of literature proving that, quite generally,
PA-graphs possess power-law degree sequences with a power-law exponent \tau>2.
We prove that the diameter of the PA-model is bounded above by a constant
times \log{t}, where t is the size of the graph. When the power-law exponent
\tau exceeds 3, then we prove that \log{t} is the right order, by proving a
lower bound of this order, both for the diameter as well as for the typical
distance. This shows that, for \tau>3, distances are of the order \log{t}. For
\tau\in (2,3), we improve the upper bound to a constant times \log\log{t}, and
prove a lower bound of the same order for the diameter. Unfortunately, this
proof does not extend to typical distances. These results do show that the
diameter is of order \log\log{t}.
These bounds partially prove predictions by physicists that the typical
distance in PA-graphs are similar to the ones in other scale-free random
graphs, such as the configuration model and various inhomogeneous random graph
models, where typical distances have been shown to be of order \log\log{t} when
\tau\in (2,3), and of order \log{t} when \tau>3
Ising model – an analysis, from opinions to neuronal states
Here we have developed a mathematical model of a random
neuron network with two types of neurons: inhibitory and excitatory. Every
neuron was modelled as a functional cell with three states, parallel to
hyperpolarised, neutral and depolarised states in vivo. These either induce
a signal or not into their postsynaptic partners. First a system including just
one network was simulated numerically using the software developed in
Python.
Our simulations show that under physiological initial conditions, the
neurons in the network all switch off, irrespective of the initial distribution
of states. However, with increased inhibitory connections beyond 85%,
spontaneous oscillations arise in the system. This raises the question
whether there exist pathologies where the increased amount of inhibitory
connections leads to uncontrolled neural activity. There has been
preliminary evidence elsewhere that this may be the case in autism and
down syndrome [1-4].
At the next stage we numerically studied two mutually coupled networks
through mean field interactions. We find that via a small range of coupling
constants between the networks, pulses of activity in one network are
transferred to the other. However, for high enough coupling there appears a
very sudden change in behaviour. This leads to both networks oscillating
independent of the pulses applied. These uncontrolled oscillations may also
be applied to neural pathologies, where unconnected neuronal systems in
the brain may interact via their electromagnetic fields. Any mutations or
diseases that increase how brain regions interact can induce this
pathological activity resonance.
Our simulations provided some interesting insight into neuronal behaviour,
in particular factors that lead to emergent phenomena in dynamics of neural
networks. This can be tied to pathologies, such as autism, Down's syndrome,
the synchronisation seen in parkinson's and the desynchronisation seen in
epilepsy. The model is very general and also can be applied to describe
social network and social pathologies
Powers of Hamilton cycles in pseudorandom graphs
We study the appearance of powers of Hamilton cycles in pseudorandom graphs,
using the following comparatively weak pseudorandomness notion. A graph is
-pseudorandom if for all disjoint and with and we have
. We prove that for all there is an
such that an -pseudorandom graph on
vertices with minimum degree at least contains the square of a
Hamilton cycle. In particular, this implies that -graphs with
contain the square of a Hamilton cycle, and thus
a triangle factor if is a multiple of . This improves on a result of
Krivelevich, Sudakov and Szab\'o [Triangle factors in sparse pseudo-random
graphs, Combinatorica 24 (2004), no. 3, 403--426].
We also extend our result to higher powers of Hamilton cycles and establish
corresponding counting versions.Comment: 30 pages, 1 figur
Giant strongly connected component of directed networks
We describe how to calculate the sizes of all giant connected components of a
directed graph, including the {\em strongly} connected one. Just to the class
of directed networks, in particular, belongs the World Wide Web. The results
are obtained for graphs with statistically uncorrelated vertices and an
arbitrary joint in,out-degree distribution . We show that if
does not factorize, the relative size of the giant strongly
connected component deviates from the product of the relative sizes of the
giant in- and out-components. The calculations of the relative sizes of all the
giant components are demonstrated using the simplest examples. We explain that
the giant strongly connected component may be less resilient to random damage
than the giant weakly connected one.Comment: 4 pages revtex, 4 figure
Seasonal effects on reconciliation in Macaca Fuscata Yakui
Dietary composition may have profound effects on the activity budgets, levelof food competition, and social behavior of a species. Similarly, in seasonally breeding species, the mating season is a period in which competition for mating partners increases, affecting amicable social interactions among group members. We analyzed the importance of the mating season and of seasonal variations in dietary composition and food competition on econciliation
in wild female Japanese macaques (Macaca fuscata yakui) on Yakushima Island, Japan. Yakushima macaques are appropriate subjects because they are seasonal breeders and their dietary composition significantly changes among the seasons. Though large differences occurred between the summer months and the winter and early spring months in activity budgets and the consumption of the main food sources, i.e., fruits, seeds, and leaves, the level
of food competition and conciliatory tendency remained unaffected. Conversely,conciliatory tendency is significantly lower during the mating season than in the nonmating season. Moreover, conciliatory tendency is lower when 1 or both female opponents is in estrous than when they are not. Thus the mating season has profound effects on reconciliation, whereas seasonal changes in activity budgets and dietary composition do not. The detrimental effects of the mating season on female social relationships and reconciliation may be due to the importance of female competition for access to male partners in multimale, multifemale societies
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