81,624 research outputs found
Strong Amplifiers of Natural Selection: Proofs
We consider the modified Moran process on graphs to study the spread of
genetic and cultural mutations on structured populations. An initial mutant
arises either spontaneously (aka \emph{uniform initialization}), or during
reproduction (aka \emph{temperature initialization}) in a population of
individuals, and has a fixed fitness advantage over the residents of the
population. The fixation probability is the probability that the mutant takes
over the entire population. Graphs that ensure fixation probability of~1 in the
limit of infinite populations are called \emph{strong amplifiers}. Previously,
only a few examples of strong amplifiers were known for uniform initialization,
whereas no strong amplifiers were known for temperature initialization.
In this work, we study necessary and sufficient conditions for strong
amplification, and prove negative and positive results. We show that for
temperature initialization, graphs that are unweighted and/or self-loop-free
have fixation probability upper-bounded by , where is a
function linear in . Similarly, we show that for uniform initialization,
bounded-degree graphs that are unweighted and/or self-loop-free have fixation
probability upper-bounded by , where is the degree bound and
a function linear in . Our main positive result complements these
negative results, and is as follows: every family of undirected graphs with
(i)~self loops and (ii)~diameter bounded by , for some fixed
, can be assigned weights that makes it a strong amplifier, both
for uniform and temperature initialization
Quasi-isometry classification of RAAGs that split over cyclic subgroups
For a one-ended right-angled Artin group, we give an explicit description of
its JSJ tree of cylinders over infinite cyclic subgroups in terms of its
defining graph. This is then used to classify certain right-angled Artin groups
up to quasi-isometry. In particular, we show that if two right-angled Artin
groups are quasi-isometric, then their JSJ trees of cylinders are weakly
equivalent. Although the converse to this is not generally true, we define
quasi-isometry invariants known as stretch factors that can distinguish
quasi-isometry classes of RAAGs with weakly equivalence JSJ trees of cylinders.
We then show that for many right-angled Artin groups, being weakly equivalent
and having matching stretch factors is a complete quasi-isometry invariant.Comment: 49 pages, 12 figures. The class of dovetail RAAGs is introduced and
the main theorem is reformulated in terms of such RAAGs. Accepted by Groups,
Geometry, and Dynamic
Scaling and Universality in City Space Syntax: between Zipf and Matthew
We report about universality of rank-integration distributions of open spaces
in city space syntax similar to the famous rank-size distributions of cities
(Zipf's law). We also demonstrate that the degree of choice an open space
represents for other spaces directly linked to it in a city follows a power law
statistic. Universal statistical behavior of space syntax measures uncovers the
universality of the city creation mechanism. We suggest that the observed
universality may help to establish the international definition of a city as a
specific land use pattern.Comment: 24 pages, 5 *.eps figure
Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions
We evaluate the virial coefficients B_k for k<=10 for hard spheres in
dimensions D=2,...,8. Virial coefficients with k even are found to be negative
when D>=5. This provides strong evidence that the leading singularity for the
virial series lies away from the positive real axis when D>=5. Further analysis
provides evidence that negative virial coefficients will be seen for some k>10
for D=4, and there is a distinct possibility that negative virial coefficients
will also eventually occur for D=3.Comment: 33 pages, 12 figure
Universal spectral statistics in Wigner-Dyson, chiral and Andreev star graphs I: construction and numerical results
In a series of two papers we investigate the universal spectral statistics of
chaotic quantum systems in the ten known symmetry classes of quantum mechanics.
In this first paper we focus on the construction of appropriate ensembles of
star graphs in the ten symmetry classes. A generalization of the
Bohigas-Giannoni-Schmit conjecture is given that covers all these symmetry
classes. The conjecture is supported by numerical results that demonstrate the
fidelity of the spectral statistics of star graphs to the corresponding
Gaussian random-matrix theories.Comment: 15 page
- …