391 research outputs found
Random subgraphs of finite graphs: I. The scaling window under the triangle condition
We study random subgraphs of an arbitrary finite connected transitive graph
obtained by independently deleting edges with probability .
Let be the number of vertices in , and let be their
degree. We define the critical threshold to be the
value of for which the expected cluster size of a fixed vertex attains the
value , where is fixed and positive. We show that
for any such model, there is a phase transition at analogous to the phase
transition for the random graph, provided that a quantity called the triangle
diagram is sufficiently small at the threshold . In particular, we show
that the largest cluster inside a scaling window of size
|p-p_c|=\Theta(\cn^{-1}V^{-1/3}) is of size , while below
this scaling window, it is much smaller, of order
, with \epsilon=\cn(p_c-p). We also obtain
an upper bound O(\cn(p-p_c)V) for the expected size of the largest cluster
above the window. In addition, we define and analyze the percolation
probability above the window and show that it is of order \Theta(\cn(p-p_c)).
Among the models for which the triangle diagram is small enough to allow us to
draw these conclusions are the random graph, the -cube and certain Hamming
cubes, as well as the spread-out -dimensional torus for
A local limit theorem for the critical random graph
We consider the limit distribution of the orders of the k largest components in the ErdÂżos-RĂ©nyi random graph inside the critical window for arbitrary k. We prove a local limit theorem for this joint distribution and derive an exact expression for the joint probability density function
Switch chain mixing times through triangle counts
Sampling uniform simple graphs with power-law degree distributions with degree exponent Ïâ(2,3) is a non-trivial problem. We propose a method to sample uniform simple graphs that uses a constrained version of the configuration model together with a Markov Chain switching method. We test the convergence of this algorithm numerically in the context of the presence of small subgraphs. We then compare the number of triangles in uniform random graphs with the number of triangles in the erased configuration model. Using simulations and heuristic arguments, we conjecture that the number of triangles in the erased configuration model is larger than the number of triangles in the uniform random graph, provided that the graph is sufficiently large
Random graph asymptotics on high-dimensional tori. II. Volume, diameter and mixing time
For critical bond-percolation on high-dimensional torus, this paper proves
sharp lower bounds on the size of the largest cluster, removing a logarithmic
correction in the lower bound in Heydenreich and van der Hofstad (2007). This
improvement finally settles a conjecture by Aizenman (1997) about the role of
boundary conditions in critical high-dimensional percolation, and it is a key
step in deriving further properties of critical percolation on the torus.
Indeed, a criterion of Nachmias and Peres (2008) implies appropriate bounds on
diameter and mixing time of the largest clusters. We further prove that the
volume bounds apply also to any finite number of the largest clusters. The main
conclusion of the paper is that the behavior of critical percolation on the
high-dimensional torus is the same as for critical Erdos-Renyi random graphs.
In this updated version we incorporate an erratum to be published in a
forthcoming issue of Probab. Theory Relat. Fields. This results in a
modification of Theorem 1.2 as well as Proposition 3.1.Comment: 16 pages. v4 incorporates an erratum to be published in a forthcoming
issue of Probab. Theory Relat. Field
The structure of typical clusters in large sparse random configurations
The initial purpose of this work is to provide a probabilistic explanation of
a recent result on a version of Smoluchowski's coagulation equations in which
the number of aggregations is limited. The latter models the deterministic
evolution of concentrations of particles in a medium where particles coalesce
pairwise as time passes and each particle can only perform a given number of
aggregations. Under appropriate assumptions, the concentrations of particles
converge as time tends to infinity to some measure which bears a striking
resemblance with the distribution of the total population of a Galton-Watson
process started from two ancestors. Roughly speaking, the configuration model
is a stochastic construction which aims at producing a typical graph on a set
of vertices with pre-described degrees. Specifically, one attaches to each
vertex a certain number of stubs, and then join pairwise the stubs uniformly at
random to create edges between vertices. In this work, we use the configuration
model as the stochastic counterpart of Smoluchowski's coagulation equations
with limited aggregations. We establish a hydrodynamical type limit theorem for
the empirical measure of the shapes of clusters in the configuration model when
the number of vertices tends to . The limit is given in terms of the
distribution of a Galton-Watson process started with two ancestors
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What Makes Some Diseases More Typical than Others? A Survey on the Impact of Disease Characteristics and Professional Background on Disease Typicality
Health professionals tend to perceive some diseases as more typical than others. If disease typicalities have implications for health professionals or health policy makersâ handling of different diseases, then it is of great social, epistemic, and ethical interest. Accordingly, it is important to find out what makes health professionals rank diseases as more or less typical. This study investigates the impact of various factors on how typical various diseases are perceived to be by health professionals. In particular, we study the influence of broad disease categories, such as somatic versus psychological/behavioral conditions, and a wide range of more specific disease characteristics, as well as the health professionalâs own background. We find that professional background strongly impacted disease typicality. All professionals (MD, RN, physiotherapists and psychologists) considered somatic conditions to be more typical than psychological/behavioral. As expected, psychologists also found psychological/behavioral conditions to be more typical than did other groups. Professions of respondents could be well predicted from their individual typicality judgments, with the exception of physiotherapists and nurses who had very similar judgment profiles. We also demonstrate how various disease characteristics impact typicality for the different professionals. Typicality showed moderate to strong positive correlations with condition severity and mortality, and only non-severe conditions were rated as atypical. Hence, studying how different disease characteristics and occupational background influences health professionalsâ perception of disease typicality is the first and important step toward a more general study of how typicality influences disease handling
The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion
For independent nearest-neighbour bond percolation on Z^d with d >> 6, we
prove that the incipient infinite cluster's two-point function and three-point
function converge to those of integrated super-Brownian excursion (ISE) in the
scaling limit. The proof is based on an extension of the new expansion for
percolation derived in a previous paper, and involves treating the magnetic
field as a complex variable. A special case of our result for the two-point
function implies that the probability that the cluster of the origin consists
of n sites, at the critical point, is given by a multiple of n^{-3/2}, plus an
error term of order n^{-3/2-\epsilon} with \epsilon >0. This is a strong
statement that the critical exponent delta is given by delta =2.Comment: 56 pages, 3 Postscript figures, in AMS-LaTeX, with graphicx, epic,
and xr package
Mean-field driven first-order phase transitions in systems with long-range interactions
We consider a class of spin systems on with vector valued spins
(\bS_x) that interact via the pair-potentials J_{x,y} \bS_x\cdot\bS_y. The
interactions are generally spread-out in the sense that the 's exhibit
either exponential or power-law fall-off. Under the technical condition of
reflection positivity and for sufficiently spread out interactions, we prove
that the model exhibits a first-order phase transition whenever the associated
mean-field theory signals such a transition. As a consequence, e.g., in
dimensions , we can finally provide examples of the 3-state Potts model
with spread-out, exponentially decaying interactions, which undergoes a
first-order phase transition as the temperature varies. Similar transitions are
established in dimensions for power-law decaying interactions and in
high dimensions for next-nearest neighbor couplings. In addition, we also
investigate the limit of infinitely spread-out interactions. Specifically, we
show that once the mean-field theory is in a unique ``state,'' then in any
sequence of translation-invariant Gibbs states various observables converge to
their mean-field values and the states themselves converge to a product
measure.Comment: 57 pages; uses a (modified) jstatphys class fil
Moment-based parameter estimation in binomial random intersection graph models
Binomial random intersection graphs can be used as parsimonious statistical
models of large and sparse networks, with one parameter for the average degree
and another for transitivity, the tendency of neighbours of a node to be
connected. This paper discusses the estimation of these parameters from a
single observed instance of the graph, using moment estimators based on
observed degrees and frequencies of 2-stars and triangles. The observed data
set is assumed to be a subgraph induced by a set of nodes sampled from
the full set of nodes. We prove the consistency of the proposed estimators
by showing that the relative estimation error is small with high probability
for . As a byproduct, our analysis confirms that the
empirical transitivity coefficient of the graph is with high probability close
to the theoretical clustering coefficient of the model.Comment: 15 pages, 6 figure
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