307 research outputs found
Degree-degree correlations in random graphs with heavy-tailed degrees
Mixing patterns in large self-organizing networks, such as the Internet, the
World Wide Web, social and biological networks are often characterized by
degree-degree {dependencies} between neighbouring nodes. One of the problems
with the commonly used Pearson's correlation coefficient (termed as the
assortativity coefficient) is that {in disassortative networks its magnitude
decreases} with the network size. This makes it impossible to compare mixing
patterns, for example, in two web crawls of different size.
We start with a simple model of two heavy-tailed highly correlated random
variable and , and show that the sample correlation coefficient
converges in distribution either to a proper random variable on , or to
zero, and if then the limit is non-negative. We next show that it is
non-negative in the large graph limit when the degree distribution has an
infinite third moment. We consider the alternative degree-degree dependency
measure, based on the Spearman's rho, and prove that it converges to an
appropriate limit under very general conditions. We verify that these
conditions hold in common network models, such as configuration model and
Preferential Attachment model. We conclude that rank correlations provide a
suitable and informative method for uncovering network mixing patterns
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
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
Ising models on power-law random graphs
We study a ferromagnetic Ising model on random graphs with a power-law degree
distribution and compute the thermodynamic limit of the pressure when the mean
degree is finite (degree exponent ), for which the random graph has a
tree-like structure. For this, we adapt and simplify an analysis by Dembo and
Montanari, which assumes finite variance degrees (). We further
identify the thermodynamic limits of various physical quantities, such as the
magnetization and the internal energy
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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 makers’ 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 professional’s 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 professionals’ perception of disease typicality is the first and important step toward a more general study of how typicality influences disease handling
Search in Complex Networks : a New Method of Naming
We suggest a method for routing when the source does not posses full
information about the shortest path to the destination. The method is
particularly useful for scale-free networks, and exploits its unique
characteristics. By assigning new (short) names to nodes (aka labelling) we are
able to reduce significantly the memory requirement at the routers, yet we
succeed in routing with high probability through paths very close in distance
to the shortest ones.Comment: 5 pages, 4 figure
Limited path percolation in complex networks
We study the stability of network communication after removal of
links under the assumption that communication is effective only if the shortest
path between nodes and after removal is shorter than where is the shortest path before removal. For a large
class of networks, we find a new percolation transition at
, where and
is the node degree. Below , only a fraction of
the network nodes can communicate, where , while above , order nodes can
communicate within the limited path length . Our analytical results
are supported by simulations on Erd\H{o}s-R\'{e}nyi and scale-free network
models. We expect our results to influence the design of networks, routing
algorithms, and immunization strategies, where short paths are most relevant.Comment: 11 pages, 3 figures, 1 tabl
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
A preferential attachment model with random initial degrees
In this paper, a random graph process is studied and its
degree sequence is analyzed. Let be an i.i.d. sequence. The
graph process is defined so that, at each integer time , a new vertex, with
edges attached to it, is added to the graph. The new edges added at time
t are then preferentially connected to older vertices, i.e., conditionally on
, the probability that a given edge is connected to vertex i is
proportional to , where is the degree of vertex
at time , independently of the other edges. The main result is that the
asymptotical degree sequence for this process is a power law with exponent
, where is the power-law exponent
of the initial degrees and the exponent predicted
by pure preferential attachment. This result extends previous work by Cooper
and Frieze, which is surveyed.Comment: In the published form of the paper, the proof of Proposition 2.1 is
incomplete. This version contains the complete proo
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
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