52 research outputs found
Private Graphon Estimation for Sparse Graphs
We design algorithms for fitting a high-dimensional statistical model to a
large, sparse network without revealing sensitive information of individual
members. Given a sparse input graph , our algorithms output a
node-differentially-private nonparametric block model approximation. By
node-differentially-private, we mean that our output hides the insertion or
removal of a vertex and all its adjacent edges. If is an instance of the
network obtained from a generative nonparametric model defined in terms of a
graphon , our model guarantees consistency, in the sense that as the number
of vertices tends to infinity, the output of our algorithm converges to in
an appropriate version of the norm. In particular, this means we can
estimate the sizes of all multi-way cuts in .
Our results hold as long as is bounded, the average degree of grows
at least like the log of the number of vertices, and the number of blocks goes
to infinity at an appropriate rate. We give explicit error bounds in terms of
the parameters of the model; in several settings, our bounds improve on or
match known nonprivate results.Comment: 36 page
Partition function zeros at first-order phase transitions: Pirogov-Sinai theory
This paper is a continuation of our previous analysis [BBCKK] of partition
functions zeros in models with first-order phase transitions and periodic
boundary conditions. Here it is shown that the assumptions under which the
results of [BBCKK] were established are satisfied by a large class of lattice
models. These models are characterized by two basic properties: The existence
of only a finite number of ground states and the availability of an appropriate
contour representation. This setting includes, for instance, the Ising, Potts
and Blume-Capel models at low temperatures. The combined results of [BBCKK] and
the present paper provide complete control of the zeros of the partition
function with periodic boundary conditions for all models in the above class.Comment: 46 pages, 2 figs; continuation of math-ph/0304007 and
math-ph/0004003, to appear in J. Statist. Phys. (special issue dedicated to
Elliott Lieb
On the Spread of Viruses on the Internet
We analyze the contact process on random graphs generated according to the preferential attachment scheme as a model for the spread of viruses in the Internet. We show that any virus with a positive rate of spread from a node to its neighbors has a non-vanishing chance of becoming epidemic. Quantitatively, we discover an interesting dichotomy: for it virus with effective spread rate λ, if the infection starts at a typical vertex, then it develops into an epidemic with probability λ^Θ ((log (1/ λ)/log log (1/ λ))), but on average the epidemic probability is λ^(Θ (1))
An theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions
We introduce and develop a theory of limits for sequences of sparse graphs
based on graphons, which generalizes both the existing theory
of dense graph limits and its extension by Bollob\'as and Riordan to sparse
graphs without dense spots. In doing so, we replace the no dense spots
hypothesis with weaker assumptions, which allow us to analyze graphs with power
law degree distributions. This gives the first broadly applicable limit theory
for sparse graphs with unbounded average degrees. In this paper, we lay the
foundations of the theory of graphons, characterize convergence, and
develop corresponding random graph models, while we prove the equivalence of
several alternative metrics in a companion paper.Comment: 44 page
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