807 research outputs found
Adaptive Replication in Distributed Content Delivery Networks
We address the problem of content replication in large distributed content
delivery networks, composed of a data center assisted by many small servers
with limited capabilities and located at the edge of the network. The objective
is to optimize the placement of contents on the servers to offload as much as
possible the data center. We model the system constituted by the small servers
as a loss network, each loss corresponding to a request to the data center.
Based on large system / storage behavior, we obtain an asymptotic formula for
the optimal replication of contents and propose adaptive schemes related to
those encountered in cache networks but reacting here to loss events, and
faster algorithms generating virtual events at higher rate while keeping the
same target replication. We show through simulations that our adaptive schemes
outperform significantly standard replication strategies both in terms of loss
rates and adaptation speed.Comment: 10 pages, 5 figure
Non-Backtracking Spectrum of Degree-Corrected Stochastic Block Models
Motivated by community detection, we characterise the spectrum of the
non-backtracking matrix in the Degree-Corrected Stochastic Block Model.
Specifically, we consider a random graph on vertices partitioned into two
equal-sized clusters. The vertices have i.i.d. weights
with second moment . The intra-cluster connection probability for
vertices and is and the inter-cluster
connection probability is .
We show that with high probability, the following holds: The leading
eigenvalue of the non-backtracking matrix is asymptotic to . The second eigenvalue is asymptotic to when , but asymptotically bounded by
when . All the remaining eigenvalues are
asymptotically bounded by . As a result, a clustering
positively-correlated with the true communities can be obtained based on the
second eigenvector of in the regime where
In a previous work we obtained that detection is impossible when meaning that there occurs a phase-transition in the sparse regime of the
Degree-Corrected Stochastic Block Model.
As a corollary, we obtain that Degree-Corrected Erd\H{o}s-R\'enyi graphs
asymptotically satisfy the graph Riemann hypothesis, a quasi-Ramanujan
property.
A by-product of our proof is a weak law of large numbers for
local-functionals on Degree-Corrected Stochastic Block Models, which could be
of independent interest
Planting trees in graphs, and finding them back
In this paper we study detection and reconstruction of planted structures in
Erd\H{o}s-R\'enyi random graphs. Motivated by a problem of communication
security, we focus on planted structures that consist in a tree graph. For
planted line graphs, we establish the following phase diagram. In a low density
region where the average degree of the initial graph is below some
critical value , detection and reconstruction go from impossible
to easy as the line length crosses some critical value ,
where is the number of nodes in the graph. In the high density region
, detection goes from impossible to easy as goes from
to , and reconstruction remains impossible so
long as . For -ary trees of varying depth and ,
we identify a low-density region , such that the following
holds. There is a threshold with the following properties.
Detection goes from feasible to impossible as crosses . We also show
that only partial reconstruction is feasible at best for . We
conjecture a similar picture to hold for -ary trees as for lines in the
high-density region , but confirm only the following part of
this picture: Detection is easy for -ary trees of size ,
while at best only partial reconstruction is feasible for -ary trees of any
size . These results are in contrast with the corresponding picture for
detection and reconstruction of {\em low rank} planted structures, such as
dense subgraphs and block communities: We observe a discrepancy between
detection and reconstruction, the latter being impossible for a wide range of
parameters where detection is easy. This property does not hold for previously
studied low rank planted structures
Self-Organizing Flows in Social Networks
Social networks offer users new means of accessing information, essentially
relying on "social filtering", i.e. propagation and filtering of information by
social contacts. The sheer amount of data flowing in these networks, combined
with the limited budget of attention of each user, makes it difficult to ensure
that social filtering brings relevant content to the interested users. Our
motivation in this paper is to measure to what extent self-organization of the
social network results in efficient social filtering. To this end we introduce
flow games, a simple abstraction that models network formation under selfish
user dynamics, featuring user-specific interests and budget of attention. In
the context of homogeneous user interests, we show that selfish dynamics
converge to a stable network structure (namely a pure Nash equilibrium) with
close-to-optimal information dissemination. We show in contrast, for the more
realistic case of heterogeneous interests, that convergence, if it occurs, may
lead to information dissemination that can be arbitrarily inefficient, as
captured by an unbounded "price of anarchy". Nevertheless the situation differs
when users' interests exhibit a particular structure, captured by a metric
space with low doubling dimension. In that case, natural autonomous dynamics
converge to a stable configuration. Moreover, users obtain all the information
of interest to them in the corresponding dissemination, provided their budget
of attention is logarithmic in the size of their interest set
A spectral method for community detection in moderately-sparse degree-corrected stochastic block models
We consider community detection in Degree-Corrected Stochastic Block Models
(DC-SBM). We propose a spectral clustering algorithm based on a suitably
normalized adjacency matrix. We show that this algorithm consistently recovers
the block-membership of all but a vanishing fraction of nodes, in the regime
where the lowest degree is of order log or higher. Recovery succeeds even
for very heterogeneous degree-distributions. The used algorithm does not rely
on parameters as input. In particular, it does not need to know the number of
communities
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