123 research outputs found
Viral Marketing On Configuration Model
We consider propagation of influence on a Configuration Model, where each
vertex can be influenced by any of its neighbours but in its turn, it can only
influence a random subset of its neighbours. Our (enhanced) model is described
by the total degree of the typical vertex, representing the total number of its
neighbours and the transmitter degree, representing the number of neighbours it
is able to influence. We give a condition involving the joint distribution of
these two degrees, which if satisfied would allow with high probability the
influence to reach a non-negligible fraction of the vertices, called a big
(influenced) component, provided that the source vertex is chosen from a set of
good pioneers. We show that asymptotically the big component is essentially the
same, regardless of the good pioneer we choose, and we explicitly evaluate the
asymptotic relative size of this component. Finally, under some additional
technical assumption we calculate the relative size of the set of good
pioneers. The main technical tool employed is the "fluid limit" analysis of the
joint exploration of the configuration model and the propagation of the
influence up to the time when a big influenced component is completed. This
method was introduced in Janson & Luczak (2008) to study the giant component of
the configuration model. Using this approach we study also a reverse dynamic,
which traces all the possible sources of influence of a given vertex, and which
by a new "duality" relation allows to characterise the set of good pioneers
Stochastic Analysis of Non-slotted Aloha in Wireless Ad-Hoc Networks
In this paper we propose two analytically tractable stochastic models of
non-slotted Aloha for Mobile Ad-hoc NETworks (MANETs): one model assumes a
static pattern of nodes while the other assumes that the pattern of nodes
varies over time. Both models feature transmitters randomly located in the
Euclidean plane, according to a Poisson point process with the receivers
randomly located at a fixed distance from the emitters. We concentrate on the
so-called outage scenario, where a successful transmission requires a
Signal-to-Interference-and-Noise Ratio (SINR) larger than a given threshold.
With Rayleigh fading and the SINR averaged over the duration of the packet
transmission, both models lead to closed form expressions for the probability
of successful transmission. We show an excellent matching of these results with
simulations. Using our models we compare the performances of non-slotted Aloha
to previously studied slotted Aloha. We observe that when the path loss is not
very strong both models, when appropriately optimized, exhibit similar
performance. For stronger path loss non-slotted Aloha performs worse than
slotted Aloha, however when the path loss exponent is equal to 4 its density of
successfully received packets is still 75% of that in the slotted scheme. This
is still much more than the 50% predicted by the well-known analysis where
simultaneous transmissions are never successful. Moreover, in any path loss
scenario, both schemes exhibit the same energy efficiency.Comment: accepted for IEEE Infocom 201
Optimal Geographic Caching In Cellular Networks
In this work we consider the problem of an optimal geographic placement of
content in wireless cellular networks modelled by Poisson point processes.
Specifically, for the typical user requesting some particular content and whose
popularity follows a given law (e.g. Zipf), we calculate the probability of
finding the content cached in one of the base stations. Wireless coverage
follows the usual signal-to-interference-and noise ratio (SINR) model, or some
variants of it. We formulate and solve the problem of an optimal randomized
content placement policy, to maximize the user's hit probability. The result
dictates that it is not always optimal to follow the standard policy "cache the
most popular content, everywhere". In fact, our numerical results regarding
three different coverage scenarios, show that the optimal policy significantly
increases the chances of hit under high-coverage regime, i.e., when the
probabilities of coverage by more than just one station are high enough.Comment: 6 pages, 6 figures, conferenc
A New Phase Transition for Local Delays in MANETs
We consider Mobile Ad-hoc Network (MANET) with transmitters located according
to a Poisson point in the Euclidean plane, slotted Aloha Medium Access (MAC)
protocol and the so-called outage scenario, where a successful transmission
requires a Signal-to-Interference-and-Noise (SINR) larger than some threshold.
We analyze the local delays in such a network, namely the number of times slots
required for nodes to transmit a packet to their prescribed next-hop receivers.
The analysis depends very much on the receiver scenario and on the variability
of the fading. In most cases, each node has finite-mean geometric random delay
and thus a positive next hop throughput. However, the spatial (or large
population) averaging of these individual finite mean-delays leads to infinite
values in several practical cases, including the Rayleigh fading and positive
thermal noise case. In some cases it exhibits an interesting phase transition
phenomenon where the spatial average is finite when certain model parameters
are below a threshold and infinite above. We call this phenomenon, contention
phase transition. We argue that the spatial average of the mean local delays is
infinite primarily because of the outage logic, where one transmits full
packets at time slots when the receiver is covered at the required SINR and
where one wastes all the other time slots. This results in the "RESTART"
mechanism, which in turn explains why we have infinite spatial average.
Adaptive coding offers a nice way of breaking the outage/RESTART logic. We show
examples where the average delays are finite in the adaptive coding case,
whereas they are infinite in the outage case.Comment: accepted for IEEE Infocom 201
Far-out Vertices In Weighted Repeated Configuration Model
We consider an edge-weighted uniform random graph with a given degree
sequence (Repeated Configuration Model) which is a useful approximation for
many real-world networks. It has been observed that the vertices which are
separated from the rest of the graph by a distance exceeding certain threshold
play an important role in determining some global properties of the graph like
diameter, flooding time etc., in spite of being statistically rare. We give a
convergence result for the distribution of the number of such far-out vertices.
We also make a conjecture about how this relates to the longest edge of the
minimal spanning tree on the graph under consideration
Connectivity in Sub-Poisson Networks
We consider a class of point processes (pp), which we call {\em sub-Poisson};
these are pp that can be directionally-convexly () dominated by some
Poisson pp. The order has already been shown useful in comparing various
point process characteristics, including Ripley's and correlation functions as
well as shot-noise fields generated by pp, indicating in particular that
smaller in the order processes exhibit more regularity (less clustering,
less voids) in the repartition of their points. Using these results, in this
paper we study the impact of the ordering of pp on the properties of two
continuum percolation models, which have been proposed in the literature to
address macroscopic connectivity properties of large wireless networks. As the
first main result of this paper, we extend the classical result on the
existence of phase transition in the percolation of the Gilbert's graph (called
also the Boolean model), generated by a homogeneous Poisson pp, to the class of
homogeneous sub-Poisson pp. We also extend a recent result of the same nature
for the SINR graph, to sub-Poisson pp. Finally, as examples we show that the
so-called perturbed lattices are sub-Poisson. More generally, perturbed
lattices provide some spectrum of models that ranges from periodic grids,
usually considered in cellular network context, to Poisson ad-hoc networks, and
to various more clustered pp including some doubly stochastic Poisson ones.Comment: 8 pages, 10 figures, to appear in Proc. of Allerton 2010. For an
extended version see http://hal.inria.fr/inria-00497707 version
How user throughput depends on the traffic demand in large cellular networks
Little's law allows to express the mean user throughput in any region of the
network as the ratio of the mean traffic demand to the steady-state mean number
of users in this region. Corresponding statistics are usually collected in
operational networks for each cell. Using ergodic arguments and Palm theoretic
formalism, we show that the global mean user throughput in the network is equal
to the ratio of these two means in the steady state of the "typical cell".
Here, both means account for double averaging: over time and network geometry,
and can be related to the per-surface traffic demand, base-station density and
the spatial distribution of the SINR. This latter accounts for network
irregularities, shadowing and idling cells via cell-load equations. We validate
our approach comparing analytical and simulation results for Poisson network
model to real-network cell-measurements
Using Poisson processes to model lattice cellular networks
An almost ubiquitous assumption made in the stochastic-analytic study of the
quality of service in cellular networks is Poisson distribution of base
stations. It is usually justified by various irregularities in the real
placement of base stations, which ideally should form the hexagonal pattern. We
provide a different and rigorous argument justifying the Poisson assumption
under sufficiently strong log-normal shadowing observed in the network, in the
evaluation of a natural class of the typical-user service-characteristics
including its SINR. Namely, we present a Poisson-convergence result for a broad
range of stationary (including lattice) networks subject to log-normal
shadowing of increasing variance. We show also for the Poisson model that the
distribution of all these characteristics does not depend on the particular
form of the additional fading distribution. Our approach involves a mapping of
2D network model to 1D image of it "perceived" by the typical user. For this
image we prove our convergence result and the invariance of the Poisson limit
with respect to the distribution of the additional shadowing or fading.
Moreover, we present some new results for Poisson model allowing one to
calculate the distribution function of the SINR in its whole domain. We use
them to study and optimize the mean energy efficiency in cellular networks
Limit theory for geometric statistics of point processes having fast decay of correlations
Let be a simple,stationary point process having fast decay of
correlations, i.e., its correlation functions factorize up to an additive error
decaying faster than any power of the separation distance. Let be its restriction to windows . We consider the statistic where denotes a score function
representing the interaction of with respect to . When depends
on local data in the sense that its radius of stabilization has an exponential
tail, we establish expectation asymptotics, variance asymptotics, and CLT for
and, more generally, for statistics of the re-scaled, possibly
signed, -weighted point measures , as . This gives the
limit theory for non-linear geometric statistics (such as clique counts,
intrinsic volumes of the Boolean model, and total edge length of the
-nearest neighbors graph) of -determinantal point processes having
fast decreasing kernels extending the CLTs of Soshnikov (2002) to non-linear
statistics. It also gives the limit theory for geometric U-statistics of
-permanental point processes and the zero set of Gaussian entire
functions, extending the CLTs of Nazarov and Sodin (2012) and Shirai and
Takahashi (2003), which are also confined to linear statistics. The proof of
the central limit theorem relies on a factorial moment expansion originating in
Blaszczyszyn (1995), Blaszczyszyn, Merzbach, Schmidt (1997) to show the fast
decay of the correlations of -weighted point measures. The latter property
is shown to imply a condition equivalent to Brillinger mixing and consequently
yields the CLT for via an extension of the cumulant method.Comment: 62 pages. Fundamental changes to the terminology including the title.
The earlier 'clustering' condition is now introduced as a notion of mixing
and its connection to Brillinger mixing is remarked. Newer results for
superposition of independent point processes have been adde
Pioneers of Influence Propagation in Social Networks
With the growing importance of corporate viral marketing campaigns on online
social networks, the interest in studies of influence propagation through
networks is higher than ever. In a viral marketing campaign, a firm initially
targets a small set of pioneers and hopes that they would influence a sizeable
fraction of the population by diffusion of influence through the network. In
general, any marketing campaign might fail to go viral in the first try. As
such, it would be useful to have some guide to evaluate the effectiveness of
the campaign and judge whether it is worthy of further resources, and in case
the campaign has potential, how to hit upon a good pioneer who can make the
campaign go viral. In this paper, we present a diffusion model developed by
enriching the generalized random graph (a.k.a. configuration model) to provide
insight into these questions. We offer the intuition behind the results on this
model, rigorously proved in Blaszczyszyn & Gaurav(2013), and illustrate them
here by taking examples of random networks having prototypical degree
distributions - Poisson degree distribution, which is commonly used as a kind
of benchmark, and Power Law degree distribution, which is normally used to
approximate the real-world networks. On these networks, the members are assumed
to have varying attitudes towards propagating the information. We analyze three
cases, in particular - (1) Bernoulli transmissions, when a member influences
each of its friend with probability p; (2) Node percolation, when a member
influences all its friends with probability p and none with probability 1-p;
(3) Coupon-collector transmissions, when a member randomly selects one of his
friends K times with replacement. We assume that the configuration model is the
closest approximation of a large online social network, when the information
available about the network is very limited. The key insight offered by this
study from a firm's perspective is regarding how to evaluate the effectiveness
of a marketing campaign and do cost-benefit analysis by collecting relevant
statistical data from the pioneers it selects. The campaign evaluation
criterion is informed by the observation that if the parameters of the
underlying network and the campaign effectiveness are such that the campaign
can indeed reach a significant fraction of the population, then the set of good
pioneers also forms a significant fraction of the population. Therefore, in
such a case, the firms can even adopt the naive strategy of repeatedly picking
and targeting some number of pioneers at random from the population. With this
strategy, the probability of them picking a good pioneer will increase
geometrically fast with the number of tries
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