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

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

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 ($dcx$) dominated by some Poisson pp. The $dcx$ 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 $dcx$ 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 $dcx$ 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

What frequency bandwidth to run cellular network in a given country? - a downlink dimensioning problem

We propose an analytic approach to the frequency bandwidth dimensioning problem, faced by cellular network operators who deploy/upgrade their networks in various geographical regions (countries) with an inhomogeneous urbanization. We present a model allowing one to capture fundamental relations between users' quality of service parameters (mean downlink throughput), traffic demand, the density of base station deployment, and the available frequency bandwidth. These relations depend on the applied cellular technology (3G or 4G impacting user peak bit-rate) and on the path-loss characteristics observed in different (urban, sub-urban and rural) areas. We observe that if the distance between base stations is kept inversely proportional to the distance coefficient of the path-loss function, then the performance of the typical cells of these different areas is similar when serving the same (per-cell) traffic demand. In this case, the frequency bandwidth dimensioning problem can be solved uniformly across the country applying the mean cell approach proposed in [Blaszczyszyn et al. WiOpt2014] http://dx.doi.org/10.1109/WIOPT.2014.6850355 . We validate our approach by comparing the analytical results to measurements in operational networks in various geographical zones of different countries

Clustering and percolation of point processes

We are interested in phase transitions in certain percolation models on point processes and their dependence on clustering properties of the point processes. We show that point processes with smaller void probabilities and factorial moment measures than the stationary Poisson point process exhibit non-trivial phase transition in the percolation of some coverage models based on level-sets of additive functionals of the point process. Examples of such point processes are determinantal point processes, some perturbed lattices, and more generally, negatively associated point processes. Examples of such coverage models are $k$-coverage in the Boolean model (coverage by at least $k$ grains) and SINR-coverage (coverage if the signal-to-interference-and-noise ratio is large). In particular, we answer in affirmative the hypothesis of existence of phase transition in the percolation of $k$-faces in the \v{C}ech simplicial complex (called also clique percolation) on point processes which cluster less than the Poisson process. We also construct a Cox point process, which is "more clustered" than the Poisson point process and whose Boolean model percolates for arbitrarily small radius. This shows that clustering (at least, as detected by our specific tools) does not always "worsen" percolation, as well as that upper-bounding this clustering by a Poisson process is a consequential assumption for the phase transition to hold.Comment: 25 pages, 1 figure. This paper complements arXiv:1111.6017. arXiv admin note: substantial text overlap with arXiv:1105.429

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

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

On comparison of clustering properties of point processes

In this paper, we propose a new comparison tool for spatial homogeneity of point processes, based on the joint examination of void probabilities and factorial moment measures. We prove that determinantal and permanental processes, as well as, more generally, negatively and positively associated point processes are comparable in this sense to the Poisson point process of the same mean measure. We provide some motivating results and preview further ones, showing that the new tool is relevant in the study of macroscopic, percolative properties of point processes. This new comparison is also implied by the directionally convex ($dcx$ ordering of point processes, which has already been shown to be relevant to comparison of spatial homogeneity of point processes. For this latter ordering, using a notion of lattice perturbation, we provide a large monotone spectrum of comparable point processes, ranging from periodic grids to Cox processes, and encompassing Poisson point process as well. They are intended to serve as a platform for further theoretical and numerical studies of clustering, as well as simple models of random point patterns to be used in applications where neither complete regularity northe total independence property are not realistic assumptions.Comment: 23 pages, 1 figure. This submission revisits and adds to ideas concerning clustering and $dcx$ ordering presented in arXiv:1105.4293. Results on associated point process in Section 3.3 are new. arXiv admin note: substantial text overlap with arXiv:1105.429