Section: New Results

Network Science

Participants : Eitan Altman, Konstantin Avrachenkov, Arun Kadavankandy, Jithin Kazhuthuveettil Sreedharan, Hlib Mykhailenko, Philippe Nain, Giovanni Neglia, Yonathan Portilla, Alexandre Reiffers-Masson.

Posting behavior in Social Networks and Content Active Filtering

In [57] , Alexandre Reiffers-Masson and Eitan Altman in collaboration with Yezekael Hayel (UAPV), model the posting behavior in Social Networks in topics which have negative externalities, and propose content active filtering in order to increase content diversity. By negative externalities, it is meant that when the quantity of posted contents about some topic increases the popularity of posted contents decreases. They introduce a dynamical model to describe the posting behavior of users taking into account these externalities. Their model is based on stochastic approximations and sufficient conditions are provided to ensure its convergence to a unique rest point. They provide a closed form expression for this rest point. Content Active Filtering (CAF) are actions taken by the administrator of the Social Network in order to promote some objectives related to the quantity of contents posted in various topics. As objective of the CAF they consider maximizing the diversity of posted contents.

Network centrality measures

Recent papers studied the control of spectral centrality measures of a network by manipulating the topology of the network. In [56] , Alexandre Reiffers-Masson, Eitan Altman and Yezekael Hayel (UAPV) extend these works by focusing on a specific spectral centrality measure, the Katz-Bonacich centrality. The optimization of the Katz-Bonacich centrality using a topological control is called the Katz-Bonacich optimization problem. The authors first prove that this problem is equivalent to a linear optimization problem. Thus, in the context of large graphs, one can use state-of-the-art algorithms. The authors provide a specific applications of the Katz-Bonacich centrality minimization problem based on the minimization of gossip propagation and make some experiments on real networks which validate the model assumptions.

Betweenness centrality is one of the basic concepts in the analysis of social networks. The initial definition for the betweenness of a node in a graph is based on the fraction of the number of geodesics (shortest paths) between any two nodes that this given node lies on, to the total number of the shortest paths connecting these nodes. This method has quadratic complexity and does not take into account indirect paths. In [45] K. Avrachenkov in collaboration with V. Mazalov (Korelian Institute of Applied Mathematical Research, Russia) and B. Tsynguev (Transbaikal State Univ., Russia) propose a new concept of betweenness centrality for weighted networks, called beta current flow centrality, based on Kirchhoff's law for electric circuits. In comparison with the original current flow centrality and alpha current flow centrality, this new measure can be computed for larger networks. The results of numerical experiments for some examples of networks, in particular, for the popular social network VKontakte as well as the comparison with PageRank method are presented.

PageRank has numerous applications in information retrieval, reputation systems, machine learning, and graph partitioning. In [44] , K. Avrachenkov and A. Kadavankandy in collaboration with L.O. Prokhorenkova and A. Raigorodskii (both from Yandex Research) study PageRank in undirected random graphs with expansion property. The Chung-Lu random graph represents an example of such graphs. The authors show that in the limit, as the size of the graph goes to infinity, PageRank can be represented by a mixture of the restart distribution and the vertex degree distribution.

Mining social networks

Social Networks became a major actor in information propagation. Using the Twitter popular platform, mobile users post or relay messages from different locations. The tweet content, meaning and location show how an event-such as the bursty one "JeSuisCharlie" happened in France in January 2015 is comprehended in different countries. In [75] , [76] researchers from UAPV and Inria (Mohamed Morchid, Yonathan Portilla, Didier Josselin, Richard Dufour, Eitan Altman, Marc El-Beze, Jean-Valère Cossu, Georges Linarès, Alexandre Reiffers-Masson), studied clustering of the tweets according to the co-occurrence of their terms, including the country, and forecasting the probable country of a non located tweet, knowing its content. First, they present the process of collecting a large quantity of data from the Twitter website. The dataset consists of 2.189 located tweets about "Charlie", from the 7th to the 14th of January. The authors then describe an original method adapted from the Author-Topic (AT) model based on the Latent Dirichlet Allocation method (LDA). They define a homogeneous space containing both lexical content (words) and spatial information (country). During a training process on a part of the sample, the authors provide a set of clusters (topics) based on statistical relations between lexical and spatial terms. During a clustering task, they evaluate the method effectiveness on the rest of the sample that reaches up to 95% of good assignments. It shows that the model is pertinent to foresee tweet location after a learning process.

Analysis of Internet Memes

Memes have been defined by R. Dawkins as cultural phenomena that propagate through non genetic ways. In [42] , Eitan Altman and Yonathan Portilla examine three very popular Internet Memes and study their impact on the society in mediterranean countries. the authors use existing software tools (such as Google Trends) as well as tools that they develop in order to quantify the impact of the Memes on the mediterranean societies. The authors obtain quite different results with the different tools they use, which they explain based on some propagation characteristic of each one of the Memes. The analysis shows the extent to which these Memes cross borders and thus contribute to the creation of a globalized culture. The authors finally identify some of the impacts of the globalization of culture.

Trend detection in social networks using Hawkes processes

In [52] , Julio Cesar Louzada Pinto and Tijani Chahed (Telecom SudParis) in collaboration with Eitan Altman propose a new trend detection algorithm, designed to find trendy topics being disseminated in a social network. The authors assume that the broadcasts of messages in the social network is governed by a self-exciting point process, namely a Hawkes process, which takes into consideration the real broadcasting times of messages and the interaction between users and topics. The authors formally define trendiness and derive trend indices for each topic being disseminated in the social network. These indices take into consideration the time between the detection and the message broadcasts, the distance between the real broadcast intensity and the maximum expected broadcast intensity, and the social network topology. The proposed trend detection algorithm is simple and uses stochastic control techniques in order to calculate the trend indices. It is also fast and aggregates all the information of the broadcasts into a simple one-dimensional process, thus reducing its complexity and the quantity of data necessary to the detection.

Study of the Youtube recommendation system

The Youtube recommendation system is one the most important view source of a video. In [54] , Yonathan Portilla, Alexandre Reiffers-Masson, Eitan Altman in collaboration with Rachid El-Azouzi (UAPV) study the role of recommendation systems in boosting the popularity of videos. The authors first construct a graph that captures the recommendation system in Youtube and study empirically the relationship between the number of views of a video and the average number of views of the videos in its recommendation list. The authors then consider a random walker on the recommendation graph, i.e. a random user that browses through videos such that the video it chooses to watch is selected randomly among the videos in the recommendation list of the previous video it watched. The authors study the stability properties of this random process and show that the trajectory obtained does not contain cycles if the number of videos in the recommendation list is small (which is the case if the computer's screen is small).

Average consensus protocols

In [22] M. El Chamie (Univ. of Washington, USA), G. Neglia and K. Avrachenkov study the weight optimization problem for average consensus protocols by reformulating it as a Schatten norm minimization with parameter $p$. They show that as $p$ approaches infinity, the optimal solution of the Schatten norm induced problem recovers the optimal solution of the original problem. Moreover, by tuning the parameter $p$ in the proposed minimization, it is possible to trade-off the quality of the solution (i.e., the speed of convergence) for communication/computation requirements (in terms of number of messages exchanged and volume of data processed). They then propose a distributed algorithm to solve the Schatten norm minimization and show that it outperforms the other distributed weight selection methods.

Estimation techniques

The estimation of a large population's size by means of sampling procedures is a key issue in many networking scenarios. Their application domains span from RFID systems to peer-to-peer networks; from traffic analysis to wireless sensor networks; from multicast networks to WLANs. In [14] , N. Accettura (Univ. of California Berkeley, USA), G. Neglia and L. A. Grieco (Politecnico di Bari, Italy) illustrate and classify in a coherent framework the main approaches proposed so far in the computer networks literature to deal with such a problem. In particular, starting from the methodologies proposed in ecological studies since the last century, they survey their counterparts in the computer network domain, finding that many lessons can be gained from this insightful investigation. Capture-Recapture techniques are deeply analyzed to allow the reader to exactly understand their pros, cons, and applicability bounds. Finally, they discuss some open issues that deserve further investigations and could be relevant to afford estimation problems in next generation Internet.

Online social networks (OSN) contain extensive amount of information about the underlying society that is yet to be explored. One of the most feasible technique to fetch information from OSN, crawling through Application Programming Interface (API) requests, poses serious concerns over the the guarantees of the estimates. In [70] J. Sreedharan and K. Avrachenkov in collaboration with B. Ribeiro (Purdue University, USA) focus on making reliable statistical inference with limited API crawls. Based on regenerative properties of the random walks, they propose an unbiased estimator for the aggregated sum of functions over edges and proved the connection between variance of the estimator and spectral gap. In order to facilitate Bayesian inference on the true value of the estimator, they derive the approximate posterior distribution of the estimate. Later the proposed ideas are validated with numerical experiments on inference problems in real-world networks.

Percolation in multilayer networks

In [79] , P. Nain and his co-authors (S. Guha and P. Basu from Raytheon BB Technologies, D. Towsley from the Univ. of Massachusetts, C. Capar from Ericsson Research, A. Swami from the US Army Research Lab.) consider multiple networks formed by a common set of users connected via $M$ different means of connectivity, where each user (node) is active, independently, in any given network with probability $q$. They show that when $q$ exceeds a threshold $q_c\left(M\right)$, a giant connected component appears in the $M$-layer network— thereby enabling faraway users to connect using 'bridge' nodes that are active in multiple network layers, even though the individual layers may only have small disconnected islands of connectivity. They show that $q_c\left(M\right)\le \sqrt{log\left(1-p_c\right)}/\sqrt{M}$, where $p_c$ is the bond percolation threshold of the underlying connectivity graph $G$, and $q_c\left(1\right)\equiv q_c$ is its site percolation threshold. The threshold $q_c\left(M\right)$ is found explicitly when $G$ is a large random network with an arbitrary node-degree distribution and numerically for various regular lattices. Finally, an intriguingly close connection between this multilayer percolation model and the well-studied problem of site-bond percolation is revealed, in the sense that both models provide a smooth transition between the traditional site and bond percolation models. This connection is used to translate analytical approximations of the site-bond critical region developed in the 1990s, which are functions only of $p_c$ and $q_c$ of the respective lattice, to excellent general approximations of $q_c\left(M\right)$.

Extreme Value Theory for Complex Networks

In [20] J. Sreedharan and K. Avrachenkov in collaboration with N. Markovich (Institute of Control Sciences, Moscow) explore the dependence structure in the sampled sequence of complex networks. They consider randomized algorithms to sample the nodes and study extremal properties in any associated stationary sequence of characteristics of interest like node degrees, number of followers, or income of the nodes in online social networks, which satisfy two mixing conditions. Several useful extremes of the sampled sequence like the kth largest value, clusters of exceedances over a threshold, and first hitting time of a large value are investigated. The dependence and the statistics of extremes is abstracted into a single parameter that appears in extreme value theory, called the Extremal Index. The authors derive this parameter analytically and also estimate it empirically. They propose the use of the Extremal Index as a parameter to compare different sampling procedures. As a specific example, degree correlations between neighboring nodes are studied in detail with three prominent random walks as sampling techniques.

Random Matrix Theory for Complex Networks

In [68] A. Kadavankandy and K. Avrachenkov in collaboration with L. Cottatellucci (Eurecom) consider an extension of Erdős-Rényi graph known in the literature as the Stochastic Block Model (SBM). They analyze the limiting empirical distribution of the eigenvalues of the adjacency matrix of a SBM. They derive a fixed point equation for the Stieltjes transform of the limiting eigenvalue empirical distribution function (e.d.f.), concentration results on both the support of the limiting e.d.f. and the extremal eigenvalues outside the support of the limiting e.d.f. Additionally, they derive analogous results for the normalized Laplacian matrix and discuss potential applications of the general results in epidemics and random walks.

In [40] , the same authors continue with the analysis of eigenvectors of a Stochastic Block Model. The eigenvalue spectrum of the adjacency matrix of a SBM consists of two parts: a finite discrete set of dominant eigenvalues and a continuous bulk of eigenvalues. They characterize analytically the eigenvectors corresponding to the continuous part: the bulk eigenvectors. For symmetric SBM adjacency matrices, the eigenvectors are shown to satisfy two key properties. A modified spectral function of the eigenvalues, depending on the eigenvectors, converges to the eigenvalue spectrum. Its fluctuations around this limit converge to a Gaussian process different from a Brownian bridge. This latter fact disproves that the bulk eigenvectors are Haar distributed.