Section: New Results

Network Science

Participants : Eitan Altman, Konstantin Avrachenkov, Mahmoud El Chamie, Julien Gaillard, Arun Kadavankandy, Jithin Kazhuthuveettil Sreedharan, Hlib Mykhailenko, Philippe Nain, Giovanni Neglia, Yonathan Portilla, Alexandre Reiffers, Vikas Singh, Marina Sokol.

Epidemic models of propagation of content

Epidemic models have received significant attention in the past few decades to study the propagation of viruses, worms and ideas in computer and social networks. In the case of viruses, the goal is to understand how the topology of the network and the properties of its nodes impact the spread of the epidemics. In [38] , E. Altman, A. Avritzer and L. Pfleger de Aguiar (Siemens Corporation, Princeton, USA), R. El-Azouzi (Univ. of Avignon), and D. S. Menasche (Federal Univ. of Rio de Janeiro, Brazil) propose rejuvenation as a way to cope with epidemics. Reformatting a computer may solve the problem of virus contamination (but it might be a costly operation) while less dramatic actions may render the computer operational again (even in the presence of the virus). In this work they evaluate the performance gain of such measures as well as sampling for early detection of viruses while these incubate. During incubation, contaminated terminals are infectious and yet, if not detected to be so, they cannot be isolated and treated.

In [60] , Y. Hayel (Univ. of Avignon), S. Trajanovski and P. Van Mieghem (Delft Univ. of Technology, The Netherlands), E. Altman, and H. Wang (Delft Institute of Applied Mathematics, The Netherlands), compare solutions involving vaccination to those that involve healing from a selfish point of view of an individual networked user. A game theoretical model is presented and the obtained equilibrium is computed for various types of topologies including the fully connected one, the bipartite graph and a community structure. A novel use of potential games is presented to compute the equilibria.

In [61] , L. Maggi and F. De Pellegrini (Create-Net , Italy), A. Reiffers, J. J. Herings (Maastricht Univ., The Netherlands) and E. Altman, study a viral diffusion of a content in a multi-community environment. Exploiting time scale separation, the authors are able to reduce the dimensionality of the problem and to compute its limiting behavior in closed form. They further study regulation and cooperative approaches for sharing the cost for fighting the spread of the infection among the communities.

Social networks can have asymmetric relationships. In the online social network Twitter, a follower receives tweets from a followed person but the followed person is not obliged to subscribe to the channel of the follower. Thus, it is natural to consider the dissemination of information in directed networks. In [44] , K. Avrachenkov in collaboration with B. Prabhu (LAAS-CNRS), K. De Turck and D. Fiems (Ghent Univ., Belgium) use the mean-field approach to derive differential equations that describe the dissemination of information in a social network with asymmetric relationships. In particular, their model reflects the impact of the degree distribution on the information propagation process. They further show that for an important subclass of their model, the differential equations can be solved analytically.

Bio-Inspired Models for Characterizing YouTube Viewcount

Bio-inspired models have long been advocated for the dissemination of content in the Internet. How good are such models and how representative are they? In [69] , C. Richier, R. El-Azouzi, T. Jimenez, G. Linares (all with Univ. of Avignon), E. Altman and Y. Portilla propose six different epidemic models. These are classified according to various criteria: (i) the size of the target population, which may be constant, or linearly increasing or infinite, (ii) the virality of the content: it is said to be viral if nodes that receive the content participate in retransmitting it (by sharing or embedding). They then collected data on the viewcounts of videos in youtube and examined how well they fit their models. They showed that their six models cover 90% of the videos with an average mean square error of less than 5%. They further studied the capability of using these models to predict the evolution of the viewcount.

Network centrality measures

Finding quickly top-k lists of nodes with the largest degrees in large complex networks is a basic problem of recommendation systems. If the adjacency list of the network is known (not often the case in complex networks), a deterministic algorithm to solve this problem requires an average complexity of $O\left(n\right)$, where $n$ is the number of nodes in the network. Even this modest complexity can be excessive for large complex networks. In [18] , K. Avrachenkov and M. Sokol in collaboration with N. Litvak (Twente Univ., The Netherlands) and D. Towsley (Univ. of Massachusetts, Amherst, USA) propose to use a random-walk-based method. They show theoretically and by numerical experiments that for large networks, the random-walk method finds good-quality top lists of nodes with high probability and with computational savings of orders of magnitude. They also propose stopping criteria for the random-walk method that requires very little knowledge about the structure of the network.

In [46] , K. Avrachenkov in collaboration with N. Litvak (Twente Univ., the Netherlands) and L. Ostroumova and E. Suyargulova (both from Yandex, Russia) address the problem of quick detection of high-degree entities in large online social networks. The practical importance of this problem is attested by a large number of companies that continuously collect and update statistics about popular entities, usually using the degree of an entity as an approximation of its popularity. They suggest a simple, efficient, and easy to implement two-stage randomized algorithm that provides highly accurate solutions for this problem. For instance, their algorithm needs only one thousand API requests in order to find the top-100 most followed users in Twitter, a network with approximately a billion of registered users, with more than 90% precision. Their algorithm significantly outperforms existing methods and serves many different purposes, such as finding the most popular users or the most popular interest groups in social networks. They show that the complexity of the algorithm is sublinear in the network size, and that high efficiency is achieved in networks with high variability among the entities, expressed through heavy-tailed distributions.

Personalized PageRank is an algorithm to classify the importance of web pages on a user-dependent basis. In [48] , K. Avrachenkov and M. Sokol in collaboration with R. van der Hofstad (EURANDOM, The Netherlands) introduce two generalizations of Personalized PageRank with node-dependent restart. The first generalization is based on the proportion of visits to nodes before the restart, whereas the second generalization is based on the proportion of time a node is visited just before the restart. In the original case of constant restart probability, the two measures coincide. They discuss interesting particular cases of restart probabilities and restart distributions. They show that both generalizations of Personalized PageRank have an elegant expression connecting the so-called direct and reverse Personalized PageRanks that yield a symmetry property of these Personalized PageRanks.

Along with K. Avrachenkov and N. M. Markovich (Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia), J. K. Sreedharan investigated distribution and dependence of extremes in network sampling processes [47] . This is one of the first studies associating extremal value theory to sampling of large networks. The work showed that for any general stationary samples from the graph (function of node samples) meeting two mixing conditions, the knowledge of bivariate distribution or bivariate copula is sufficient to derive many of its extremal properties. The work proved the usage of a single parameter to find many relevant extremes in networks like order statistics, first hitting time, mean cluster size etc. In particular, correlation in degrees of adjacent nodes are modelled and different random walks, such as PageRank, are studied in detail. This work has been done in the context of Inria Alcatel-Lucent Bell Labs joint laboratory's ADR “Network Science” (see § 7.1.2 ).

Influence maximization in complex networks

Efficient marketing or awareness-raising campaigns seek to recruit a small number, $w$, of influential individuals—where $w$ is the campaign budget—that are able to cover the largest possible target audience through their social connections. In [43] K. Avrachenkov and G. Neglia in collaboration with P. Basu (BBN Technologies, US), B. Ribeiro (CMU, US) and D. Towsley (Univ. of Massachusetts, Amherst, USA) assume that the topology is gradually discovered thanks to recruited individuals disclosing their social connections. They analyze the performance of a variety of online myopic algorithms (i.e. that do not have a priori information on the topology) currently used to sample and search large networks. They also propose a new greedy online algorithm, Maximum Expected Uncovered Degree (MEUD). Their proposed algorithm greedily maximizes the expected size of the cover, but it requires the degree distribution to be known. For a class of random power law networks they show that MEUD simplifies into a straightforward procedure, denoted as MOD because it requires only the knowledge of the Maximum Observed Degree. This work has been done in the context of THANES Joint team (see § 8.3.1.1 ) and Inria Alcatel-Lucent Bell Labs joint laboratory's ADR “Network Science” (see § 7.1.2 ).

In [66] G. Neglia, in collaboration with X. Ye (Politecnico di Torino, Italy), M. Gabielkov and A. Legout (from the Diana team) consider how to maximize users influence in Online Social Networks (OSNs) by exploiting social relationships only. Their first contribution is to extend to OSNs the model of Kempe, Kleinberg and Tardös on the propagation of information in a social network and to show that a greedy algorithm is a good approximation of the optimal algorithm that is NP- hard. However, the greedy algorithm requires global knowledge, which is hardly practical. Their second contribution is to show on simulations on the full Twitter social graph that simple and practical strategies perform close to the greedy algorithm.

Clustering

Clustering of a graph is the task of grouping its nodes in such a way that the nodes within the same cluster are well connected, but they are less connected to nodes in different clusters. In [45] K. Avrachenkov, M. El Chamie and G. Neglia propose a clustering metric based on the random walks' properties to evaluate the quality of a graph clustering. They also propose a randomized algorithm that identifies a locally optimal clustering of the graph according to the metric defined. The algorithm is intrinsically distributed and asynchronous. If the graph represents an actual network where nodes have computing capabilities, each node can determine its own cluster relying only on local communications. They show that the size of clusters can be adapted to the available processing capabilities to reduce the algorithm's complexity.

Average consensus protocols

In [54] , [82] , M. El Chamie in collaboration with J. Liu and T. Başar (Univ. of Illinois at Urbana Champaign, USA) studies the performance of a subclass of distributed averaging algorithms where the information exchanged between neighboring nodes (agents) is subject to deterministic uniform quantization. They give the convergence properties of linear averaging due to such quantization (which is a practical concern for many applications) that cause nonlinearity in the system. This is the first attempt to solve the exact model.

In [53] , M. El Chamie in collaboration with T. Başar (Univ. of Illinois at Urbana Champaign, USA) considers optimal design strategies in consensus protocols for networks vulnerable to adversarial attacks. They provide a game theoretical model for the problem of a network with an adversary corrupting the control signal with noise. They derive the optimal strategies for both players (the adversary and the network designer) of the resulting game using a saddle point equilibrium solution in mixed strategies.