Section: New Results

Network Science

Participants : Eitan Altman, Konstantin Avrachenkov, Mahmoud El Chamie, Philippe Nain, Giovanni Neglia, Marina Sokol.

Epidemic models of propagation of content

E. Altman and P. Nain have studied in [96] in collaboration with A. Shwartz (Technion, Israel) and Y. Xu (Univ. Avignon/LIA) the efficiency of the existing methods for reducing availability of non-authorized copyrighted content for free download on the Internet. To model the propagation of the content, they used both branching processes as well as several epidemic models. One of the important finding is that the greatest impact of measures against unauthorized download is obtained whenever some parameter that describes the virality of the content is close to some critical value (which is computed in this work).

Control and game models for malware attack

In collaboration with M. H. R. Khouzani (Ohio State Univ., USA) and S. Sarkar (Univ. of Pennsylvania, USA), E. Altman has used in [31] ,[33] , [32] , optimal control theory to study malware attack in networks. The structure of optimal policies is obtained by using the Pontryagin maximum principle. In the first two references, optimal defense policies are studies in the goal of protecting the network. In the third work, the worst case behavior of the attack is identified using control theory. The authors then study in [34] the combined problem of identifying the defensive control that achieves the best performance under the worst possible malware attack. This is done through a zero-sum game context.

Time random walks on time varying graphs

In collaboration with D. Figueiredo (Federal Univ. of Rio de Janeiro, Brazil), B. Ribeiro and D. Towsley (both from the Univ. of Massachusetts at Amherst, USA), P. Nain has studied the behavior of a continuous time random walk (CTRW) on a stationary and ergodic time varying dynamic graph [57] . Conditions have been established under which the CTRW is a stationary and ergodic process. In general, the stationary distribution of the walker depends on the walker rate and is difficult to characterize. However, the stationary distribution has been characterized in the following cases: i) the walker rate is significantly larger or smaller than the rate in which the graph changes (time-scale separation), ii) the walker rate is proportional to the degree of the node that it resides on (coupled dynamics), and iii) the degrees of nodes belonging to the same connected component are identical (structural constraints). Examples are provided that illustrate these theoretical findings.

Quick detection of central nodes

In [50] K. Avrachenkov and M. Sokol, together with N. Litvak (Twente Univ., The Netherlands) and D. Towsley (Univ. of Massachusetts at Amherst, USA) propose a random walk based method to quickly find top $k$ lists of nodes with the largest degrees in large complex networks. The authors 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 which requires very little knowledge about the structure of the network.

Graph-based semi-supervised learning

In [48] K. Avrachenkov and M. Sokol, together with P. Gonçalves (Inria project-team Reso ) and A. Mishenin (St. Petersburg State Univ., Russia) develop a generalized optimization framework for graph-based semi-supervised learning. The framework gives as particular cases the Standard Laplacian, Normalized Laplacian and PageRank based semi-supervised learning methods. The authors provide new probabilistic interpretation based on random walks and characterize the limiting behaviour of the methods. The random walk based interpretation allows one to explain differences between the performances of methods with different smoothing kernels. It appears that the PageRank based method is robust with respect to the choice of the regularization parameter and the labelled data. The theoretical results are illustrated with two realistic datasets, characterizing different challenges: “Les Misérables” characters social network and Wikipedia hyper-link graph. It appears that the PageRank based method can classify the Wikipedia articles with very good precision and perfect recall employing only the information about the hyper-text links.

In [47] K. Avrachenkov and M. Sokol, together with P. Gonçalves (Inria project-team Reso ) and A. Legout (Inria project-team Planete ) apply the theoretical results of [48] to classification of content and users in BitTorrent. The general intuition behind the application of the graph based semi-supervised learning methods is that the users with similar interests download similar contents. PageRank based semi-supervised learning method was chosen as it scales well with very large volumes of data. The authors provide recommendations for the choice of parameters in the PageRank based semi-supervised learning method, and show, in particular, that it is advantageous to choose labelled points with large PageRank score.

Optimal weight selection in average consensus protocols

In average consensus protocols, nodes in a network perform an iterative weighted average of their estimates and those of their neighbors. The protocol converges to the average of initial estimates of all nodes found in the network. The speed of convergence of average consensus protocols depends on the weights selected on links (to neighbors). In [92] K. Avrachenkov, M. El Chamie and G. Neglia address how to select the weights in a given network in order to have a fast speed of convergence for these protocols. They approximate the problem of optimal weight selection by the minimization of the Schatten p-norm of a matrix with some constraints related to the connectivity of the underlying network. They then provide a totally distributed gradient method to solve the Schatten norm optimization problem. By tuning the parameter p in the proposed minimization, it is possible to simply 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). Simulation results on random graphs and on real networks show that this approach provides very good performance already for values of p that only needs limited information exchange. The weight optimization iterative procedure can also run in parallel with the consensus protocol and form a joint consensus–optimization procedure.

Reducing communication overhead of average consensus protocols

The average consensus protocol converges only asymptotically to consensus and implementing a termination algorithm is challenging when nodes are not aware of some global information (e.g. the diameter of the network or the total number of nodes). In [93] K. Avrachenkov, M. El Chamie and G. Neglia propose a totally distributed algorithm for average consensus where nodes send more messages when they have large differences in their estimates, and reduce their message sending rate when the consensus is almost reached. The convergence of the system is guaranteed to be within a predefined margin from the true average and the algorithm gives a trade-off between the precision of consensus and the number of messages send in the network. The proposed algorithm is robust against nodes changing their estimates and can also be applied in dynamic networks with faulty links.