Server resource dimensioning and routing of service function chain in NFV network architectures

The Network Function Virtualization (NFV) technology aims at virtualizing the network service with the execution of the single service components in Virtual Machines activated on Commercial-off-the-shelf (COTS) servers. Any service is represented by the Service Function Chain (SFC) that is a set of VNFs to be executed according to a given order. The running of VNFs needs the instantiation of VNF instances (VNFI) that in general are software components executed on Virtual Machines. In this paper we cope with the routing and resource dimensioning problem in NFV architectures. We formulate the optimization problem and due to its NP-hard complexity, heuristics are proposed for both cases of offline and online traffic demand. We show how the heuristics works correctly by guaranteeing a uniform occupancy of the server processing capacity and the network link bandwidth. A consolidation algorithm for the power consumption minimization is also proposed. The application of the consolidation algorithm allows for a high power consumption saving that however is to be paid with an increase in SFC blocking probability

Poincare-Einstein Holography for Forms via Conformal Geometry in the Bulk

We study higher form Proca equations on Einstein manifolds with boundary data along conformal infinity. We solve these Laplace-type boundary problems formally, and to all orders, by constructing an operator which projects arbitrary forms to solutions. We also develop a product formula for solving these asymptotic problems in general. The central tools of our approach are (i) the conformal geometry of differential forms and the associated exterior tractor calculus, and (ii) a generalised notion of scale which encodes the connection between the underlying geometry and its boundary. The latter also controls the breaking of conformal invariance in a very strict way by coupling conformally invariant equations to the scale tractor associated with the generalised scale. From this, we obtain a map from existing solutions to new ones that exchanges Dirichlet and Neumann boundary conditions. Together, the scale tractor and exterior structure extend the solution generating algebra of [31] to a conformally invariant, Poincare--Einstein calculus on (tractor) differential forms. This calculus leads to explicit holographic formulae for all the higher order conformal operators on weighted differential forms, differential complexes, and Q-operators of [9]. This complements the results of Aubry and Guillarmou [3] where associated conformal harmonic spaces parametrise smooth solutions.Comment: 85 pages, LaTeX, typos corrected, references added, to appear in Memoirs of the AM

A PARTAN-Accelerated Frank-Wolfe Algorithm for Large-Scale SVM Classification

Frank-Wolfe algorithms have recently regained the attention of the Machine Learning community. Their solid theoretical properties and sparsity guarantees make them a suitable choice for a wide range of problems in this field. In addition, several variants of the basic procedure exist that improve its theoretical properties and practical performance. In this paper, we investigate the application of some of these techniques to Machine Learning, focusing in particular on a Parallel Tangent (PARTAN) variant of the FW algorithm that has not been previously suggested or studied for this type of problems. We provide experiments both in a standard setting and using a stochastic speed-up technique, showing that the considered algorithms obtain promising results on several medium and large-scale benchmark datasets for SVM classification

A polynomial eigenvalue approach for multiplex networks

We explore the block nature of the matrix representation of multiplex networks, introducing a new formalism to deal with its spectral properties as a function of the inter-layer coupling parameter. This approach allows us to derive interesting results based on an interpretation of the traditional eigenvalue problem. More specifically, we reduce the dimensionality of our matrices but increase the power of the characteristic polynomial, i.e, a polynomial eigenvalue problem. Such an approach may sound counterintuitive at first glance, but it allows us to relate the quadratic problem for a 2-Layer multiplex system with the spectra of the aggregated network and to derive bounds for the spectra, among many other interesting analytical insights. Furthermore, it also permits to directly obtain analytical and numerical insights on the eigenvalue behavior as a function of the coupling between layers. Our study includes the supra-adjacency, supra-Laplacian, and the probability transition matrices, which enable us to put our results under the perspective of structural phases in multiplex networks. We believe that this formalism and the results reported will make it possible to derive new results for multiplex networks in the future.Comment: 15 pages including figures. Submitted for publicatio

On degree-degree correlations in multilayer networks

We propose a generalization of the concept of assortativity based on the tensorial representation of multilayer networks, covering the definitions given in terms of Pearson and Spearman coefficients. Our approach can also be applied to weighted networks and provides information about correlations considering pairs of layers. By analyzing the multilayer representation of the airport transportation network, we show that contrasting results are obtained when the layers are analyzed independently or as an interconnected system. Finally, we study the impact of the level of assortativity and heterogeneity between layers on the spreading of diseases. Our results highlight the need of studying degree-degree correlations on multilayer systems, instead of on aggregated networks.Comment: 8 pages, 3 figure