Current estimates of mobile data traffic in the years to come foresee a 1,000 increase of mobile data traffic in 2020 with respect to 2010, or, equivalently, a doubling of mobile data traffic every year. This unprecedented growth demands a significant increase of wireless network capacity. Even if the current evolution of fourth-generation (4G) systems and, in particular, the advancements of the long-term evolution (LTE) standardization process foresees a significant capacity improvement with respect to third-generation (3G) systems, the European Telecommunications Standards Institute (ETSI) has established a roadmap toward the fifth-generation (5G) system, with the aim of deploying a commercial system by the year 2020 [1]. The European Project named ?Mobile and Wireless Communications Enablers for the 2020 Information Society? (METIS), launched in 2012, represents one of the first international and large-scale research projects on fifth generation (5G) [2]. In parallel with this unparalleled growth of data traffic, our everyday life experience shows an increasing habit to run a plethora of applications specifically devised for mobile devices, (smartphones, tablets, laptops)for entertainment, health care, business, social networking, traveling, news, etc. However, the spectacular growth in wireless traffic generated by this lifestyle is not matched with a parallel improvement on mobile handsets? batteries, whose lifetime is not improving at the same pace [3]. This determines a widening gap between the energy required to run sophisticated applications and the energy available on the mobile handset. A possible way to overcome this obstacle is to enable the mobile devices, whenever possible and convenient, to offload their most energy-consuming tasks to nearby fixed servers. This strategy has been studied for a long time and is reported in the literature under different names, such as cyberforaging [4] or computation offloading [5], [6]. In recent years, a strong impulse to computation offloading has come through cloud computing (CC), which enables the users to utilize resources on demand. The resources made available by a cloud service provider are: 1) infrastructures, such as network devices, storage, servers, etc., 2) platforms, such as operating systems, offering an integrated environment for developing and testing custom applications, and 3) software, in the form of application programs. These three kinds of services are labeled, respectively, as infrastructure as a service, platform as a service, and software as a service. In particular, one of the key features of CC is virtualization, which makes it possible to run multiple operating systems and multiple applications over the same machine (or set of machines), while guaranteeing isolation and protection of the programs and their data. Through virtualization, the number of virtual machines (VMs) can scale on ?demand, thus improving the overall system computational efficiency. Mobile CC (MCC) is a specific case of CC where the user accesses the cloud services through a mobile handset [5]. The major limitations of today?s MCC are the energy consumption associated to the radio access and the latency experienced in reaching the cloud provider through a wide area network (WAN). Mobile users located at the edge of macrocellular networks are particularly disadvantaged in terms of power consumption and, furthermore, it is very difficult to control latency over a WAN. As pointed out in [7]?[9], humans are acutely sensitive to delay and jitter: as latency increases, interactive response suffers. Since the interaction times foreseen in 5G systems, in particular in the so-called tactile Internet [10], are quite small (in the order of milliseconds), a strict latency control must be somehow incorporated in near future MCC. Meeting this constraint requires a deep ?rethinking of the overall service chain, from the physical layer up to virtualization.

The goal of this paper is to establish the fundamental tools to analyze signals defined over a topological space, i.e. a set of points along with a set of neighborhood relations. This setup does not require the definition of a metric and then it is especially useful to deal with signals defined over non-metric spaces. We focus on signals defined over simplicial complexes. Graph Signal Processing (GSP) represents a special case of Topological Signal Processing (TSP), referring to the situation where the signals are associated only with the vertices of a graph. Even though the theory can be applied to signals of any order, we focus on signals defined over the edges of a graph and show how building a simplicial complex of order two, i.e. including triangles, yields benefits in the analysis of edge signals. After reviewing the basic principles of algebraic topology, we derive a sampling theory for signals of any order and emphasize the interplay between signals of different order. Then we propose a method to infer the topology of a simplicial complex from data. We conclude with applications to real edge signals and to the analysis of discrete vector fields to illustrate the benefits of the proposed methodologies.

Mobile cloud computing is offering a very powerful storage and computational facility to enhance the capabilities of resource-constrained mobile handsets. However, full exploitation of the cloud computing capabilities can be achieved only if the allocation of radio and computational capabilities is performed jointly. In this paper, we propose a method to jointly optimize the transmit power, the number of bits per symbol and the CPU cycles assigned to each application in order to minimize the power consumption at the mobile side, under an average latency constraint dictated by the application requirements. We consider the case of a set of mobile handsets served by a single cloud and we show that the optimization leads to a one-to-one relationship between the transmit power and the percentage of CPU cycles assigned to each user. Based on our optimization, we propose then a computation scheduling technique and verify the stability of the computations queue. Then we show how these queues are affected by the degrees of freedom of the channels between mobile handsets and server.

In Part I of this paper, we proposed and analyzed a novel algorithmic framework for the minimization of a nonconvex objective function, subject to nonconvex constraints, based on inner convex approximations. This Part II is devoted to the (nontrivial) application of the framework to the following relevant large-scale problems ranging from communications to machine learning: 1) (generalizations of) the rate profile maximization in MIMO interference broadcast networks; 2) the max-min fair multicast multigroup beamforming problem in a multicell environment; and 3) a general nonconvex constrained bi-criteria formulation for k-sparse variable selection in statistical learning; the two criteria are a nonconvex loss objective function, measuring the fitness of the model to data, and the latter is a nonconvex sparsity-inducing constraint in the general form of difference-of-convex (DC) functions, which allows to accomodate in a unified fashion convex and nonconvex surrogates of the 10 function. The proposed algorithms outperform current state-of-the-art schemes for 1)-3) both theoretically and numerically. For instance, they are the first distributed schemes for the class of problems 1) and 2); and they also lead to subproblems enjoying closed form solutions.

Distributed consensus algorithms have recently gained large interest in sensor networks as a way to achieve globally optimal decisions in a totally decentralized way, that is, without the need of sending all the data collected by the sensors to a fusion center. However, distributed algorithms are typically iterative and they suffer from convergence time and energy consumption. In this paper, we show that introducing appropriate asymmetric interaction mechanisms, with time-varying weights on each edge, it is possible to provide a substantial increase of convergence rate with respect to the symmetric time-invariant case. The basic idea underlying our approach comes from modeling the average consensus algorithm as an advection-diffusion process governing the homogenization of fluid mixtures. Exploiting such a conceptual link, we show how introducing interaction mechanisms among nearby nodes, mimicking suitable advection processes, yields a substantial increase of convergence rate. Moreover, we show that the homogenization enhancement induced by the advection term produces a qualitatively different scaling law of the convergence rate versus the network size with respect to the symmetric case.