The edge cloud: A holistic view of communication, computation and caching

i i “Bo ok” — 2018/2/5 — 1:55 — page 1 — #1 i i i i i i The edge cloud: A holistic view of communication, computation and caching Sergio Ba rba rossa , a, ∗ , Stefania Sa rdellitti ∗ , Elena Ceci ∗ and Mattia Merluzzi ∗ ∗ Sapienza University of R ome, Dept. of Information Engine ering, Ele ctr onics, and T ele communic ations, via Eudossiana 18, 00184, R ome, Italy. a Corr esponding: sergio.barbarossa@uniroma1.it ABSTRA CT The ev olution of communication net works sho ws a clear shift of fo cus from just impro ving the comm unications asp ects to enabling new imp ortan t services, from Industry 4.0 to automated driving, virtual/augmen ted reality , In ternet of Things (IoT), and so on. This trend is eviden t in the roadmap planned for the deplo yment of the fifth generation (5G) communication netw orks. This ambitious goal requires a paradigm shift to w ards a vision that lo oks at communication, computation and cac hing (3 C ) resources as three comp onents of a single holistic system. The further step is to bring these 3 C resources closer to the mobile user, at the e dge of the net work, to enable very low latency and high reliability services. The scope of this c hapter is to show that signal processing techniques can pla y a key role in this new vision. In particular, we motiv ate the joint optimization of 3 C resources. Then w e sho w ho w graph-based representations can play a key role in building effective learning metho ds and devising innov ative resource allocation techniques. Keyw ords: 5G netw orks, wireless communications, graph-based learning 0.1 INTRODUCTION The ma jor goal of next generation (5G) comm unication net w orks is to build a communication infrastructure that will enable new business opp ortunities in div erse sectors, or vertic als , such as automated driving, e-health, vir- tual/augmen ted realit y , Internet of Things (IoT), smart grids, and so on [1], [2]. These services hav e very different specifications and requiremen ts 1 i i “Bo ok” — 2018/2/5 — 1:55 — page 2 — #2 i i i i i i 2 Chapter Title in terms of latency , reliability , data rate, num b er of connected devices, and so on. Thinking of enabling such div erse services using a common commu- nication platform migh t then lo ok like a crazy idea. But, in realit y , if the system is prop erly designed, reusing a common infrastructure for different purp oses might induce a significan t economic adv antage. The k ey idea for making this p ossible is to use virtualization [3] and implemen t network slicing [4]. Through virtualization, many netw ork functionalities are implemented in softw are through virtual machines that can b e instan tiated and mo v ed up on request [5]. Building on virtualization, netw ork slicing partitions a physic al net w ork into multiple virtual net w orks, eac h matched to its sp ecific requiremen ts and constraints, thus enabling op erators to provide netw orks on an as-a-service basis, while meeting a wide range of use cases in parallel. This new realit y , sometimes called fourth industrial revolution, can b e realized by a new architecture able to meet adv anced requirements, esp ecially in terms of latency (b elow 5 ms), reliability (around 0.99999), cov erage (up to 100 devices /m 2 ), and data rate (more then 10 Gbps). A t the physical la yer, 5G builds on a significan t increase of system capacit y b y incorp orating massiv e MIMO techniques, dense deplo yment of radio access p oin ts, and wider bandwidth. All these strategies are facilitated by the in tro duction of millimeter w av e (mmW a v e) communications [6], [7], [8]: mmW av es make p ossible the reduction of the antenna size, thus enabling the use of array with man y elemen ts, as required in massive MIMO; dense deploymen t is also facilitated b ecause mmW av es give rise to a stronger intercell attenuation; finally , increasing the carrier frequency facilitates the usage of wider band- widths. Ho w ev er, the significant impro v emen t achiev able at the physical la y er could b e still insufficient to meet the challenging and div erse requirements of v ery low latency and ultra reliability . A further improv ement comes from a paradigm shift that puts applications at the center of the system design. Net work F unction Virtualization (NFV) and Multi-access Edge Computing (MEC) [9] are the key to ols of this application-cen tric netw orking. In partic- ular, MEC plays the k ey role of bringing cloud-computing resources at the edge of the netw ork, within the R adio Access Net work (RAN), in close prox- imit y to mobile subscrib ers [9], [10]. MEC is particularly effectiv e to deliv er con text-aw are services or to enable computation offloading from resource-po or mobile devices to fixed servers or to p erform intelligen t cac he pre-fetching, based on lo cal learning of the most p opular con ten ts across space and time. Giv en this persp ective, the goal of this c hapter is to show that graph-based metho ds can pla y a significant role in optimizing resource allo cation or deriving new learning mechanisms. The organization of this c hapter is the follo wing. In Section 0.2 w e present the edge-cloud architecture and w e motiv ate the holistic approach that lo oks at 3 C resources as a common p o ol of resources to b e handled jointly with the goal of ac hieving, on the user side, a satisfactory qualit y of exp erience and, on the netw ork side, a balanced and efficien t use i i “Bo ok” — 2018/2/5 — 1:55 — page 3 — #3 i i i i i i 0.2 Holistic view of communication, computation and caching 3 of resources. Then, in Section 0.3, we will fo cus on the join t optimization of computation and comm unication resources, with sp ecific attention to com- putation offloading in the edge-cloud. In Section 0.4, we will concentrate on the joint optimization of caching ad communication. Differently from storage, whic h is fundamentally static , caching is inheren tly dynamic , so that cache memories are pre-fetched when and where needed, and then released. In b oth cases of join t optimization, the goal is to bring resources, either computation (virtual mac hines) or cache, as close as possible to the end user, to enable truly lo w latency and lo w energy consumption services. After presenting this holistic view, w e will mov e in Section 0.5 to presen t some learning mec hanisms based on graph signal pro cessing. In particular, we show how to reconstruct the radio environmen t map (REM), whic h enables a cognitive usage of the radio resources. Then, building again on graph representations, in Section 0.6, w e show how to achiev e an optimal resource allo cation across a netw ork while b eing robust to link failures. The prop osed approach is based on a small p erturbation analysis of net w ork top ologies affected b y sp oradic edge failures. Finally , in Section 0.7 we dra w some conclusions and suggest some p ossible further dev elopmen ts. 0.2 HOLISTIC VIEW OF COMMUNICA TION, COMPUT A TION AND CA CHING The new inf rastructure pro vided b y next comm unication net works can b e seen as a truly distributed and p erv asiv e computer that pro vides very differen t ser- vices to mobile users with sufficien tly goo d qualit y of exp erience. The physical resources comp osing this p erv asive computer are cac he memories, computing mac hines, and comm unication c hannels. The system should serv e the end user, either a mobile subscrib er or a car or the comp onent of a pro duction pro cess with, ideally , zero latency , which means an end-to-end latency smaller than the user p erception capabilit y or than the maxim um v alue ensuring prop er con trol, lik e breaking time in automated driving. T o enable this vision, at the ph ysical la y er, the net w ork will support a m uch higher system (or area) capac- it y (bits/sec/km 2 ). In 5G systems, a 1 , 000-fold increase of system capacity is planned, exploiting mmW a v e communications, massive MIMO, and dense deplo yment of access p oints. How ev er, in spite of this enormous improv ement in system capacity , the zero-latency ideal could still b e far to be obtained b ecause it is v ery complicated, if not imp ossible, to control latency ov er a wide area net w ork. F or this reason, the next step is to bring computation and cac he resources as close as p ossible to the end user, where proximit y is actu- ally measured in terms of service time. This creates a new eco-system, called edge-cloud, whose architecture is sketc hed in Fig. 0.1. In this system, within a macro-cell serv ed b y one base station, we hav e m ultiple millimeter-w a v e i i “Bo ok” — 2018/2/5 — 1:55 — page 4 — #4 i i i i i i 4 Chapter Title access p oints (AP), co v ering muc h smaller areas. Each AP is endow ed with computation and caching capabilities, to enable mobile users to get proximit y access to cloud functionalities. This makes p ossible to provide cloud services with very lo w latency and high data rate, while at the same time keeping data traffic and computation as local as p ossible. Of course, the computing and cac hing capabilities of lo cal MEC servers are significantly lo w er than a typical cloud, but they also serv e a limited num b er of requests and, whenev er their resources are insufficien t, they ma y interact with nearby MEC servers, under the sup ervision of a MEC orchestrator. In this system, mobile applications macrocell Radio Access Point mmW radio AP + MEC server Cloud-5G centralized control Mobile ed ge orchestra t o r : Data plane  : Control plane  : mmW backhaul  : wired backhaul  Mobile edge orchestrator Figure 0.1 Edge-cloud a rchitecture. are handled by virtual machines (or containers) instantiated at the edge of the netw ork, close to the end user. The edge is either the ensem ble of netw ork access p oints, as in Multi-access Edge Computing (MEC) [11], or it might ev en include the mobile terminals as well, as in fo g c omputing [12]. Similarly , con ten ts mo v e dynamically when and where it is more con v e- nien t to ha v e them. Cac hing can in fact be seen as a non-c ausal communi- cation, where conten t mo v e b efore they are actually requested, to minimize the downloading time. In this framework, it makes sense to allo cate 3 C re- sources jointly , with the ob jectiv e of guaranteeing some ultimate user quality of experience. Assuming such a holistic view p ersp ective, the first imp ortant question is wh y using a c ommon platform, call it 5G or the generations to come next, to accommo date services having so different requirements, like IoT, virtual reality or automated driving. This is indeed one of the main c hallenges faced by 5G systems. The approach prop osed in the 5G roadmap is network slicing [5]. A net work slice is a virtual netw ork that is implemen ted on top of a ph ysical net work in a wa y that creates the illusion to the slice tenan t of op erating its i i “Bo ok” — 2018/2/5 — 1:55 — page 5 — #5 i i i i i i 0.3 Joint optimization of communication and computation 5 o wn dedicated ph ysical netw ork. Optimizing net w ork slicing is a first imp ortant application of graph-based represen tations, at a high level. In fact, a mathematical formulation of net w ork slicing has b een recently prop osed in [13], where the comm unication netw ork is represented as a graph G = ( V , E ), where V is the set of no des and E is the set of directed links. There is a subset of function no des, enabled with NFV functionalities, that can provide a service function f . In general, there are K flo ws, eac h requesting a distinct service. The requirement of each service k is represen ted as a service function chain F ( k ) consisting of a set of functions that hav e to b e performed in the predefined order throughout the netw ork. Zhang et al. in [13] formulated the slicing problem as the optimal allo cation of service functions across the NFV-enabled no des, while minimizing the total flo w in the netw ork. The problem is a mixed binary linear program, which is NP-hard. Nevertheless, the authors of [13] prov ed that the problem can b e relaxed with p erformance guarantees. This is indeed a very in teresting application of a graph-theoretic formulation of a very high-level problem. In the following tw o sections, we will fo cus on the joint optimization of pairs of 3 C resources, namely comm unication and computation in Section 0.3 and comm unication and caching in Section 0.4. 0.3 JOINT OPTIMIZA TION OF COMMUNICA TION AND COMPUT A TION Smartphones ha v e really explo ded in their usage and capabilities, placing sig- nifican t demand up on battery usage. Unfortunately , adv ancements in battery tec hnology ha v e not kept pace with the demands of users and their smart- phones. One approach to o vercome the battery energy limitations is to offload computations from mobile devices to fixed devices. Computation offloading ma y b e conv enien t for the follo wing reasons [14], [15]: i) to sa v e energy and then prolong the battery lifetime of hand-held devices; ii) to enable simple de- vices, like inexp ensive sensors, to run sophisticated applications; iii) to reduce latency . F rom a user p ersp ective, one of the parameters mostly affecting the qualit y of exp erience is the end-to-end (E2E) latency , i.e. the time necessary to get the result of running an application. In case of offloading, this latency includes: i) the time to send bits from the mobile device to the fixed server to enable the program; ii) the time to run the application remotely; iii) the time to get the result bac k. It is precisely this E2E latency that couples comm u- nication and computation resources and then motiv ates the joint allocation of these resources. W e recall now the approac h prop osed in [16] and later expanded in [14] and [17]. W e first consider the case where multiple users are served by a single AP/MEC pair. Then, we will mov e to the more challenging case where i i “Bo ok” — 2018/2/5 — 1:55 — page 6 — #6 i i i i i i 6 Chapter Title m ultiple users are served by multiple AP’s and MEC servers. In the first case, the assignment of each UE to a pair of AP and MEC is supp osed to b e giv en; in the second case, the assignmen t is part of the optimization problem. In b oth cases, for economical reasons asso ciated to promoting their capillary deplo yment, the computational capabilities of MEC servers are enormously smaller than a typical cloud. This implies that the num b er of cores p er serv er is v ery limited or, in other w ords, that the a v ailable cores in a MEC serv er m ust op erate in a multi-tasking mo de to accommo date the requests of m ultiple users. This means that a serv er running K applications for as many mobile users will allo cate a certain p ercen tage β k of its CPU time to the users that are b eing served concurrently . If F S denotes the n um b er of CPU cycles/sec that the serv er can run, the p ercen tage of CPU cycles/sec assigned to the k -th user is then f k = β k F S . Multiple users serve d by a single AP/MEC p air W e start by considering K user equipmen ts (UE) assigned to a single AP and a single MEC. The decision to offload a computation from the mobile device to the MEC server dep ends on the characteristics of the application to b e of- floaded. Not all applications are equally amenable to offloading. The decision should take in to account all sources of energy consumption in a smartphone, lik e display , netw ork, CPU, GPS, camera, and so on. Profiling energy con- sumption of applications running on smartphones, rather than on a general purp ose computer, is not an easy task b ecause of asynchr onous p ower b ehavior , where the effect on a comp onent’s p ow er state due to a program entit y lasts b ey ond the end of that program entit y [18]. The signal pro cessing communit y could provide a significan t con tribution to this researc h field by optimizing app dev elopments taking into account the associated energy profiling for a class of smartphone op erating systems, e.g. OS, Android, and so on, and a class of applications. In this chapter, we do not dig in to these asp ects. W e rather concen trate on the joint optimization of radio and computational resources asso ciated to computation offloading, in a multiuser con text. F rom this p oint of view, we simplify the classification of applications b y identifying a few most significan t parameters, as relev ant for computation offloading. F or eac h user k , we consider: i) the num b er b k of bits to be transmitted from the mobile user to the server to transfer the program execution; ii) the num b er of CPU cycles w k necessary to run the application to b e offloaded. W e denote by L k the E2E latency requested from UE k . The ov erall latency T k exp erienced by the k -th UE for offloading an application is the sum of three terms: i) the time T tx k necessary to transmit all bits to the serv er to enable the transfer of program execution; ii) the time T exe k for the serv er to run the application; iii) the time T rx k to get the result back to the UE. In form ulas, T k = T tx k + T exe k + T rx k . (0.1) i i “Bo ok” — 2018/2/5 — 1:55 — page 7 — #7 i i i i i i 0.3 Joint optimization of communication and computation 7 This equation, in its simplicity , shows that enforcing an E2E latency con- strain t induces a coupling b etw een comm unication and computation resources. F rom a user-centric p ersp ective, the goal might either b e to minimize the E2E latency , under a maximum transmit p ow er constraint or, b y dualit y , to minimize the transmit pow er necessary to guarantee a desired latency . W e follo w this latter approach, but clearly the tw o strategies can b e in terc hanged. Let us no w express the single con tributions in (0.1) in terms of the parameters to be optimized. The first contribution is the time T tx k to transmit b k bits from the UE to the AP: T tx k ( p k ) = c k r k ( p k ) (0.2) where c k = b k /B , B is the bandwidth and r k ( p k ) is the spectral efficiency o v er the c hannel betw een UE and AP , whic h is equal to r k ( p k ) = log 2 (1 + α k p k ) (0.3) where p k is the transmit pow er of UE k ; α k = | h k | 2 / ( d γ k σ 2 n ) is a an equiv a- len t channel co efficient that incorp orates the channel co efficient h k , the noise v ariance σ 2 n , the distance d k b et ween UE and AP , and the channel exp onent factor γ . The second contribution in (0.1) is the execution time at the serv er, whic h is equal to T exe k = w k /f k . F rom the user persp ective, the third term in (0.1) do es not imply a transmit p o wer, but only the energy to pro cess the receiv ed data. This term is typically muc h smaller than the first term and in the following deriv ations we will assume it to b e a fixed term incorporated in the o v erall latency . W e are now ready to formulate the computation offloading optimization problem in terms of the transmit p ow ers p k and the CPU percentages f k , k = 1 , . . . , K : min p , f s K X k =1 p k , [ P . 1 ] s.t. c k log 2 (1 + p k α k ) + w k f k ≤ L k , k = 1 , . . . , K 0 < p k ≤ P T , f k > 0 , k = 1 , . . . , K K X k =1 f k ≤ F S (0.4) where p = ( p 1 , . . . , p K ) and f s = ( f 1 , . . . , f K ). This is a con vex problem that can b e easily solved. In particular, the i i “Bo ok” — 2018/2/5 — 1:55 — page 8 — #8 i i i i i i 8 Chapter Title optimal computational rates can b e expressed in closed form as [19]: f k = √ w k η k P K k =1 √ w k η k F S , (0.5) where η k are co efficients that dep end on the channel co efficients. This simple form ula sho ws ho w the allocation of computational resources depends not only on computational asp ects, but also on the c hannel state. Note also that the ab o ve formula con trasts with the prop ortional allo cation of computational rates that would hav e b een p erformed in a conv entional system, i.e. f k = w k P K k =1 w k F S . (0.6) A further substantial improv emen t to computation offloading comes from the introduction of mmW a ve links. Merging MEC with an underlying mmW av e physical lay er creates indeed a unique opp ortunity to bring IT services at the mobile user with very low latency and v ery high data rate. This merge is indeed one of the main ob jectives of the join t Europ e/Japan H2020 Pro ject called 5G-MiEdge (Millimeter-wa ve Edge Cloud as an Enabler for 5G Ecosystem) [20]. The challenge coming from the use of mmW av e links is that they are more prone to blo c king even ts [21], which may jeopardize the b enefits of computation offloading. A p ossible wa y to coun teract blo c king ev ents in a MEC system using mmW av e links was prop osed in [22], [19]. Multiple users serve d by multiple AP’s and multiple MEC servers Let us consider now a more complex scenario, where multiple users may get radio access through multiple AP’s and multiple MEC’s. Besides resource allo cation, our goal no w is to find also the optimal asso ciation b etw een UE’s, AP’s and MEC servers. W e consider a system comp osed of N b small cell access p oin ts, N c MEC servers, and K mobile UE’s. Within the edge-cloud scenario depicted in Fig. 0.1, the asso ciation of a mobile user to an access p oint do es not necessarily follow the same principles of current systems, where a mobile user gets access to the base station with the largest signal-to-noise ratio. In the edge-cloud scenario depicted in Fig. 0.1, the association of a UE to a pair of AP and MEC server dep ends not only on radio channel parameters, but also on the av ailability of computational resources at the MEC server. F urthermore, a UE can get radio access from a certain AP , but its application can run elsewhere, not necessarily on the nearest MEC, dep ending on the a v ailability of computational resources. Actually , since the applications run as virtual machines (VM), w e can think of migrating these VM’s in order to follo w the user. The orc hestration of MEC servers in order to provide seamless service contin uit y to mobile users is an item that has been recently included in the standardization activities of ETSI, within the MEC study i i “Bo ok” — 2018/2/5 — 1:55 — page 9 — #9 i i i i i i 0.3 Joint optimization of communication and computation 9 group [23]. Migrating VM’s is not an easy task, b ecause the instantiation of a VM requires times that are too large with resp ect to some of the latency requiremen ts foreseen in 5G. This has motiv ated significant research efforts in in vestigating light forms of virtual mac hines, named c ontainers , that do not need the instantiation of the whole op erating system, but only of a restricted k ernel [24]. Here, w e do not consider the migration of VM’s, but we do consider the p ossibilit y of letting a UE get access under one AP , while having its applica- tion run in an MEC lo cated elsewhere. In this case, we need to incorp orate in the E2E latency the delay along the backhaul link connecting AP and MEC. In particular, we denote by T B nm the latency b etw een access p oint n and MEC serv er m . F ollowing an approach similar to what we prop osed in [25], we gener- alize now the resource allo cation problem by incorp orating binary v ariables a knm ∈ { 0 , 1 } that assume a v alue a knm = 1 if user k gets radio access through AP n to hav e its application running on MEC server m , and a knm = 0 oth- erwise. F or the sake of simplicit y , we assume that each user is served b y a single base station and a single cloud. Our goal no w is to find the optimal assignmen t rule, together with the optimal transmit p ow ers p k and the com- putational rates f mk assigned b y MEC s erv er m to UE k . As in the previous section, our goal is to minimize the o v erall UE p ow er consumption, under a latency constrain t. The resulting optimization problem is: min p , f , a f ( p , a ) , K X k =1 N b X n =1 N c X m =1 p k a knm ( P ) s.t. i) g knm ( p k , f mk , a knm ) ≤ L k , ∀ k , n, m ii) p k ≤ P k , p k ≥ 0 , ∀ k iii) h m ( f , a ) , K X k =1 N b X n =1 a knm f mk ≤ F m , ∀ m, f ≥ 0 iv) N b X n =1 N c X m =1 a knm = 1 , a knm ∈ { 0 , 1 } , ∀ k , n, m (0.7) where f := ( f mk ) ∀ m,k , a := ( a knm ) ∀ k,n,m , and g knm ( p k , f mk , a knm ) , a knm  c k r kn ( p k ) + w k f mk + T B nm  with r kn ( p k ) = log 2 (1 + α kn p k ) denoting the sp ectral efficiency of UE k accessing AP n and α kn the equiv alen t channel co efficient b et ween UE k and AP n . i i “Bo ok” — 2018/2/5 — 1:55 — page 10 — #10 i i i i i i 10 Chapter Title The ob jectiv e function is the total transmit p ow er consumption from the mobile users. The constraints ha v e the following meaning: i) the o verall latency for eac h user k must b e less than the maximum v alue L k ; ii) the total p o wer sp ent by eac h user m ust b e low er than a fixed total p ow er budget P k ; iii) the sum of the computational rates f mk assigned by each serv er cannot exceed the serv er computational capability F m ; iv) each mobile user should be serv ed b y one AP/MEC pair; this is enforced b y imp osing N b X n =1 N c X m =1 a knm = 1, for eac h k , together with a knm ∈ { 0 , 1 } . Unfortunately , problem P is a mixed-binary problem and is, in general, NP-hard. T o ov ercome this difficulty , as w e suggested in [26], we relax the binary v ariables a knm to be real v ariables in the in terv al [0 , 1] and adopt a sub- optimal successive con vex approximation strategy [27], [25], able to conv erge to local optimal solutions. Additionally , to driv e the assignmen t v ariables a knm to contain only one v alue equal to one and all others to zero, for each k , we incorporate a further constraint recen tly suggested in [13]. The p enalty metho d in [13] is based on the fact that the following problem min a k k a k +  1 k p p , N b X n =1 N c X m =1 ( a knm +  ) p s.t. k a k k 1 = 1 , a knm ∈ [0 , 1] , ∀ n, m (0.8) with a k = ( a knm ) ∀ n,m and p ∈ (0 , 1),  > 0, admits an optimal solution that is binary , i.e. only one element is one and all the others are zero. The optimal solution is c ,k = (1 +  ) p + ( N b N c − 1)  p . Therefore, b y relaxing the binary v ariables a knm so that they b elong to the following conv ex set A = { ( a k ) k ∈I : a knm ∈ [0 , 1] , N b X n =1 N c X m =1 a knm = 1 , ∀ k , n, m } , where I denotes the set of K users, w e form ulate the following relaxed opti- mization problem [26]: min p , f , a f P σ ( p , a ) , f ( p , a ) + σ P  ( a ) ( P σ ) s.t. i) g knm ( p k , f mk , a knm ) ≤ L k , ∀ k , n, m ii) h m ( f , a ) , K X k =1 N b X n =1 a knm f mk ≤ F m , ∀ m, f ≥ 0 iii) p k ≤ P k , p k ≥ 0 , ∀ k ∈ I , a ∈ A (0.9) i i “Bo ok” — 2018/2/5 — 1:55 — page 11 — #11 i i i i i i 0.3 Joint optimization of communication and computation 11 2.4 2.6 2.8 3 3.2 3.4 x 10 −3 −15 −10 −5 0 5 10 15 20 25 T ot a l tr a n sm i t p ower [d B m ] L [ s ec ] PSCA Exhaustive search SNR−based association, joint SNR−based association, disjoint Figure 0.2 Overall UE transmit p ow er consumption vs. L . where σ > 0 is the p enalty parameter, and P  ( a ) , K X k =1 k a k +  1 k p p − c ,k . (0.10) It is imp ortant to emphasize that this penalty is differentiable with resp ect to the unknown v ariables. Ev en by relaxing the binary v ariables a , problem in (0.9) is still non-conv ex, since the ob jectiv e function and the constraints i), ii) are non conv ex. In [26], w e prop osed a Successive Con v ex Approximation (SCA) technique, inspired by [27], to devise an efficient iterative p enalty SCA appro ximation algorithm (PSCA) con v erging to a local optimal solution of (0.9). W e omit the details here, but w e report some numerical results. T o test the effectiveness of the prop osed offloading strategy , in Fig. 0.2 w e rep ort the optimal total transmit p ow er consumption vs. the maximum la- tency L k . W e consider a net w ork comp osed of K = 4 users, a num ber of base stations equal to the num ber of clouds, i.e. N b = N c = 2. The other param- eters are set as follows: F 1 = 2 . 7 · 10 9 , F 2 = 6 · 10 8 , P k = 2 · 10 − 1 , p = 0 . 025. F rom Fig. 0.2, we may observe that the PSCA algorithm provides results v ery close to the exhaustiv e searc h algorithm whose complexity is exp onen- tial. Additionally , we consider as a comparison term the SNR-based asso cia- tion method, in b oth cases where the radio and computational resources are optimized jointly or disjoin tly . It can b e noted that the PSCA algorithm yields considerable p ow er savings compared to metho ds based on SNR only , since it tak es adv antage of the optimal assignment of eac h user to a cloud through the most con v enien t base station. i i “Bo ok” — 2018/2/5 — 1:55 — page 12 — #12 i i i i i i 12 Chapter Title 0.4 JOINT OPTIMIZA TION OF CACHING AND COMMUNICA TION Cac hing p opular conten ts in storage disks distributed across the netw ork yields significant adv an tages in terms of reduction of downloading times and limitation of data traffic. Cac hing can b e seen as a non-c ausal communica- tion, where p opular conten ts mo v e throughout the netw ork in the off-p eak hours to anticipate the users’ requests. Clearly an effectiv e caching strategy builds significantly on the ability to learn and predict users’ b ehaviors. This capabilit y lies at the foundation of pr o active c aching [28] and it motiv ates the need to merge future net w orks with big data analytics [29]. An alternativ e approac h to proactive caching based on reinforcemen t learning to learn file p opularit y across time and space was recently prop osed in [30]. Another important pillar of future net works is Information-Cen tric Net- w orking (ICN), a relativ ely nov el paradigm concerning the distribution of con tents throughout the net work in a manner muc h more efficient than con- v entional Internet [31]. Differen t from what happ ens in the In ternet, where con tents are retriev ed through their address, in ICN, information is retrieved b y name d conten ts [31]. In the ICN framework, netw ork entities are equipp ed with storage capabilities and conten ts mov e throughout the netw ork to serve the end user in the b est p ossible w a y [32]. The con tent placement problem, incorp orating num b er of conten t copies and their lo cations in order to mini- mize a cost function capturing access costs (dela y , bandwidth) and/or storage costs, has b een formulated as a mixed integer linear program (MILP), sho wn to b e NP-Hard [33]. In the case where global knowledge of user requests and net work resources is a v ailable, an In teger Linear Programming (ILP) formula- tion was given in [32], yielding the maximum effic iency gains. In this section w e recall and extend the form ulation of [32] to incorp orate the cost of in- efficien t storage of non-p opular conten ts. Consider an information netw ork G = ( V , E , K ), comp osed of a set of no des V , a set of links E , and a set of information ob jects K , as depicted in Fig. 0.3. A con tent file can b e stored (p ermanen tly or temp orarily) ov er the no des of this graph or trav el through its edges. Some con ten ts reside p ermanently o v er some rep ository no des (e.g., the disks in Fig. 0.3). In all other no des (e.g., the circles in Fig. 0.3), conten ts ma y app ear and disapp ear, according to users’ requests and net w ork resource allo cation. W e suppose, for simplicity , that all conten ts are sub divided in to ob jects of equal size. Each ob ject is then identified by an index k ∈ K . Each no de is characterized by a storage capability and every edge is characterized b y a transp ort capacity . Time is considered slotted and ev ery slot has a fixed duration ∆ τ . At time slot n , each no de u ∈ V hosts, as a rep ository , a set of information ob jects K u [ n ] ∈ K and requests, as a consumer, a set of informa- i i “Bo ok” — 2018/2/5 — 1:55 — page 13 — #13 i i i i i i 0.4 Joint optimization of caching and communication 13 u ! v ! p ! uv ! up ! r ! q ! w ! Figure 0.3 Info rmation net w ork. tion ob jects Q u [ n ] ∈ K . Let q [ n ] ∈ { 0 , 1 } |V ||K| b e the request arriv al pro cess suc h that q u [ k , n ] = 1 if no de u requests ob ject k at time n , and q u [ k , n ] = 0 otherwise. Giv en this graph, w e define a v ertex signal ov er its nodes and an edge signal o v er its edges. The vertex signal s u [ k , n ] is a binary signal defined as: s u [ k , n ] = ( 1 , if conten t k , at time n, is stored on no de u 0 , otherwise , u ∈ V . The amoun t of conten t stored on no de u , at time n , is then S u [ n ] := P k s u [ k , n ]. Similarly , we can define an edge signal as a binary signal, defined on each edge, as t uv [ k , n ] = ( 1 , if conten t k , at time n, is transp orted ov er link uv 0 , otherwise , uv ∈ E . The amount of conten t transp orted o ver link uv at time n , is then T uv [ n ] := P k t uv [ k , n ]. T ypically , eac h con tent may b e host on ev ery no de and m o v ed whenev er useful. The storage and capacity constraints limit the v ariability of b oth S u [ n ] and T uv [ n ] as 0 ≤ S u [ n ] ≤ S u , 0 ≤ T uv [ n ] ≤ T uv , (0.11) where S u is the storage capabilit y of no de u , whereas T uv is the transport capacit y of link uv . The state of the netw ork, at time slot n , is repre- sen ted b y the v ector x [ n ] := [ s [ n ]; t [ n ]], with s [ n ] := ( s u [ k , n ]) ∀ u,k and t [ n ] := ( t uv [ k , n ]) ∀ k,uv ∈E . In principle, a conten t k ∈ K ma y b e cached, at any time slot n , in more then one location. How ev er, there is a cost in k eeping a con ten t in one place, if is not utilized. The goal of dynamic c aching is to find the state v ector x [ n ] that i i “Bo ok” — 2018/2/5 — 1:55 — page 14 — #14 i i i i i i 14 Chapter Title minimizes an o verall cost function that includes the cost for caching and the cost for transp ortation, under constraints dictated by the storage capability , the transp ort capacity , and the users’ requirements in terms of latency to get access to their desired conten ts. The fundamental difference b etw een c aching and stor age is that storage is in trinsically static , whereas caching is fundamentally dynamic . This means that cached conten ts mov e throughout the netw ork, app ear in some nodes and disapp ear from others. There are only some r ep ository no des (e.g., no des p, q , and r in Fig. 0.3) that k eep a p ermanent record or hav e fast access to a con tent delivery net work. The assumption is that each conten t is host in at least one rep ository no de. The basic question about cac hing is then to decide, dynamically , depe nding on the users’ requests, when and where to place all conten ts, ho w to mov e them, and when to drop con ten ts to sa v e memory . The decision for cac hing an ob ject k at node u , at time slot n , m ust result from a trade-off betw een the cost for storing for a certain amount of time and the cost for transp orting the con ten t from its current lo cation to the netw ork access p oin t nearest to the user who requested it. The cost asso ciated to storing a conten t k on no de u during T consecutiv e time slots, in the time window [ n 0 − T + 1 , n 0 ], is E st = n 0 X n = n 0 − T +1 X k ∈K X u ∈V s u [ k , n ] c u [ k ] , (0.12) where c u [ k ] is the energy cost for keeping conten t k on no de u p er unit of time. This unit time cost dep ends on the p opularit y of con ten t k in a neighborho o d of node u . F or instance, w e can set c u [ k ] = c 0 1 + P u [ k ] /P 0 (0.13) where P u [ k ] is the p opularity of conten t k at no de u and c 0 is the (energy) cost for k eeping a con tent ob ject with zero p opularit y and P 0 is the popularity lev el that justifies halving the cost for cac hing p er unit of time, with resp ect to zero-p opularity conten ts. The introduction of the cost co efficien ts c u [ k ] is what makes the formulation c ontext-awar e . In fact, the p opularity P u [ k ] may v ary across the net w ork. The cost asso ciated to conten t transp ortation is E tr = n 0 X n = n 0 − T +1 X k ∈K X uv ∈E t uv [ k , n ] c uv [ k ] , (0.14) where c uv [ k ] is the energy cost for transporting ob ject k ov er link uv . In general, when user u makes a request of conten t k , w e may asso ciate to that request a maximum delivery time, which w e call D u [ k ]. W e also denote by i i “Bo ok” — 2018/2/5 — 1:55 — page 15 — #15 i i i i i i 0.4 Joint optimization of caching and communication 15 N u the neighborho o d of no de u , i.e., the set of no des that are one hop aw ay from no de u , and by x T := [ x [ n 0 − T + 1]; . . . ; x [ n 0 ]] the state v ector during T consecutiv e time slots. The dynamic caching optimization problem can then b e form ulated as ˆ x T = arg min x T ( E st ( x T ) + E tr ( x T )) (0.15) sub ject to the follo wing constrain ts ( a ) q u [ k , n ] ≤ s u [ k , n ] + X v ∈N u D u [ k ] X j =0 t v u [ k , n + j ] ( b ) s u [ k , n ] ≤ s u [ k , n − 1] + X v ∈N u t v u [ k , n − 1] ( c ) t v u [ k , n ] ≤ s v [ k , n − 1] + X w ∈N v t wv [ k , n − 1] ( d ) s u [ k , n ] = 1 , ∀ k ∈ K u [ n ] , s u [ k , 0] = 0 , k / ∈ K u [ n ] ( e ) S u [ n ] ≤ S u ( f ) T uv [ n ] ≤ T uv ( g ) s u [ k , n ] ∈ { 0 , 1 } , t uv [ k , n ] ∈ { 0 , 1 } , (0.16) ∀ u ∈ V , v u ∈ E , k ∈ K , n ∈ [ n 0 − T + 1 , n 0 ]. The abov e constrain ts reflect the storage and flow constraints [32]: (a) ensures that if ob ject k is requested by node u at time slot n , then k either is in the cac he of no de u at time n or needs to b e received by no de u from a neigh b or no de v ∈ N u within D u [ k ] time slots; (b) assures that if k is b eing cached at no de u at time n , then k either was in the cac he of u at time n − 1 or w as receiv ed b y no de u from a neighbor no de v ∈ N u at time n − 1; (c) assures that if ob ject k is receiv ed by no de u from a neigh b or no de v ∈ N u at time n , then k either was in the cac he of v at time n − 1 or w as received b y no de v from a neighbor no de w ∈ N v at time n − 1; (d) describes the initial condition constraints that assure that each no de u alw ays stores the ob jects that it hosts as a rep ository , K u [ n ], and at n = 0 nothing else; (e) and (f ) define the storage and transp ort capacit y constrain ts; (g) states the binary nature of the netw ork configuration (storage and trans- p ort) v ariables. T o simplify the solution of the ab o v e problem, w e let the entries of v ec- tor x T to be real v ariables in [0 , 1]. A numerical example resulting from our relaxed form ulation is sho wn in Fig. 0.4 where w e illustrate the optimal trans- p ort energy vs. the arriv al request rate. W e consider a netw ork comp osed of i i “Bo ok” — 2018/2/5 — 1:55 — page 16 — #16 i i i i i i 16 Chapter Title 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 100 200 300 400 500 600 700 Request arrival rate Average optimal transport energy ILP, D u [ k ] = 0 ILP, D u [ k ] = 3 ILP, D u [ k ] = 6 Sh o rtes t p a th alg o rithm Figure 0.4 Average optimal transp ort energy vs. the request arrival rate. |V | = 10 no des and |K | = 4 information ob jects to b e transp orted, by setting T = 25, M τ = 1s, T uv = 2Mb, and S u = 4. W e considered, for simplicity , no kno wledge of p opularity and same transp ortation costs o v er all links. T o b et- ter ev aluate the effect of the transport energy , w e neglected the storage energy E st term in the in teger linear program (ILP) (0.15), b y assuming that only three rep ository no des store the information ob jects for all time. As a b ench- mark method, we consider the shortest path algorithm, which at eac h request forw ards the desired conten t along the shortest path. It can b e noted that the relaxed ILP metho d yields a considerable p erformance gain with resp ect to the shortest path algorithm: moreov er, the improv emen t gro ws as the maxi- m um deliv ery time D u [ k ] (set equal for each k ) increases, due to the greater degrees of freedom of the algorithm. 0.5 GRAPH-BASED RESOURCE ALLOCA TION Enabling proactiv e resource allo cation strategies is a k ey feature of 5G net- w orks. Proactivit y is rooted on the capability to predict users’ b ehavior. Proactiv e caching is one example where the prediction is based on learning the p opularit y matrix. But of course caching is not the only net work asp ect that can benefit from learning. Radio co verage is one more case where learning maps of the radio en vironmen t may b e useful to ensure seamless connectivity to moving users, p ossibly keeping the smallest num ber of access p oints active to sav e energy . This requires prediction of users’ mobility and the capabilit y i i “Bo ok” — 2018/2/5 — 1:55 — page 17 — #17 i i i i i i 0.5 Graph-based resource allo cation 17 to build Radio Environmen t Maps (REM) [34]. Building a REM is also a k ey step to enable cognitive radio [34], [35], [36]. Balancing data traffic across the netw ork is another problem that could tak e adv an tage of the capability to predict data flows exploiting spatio-temp oral correlation (low-rank) [37], [38]. 0.5.1 RADIO ENVIRONMENT MAP In this section, we sho w how graph-based represen tations can be useful to build a REM from sp oradic measuremen ts. Graph-based representations play a key role in many machine learning tec hniques, as a wa y to formally tak e in to accoun t all similarities among the en tities of an interconnected system. In the signal pro cessing communit y , there is a growing interest in metho ds for pro cessing signals defined ov er a graph, or graph signal pro cessing (GSP), for short [39]. W e show now an application of GSP to recov ering the REM in a urban environmen t from sp oradic measurements collected by mobile devices. The goal is to reconstruct the field o ver an ideal grid, built according to the city map, starting from observ ations taken ov er a subset of no des. W e use a graph- based approach to iden tify patterns useful for the ensuing reconstruction from sparse observ ations. More sp ecifically , given a set of N points in space, whose co ordinate vectors are r i and denoting with E i the field measured at no de i , w e define the co efficients of the adjacency matrix A as a ii = 0 and a i,j = ( e − | E i − E j | 2 2 σ 2 , if || r i − r j || 2 ≤ R 0 0 , otherwise , i 6 = j, where σ and R 0 are tw o parameters used to assess the similarity of t w o no des: σ is a v ariable used to establish the interv al of v alues in the e.m. field within whic h t wo no des are assumed to sense a similar v alue; R 0 is the distance within whic h tw o no des are assumed to b e neighbors. Building matrix A requires some prior information on the field that can b e either acquired through time from measurements or it ma y b e inferred from ray-tracing to ols. F rom the adjacency matrix A , we build the Laplacian matrix L = D − A (0.17) where D is the diagonal matrix whose i -th en try is the degree of no de i : d i = P N j =1 a ij . T aking the eigendecomp osition of L L = UΛU T (0.18) w e hav e a w ay to identify the principal comp onents of the field. It is well kno wn from sp ectral graph theory [40], in fact, that the eigenv ectors asso ciated to the smallest eigenv alues of L identify clusters, i.e., w ell connected comp onen ts. Hence, the eigen vectors associated to the smallest eigen v alues of the Laplacian matrix built according to the abov e method are useful to identify patterns in the e.m. field. Denoting with u k the eigenv ector asso ciated to the k -th i i “Bo ok” — 2018/2/5 — 1:55 — page 18 — #18 i i i i i i 18 Chapter Title eigen v alue, the useful signal x can then b e mo deled as the sup erp osition of the K principal eigenv ectors: x = K X k =1 u k s k := U K s , (0.19) with K < N to b e determined from measuremen ts and U K := [ u 1 , . . . , u K ]. In the GSP literature, a signal as in (0.19), with K < N , is called a b and- limite d signal o v er the graph. In general, a real signal is nev er p erfectly ban- dlimited, but it can b e approximately bandlimited. Having a band-limited mo del is instrumen tal to establish the condition for the recov ery of the entire signal from a subset of samples [41]. In a real situation, it is typical to ha v e several access p oin ts whose radio co verage areas ov erlap. F or eac h access p oint, we can build a dictionary using the metho d describ ed ab o v e, using for the e.m. field a ray-tracing algorithm. W e denote by U ( m ) K the dictionary built when only AP m is active. A t any giv en time frame, only a few AP’s are active. Therefore, the o verall map can b e written as x = M X m =1 K X k =1 u k s ( m ) k := M X m =1 U ( m ) K s ( m ) := Us , (0.20) where M is the num ber of AP’s cov ering the area of interest (not all of them necessarily activ e at the same time), U := ( U (1) K , . . . , U ( M ) K ) and s := ( s (1) ; . . . ; s ( M ) ) is sparse. The observed signal typically consists in a limited n umber of measuremen ts collected along the grid. W e ma y write the observed signal as: y = Σ M X m =1 U ( m ) K s ( m ) = ΣUs , (0.21) where Σ is a diagonal selection matrix, whose i -th entry is one if no de i is observ ed, and zero otherwise. The reco very of the ov erall radio cov erage map can then b e formulated as a sparse recov ery problem. W e used Basis Pursuit (BP), whic h implies solving the following conv ex problem: ˆ s = arg min s k s k 1 s.t. y = ΣUs (0.22) and then we used ˆ x = U ˆ s . An example of reconstruction using BP is shown in Fig.0.5. The grid is comp osed of N = 547 no des and the n um b er M of AP’s cov ering the city area illustrated in the figure is 4. The AP’s are located in the south-east, north- i i “Bo ok” — 2018/2/5 — 1:55 — page 19 — #19 i i i i i i 0.5 Graph-based resource allo cation 19 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 -4 Figure 0.5 Example of reconstructed e.m. field. east, north-w est and south-west side of the examined area. The num ber of measuremen ts is 115. Measurement noise is considered negligible. W e as- sumed a bandwidth K = 40, equal for all AP’s. The background (con tin uous) color is the map ground-truth, obtained using the ra y-tracing to ol Remcom Wireless InSite 2.6.3 [42]. The colors on eac h vertex of the grid represent the reconstructed v alue. Comparing each no de color with the background, we can testify the go o dness of the metho d to reconstruct the ov erall map. The Normalized Mean Square Error (NMSE), measured as the square norm of the error, normalized by the square norm of the true signal, in this example, is N M S E = 0 . 018. The qualit y of the reconstruction dep ends on the num b er of measuremen ts and on the assumption on the bandwidth. Clearly , the larger is the bandwidth, the better is the reconstruction, but the larger is also the n umber of measurements to b e taken to enable the reconstruction. This sug- gests that the choice of the bandwidth must come from a trade-off b etw een accuracy and complexity . i i “Bo ok” — 2018/2/5 — 1:55 — page 20 — #20 i i i i i i 20 Chapter Title 0.5.2 MA TCHING USERS TO 3 C RESOURCES In Section 0.3 we motiv ated the use of a joint allo cation of computation and comm unication resources in computation offloading. W e also incorporated the assignmen t rule b etw een UE, AP , and MEC within the ov erall optimization problem. The resulting formulation yields b etter p erformance than a disjoint form ulation, how ev er it is also computationally demanding b ecause it in v olv es the solution of a mixed-integer programming problem. A p ossible wa y to ov ercome this difficulty is to simplify the rule for asso- ciating UE’s to AP and MEC. One p ossibility is to resort to matching the ory , a low complexit y to ol used to solve the combinatorial problem of matching pla yers from differen t sets, based on their preferences. Matching theory can be seen as the problem of finding a bipartite graph connecting tw o sets, dep end- ing on the preference lists. Matc hing theory has already b een prop osed in [43] for resource allo cation in multi-tiered wireless heterogeneous architectures, with applications to cognitiv e radio netw orks, heterogeneous small-cell-based net works and Device-to-Device communications (D2D). In [44], a multi-stage matc hing game is used in the C-RAN context to assign Radio Remote Heads (RRH), Base Band Units (BBU) and computing resources for computation of- floading, aimed at minimizing the refusal ratio, i.e. the prop ortion of offload- ing tasks that are not able to meet their deadlines. A well-kno wn matching problem is the college admission game presented in [45], where a Deferred- Acceptance (DA) algorithm is prov ed to conv erge to a stable match ing with extremely low complexity . The key initial step of matching theory is to estab- lish a preference rule. F or instance, in [46] the users’ preferences are defined as the R -factor, which captures b oth Pac ket Success Rate (PSR) and wireless dela y . Ho wev er, as p ointed out in [46], the complexit y of this algorithm in- creases considerably when dealing with in terdep enden t preferences, i.e. when the preference of a user is affected by the acceptance of the others. This is indeed the case of user association in wireless netw orks, b ecause, contin uing in the example defined abov e, the R factor of a user changes as other users get accepted by the same AP . T o ov ercome this problem, the authors of [46] divide the game into tw o interdependent subgames: 1. An admission matching game with R -factor guarantees, dep ending on the maxim um dela y exp erienced at each access p oint; 2. A coalitional game among access p oints, where the coalitions are sets of AP’s and asso ciated users. In particular, a user assigned to a certain AP a through the first subgame, could prefer to b e matched to another AP b , since the utilit y functions change as users get admitted. Then, a user k requests to b e transferred from a to b if it improv es its R -factor. The transfer is acce pted if and only if: 1. The access point b do es not exceed its quota (maximum n umber of admitted i i “Bo ok” — 2018/2/5 — 1:55 — page 21 — #21 i i i i i i 0.6 Net w o rk reliabilit y 21 users); 2. The so cial welfare (sum of the R -factors of the tw o coalitions) is increased. Starting from an initial partition (sets of coalitions) obtained with the de- ferred acceptance algorithm, the algorithm in [46] conv erge to a final partition that is also Nash-stable. In the holistic view of 3 C resources, other utility functions can b e used to take into account all the three asp ects of 3 C : com- m unication, computation, and cac hing. F or instance, additional parameters to b e tak en into accoun t are the computational load on MEC servers in case of computation offloading and the amount of storage for cac hing. One more example where graph theory can b e used is load balancing. In fact, especially in view of the dense deplo yment of access p oints, there is a high probabilit y that the load, either data rate, computational load or storage, can b e highly unbalanced throughout the netw ork [47]. One possibility to balance the situation is to split the netw orks in many non-o v erlapping clusters. A cluster head is then elected in eac h cluster and it enforces a balance within the cluster. Then, balancing across clusters is achiev ed by repeated clustering and balancing steps. A p ossible wa y to do clustering is to use sp ectral clustering, whic h starts from the creation of a similarity (adjacency) matrix. In this case, as suggested in [48], it could b e useful to include in the construction of the adjacency matrix a dissimilarity measure that assesses ho w m uch tw o nodes are un balanced. In this w a y , the ensuing clustering tends to put together no des that are close but unbalanced so that the resulting in-cluster balancing will be more effective. 0.6 NETW ORK RELIABILITY The edge-cloud architecture describ ed in Section 0.2 clearly builds on the reli- abilit y of the net w ork connectivit y . How ever, in practice, the presence of a link b et ween a pair of no des is sub ject to random changes. In a wireless commu- nication system, for instance, it is typical to hav e random link failures due to fading. With mmW av e communications, link failures are typically even more pronounced because of blo c king due to obstacles b et ween transmit and receiv e devices. The goal of this section is to build on graph-based representations to assess the effect of random failure on a limited n umber of edge on macroscopic net work parameters, suc h as, for example, connectivity . W e build our study on a small p erturbation analysis of the eigendecomp osition of the Laplacian matrix describing the graph, as suggested in [49]. An outcome of our analysis is the identification of the most critical links, i.e. those links whose failure has a ma jor effect on some netw ork macroscopic features, such as connectivit y . A small p erturbation analysis of the eigen-decomp osition of a matrix is a classical problem that has b een studied since a long time, see, e.g. [50], [51]. i i “Bo ok” — 2018/2/5 — 1:55 — page 22 — #22 i i i i i i 22 Chapter Title In this section we fo cus on the small p erturbation analysis of the eigendecom- p osition of a perturb ed Laplacian L + δ L , incorp orating an original graph Laplacian L plus the addition or deletion of a small p ercentage of edges. W e consider a graph composed of N vertices, so that the dimension of L is N × N . W e denote b y ˜ λ i = λ i + ∆ λ i the p erturb ed i -th eigenv alue and by ˜ u i = u i + ∆ u i the associated p erturb ed eigenv ector. If only one link fails, let us say link m , the p erturbation matrix can b e written as δ L ( m ) = − a m a T m , where a m = [ a m 1 · · · a m n ] T is a column vector of size N that has all entries equal to zero, except the tw o elements a m ( i m ) = 1 and a m ( f m ) = − 1, where i m and f m are the initial and final v ertices of the failing edge m . In case of addition of a new edge, the p erturbation matrix is simply the opp osite of the previous expression, i.e. δ L ( m ) = a m a T m . It is straightforw ard to see that the p erturbation of the Laplacian matrix due to the simultaneous deletion of a small set of edges is simply δ L = − P m ∈E p a m a T m where E p denotes the set of p erturb ed edges. The p erturb ed eigenv alues and eigenv ectors ˜ λ i and ˜ u i , in the case where all eigenv alues are distinct and the p erturbation affects a few p ercen tage of links, are related to the unperturb ed v alues λ i and u i b y the follo wing form ulas [50]: ˜ λ i ' λ i + u T i δ L u i (0.23) ˜ u i ' u i + X j 6 = i u T j δ L u i λ i − λ j u j . (0.24) In particular, the p erturbations due to the failure of a generic link m on the i -th eigen v alue and asso ciated eigen v ector are: ∆ λ i ( m ) = u T i δ L ( m ) u i = − u T i a m a T m u i = = −|| a T m u i || 2 = − [ u i ( f m ) − u i ( i m )] 2 (0.25) and ∆ u i ( m ) = X j 6 = i u T j δ L ( m ) u i λ i − λ j u j = − X j 6 = i u T j a m a T m u i λ i − λ j u j = X j 6 = i [ u j ( i m ) − u j ( f m )][ u i ( f m ) − u i ( i m )] λ i − λ j u j . (0.26) Within the limits of v alidit y of first order p erturbation analysis, the o v erall p erturbation resulting from the deletion of multiple edges is the sum of all the p erturbations occurring on single edges: ∆ λ i = X m ∈E p ∆ λ i ( m ) , (0.27) where E p denotes the set of perturb ed edges. In their simplicit y , the ab ov e i i “Bo ok” — 2018/2/5 — 1:55 — page 23 — #23 i i i i i i 0.6 Net w o rk reliabilit y 23 form ulas capture some of the most relev ant asp ects of p erturbation and their relation to graph top ology . In fact, it is kno wn from sp ectral graph theory , see e.g., [40], that the entries of the Laplacian eigenv ectors asso ciated to the smallest eigen v alues tend to be smo oth and assume the same sign o v er v er- tices within a cluster, while they can v ary arbitrarily across differen t clusters. T aking into account these prop erties, the ab o v e p erturbation form ulas (0.23)- (0.26) giv e rise to the following interpretations: 1. the edges whose deletion causes the largest p erturbation are in ter-cluster edges; 2. given a connected graph, the eigenv ector asso ciated to the n ull eigen v alue do es not induce any p erturbation on any other eigenv alue/eigenv ector, b e- cause it is constant; 3. the eigenv ector p erturbation is larger for quantities (either eigen v alues or eigen vectors) asso ciated to eigen v alues very similar to eac h other (recall that form ulas (0.23) and (0.24) hold true only for distinct eigenv alues). 0.6.1 A NEW MEASURE OF EDGE CENTRALITY Based on the ab ov e deriv ations, w e propose a new measure of edge central- it y , whic h we call p erturb ation c entr ality . W e assume a connected undirected graph. If we denote by K the num b er of clusters in the graph and by ∆ λ i ( m ) the perturbation of the i -th eigenv alue due to the deletion of edge m , we define the topology p erturbation centralit y of edge m as follo ws [49]: p K ( m ) := K X i =2 | ∆ λ i ( m ) | . (0.28) The summation starts from i = 2 simply b ecause, from (0.23), the p erturba- tion induced by the deletion of any edge on the smallest eigenv alue is null. The ab ov e parameter p K ( m ) assigns to each edge the p erturbation that its deletion causes to the ov erall net w ork connectivit y , measured as the sum of the K smallest eigen v alues of the Laplacian matrix [40]. This parameter is particularly relev ant in case of mo dular graphs, i.e. graphs evidencing the presence of clusters. In such a case, it is well known from sp ectral clustering theory [40] that the smallest eigenv alues of the Laplacian carry information ab out the num be r of clusters in a graph. In Fig. 0.6 w e report an example of mo dular graph, obtained b y connecting t wo clusters through a few edges. The p erturb ation c entr ality is enco ded in the color in tensit y of each edge. It is in teresting to see that the edges with the dark est color are, as exp ected, the ones connecting the tw o clusters. i i “Bo ok” — 2018/2/5 — 1:55 — page 24 — #24 i i i i i i 24 Chapter Title Figure 0.6 Example of p erturbation centrality measure. 0.6.2 APPLICA TION: ROBUST INF ORMA TION TRANSMISSION O VER WIRELESS NETW ORKS No w w e apply our statistical analysis to optimize the resource (p ow er) allo- cation o ver a wireless netw ork in order to mak e the net work robust against random link failures. W e consider a wireless comm unication netw ork with M links, where each link is sub ject to a random failure b ecause of fading or blo c king. Ev ery edge is characterized by an outage probability P out ( m ) , m = 1 , . . . , M . W e supp ose the failure ev ents o ver differen t links to b e independent of each other. W e consider first a single-input-single-output (SISO) Rayleigh flat fading channel for eac h link. In suc h a case, the c hannel coefficient h is a complex Gaussian random v ariable (r.v.) with zero mean and circularly sym- metric. Hence, the r.v. α = | h | 2 has an exp onen tial distribution. Denoting with F n ( x ; λ ) the cum ulativ e distribution function (CDF) of a gamma random v ariable x of order n , with parameter λ , the CDF of α can then be written as F 1 ( α ; λ ). W e also denote with C = log 2 (1 + | h | 2 ρ ) the link capacity (in bits/sec/Hz), where ρ = P T ( m ) σ 2 n r 2 m is the signal-to-noise ratio (SNR), P T ( m ) is the transmitted p ow er ov er the m -th link, σ 2 n is the noise v ariance, and r m the distance co v ered by link m . Denoting b y R the data rate, the outage i i “Bo ok” — 2018/2/5 — 1:55 — page 25 — #25 i i i i i i 0.6 Net w o rk reliabilit y 25 probabilit y P out ( m ) is defined as: P out ( m ) = P r { C < R } = P r { log 2 (1 + | h | 2 ρ ) < R } (0.29) = P r {| h | 2 < 2 R − 1 ρ } = Z 2 R − 1 ρ 0 λe − λα dα = F 1  2 R − 1 ρ ; λ  = 1 − e − λ ρ (2 R − 1) . Since the CDF of α is inv ertible, it is useful to introduce its inv erse. In particular, if y = F n ( x ; λ ), we denote its inv erse as x = F − 1 n ( y ; λ ). Expression (0.29) can then b e inv erted to deriv e the transmit p ow er P T ( m ) as a function of the outage probability: P T ( m ) = − λσ 2 n r 2 m (2 R − 1) log(1 − P out ( m )) = σ 2 n r 2 m (2 R − 1) F − 1 1 ( P out ( m ); λ ) . (0.30) The small p erturbation statistical analysis derived ab ov e can b e used to for- m ulate a robust net work optimization problem. W e assess the net work robust- ness, in terms of connectivity , as the ability of the net w ork to give rise to small c hanges of connectivity , as a consequence of a small n umber of edge failures. The net w ork connectivit y is measured b y the second smallest eigen v alue of the Laplacian, also known as the graph algebr aic c onne ctivity . This parameter is kno wn to provide a bound for the graph conductance [52]. Our goal now is to ev aluate the transmit p o w ers P T ( m ), or equiv alen tly , through (0.30), the outage probabilities, that minimize the av erage p erturbation of the algebraic connectivit y , sub ject to a cost function on the total transmit pow er P T max of the ov erall netw ork. In formulas, we wish to solve the following optimization problem: min P out X m ∈E E {| ∆ λ 2 ( m ) |} s.t. P m ∈E P T ( m ) ≤ P T max P out ( m ) ∈ [0 , 1] , ∀ m ∈ E . Using equation (0.25) and (0.30), w e can rewrite the optimization problem explicitly in terms of the outage probabilities P out ( m ) as: min P out X m ∈E P out ( m )[ u 2 ( i m ) − u 2 ( f m )] 2 s.t. ( Q ) P m ∈E r 2 m F − 1 1 ( P out ( m ); λ ) ≤ C max P out ( m ) ∈ [0 , 1] , ∀ m ∈ E i i “Bo ok” — 2018/2/5 — 1:55 — page 26 — #26 i i i i i i 26 Chapter Title where C max := P T max σ 2 n (2 R − 1) . Problem ( Q ) is non-con vex because the constraint set is not con vex. How ever, if we p erform the change of v ariable t m := 1 /F − 1 1 ( P out ( m ); λ ) = − λ/ log (1 − P out ( m )) , m = 1 , . . . , M , the first constraint b ecomes linear. The ob jectiv e function b ecomes non-conv ex. How ev er, if we limit the v ariabilit y of the un- kno wn v ariables to the set t m ≥ λ/ 2 , ∀ m , the ob jectiv e function b ecomes con vex, so that the original problem conv erts in to the follo wing con v ex prob- lem: min t P m ∈E F 1 ( 1 t m ; λ ) | ∆ λ 2 ( m ) | = P m ∈E (1 − e − λ t m ) | ∆ λ 2 ( m ) | s.t. P m ∈E r 2 m t m ≤ C max ( Q 1 ) t m ≥ λ 2 , ∀ m ∈ E . (0.31) W e can now generalize the previous form ulation to the Multi-Input Multi- Output (MIMO) case, assuming multiple indep endent Rayleigh fading c han- nels. One fundamental property of MIMO systems is the div ersity gain, whic h mak es them more robust against fading with resp ect to SISO systems [53]. In fact, differen t p erformance can b e obtained dep ending on the num ber of an tennas on the transmitting sides n T and receiving sides n R exploiting the div ersity gain. In a MIMO system whit n = n T × n R statistically independent c hannels, denoting b y h ij the co efficient b etw een the i -th transmit and the j - th receive antenna, the p df of the random v ariable α := P n T i =1 P n R j =1 | h ij | 2 is the Gamma distribution: P A ( α ) = λ n ( n − 1)! α n − 1 e − λα (0.32) and we denote by F n ( α ; λ ) its cumulativ e distribution function (CDF), with parameters n and λ . Pro ceeding similarly to the SISO case, the optimization problem can b e formulated as min t P m ∈E F n ( 1 t m ; λ ) | ∆ λ 2 ( m ) | s.t. P m ∈E r 2 m t m ≤ C max ( Q 2 ) t m ≥ λ/ ( n + 1) , ∀ m ∈ E (0.33) where the constraint on the v ariables t m has b een introduced to make the problem conv ex. Indeed, problem Q 1 is a special case of problem Q 2 , when n = 1. An interesting result ab out the conv exity of problem Q 2 is that the b ounding region increases with the num b er of indep endent channels. As a numerical example, we considered a connected netw ork comp osed by t wo clusters, with a total of |E | = 761 edges and four bridge edges b etw een the tw o cluste rs. F or the sak e of simplicity , w e assumed the same distances r m o ver all links. In Fig. 0.7, we compare the exp ected p erturbations of the i i “Bo ok” — 2018/2/5 — 1:55 — page 27 — #27 i i i i i i 0.7 Conclusions 27 algebraic connectivity , normalized to the nominal v alue λ 2 , obtained using our optimization pro cedure or using the same pow er ov er all links, assuming the same ov erall p ow er consumption. W e rep ort the result for b oth SISO and MIMO cases. F rom Fig. 0.7, we can observe a significant gain in terms of the total p o w er necessary to ac hieve the same exp ected p erturbation of the net work algebraic connectivity . W e can also see the adv antage of using MIMO comm unications, at least in the case of statistically indep enden t links. 100 200 300 400 500 600 700 800 Total Power 10 -5 10 -4 10 -3 10 -2 10 -1 E{| 2 |}/ 2 Optimized SISO Optimized MIMO SISO MIMO Figure 0.7 Exp ected perturbation of algebraic connectivity vs. total p ow er. 0.7 CONCLUSIONS In this chapter we hav e describ ed some of the asp ects of the edge-cloud archi- tecture, a framework prop osed to bring cloud and communication resources as close as p ossible to mobile users to reduce latency and achiev e a more effi- cien t usage of the av ailable energy . F rom the edge-cloud p ersp ective, we hav e motiv ated a holistic view that aims at optimizing the allo cation of comm u- nication, computation and caching resources join tly . Within this framew ork, graph-based representations play a k ey role. In this c hapter, we considered just a few cases where these represen tations can provide a v alid and innov ative to ol for an efficien t deplo ymen t of the edge-cloud system. As it happens in most engineering problems, big p otentials come with big challenges. One of i i “Bo ok” — 2018/2/5 — 1:55 — page 28 — #28 i i i i i i 28 Chapter Title these is complexit y . T o take full adv an tage of graph representations, there is the need for devising efficient distributed computational to ols to analyze graph-based signals. F urthermore, we b elieve that graph representations are only the b eginning of the story , as they are built incorp orating only pairwise relations. More sophisticated to ols may b e en visaged by enlarging the hori- zon to include m ulti-wa y relations, using for example simplicial complexes or h yp ergraphs, as suggested in [54], or multila yer net work representations [55], [56]. 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