Nicky Hallett, The Senses in Religious Communities, 1600-1800. Early Modern 'Convents of Pleasure', Women and Gender in the Early Modern World series, Farnham: Ashgate, 2013, pp. xii + 249, £ 65, ISBN: 978-1-4094-4946-1 (hardcover). This is an original, unusual, at times challenging and always thoughtprovoking book, a new landmark in the current developments made by scholars in the study of early modern English nuns. The fates of Catholic women who wished to enter the religious life have been all but ignored even in such seminal works as John Bossy's English Catholic Community or Peter Guilday's Catholic Refugees on the Continent. over the last decade however, thanks mainly to Claire Walker's research and to Caroline Bowden's personal and collective projects, much has been done to correct this imbalance. Manuscript sources have been located across Europe and England, most have been catalogued and many digitised as part of the AHRC-funded project entitled Who were the nuns? A prosopographical study of the English convents in exile, 1600-1800. The resulting six volumes of primary sources published as English Convents in Exile, 1600-1800 by Pickering & Chatto (2012-2014) offer readers a broad overview of the incredible richness of material to document the life writing, history writing, reading habits, spiritual life, everyday life, or even the political involvement of nuns in their local and national contexts. Nicky Hallett participated in this endeavour by editing volume 3 on Life Writing, and she had previously contributed two of her own volumes of edited sources, both published in 2007 by Ashgate: Lives of Spirit: English Carmelite Self-Writing of the Early Modern Period and Witchcraft, Exorcism and the Politics of Possession in a Seventeenth Century Convent: 'How Sister Ursula was once bewiched and Sister Margaret twice'.

In the work under review, Hallett continues her exploration of the Carmelite communities founded at Antwerp (1619) and Lierre (1648), but this time she offers new insights into the nuns' experiences of contemplative life through the conceptual lens of the senses. The book opens most clearly with those questions: 'What did it mean to be sensate in an early modern convent? How did it feel to be alive -or dead -in a contemplative community?' (p. 1). To answer these questions, the study is developed around a general introduction followed by eight chapters, the last one being 'A Conclusion of Sorts'. Chapters 1 and 2 contextualise and introduce chapters 3 to 7, which are dedicated each to one of the five senses in turn: touch, taste, hearing, smell and sight respectively.

After a scholarly and informative general introduction, readers may find the first two chapters puzzling. Chapter 1, enticingly entitled 'In Which Mrs Eyre Protects the Impressionable Souls of her Tender Daughters', does not offer a clear sense of direction so much as suggestions to map out the sensory contexts in which young Catholic Englishwomen grew up. Through the example of Mary Eyre, née Bedingfield, who took her three daughters to the Carmelite convent of Lierre, Nicky Hallett shows that taking the veil was, for many young women, a safe and wise option in a generally 'sensory-censorious' culture (p. 27). She evokes the crucial importance of conduct books and makes the distinction between interior and exterior senses in the spiritual writings of nuns and of the authors they read. Bodily senses were to be disciplined and transcended, in order to allow the contemplative soul to feel interiorly. Chapter 2, 'Becoming Behaviour: Two Cases of Sensational Reading' is shorter and continues the author's background-building by exploring the activity of reading with regards to the senses. It highlights the importance of the nuns' medieval heritage (particularly of authors such as Julian of Norwich, John of the Cross, or Teresa of Jesus), the influence of contemporaneous writers such as Alfonso Rodriguez, and even of secular philosophers such as Hobbes and Bacon. The reader may find it difficult to pinpoint the focus of this chapter which does not solely dwell upon the convents' reading habits per se but chooses to evoke some fascinating cases of demonic possession and exorcism, in which the bodies of the afflicted nuns became like living books to their sisters.

After these somewhat unusual first two chapters, the next five appear more traditional. In chapter 3, 'Titillation and Texture: The Sixth Sense of "Handsome Handid Nuns"', the reader explores the sensory experiences of nuns through their manual work and the sense of touch, which was regarded as meanest of the senses. yet Carmelites were particularly preoccupied with this sense, which allowed them at times to verify other senses, validating or invalidating the impressions given by sight, smell or sound. Here, Hallett explores textures, for instance the softness and sensuousness of the clothes worn during clothing ceremonies, followed by the roughness of the conventual habit. She evokes the importance of pain in convents where the imitation of Christ was a model. She also, imaginatively -and unexpectedly for a chapter on touch -develops an analysis of smell, which she interprets as a remote form of touch through vignettes concerning the odours emanating from saintly corpses.

Taste is studied next, as another form of touch; in chapter 4, 'of Taste and Tongue: "a very slippery member"', readers will learn about the humoural theories related to different types of food and about the penitential practices related to food and drink in English Carmelite convents. The author, however, chooses to go further and to integrate the role of the tongue as the organ of speech; she develops an analysis of selfcontrol, silence and the regulation of speech which, stricto sensu, may be seen as a little outside the remit of this chapter on the sense of taste, but which brings fascinating insights into early modern conventual life.

In chapter 5, 'Still Small Voices: Sounds, Sibilance and Silence in Early Modern Convents', the study moves on to the sense of hearing: in enclosed communities, separated from the world by walls and grilles, the nuns were never to be seen but they were heard, as 'disembodied voices', during Mass (p. 147). And on the inside, what did the nuns hear? This chapter explores the importance of the convents' bells, the role of silence and the place of music, and it conjures up the special atmosphere of enclosed religious communities through the ages. It concludes with a very brief evocation of inner hearing, those sounds heard spiritually by ecstatic nuns during raptures, and of miraculous sounds, unexplained yet heard physically 'with their corporeal eares', often by several witnesses (p. 155). Sadly, the chapter closes on the mere evocation of these intriguing elements, full of such promise of fascinating insights; this is one of the instances when the reader might wish to know more of the author's personal interpretation of the phenomena she conjures up.

Chapter 6, 'of Smell and Space: "Evaporating Subjects"', is perhaps the most sensuous or sensory of all the chapters. Smell, Nick Hallett tells us, 'is an especially intimate sense' (p. 162), and this particular chapter reflects this intimacy beautifully. It shows how the experience of smell was both individually formative and communally essential, since convents at times built their spiritual identity around their shared common experience of smells, often the sweet odours of saintly, uncorrupted bodies. In chapter 7, 'Eagle-eyed Nuns: Envisioning Vision in Contemplative Communities', the book reaches what was usually known as the highest, the most noble of the five senses. Vision was to be the particular object of the nuns' vigilance, as it was linked to the vice of curiosity: the eyes were therefore disciplined, downcast and chaste. But they could also be paths to spiritual inspiration, and nuns used pictures to help their souls to soar in contemplation. Here, Nicky Hallett draws a distinction between various forms of pictures, namely the printed or painted images used to represent sanctity in the convent, the imaginary images or evocations used during spiritual exercises and the sights seen inwardly during mystical visions or revelations. In this captivating chapter, the author draws extensively upon Teresa of Avila's writings on sight and vision, thereby giving readers a contextualised sense of what English Carmelites would have believed about 'seeing'.

Finally in chapter 8, 'Sensate Certainty: A Conclusion of Sorts', Nicky Hallett considers the trustworthiness of the senses. Citing numerous spiritual and philosophical writings on the subject, she situates the perception of early modern English Carmelites in the historiography of their time to show at once both their indebtedness to previous authors and their originality, at the periphery of the canon. This conclusion opens up important questions about historicising and periodization, and about the rapport of communities at the margins with more central, canonical sources.

This book is highly original, atypical, and it is at once a delight and a challenge. Much of the delight comes from the chosen topic, which offers rare and intimate insights into the lives of individuals and communities, lives which have remained very private and mostly unknown for centuries. But it also derives from Nicky Hallett's own distinctive writing style; a quick glance at the table of contents will reveal the author's inimitable cachet, allying quotations from the nuns' papers with her own imaginative and graceful captions, always so richly alliterated. This technique in fact echoes the body of the work, in which Hallett cites the convents' papers at length and comments only lightly at times, as if to allow the nuns' voices to speak for themselves, in the first person, about their own sensory experiences and their lived contemplative lives. The length of the footnotes shows that the author has preferred to relegate some of her interpretative work to the periphery of the text, as one would do in an edition of primary sources. And therein lies the challenge. Indeed, this authorial decision leads, at times, to some repetition in the cited material across various chapters and may leave the reader somewhat hungry for more analysis, more explanatory material; yet it is in keeping with the overall feel of the book, in which the author seems to strive towards the evocative rather than the definitive. In this sense, this study may be described as historical in content yet highly poetic in form. In a fittingly reflexive manner, it echoes the nuns' own literary productions: it is a precious, well-informed, beautifully researched addition to the existing canonical historiography, yet it quite deliberately remains at the margins, both through its subject and its form. It is audacious, scholarly and lyrical at once, it defies the rigid pigeon-holes of genres, and engages readers fully by not always giving them what they expected when and where they would have expected it. It is also a landmark in the developing scholarship on early modern convents in exile, since -to my knowledge -this is the first monograph to take the study of these nuns beyond its traditional remit and make it into an academic subject for the broader history of ideas, where it can be analysed more conceptually.

LERMA, Aix-Marseille Université

laurenCe lux-sterritt


Hugo Méric
email: hmeric@niclabs.cl

Jérôme Lacan
email: jerome.lacan@isae.fr

Capacity-approaching signals for the scalar Gaussian channel

We consider the scalar Gaussian channel with power constraint P . A gap exists between the channel capacity and the highest achievable rate of equiprobable uniformly spaced signal. Several approaches enable to overcome this limitation such as constellations with non-uniform probability or constellation shaping. In this short paper, we focus on constellation shaping. We present a method to construct one-dimensional constellations with equiprobable signaling that achieve the Gaussian capacity as the number of signal points goes to infinity.

I. INTRODUCTION

We consider the scalar additive white Gaussian noise (AWGN) channel with signal-to-noise ratio (SNR) P/N 0 , where P is the average power constraint of the input signal and N 0 the noise variance. Let snr = P/N 0 , the Gaussian channel capacity is

C = 1 2 log 2 (1 + snr) bits/dimension, (1)
and the optimal input distribution is Gaussian with zero mean and variance P [START_REF] Cover | Elements of Information Theory[END_REF].

Approaching the Gaussian capacity is a challenging problem in digital communications systems. Indeed the optimal input distribution is continuous while in practice the signal is chosen from a finite constellation of points. Several approaches enable to overcome this limitation such as constellation shaping or constellations with non-uniform probabilities [START_REF] Kschischang | Optimal nonuniform signaling for Gaussian channels[END_REF]. In this letter, we focus on the former solution as it is more attractive for practical systems that mainly use equiprobable signaling.

Sun and van Tilborg were the first to present a sequence of random variables equiprobably distributed over a finite support that achieves the Gaussian capacity (in one dimension) as the signal set cardinality tends to infinity [START_REF] Sun | Approaching capacity by equiprobable signaling on the Gaussian channel[END_REF]. Until then, the traditional rule of designing constellations for the AWGN channel was to maximize the squared Euclidean distance between the signal points under the power constraint. In one dimension, this results in uniformly spaced signal sets that, when combined with equiprobable signaling, exhibit a gap of πe 6 ≈ 1.53 dB with the Gaussian capacity at large SNR [START_REF] Forney | Efficient modulation for band-limited channels[END_REF]. Sun and van Tilborg shows how to close this gap in one dimension. In higher dimensions, this shaping gain can be obtained by choosing the signal points on an N -dimensional rectangular lattice from within an N -sphere rather than an Ncube [START_REF] Forney | Efficient modulation for band-limited channels[END_REF]. However this is impractical when N increases.

The result in [START_REF] Sun | Approaching capacity by equiprobable signaling on the Gaussian channel[END_REF] was extended by Schwarte that provided sufficient conditions for equiprobable input distributions with finite support to approach the Gaussian capacity in any dimension [START_REF] Schwarte | Approaching capacity of a continuous channel by discrete input distributions[END_REF]. Similar conditions are given in [START_REF] Wu | Functional properties of MMSE[END_REF]Theorem 9].

A recent work by Wu and Verdú studied the AWGN channel capacity with finite signal set [START_REF]The impact of constellation cardinality on Gaussian channel capacity[END_REF]. They showed that as the input signal cardinality grows, the constellation capacity approaches the Gaussian capacity exponentially fast. They also introduced a family of constellation, based on the Hermite polynomials roots, achieving exponential convergence. The resulting modulations combine constellation shaping with nonuniform probabilities.

If the channel inputs are also subject to peak power constraints, the capacity and the optimal input distribution were studied for the scalar Gaussian channel in [START_REF] Smith | The information capacity of amplitude-and varianceconstrained scalar Gaussian channels[END_REF]. Smith showed that the capacity is achieved by a unique discrete random variable taking on a finite number of values. It was later extended to the quadrature Gaussian channel [START_REF] Shamai | The capacity of average and peakpower-limited quadrature Gaussian channels[END_REF]. In two dimensions, the optimal distribution is discrete along the radial direction and continuous in the angular direction.

A very interesting paper by Huang and Meyn studies the structure of capacity-achieving input distributions [START_REF] Huang | Characterization and computation of optimal distributions for channel coding[END_REF]. One of the result is that discrete distribution can nearly reach the capacity even if the optimal distribution is continuous.

In this work, our main contribution (presented in Section II) is the construction of one dimensional constellations with equiprobable signaling that achieve the Gaussian capacity as the number of constellation points goes to infinity. To that end, we rely on a shaping function to produce signal sets with a Gaussian shape, the key ingredient to approach the AWGN channel capacity. This is an alternative to the previous solutions [START_REF] Sun | Approaching capacity by equiprobable signaling on the Gaussian channel[END_REF], [START_REF] Schwarte | Approaching capacity of a continuous channel by discrete input distributions[END_REF]. We illustrate our result by evaluating the performance of a capacity-approaching constellation obtained from our proposal. We conclude the paper by discussing several research directions in Section III.

II. CAPACITY-APPROACHING CONSTELLATIONS

A. Preliminary remarks

Approaching the AWGN channel capacity with equiprobable Gaussian-shaped discrete distributions was first proposed by Sun and van Tilborg [START_REF] Sun | Approaching capacity by equiprobable signaling on the Gaussian channel[END_REF]. Based on this result, we recently introduced the concept of shaping function that aims to give a Gaussian shape to a finite constellation of points [START_REF] Méric | Approaching the Gaussian channel capacity with APSK constellations[END_REF]. We 978-1-5090-0314-3/16/$31.00 c 2016 IEEE proposed the Box-Muller transform as shaping function for the quadrature channel, resulting in capacity-achieving amplitude and phase-shift keying constellations [START_REF] Méric | Approaching the Gaussian channel capacity with APSK constellations[END_REF].

Our current work focuses on the one-dimensional case and provides a general method to generate capacity-achieving signal sets with equiprobable signaling. Our choice of the shaping function relies on the following lemma: Lemma 1. Let U be the continuous uniform distribution in the interval (0, 1) and Φ P the cumulative density function of the normal distribution with zero mean and variance P , then Φ -1 P (U ) ∼ N (0, P ).

Based on the previous lemma, our idea is to consider Φ -1 P as the shaping function for the one-dimensional case.

B. Main result

The following theorem explains how to generate capacityapproaching signal sets for the scalar AWGN channel. It is the main contribution of this letter. Theorem. Let (U n ) n 1 be a sequence of discrete random variables that converge weakly to U , the continuous uniform distribution over (0, 1), such that sup n Var(Φ -1 P (U n )) < ∞. The signal defined by X n = Φ -1 P (U n ) achieves the AWGN channel capacity with power constraint P as n → ∞. More formally,

I(X n ; X n + N ) -→ n→∞ 1 2 log 2 (1 + snr), (2)
where I denotes the mutual information [START_REF] Cover | Elements of Information Theory[END_REF] and N ∼ N (0, N 0 ) is the Gaussian noise with variance N 0 . Proof: The idea is to show that the sequence (X n ) n 1 satisfies the two conditions given in [6, Theorem 9].

The first condition requires that (X n ) n 1 verifies sup n Var(X n ) < ∞, which is true by hypothesis.

Next we prove that (X n ) n 1 converges weakly to the distribution of a Gaussian variable with zero mean and variance P .

By assumption, we have that U n L -→ U (weak convergence). The function Φ -1 P is continuous on (0, 1), thus the continuous mapping theorem can be applied [START_REF] Billingsley | Convergence of probability measures[END_REF]. We obtain

X n = Φ -1 P (U n ) L -→ Φ -1 P (U ) . (3)
Lemma 1 ensures that Φ -1 P (U ) ∼ N (0, P ). Thus the signal defined by X n = Φ -1 P (U n ) satisfies both conditions in [6, Theorem 9]. We conclude that the signal (X n ) n 1 approaches the Gaussian channel capacity with power constraint P as n → ∞.

In the theorem, the assumption sup n Var(Φ -1 P (U n )) < ∞ is required to control the signal energy. This prevents to generate signals that approach the Gaussian distribution but may not have a finite variance. Another consequence is that U n can not take the values 0 or 1, as in that case Φ -1 P is not defined.

C. Application

To illustrate our result, we construct a capacity-achieving constellation with equiprobable signaling and evaluate its performance on the AWGN channel. Let n 1 be an integer and N n = {0, 1, 2, . . . , n -1} the set of integers from 0 to n -1. We consider the discrete set

S n = 1 2n + k n k ∈ N n , (4)
that verifies |S n | = n and S n ⊂ (0, 1). We introduce U n the discrete random variables uniformly distributed on S n and the proposed signal is

X n = Φ -1 P (U n ).
As U n is uniformly distributed on S n , the random variable X n is uniformly distributed on a set C n of n points. The points in C n exhibit a Gaussian shape as depicted in Fig. 1 for n = 10 and P = 4. This illustrates the objective of the shaping function: applying the shaping function to a set of uniformly spaced points generates a Gaussian-shaped constellation.

In order to show that the proposed constellations achieve the AWGN channel capacity, we prove that the two assumptions in the theorem are satisfied. The following lemma studies the convergence of (U n ) n 1 . Lemma 2. The sequence (U n ) n 1 converges weakly to U , the uniform distribution over (0, 1).

Proof: The characteristic function of U n is

E e itUn = 1 n n k=0 e it( 1 2n + k n ) (5) = e it 2n n × 1 -e it 1 -e it n -→ n→∞ e it -1 it = E e itU . ( 6
)
Using the continuity theorem [13, Theorem 26.3], we conclude that (U n ) n 1 converges weakly to U . Next we study the variance of the sequence

(X n ) n 1 . Lemma 3. (X n ) n 1 verifies sup n Var(X n ) < ∞. Proof: The variance of X n is equal to Var(X n ) = E[X 2 n ] -E[X n ] 2
. By construction of S n , X n is symmetrically distributed about the origin (see Fig. 1). Combined with the equiprobable signaling, this results in E[X n ] = 0.

Then we consider the term E[X 2 n ] in the variance. First it is important to note that (Φ -1 P ) 2 is Riemann integrable over Fig. 2: Gap to the Gaussian capacity for the uniformly spaced signal with equiprobable signaling and the proposed constellation (0, 1). Indeed, we have

1 0 Φ -1 P (t) 2 dt = R u 2 Φ P (u)du = P, (7)
where we use the substitution u = Φ -1 P (t). By definition,

E[X 2 n ] = n-1 k=0 1 n Φ -1 P 1 2n + k n 2 . ( 8
)
The right term in ( 8) is a Riemann sum, therefore

E[X 2 n ] -→ n→∞ 1 0 Φ -1 P (t) 2 = P. (9)
This concludes the proof that sup n Var(X n ) < ∞.

Previous lemmas show that the sequence (U n ) n 1 satisfies both assumptions in the theorem. Thus, the Gaussianshaped signal with equiprobable signaling X n asymptotically approaches the AWGN channel capacity. Fig. 2 depicts the gap to Gaussian capacity for the uniformly spaced signal with equiprobable signaling and the proposed constellation for various SNR values and 2 n 35. As expected, the capacity gap vanishes for the proposed constellation as n → ∞. However the convergence speed remains an open question.

III. CONCLUDING REMARKS

Based on a shaping function, we introduce a constellation design enabling to achieve the Gaussian capacity as the signal cardinality grows to infinity. In that sense, the proposed constellations are better than uniformly spaced signals with equiprobable signaling. We illustrate our result by presenting and evaluating the performance of a sequence of capacityapproaching signals. It is important to note that the design is not unique as modifying the set S n , with respect to the assumptions given in the theorem, may generate other constellations that asymptotically achieve the Gaussian capacity. Our last remark is to point out that our solution can produce capacity-approaching constellations in higher dimensions. For instance in two dimensions, we can take two capacityachieving signals (for the one-dimensional case) transmitted on two quadrature carriers. Future work will focus on three main directions. First, we plan to study the convergence speed of the proposed constellations. For the scalar AWGN channel with input cardinality n, Wu and Verdú showed that the achievable rate approaches exponentially fast the Gaussian capacity as n grows [START_REF]The impact of constellation cardinality on Gaussian channel capacity[END_REF]. However their solution combine constellation shaping and non-uniform probabilities, while we restrict our work to constellation shaping which is more attractive from a practical point of view. Then, we will seek the optimal constellations in terms of capacity (for a given SNR) or convergence speed. Finally, we will investigate if the representation of the input signal X n using U n may accelerate the decoding process.

Fig. 1 :

 1 Fig.1: Proposed Gaussian-shaped constellation for n = 10 and P = 4. On the vertical axis, the squares represent the support of U n . On the real axis, the circles represent the support of Φ -1 P (U n ), in other words the constellation C n .


Francis Young, English Catholics and the Supernatural, 1553-1829. Catholic Christendom, 1300-1700. Farnham: Ashgate, 2013, xii + 308 pp. ISBN 978-1-4094-5565-3. 96,32 € In this monograph, Francis Young acts upon one of the key observations he had made in an article on exorcism in the early modern English Catholic mission: there appeared to be no full-length studies of the attitudes of English Catholics to the supernatural. Scholars had offered illuminating analyses of the Protestant views of Catholic involvement with the supernatural, but nothing had been published to document at length how Catholics themselves perceived and experienced supernatural phenomena in their daily lives, or to determine the part played by supernatural beliefs in the 'old' faith. Steering away from an analysis of Protestant polemics about papist beliefs and practices, this study finally addresses the attitudes of English Catholics (at home and in exile) to the broad, multifaceted and rather nebulous concept of the supernatural.

The book is ambitious in covering a period which ranges from the accession of Mary Tudor in 1553 to the political Emancipation of 1829. Yet this time-span is one of its strengths, since it allows the author to develop his study over the longue durée and to nuance the changes which affected English Catholic beliefs from the Reformation to the Enlightenment. This approach, however, poses some difficulties in terms of the construction of the book. The author chose to dedicate two distinct chapters (chapter 1 and 2) to the Catholic treatment of 'superstition', dealing first with early modern attitudes, and second with the shifts which occurred during the Enlightenment period. The same division between early modern and enlightenment occurs in chapters 4 and 5, where the author considers the English Catholic approach to witchcraft and magic. The subject of ghosts and apparitions, however, warrants only a single chapter (chapter 3) in which the authors deals with ghost stories from the sixteenth to the eighteenth century; the topic of the devil and exorcism is also dealt with in one single chapter (chapter 6), and concluded the study.

The subject of English Catholic attitudes to the supernatural is as ambitious as the time-span chosen, and Francis young offers no clear-cut answers here. What he does do, however, is much more interesting: through remarkably detailed case studies, he demonstrates that there were many approaches to the supernatural. Throughout the study, it becomes increasingly clear that credulity and skepticism existed concomitantly, and right from the beginning of the period. The author demonstrates that some of the conclusions published in Keith Thomas's ground-breaking Religion and the Decline of Magic (1971) were somewhat over-simplified and deserve nuancing. He makes use of Latin primary sources which, as he correctly argues, have been too long overlooked by historians of English Catholicism. Indeed, Latin was the language of choice by many Catholics. Decades of neglect have skewed our perception of Catholic attitudes and beliefs, silencing the authors who published in Latin.

Francis Young shows that, in the early modern period, Jansenists and Jesuits, or Cisalpines and Ultramontanes did hold the same views in the subject of the supernatural. The boundary between faith and superstition a point of debate in which some argued for skepticism and whilst others embraced elements of the preternatural more willingly. In the same way as the English Catholic community should be viewed as plural rather than monolithic, its attitudes to the supernatural varied greatly, not only over time but also depending on regional folklore or social and cultural factors. Whilst this conclusion should hardly come as a surprise, Young's archival work provides lively vignettes which make the book thoroughly engaging and a pleasure to read.

The major contribution of the monograph is to be found in its passages on the specific Englishness of Catholic attitudes to the supernatural. From the various case studies presented, Young posits that the beliefs and practices of English Catholics did not differ greatly from their Protestant compatriots. When ghosts were concerned, Catholics and Protestants reacted in similar ways, and constructed comparable stories, with comparable embellishments added in the Romantic age. Surprisingly, the quintessential disagreement about the existence of purgatory did not seem to impact attitudes towards ghost-related folklore. When considering witchcraft, the same conclusion applies. Catholics and Protestants held compatible views: the importance of the pact with the devil was eroded in England, the Sabbath played little or no part in the national imagination, and witchcraft accusations were no more prominent in Catholic than in Protestant communities. If anything, belief and practice regarding witchcraft present a certain Englishness which differentiates them from attitudes on the Catholic Continent. Thus, although anti-Catholic polemicists often heavily implied a quasi-organic link between witchcraft and what they described as an obscurantist, idolatrous faith, it seems that English Catholics could be as inclined to skepticism as their Protestant neighbours.

On the subject of exorcism, however, this argument is a little less valid. This is the only topic in this book which has previously been tackled by other scholars. Alexandra Walsham, for instance, argued most convincingly that the practice of exorcism served a pragmatic purpose in the mission. Its aim was to convince crowds, through somewhat sensationalist shows, that the Catholic clergy possessed really and effective power against the forces of evil. Exorcism was an effective tool in the confessional propaganda of the English mission, and exorcists have so far been shown as zealous proselytizing agents of the Counter-Reformation. Francis Young tries to nuance this view by emphasizing a certain continuity with long-held beliefs in the correspondence of physical and spiritual ailments; yet it seems quite indisputable that exorcism was clearly a practice where Protestant and Catholic beliefs differed greatly in England. Some passages in this book are less convincing than others. For instance, whilst it made sense to avoid complicated conceptual definitions and to propose a broad working understanding of the supernatural as comprising all preternatural and supernatural phenomena, it was curious to see the author discuss popular belief in the Benedictine knowledge of herbal remedies as actual 'superstition'. The main criticism however has to be the lack of a clear conclusion: the quality of the work presented over more than two hundred pages deserved as much. Despite this regrettable oversight, this monograph remains a thought-provoking and appealing read, which offers a rich, well-documented and intelligent analysis of a so-far neglected subject.

Laurence Lux-Sterritt Aix Marseille Univ, LERMA, Aix-en-Provence, France


Bachir Habib
email: bachirhabib@usek.edu.lb

Badih Baz
email: badihbaz@usek.edu.lb

Hardware MIMO channel simulator for cooperative & heterogeneous 5G networks with VLC signals

Keywords: Hardware Simulator, FPGA, MIMO, Wireless Heterogeneous Systems, Cooperative Networks

This paper presents the design of an 8x8 Multiple-Input Multiple-Output (MIMO) hardware simulator for cooperative time-varying propagation channels (based on relays) using heterogeneous systems (LTE-A and 802.11ac) with Visible Light Communications (VLC) signals. The simulator reproduces a desired radio channel scenario and makes it possible to test "on table" different systems. It uses different impulse responses to cover many types of channels by merging TGn models, 3GPP-LTE models, outdoor-to-indoor measurements results and VLC models. An algorithm is introduced to convert the measured impulse responses from 200 MHz sample frequency to 50 MHz to be compatible with LTE-A signals. A specific architecture of the simulator digital block is presented to characterize 5G scenarios. The architecture is designed on a Xilinx Virtex-VII XC7VX690T Field Programmable Gate Array (FPGA). Its accuracy and FPGA occupation are analyzed.

Introduction

MIMO techniques [START_REF] Behbahani | Optimizations of a MIMO relay network[END_REF], cooperative networks and heterogeneous systems [START_REF] Kwadwo Agyapong | Design Considerations for a 5G Network Architecture[END_REF][START_REF]LTE Advanced: Heterogeneous Networks[END_REF] became essential in the design of the 5G communication systems. Moreover, VLC technologies [START_REF] Cossu | VLC-signals distribution over GI-POF for in-home wireless networks[END_REF][START_REF] Kim | VLC-TDOA Using Sinusoidal Pilot Signal[END_REF][START_REF] Yu | Using Delta-Sigma Modulators in Visible Light OFDM Systems[END_REF][START_REF] Arnon | Visible Light Communication[END_REF] (LED or laser based), that are under development, show that they could be more than 100 times faster than current Wi-Fi. Therefore, all these technologies have to be considered in the design of any channel simulator platform that makes it possible to predict the received signal for any transmitted signal for a defined environment. With continuing increase of the FPGA capacity, entire baseband simulators can be mapped onto faster FPGAs for more efficient prototyping and testing [START_REF]Xilinx: FPGA, CPLD and EPP solutions[END_REF][START_REF] Murphy | An FPGA based rapid prototyping platform for MIMO systems[END_REF][START_REF]Wireless Channel Emulator[END_REF][START_REF]Baseband Fading Simulator ABFS, Reduced costs through baseband simulation[END_REF]. Several architectures of the digital block of a hardware simulator have been studied [START_REF] Murphy | An FPGA based rapid prototyping platform for MIMO systems[END_REF][START_REF] Fouladi Fard | A single FPGA filterbased multipath fading emulator[END_REF]. Typically, radio channels are simulated using Finite Impulse Response (FIR) filters [START_REF] Fouladi Fard | A single FPGA filterbased multipath fading emulator[END_REF][START_REF] Eslami | Design and implementation of a scalable channel Emulator for wideband MIMO systems[END_REF]. The Fast Fourier Transform (FFT) module with algebraic product can also be used. However, the FFT operates correctly only for signals not exceeding the FFT size. Thus, new freque has been presented and tested in [1 time domain architecture is better in terms of occupation on FPGA, output error and latency. Thus, in this paper, only the time do Impulse responses must be implemented in the simulator to test a desired environment. They can be obtained from standard models, as the TGn 802.11n [1 (for indoor channels) and the LTE models [1 measurements conducted with a MIMO chan cover outdoor-to-indoor have been obtained regarding time channel sounding equipment [1 channels can be obtained using Rayleigh fading The main contributions of the paper are:

• An algorithm is presented to switch between several outdoor environm make it possible to simulate heterogeneous radio propagation channels. • An algorithm is used for t MHz sampling frequency ( • Two new 5G scenario switch between WLAN and mobile networks. networks by using cell phones in the environment as relays. • Simulation of 5G 8x8 MIM • The new simulator architecture is implemented on a better precision compared to [START_REF] Habib | MIMO hardware simulator: new digital block design in frequency domain for streaming signals[END_REF][START_REF] Habib | MIMO hardware simulator: digital block design for 802.11ac applications with TGn channel model test[END_REF][START_REF] Habib | Digital block design of MIMO hardware simulator for LTE applications[END_REF] due to the use of 16 bits DAC/ADC. The rest of this paper is organized as follows. Section II presents the channel characteristics. Section III describes the architecture and its hardware implementation. In Section IV, the architecture are presented. Lastly, Section V give exceeding the FFT size. Thus, new frequency architecture avoiding this limitation has been presented and tested in [START_REF] Habib | MIMO hardware simulator: new digital block design in frequency domain for streaming signals[END_REF]. However, it has been shown [START_REF] Habib | MIMO hardware simulator: digital block design for 802.11ac applications with TGn channel model test[END_REF][START_REF] Habib | Digital block design of MIMO hardware simulator for LTE applications[END_REF] that the time domain architecture is better in terms of occupation on FPGA, output error , in this paper, only the time domain architecture is considered. Impulse responses must be implemented in the simulator to test a desired environment. They can be obtained from standard models, as the TGn 802.11n [START_REF] Erceg | TGn Channel Models[END_REF] and the LTE models [START_REF]Advanced design system -LTE channel model[END_REF] (for outdoor channels), or by using measurements conducted with a MIMO channel sounder realized at IETR [START_REF] Farhat | A dual band MIMO channel sounder at 2.2 and 3.5 GHz. IMTC[END_REF] to indoor channels. In MIMO context, little experimental results have been obtained regarding time-variations due to several limitations of the unding equipment [START_REF] Farhat | A dual band MIMO channel sounder at 2.2 and 3.5 GHz. IMTC[END_REF]. However, theoretical models of time-varying channels can be obtained using Rayleigh fading and Kronecker method [START_REF] Almers | Survery of channel and Radio propagation models for wireless MIMO systems[END_REF][START_REF] Schumacher | From antenna spacing to theoretical capacities -guidelines for simulating MIMO systems[END_REF]. The main contributions of the paper are:

The work structure with

An algorithm is presented to switch between several outdoor environments and make it possible to simulate heterogeneous radio propagation channels.

used for the measured impulse responses to switch from 200 MHz sampling frequency (f s ) to 50 MHz and make it compatible with LTE-A.

new 5G scenarios are proposed. They consider heterogeneous networks that between WLAN and mobile networks. They also consider cooperative networks by using cell phones in the environment as relays. Simulation of 5G 8x8 MIMO systems using VLC Li-Fi signals is considered.

architecture is implemented on a Virtex VII FPGA. It has better precision compared to [START_REF] Habib | MIMO hardware simulator: new digital block design in frequency domain for streaming signals[END_REF][START_REF] Habib | MIMO hardware simulator: digital block design for 802.11ac applications with TGn channel model test[END_REF][START_REF] Habib | Digital block design of MIMO hardware simulator for LTE applications[END_REF] due to the use of 16 bits DAC/ADC. with the contributions is shown in Fig. 1. s paper is organized as follows. Section II presents the channel characteristics. Section III describes the architecture and its hardware implementation. In Section IV, the architecture accuracy is analyzed and results

. Lastly, Section V gives concluding remarks and prospects. Fig. 2 and Table 1 describe the two scenarios in details. In both scenarios, the person driving his vehicle moves from an urban environment to a highway. In this context, another base station is used as a relay. The signal transmitted between the 2 base stations (in LOS) is done using Li-Fi red laser to have fast communication and to surpass the frequency capacity spectrum. On the highway he exits the covered area by the base station. Therefore, the cooperative networks concept is used. Another person's cell phone between the transmitter (Tx) and the receiver (Rx) is used as a relay. It receives the signal, amplify it and retransmit it.

In the first scenario, the person (in red in Fig. 1) walks to his home. When he enters, he will receive an outdoor-to-indoor signal. Therefore, the channel will be simulated using measurements obtained by a campaign realized to retrieve outdoorto-indoor impulse responses. Later, the system switches to LED Li-Fi once receiving a powerful signal using heterogeneous systems handover techniques [START_REF]LTE Advanced: Heterogeneous Networks[END_REF].

In the second scenario, the person (in black in Fig. 1) switches to 802.11ac signal. This case is considered to simulate heterogeneous systems without VLC signals.

For LOS communications, as in the case of VLC Li-Fi red laser signal, the impulse responses are computed by using the relation d -γ between the average received power and the distance to determine the attenuation of the signal, where γ is the path loss exponent and it is equal to 2. The delay of the signal is computed by considering the speed of light as a reference.

The models used are the LTE Extended Vehicular A (EVA), the LTE Extended Typical Urban (ETU), the LTE Extended Pedestrian A (EPA) [START_REF]Advanced design system -LTE channel model[END_REF] and the TGn model D, C and B [START_REF] Erceg | TGn Channel Models[END_REF]. v is the mean environmental speed, f ref is the refresh frequency between two successive MIMO profiles, t is the duration of movements in the considered environment, d is the traveled distance and N p = t× f ref is the number of profiles in each environment. f d is the Doppler frequency. f ref is chosen greater than 2f d to respect the Nyquist-Shannon sampling theorem.

Update Measurements to LTE-A and environment switch Algorithms

The measurements, using the MIMO channel sounder, were made with f s =200 MHz. To use these measurements with LTE-A signals, many steps have to be done.

To pass from baseband complex impulse responses to real impulse responses between [∆, ܤ ∆], the following formula is used where f c =∆+B/2:

݄ሺݐሻ ൌ ݄ ሺݐሻ ݂ߨ2ሺݏܿ ݐሻ െ ݄ ሺݐሻ ݊݅ݏሺ2ߨ݂ ݐሻ (1)
The obtained real impulse responses must be limited between 0 and -20 dB to select its significant part. Moreover, to decrease the number of non-null taps of the impulse responses, an algorithm is used which consists of detecting the taps considered as points of change for the sign of the slope of the curve. The switch between the environments must be made in continuous mode. Between E 10 and E 11 for example, the person speed decelerate from 2 (km/h) to 1.4 (km/h).To pass from the RP of the last tap h ij in E 10 (RP

- - × - + = N RP RP i RP RP f i , 1 ) 1 ( 1 1 - - × - + = N f f i f f ref f ref ref i ref ( 2
)
where i is an integer that varies from 1 to N. N = 80, in fact, 80/((20+15)/2) = 4.5 s is the needed time to switch between the impulse responses which is sufficient to pass from v = 2 km/h (E 10 ) to v= 1.4 km/h (E 11 ). Fig. 4 presents the result of the 80 impulse responses (h) while switching between channels.

Channel Real Time Variation using Rayleigh and Kronecker Method

After computing the impulse responses, this sub-section presents a method to vary the channel. The Rayleigh fading is used of each SISO channel. For 2×2 MIMO fading channel, Kronecker method [START_REF] Schumacher | From antenna spacing to theoretical capacities -guidelines for simulating MIMO systems[END_REF] is considered. It can be characterized by the relative power P c (resp. P s ) of constant (resp. scattering) channel components corresponding to LOS (resp. NLOS) paths. The ratio P c /P s is the Rice K-factor. The same concept is applied to 8x8 MIMO. Assuming that all the elements of the MIMO channel matrix H are Rice distributed, it can be expressed for each tap by:

v s F c H P H P H . . + = with , 1 . + = K K P P c 1 1 . + = K P P s (3)
where H F and H V are the constant and the scattered matrices respectively. The total received power is P = P c + P s . K = 0 to obtain Rayleigh fading. For 2 Tx/Rx:

      = 22 21 12 11 . X X X X P H (4)
To correlate the X ij elements, a product-based model is used. This model assumes that the correlation coefficients are independently derived at each end of the link:

T t iid r R H R X ) ) .(( . ) ( 2 / 1 2 / 1 = where       = 1 1 * α α t R ,       = 1 1 * β β r R ( 5
)
H iid is a matrix of independent zero means, unit variance, complex Gaussian random variables. R t and R r are the Tx/Rx correlation matrices. The correlation coefficients ߙ/ ߚ are expressed byߩ ൌ ܴ ௫௫ ሺܦሻ ݆. ܴ ௫௬ ሺܦሻ, D = 2πd/λ, d = 0.5λ the distance between antennas with wavelength λ. The Power Angular Spectrum (PAS) matches the Laplacian distribution with σ the standard deviation. Thus:

∫ - = π π ϕ ϕ ϕ d PAS D D R xx ). ( )). sin( . cos( ) ( , ∫ - = π π ϕ ϕ ϕ d PAS D D R xy ). ( )). sin( . sin( ) ( (6)

Digital Block Design of the Hardware Simulator

As the development board has 8 Analog-to-Digital Converter ADC (FMC168) and 8 Digital-to-Analog Converter DAC (FMC216), it will be connected to 8 downconversion and 8 up-conversion RF units. 64 FIR filters are considered to simulate the 8×8 MIMO channel. However, as the scenarios contain one relay (intermediate base station or cell phone), the number of FIR filters will be 64*2=128 (Fig. 5). The FIR width and the multipliers number are determined by the taps of each SISO channel. In a scenario, the entire SISO channels have to be considered. To use the limited number of multipliers of the FPGA, a solution is proposed to control the change of delays by connecting each multiplier to the corresponding cells of the shift register and RAM. Thus, the number of multipliers in the FIR filters is equal to the maximum number of taps of all SISO channels. 128 FIR filters, each one with 24 multipliers are considered. Fig. 6 presents a FIR filter 250 with 24 multipliers for one SISO channel. We have developed our own FIR filter instead of using Xilinx filter to make it possible to reload the filter coefficients. Due to the use of a 16-bit DAC, the final output must be truncated. The best solution is the sliding window truncation presented in Fig. 7 which uses the 16 most significant bits.The number of bits at the output before the truncation is:

T MIMO x h y n n n n n + + + = where   T MIMO N n 2 log = ,   T T N n 2 log = (7)
where n y is the number of bits at the output before truncation, n h is the number of bits at the impulse response, which is equal to 16 bits, n x = 16 is the number of bits of the input signal and N T is the number of non-null taps of impulse responses.

n(i) n(i-1) n(i-2) -----------------------------n(i-249) h(0) h(1) h(2) ----------------------------- h(249) h(0) h(1) h(2) ------------------------------ h (

0001010011001100001…

Truncation DAC Fig. 8 shows the Xilinx Virtex-VII XC7VX690T [START_REF]Xilinx: FPGA, CPLD and EPP solutions[END_REF] used for the implementation of the architecture. The simulations are made with ISE [START_REF]Xilinx: FPGA, CPLD and EPP solutions[END_REF] and ModelSim [START_REF]ModelSim -Advanced Simulation and Debugging[END_REF]. It also shows the device utilization for 128 FIR filters of 24 multipliers each. However, the total number of RAM blocks on the FPGA is 2,940. Therefore, another method is used to save the impulse responses. They are saved on the computer and refreshed each f ref .

In this way, only 2 profiles are needed to be saved in the RAM blocks of the FPGA. In this case, the total number of taps is: 2*128*24*2=12,288 taps of 16 bits = 196608 total bits = 11 RAM blocks < 2,940.

For a MIMO profile, 24×128 = 3072 words of 16 bits = 6144 bytes have to be transmitted. Thus, the transfer speed is 6144×f ref Bps.

Results and Analysis

An input Gaussian signal x(t) is considered for all inputs. To obtain a signal that respects the bandwidth, x(t) centered between [ ∆ , ∆ +B] is obtained by:

              ∆ + → ⇒ ≤ ≤ = - - t B t x t x Wt t e x t x x x m t m . 2 . 2 cos ). ( ) ( , 0 , ) ( 2 2 2 ) ( π σ (8)
In this work, ߪ ݔ =3/ߨ B is considered. m x is chosen equal to 20T s >3 for both 802.11ac and LTE-A signals. Moreover, ∆ <<B is chosen equal to 2 MHz. The

ADC and DAC converters of the development board have a full scale

[-V m ,V m ], with V m = 1 V. For the simulations, x m = V m /2 is considered.
The first theoretic output signals y 1 (t) of the 8×8 MIMO channel is computed by:

) ( * ) ( ... ) ( * ) ( ) ( * ) ( ) (
∑ ∑ = = = 8 1 8 1 ) ( * ) ( * ) ( ) ( 1 2 ∑ = - = Nt k s k k T i t x i h h x 1 ) ( ). ( * (11)
The formula of the theoretical y k (t) becomes:

∑∑ ∑∑

= = = =               + - = 8 1 1 8 1 1 ) ) ( ( ). ( ). ( ) ( 1 2 b Nq q a Np p s p q a p ab E q bk E k T i i t x i h i h t y (12)
where N p and N q are the number of taps of the impulse response and h(i n ) is the RP of the n th path with the delay i n T s .

In order to determine the accuracy, a comparison is made between the theoretical/Xilinx output signals. Fig. 9 presents a snapshot of the Xilinx output signal using Brutal (BT) and Sliding (ST) Truncations, with the theoretical signal, the relative error and the SNR for y 1 (t) using measured outdoor-to-indoor data (left) and 3GPP-LTE model ETU (right) with and x LTE (t) at the input. Fig. 10 presents a snapshot of the result for y 1 (t) using TGn model B (left) with x WLAN (t) and VLC LED Li-Fi (right). The relative error and SNR are computed by: , 100 ) (

) ( ) ( ) ( × - = i y i y i y i RE k k qk k [ ] dB i y i y i y i SNR k qk k k ) ( ) ( ) ( log 20 ) ( 10 - = (13)
where y qk is the vectors containing the samples of the Xilinx output signals. The table shows the benefit of a ST in the case of using real input signals that respects the band [ ]

B + ∆ ∆,
. The previous figures of the relative errors show that the ST provides low variations (around zero %) for the relative error where the output signal is high. However, the BT presents high variations.

Conclusion

In this paper, tests have been made for scenario models that switch between different environments to simulate heterogeneous propagation environments and cooperative networks. Moreover, VLC signals are used. An algorithm has been proposed to switch between the environments in a continuous manner and another algorithm is introduced to convert the measured impulse responses from 200 MHz sample frequency to 50 MHz to be compatible with LTE-A signals. In order to simulate a time-varying channel, Rayleigh fading method is used to vary the relative power of the taps of the channel impulse response. The architecture of the 8x8 MIMO simulator has been implemented on FPGA Virtex VII and analyzed.

This work can be completed by developing RF cards for the new standards and for heterogeneous systems. The hardware simulator realized considers the variation of the channel on small scale. Thus, variation on large scale must be considered. For the future cooperative communication systems, the Tx/Rx are in motion, the V2V concept has to be considered. Finally, the Tx/Rx distance has to be considered and simulation must be made not just for scenario models but also for real scenarios.

Fig. 1 .

 1 Fig. 1. Work structure.

Fig. 2 .

 2 Fig. 2. Proposed scenarios.

→VLC

  Li-Fi red laser →LTE-A, fs=50 MHz →VLC Li-Fi red LED →802.11ac, fs=165 MHz •Wi-Fi 802.11ac pass from f s = 200 MHz to 50 MHz, an algorithm is proposed. It consists on replacing each 4 successive taps (i, i+1, i+2, i+3), which have a sampling period T s = 5 ns (f s = 200 MHz), by one tap placed at i+1. Thus, a new sampling period T s = 20 ns (f s = 50 MHz) is obtained. This new tap placed at i+1 is the sum of the linear Relative Powers (RP) of the 4 taps. A normalization algorithm is preceded on the RP to increase the simulator output signals precision. The results are obtained for 8×8 MIMO system. Fig. 3 presents the real impulse response on 2048T s , the impulse response used by the simulator after discrimination, normalization and limitation between [0, -20 dB] of the measured impulse responses with f s = 200 MHz and the impulse responses compatible with LTE-A with f s = 50 MHz, for h 11 . Moreover, the number of multipliers also decreases. In fact, it is divided by 4 due to the transition from f s = 200 MHz to f s = 50 MHz.

Fig. 3 .

 3 Fig. 3. Data Measured (left), used with f s =200 MHz (middle), used with f s =50 MHz (right).

Fig. 4 .

 4 Fig. 4. Switching between TGn model B to TGn model C.

Fig. 5 .

 5 Fig. 5. 8x8 MIMO system with relay.

Fig. 6 .Fig. 7 .

 67 Fig. 6. FIR 250 with 24 multipliers for one SISO channel using h 11 .

Fig. 8 .

 8 Fig. 8. Virtex-VII FPGA board from Xilinx and FPGA occupation for 128 FIR filters.The number of DSP multipliers is equal to: 24 taps * 128 FIR filters = 3072. The impulse responses profiles are saved in the RAM memory blocks. The total number of profiles in the scenarios is 745,506. Each profile contains 128 channels each with 24 taps. For each tap the delay in time and the signal power must be known. Therefore, the total number of taps is: 745,506*128*24*2= 4,580,388,864 taps of 16 bits each = 73,286,221,824 total bits. The capacity of each RAM block is 18Kb. Thus, the number of blocks needed is: 73,286,221,824/18K = 4,071,457.

Fig. 9 .

 9 Fig. 9. Snapshot results for y 1 using measured outdoor-to-indoor data (left) and 3GPP-LTE model ETU (right).

Fig. 10 .

 10 Fig. 10. Snapshot results for y 1 using TGn model B (left) and VLC LED Li-Fi (right).

Fig. 11

 11 Fig.11presents the Global SNR variation in time for all profiles of y 1 (t) and y 2 (t) of four different environments of the scenarios. For the ETU environment only 1,000 profiles of 90,000 are presented.

Fig. 11 .

 11 Fig. 11. Global SNR variation in time for y 1 and y 2 .

Table 1 .

 1 Scenarios description.

	2	14,400

Scenario/Env. Model v (km/h) fc (GHz) fd (Hz) fref (Hz) t (s) d (km) Np 1/ E1 E3 Li-Fi laser*ETU 0 ; 40 472T;1.8 0 ; 66 0 ; 150 600 6.6 90,001 1/ E1 E4 Li-Fi laser*EVA 0 ; 80 472T;1.8 0 ; 133 0 ; 275 600 13.3 165,001 1/ E5 E6 EPA*EVA 4 ; 80 1.8 ;1.8 18.3;133 40 ; 75 300 6.6 94,500 1/ E5 E8 EPA*EPA 4 ; 4 1.8 ;1.8 18.3;18.3 40 ; 40 180 0.2 14,400 1/ E2 E12 Li-Fi laser*EPA 0 ; 4 472T;1.8 0 ; 18.3 0 ; 40 180 0.2 7,201 1/ E2 E13 Li-Fi laser*Measures 0 ; 2 472T;1.8 0 ; 9.1 0 ; 20 60 0.03 1,201 1/ E14 Li-Fi LED 1.4 472T 6.5 15 30 0.01 450 2/ E1 E3 Li-Fi laser*ETU 0 ; 40 472T;1.8 0 ; 66 0 ; 150 600 6.6 90,001 2/ E1 E4 Li-Fi laser*EVA 0 ; 80 472T;1.8 0 ; 133 0 ; 275 600 13.3 165,001 2/ E5 E6 EPA*EVA 4 ; 80 1.8 ;1.8 18.3;133 40 ;275 300 6.6 94,500 2/ E5 E7 EPA*EPA 4 ; 4 1.8 ;1.8 18.3;18.3 40 ; 40 180 0.

Table 1 .

 1 1 ) to the first h ij tap in E 11 (RP f ), a relation is proposed that varies the RP for each f ref . f ref also changes as in It passes from 20 (

Table 2

 2 shows the average total global SNR for different environments. The results are given with BT and ST.

Table 2 .

 2 Global SNR (dB) for different environments of the scenarios.

		Outdoor-to-indoor	ETU	TGn model B	VLC LED
	BT	34.32	49.15	47.12	52.38
	ST	61.77	72.68	73.38	75.31


Vasileios Karyotis

Eleni Stai
email: eleni.stai@epfl.ch

Symeon Papavassiliou

On the Evolution of Complex Network Topology under Network Churn

Keywords: Network evolution, Complex networks, Social network analysis metrics, Future Internet, Edge churn

The future Internet is becoming more diverse, incorporating heterogeneous access networks. The latter are characterized by numerous devices that join/leave the network dynamically, creating intense churn patterns. New approaches to analyze and quantify churn-induced network evolution are required. In this paper, we address such need by introducing a new analysis framework that maps network evolution into trajectories in multi-dimensional vector spaces. Each network instance is characterized by a feature vector, indicating network properties of interest. To demonstrate the potentials of this approach, we exemplify and study the effect of edge churn on various complex topologies, frequently emerging in various communications environments. We investigate via simulation the impact of network evolution, by quantifying its effect on key network analysis metrics, such as the clustering coefficient and the plethora of centrality metrics, employed at large for analyzing topologies and designing applications. The proposed framework aspires to establish more holistic and efficient complex network control.

Introduction

Future networks are expected to be highly heterogeneous, consisting of different complex topologies with diverse characteristics and behaviors. For example, 5G is expected to consist of a wired-wireless broadband core, while assuming wireless heterogeneous access networks at the periphery, characterized by numerous simple devices, e.g., Internet of Things (IoT) [START_REF] Al-Fuqaha | Internet of Things: A Survey on Enabling Technologies, Protocols, and Applications[END_REF], fog networks [START_REF] Bonomi | Fog Computing and its Role in the Internet of Things[END_REF], etc. Due to pushing cloud services towards the access network [START_REF] Bonomi | Fog Computing and its Role in the Internet of Things[END_REF] and a multitude of factors, such as operational environment, user trends and device heterogeneity, it is anticipated that the on/off behavior (churn) of network nodes and corresponding communication links between them will become critical for the feasibility and scaling of future resource allocation/optimization mechanisms, signalling, etc. For this reason, network evolution as a consequence of network churn 3 , will be one of the key dynamic processes that will impact and possibly determine the success of future network design/planning efforts. Network design and optimization will need to take into account the anticipated churn rates in order to provide services at low cost and with predictable resource demands.

The impact of network churn on network structure, management and application services has been more or less neglected in the literature. Only a few scattered works have addressed it under targeted settings. The authors in [START_REF] Kar | Dynamic Node Activation in Networks of Rechargeable Sensors[END_REF] have focused on sensor network churn induced by energy depletion, addressing it via a queuing framework. Churn of users in online social networks, i.e., how users enter-leave the system have been studied in [START_REF] Stutzbach | Understanding Churn in Peer-to-peer Networks[END_REF][START_REF] Karnstedt | Churn in Social Networks[END_REF]. In [START_REF] Holzer | Monitoring Churn in Wiireless Networks[END_REF] a mechanism for network robustness against node churn in single-hop wireless networks was developed, while [START_REF] Karyotis | Macroscopic Malware Propagation Dynamics for Complex Networks With Churn[END_REF] addressed the impact of node churn on malware spreading under random attacks. The scope of our work is broader, aiming at extending the study of churn-induced network evolution to arbitrary complex networks and eventually more general churn mechanisms.

More specifically, in this paper, we first introduce a general analysis framework for evolving networks, and then use it to study the impact of churn on various complex network topologies that emerge as components in current and future networks. The proposed framework maps topology evolution into multidimensional vector trajectories and quantifies the similarity of different topologies. We analyze the impact of evolution on random, scale-free and small-world topologies via various graph metrics typically employed for social/complex network analysis, i.e., average values of node degree, path length, clustering coefficient and centrality measures, thus obtaining a clearer picture of the impact anticipated by network churn. The latter is considered in the form of random edge (link) churn and various types of preferential edge churn, namely churn based on degree, closeness and betweenness centrality metrics. We analyze the impact of churn on each topology and compare their evolution cumulatively, thus obtaining useful knowledge for efficient network design and planning.

The rest of this paper is structured as follows. Section 2 summarizes related work and background on complex networks, analysis metrics and churn, and explains the contribution of this paper. Section 3 introduces the proposed network evolution analysis framework, while Section 4 describes the considered edge churn processes. Section 5 provides extensive simulation results on synthetic complex network topologies analyzing the impact of churn on their evolution. Finally, Section 6 concludes the paper and provides directions for future research.

Network Evolution and Complex Network Analysis Metrics

Network Evolution and Churn

Networks in general, and computer networks in particular, are characterized by various forms of topological and operational evolution [START_REF] Dorogovtsev | Evolution of Networks: From Biological Nets to the Internet and WWW[END_REF]. For instance, different traffic patterns emerge as networks grow, or as user habits change over time. Network evolution can involve variations of nodes/users, or most frequently connection links (edge) variations. With respect to the latter, the authors in [START_REF] Papavassiliou | A Continuum Theory based Approach to the Modeling of Dynamic Wireless Sensor Networks[END_REF] proposed a continuum model for dynamic wireless networks, assuming that communications links change in a continuous fashion. They formulated network evolution through differential equations and obtained the long-term behavior of the average node degree. The work in [START_REF] Leskovec | Graphs Over Time: Densification Laws, Shrinking Diameters and Possible Explanations[END_REF] studied the evolution of real networks, implicitly focusing on power-law (scale-free) topologies, with respect to the metrics of the average path length, network diameter and node degree. The evolution was analyzed under the regime of network growth, which is realistic for commercial communications networks. In [START_REF] Stai | Topology Enhancements in Wireless Multihop Networks: A Top-Down Approach[END_REF][START_REF] Karyotis | Evolutionary Dynamics of Complex Communications Networks[END_REF] network evolution models for wireless multihop networks were considered, and especially, a methodology for adding features typical of social networks (small-world characteristics, etc.) into multihop networks via network churn evolution mechanisms was proposed. Compared to the above works addressing various facets of network evolution, our work attempts a more holistic consideration. More specifically, in contrast to [START_REF] Papavassiliou | A Continuum Theory based Approach to the Modeling of Dynamic Wireless Sensor Networks[END_REF], we consider more network analysis metrics and various complex network topologies for random and preferential edge churn. Like [START_REF] Leskovec | Graphs Over Time: Densification Laws, Shrinking Diameters and Possible Explanations[END_REF], we focus on the impact of churn on complex topologies that emerge at various capacities in real networks, but in a broader setting where more topologies and analysis metrics are considered. Finally, compared to [START_REF] Stai | Topology Enhancements in Wireless Multihop Networks: A Top-Down Approach[END_REF][START_REF] Karyotis | Evolutionary Dynamics of Complex Communications Networks[END_REF], this paper focuses on studying how diverse topology transformations impact the network properties, rather than imposing specific churn-based network evolution as in [START_REF] Stai | Topology Enhancements in Wireless Multihop Networks: A Top-Down Approach[END_REF][START_REF] Karyotis | Evolutionary Dynamics of Complex Communications Networks[END_REF]. In essense, this work aspires to pave the way for a more general topology control framework, by enabling controlled network evolution towards desired topologies, for arbitrary initial complex topologies and several analysis metrics, as each application setting specifies.

Complex and Social Network Analysis Metrics

Various metrics have been employed for complex and social network analysis. The most significant ones have been the node degree and the associated degree distribution, the average path length, the clustering coefficient (CC) and the variants of centrality measures [START_REF] Karyotis | Evolutionary Dynamics of Complex Communications Networks[END_REF][START_REF] Newman | The structure and function of complex networks[END_REF]. Some of these metrics refer to network nodes independently, others to the overall network, and some to both. Thus, the CC and centrality can be defined for individual nodes and also computed as network averages. Node degrees refer to individual nodes, but the node degree distribution uniquely characterizes a specific type of network topology. The average path length is a network wide metric, as well.

The clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. For a network node, the local CC is given by the number of links between the nodes within its neighborhood divided by the number of links that could possibly exist among them [START_REF] Karyotis | Evolutionary Dynamics of Complex Communications Networks[END_REF]. Centrality is a measure of node importance in a network. Since the latter is subjective in many application settings, there exist various centrality metrics, e.g., degree centrality (a normalized version of node degree), closeness centrality, betweenness centrality, eigenvector centrality, etc. [START_REF] Karyotis | Evolutionary Dynamics of Complex Communications Networks[END_REF]. In this paper we focus on degree, betweenness and closeness centrality metrics, but other variations/definitions of centrality can be used in a similar fashion. Node betweenness centrality is defined as a normalized sum of the percentages of the number of shortest paths between all pairs of vertices that pass through that node. Various approaches have been employed for computing efficiently such type of centrality, including variants of its definition [START_REF] Brandes | On Variants of Shortest-Path Betweenness Centrality and their Generic Computation[END_REF]. Node closeness centrality is typically defined as the normalized reciprocal of the sum of distances of a node from all other nodes in the network.

Analyzing Network Evolution with Network Feature Vector

Network topologies are typically represented as graphs bearing structural features, characteristic of the interactions among nodes/users. In this section, we introduce a framework for tracking and analyzing network evolution, based on the observation of graph properties and other metrics of interest.

For that, we define the notion of network feature vector. Assume a set of m parameters of interest of a network graph, each of which is denoted as g i , 1 ≤ i ≤ m. The number m of available/employed network metrics can be arbitrary but finite, and mainly depends on the application setting and the objectives of network evolution analysis. For instance, the metrics mentioned in Section 2.2 are sufficient for an overall study of network evolution or node importance, while additional features are required for the study of resource allocation.

In the general case, the metrics can be split into node-oriented and edgeoriented. The first are characteristic of node properties, e.g., node centrality or node clustering coefficient. The second are characteristic of edge properties, e.g., edge weight distribution or average weight of the links, etc. Thus, the set of parameters employed can be split in two subsets, one with node-related features {g

(n) i , 1 ≤ i ≤ m 1 } and a second with edge-related metrics {g (e) i , 1 ≤ i ≤ m 2 }, m 1 + m 2 = m.
With this in mind we can define the network feature vector, consisting of all the employed network metrics:

g = [g (n) 1 g (n) 2 . . . g (n) m1 node-related features | g (e) 1 g (e) 2 . . . g (e) m2 ] T edge-related features (1)
If the topology varies with time, at least some of the features will be time-varying and a time-dependence can be considered. In this paper, we will demonstrate the framework with a small number of parameters and we will not maintain the aforementioned distinction of features to node-related and edge-related. Thus, we will employ a more compact form of the feature vector g(t) = [g 1 (t) g 2 (t) . . . g m (t)] T . Also, for simplicity and without loss of generality, we will drop the time variable whenever the time-dependence is clear. Through the feature vector g, a specific network topology with its properties is mapped to a vector (point) in a metric space. As the topology evolves, so does the point in the metric space and the direction of the associated position vector g(t), i.e., the angle coordinates and measure of g(t) evolve in time. The dimension of the metric space, denoted as network feature space, depends on the number of parameters considered, so that different network feature spaces correspond to studying different properties of network evolution.

The time-varying feature vector can be used to assess the similarity of different topologies, namely quantify how "close" the final topology is to the initial, after a series of modifications. It can be also used to assess the similarity of different types of topologies. Since by the above mapping a topology snapshot corresponds to a point in space, various distance or similarity metrics can be employed to quantify the distance/similarity between topologies. In the special case that each component (metric) of the feature vector is independent of the rest, the network feature vector can be cast as a probability density function, and thus, in addition to distance, entropy-like measures can be employed as well [START_REF] Cha | Comprehensive Survey on Distance/similarity Measures between Probability Density Functions[END_REF]. With respect to the metrics employed in this work, and since the relation of clustering coefficient and centrality metrics among them and with other metrics has not been absolutely clarified, we will employ only distance and similarity metrics.

Distance metrics are more adequate to quantify the "magnitude" of network evolution, corresponding to the magnitude change of the network feature vector. On the other hand, inner product metrics (e.g., cosine metric) depict the "direction" of change (e.g., if a network changes character drastically or to a lesser degree) corresponding to the directional change of the network feature vector.

In this paper, and in order to demonstrate the potentials of the topology evolution framework and the role of distance/similarity metrics, we will show one simple representative metric from each category. Specifically, we will employ the Euclidean distance and cosine metrics, to quantify the magnitude and direction of change, respectively, of two topology instances, g(t 1 ) and g(t 2 ):

d e = m i=1 |g i (t 1 ) -g i (t 2 )| 2 (2)
s c = m i=1 g i (t 1 )g i (t 2 ) m i=1 g 2 i (t 1 ) m i=1 g 2 i (t 2 ) (3)
It should be noted that g(t) corresponds to the instance of a topology at time t, as explained above. g(t ) denotes another instance, namely the evolution of the topology up to time t > t. Furthermore, it is noted that since the Euclidean distance is a special case of the Minkowski distance family [START_REF] Cha | Comprehensive Survey on Distance/similarity Measures between Probability Density Functions[END_REF] and the cosine similarity a special case of the inner product family of similarities [START_REF] Cha | Comprehensive Survey on Distance/similarity Measures between Probability Density Functions[END_REF], the results provided are indicative of the trends one would obtain by any other member of the Minkowski and inner product families, apart from some scaling factors.

Another important observation refers to the dimension of the network feature space. It should be stressed that this is defined by the number of metrics employed in the feature vector definition and not the number of nodes in the network. The dimension of the feature vector (not its values) is independent from the size and order of the network graph.

One of the most fascinating potentials of the introduced network feature vector is the control capability over topology evolution. As explained above, different instances in time of an evolving topology correspond to a trajectory of a vector in a metric space. By properly defining a cost function of the form

J = h(g(t f ), t f ) + t f t0 k(g(t), u(t), t)dt
, where [t 0 , t f ] is the observation time interval, h(•), k(•) properly defined continuous functions and u(t) a control function related to the real mechanics of network evolution that determines the evolution of the topology through the system of equation ġ(t) = a(g(t), u(t), t), where a(•) determines the relation of the network feature with each control, one can potentially develop an optimal control problem on g(t) and exploit the constraints and controls for optimally balancing trade-offs relevant to network evolution and the benefit-cost relations of network processes emerging. The type of cost J, controls u(t) and relation between network feature vector-controls a(•), are application/network dependent and define the type of control problem emerging. They also determine the required solution methodology to be employed.

In the following, we demonstrate the impact of edge churn on network topology, by exploiting the capabilities of the framework described above to quantify and provide intuition on the similarity between instances of an evolving topology and of different networks among them.

Network Churn and Network Evolution

In this work, we will focus on edge churn for relational complex networks, i.e., variation of links between nodes. In the latter, any link can be added or deleted (no constraint on link formation). Examples of such networks with edge churn are peer-to-peer and online social network, where users enter/leave the network arbitrarily and form relational links arbitrarily with other users. This would not be the case in lattice and random geometric networks, where spatial constraints would apply for the links that could be added/deleted. However, even in these cases, network evolution could be analyzed similarly. We restrict ourselves to relational graphs only in order to directly compare the results between them and focus on the churn effect, rather than take into account evolutionary features induced by the type of analyzed network.

We consider a generic edge churn mechanism and several variations of it, as described in the forthcoming subsection. Edge churn is a combined effect in complex networks, including environmental aspects (wireless access cases), user behavior (turning on/off devices), service patterns (connectionless, serviceoriented, etc.). A general trend discovered in [START_REF] Leskovec | Graphs Over Time: Densification Laws, Shrinking Diameters and Possible Explanations[END_REF] is that of network "densification", i.e., edges are added with higher rate than edges deleted. Thus, we will also consider a slightly higher edge addition than deletion rate, reducing also in this way the probability of rendering the network topology disconnected. Furthermore, by considering the churn-based constructive nature of many complex networks, such as the ones included in this work (especially the scale-free and small-world ones), we further study the effect of edge churn mechanisms on their structure, thus obtaining clearer picture of its impact on their evolution, while exemplifying the proposed framework.

Edge churn is performed successively in multiple steps, in a constructive process [START_REF] Karyotis | Evolutionary Dynamics of Complex Communications Networks[END_REF]. We assume that at each step, only one of the sub-processes takes place, as described below:

-Edge Addition: With probability p, 0 ≤ p ≤ 1, we add one new link to a selected pair of nodes that are not currently linked. -Edge Deletion: With probability r, 0 ≤ r ≤ 1, we delete one selected edge from an already connected node pair. -No action: With probability 1 -p -r, neither edge addition, nor edge deletion are performed.

Note that it should hold p + r ≤ 1, i.e., the vector [p r 1 -p -r] is a probability distribution over the above processes.

The selection mechanism for which edges to add/delete determines diverse edge churn types. We consider random edge churn (RC) that corresponds to selecting randomly and uniformly which edge to add/delete, as well as which process to perform. We also consider preferential edge churn (PA), which selects nodes and edges for edge churn preferentially with respect to a selected feature (i.e., centrality metric). We consider three types of preferential edge churn. In the first, new edges are preferentially attached to nodes with high degree centrality and deleted from nodes with low degree centrality. The second and the third variants are similar, but instead of degree centrality, we use betweenness and closeness centrality correspondingly. Also, we examine the inverse of these edge churn types, i.e., the inverse of the first type corresponds to adding new edges preferentially to nodes with low degree centrality and deleting edges from nodes with high degree. The reason of applying such preferential attachment (PA) edge churn mechanisms is to study how network evolution based on a particular network feature (part of the network feature vector) impacts the other features and the behavior/structure of each network as a whole. It is also in line with network churn processes as the ones described above in peer-to-peer or online social networks, and IoT networks within the fog computing paradigm.

Numerical Evaluation

In this section, we exemplify the application of the proposed framework via numerical results obtained by simulating RC and PA edge churn in various complex networks. Initially, we explain the models for the complex networks employed and then present and analyze the obtained results.

Relational Complex Networks

We consider three different types of relational graphs, random graphs (RG), scale-free (SF) and small-world (SW). The simplest model of RG is the Gilbert G(N, p), where every possible edge occurs independently with probability 0 < p < 1. This is a model extensively used for peer-to-peer networks and other spontaneously forming networks. We consider the Erdős-Rényi (ER) model, G(N, E), where a graph is chosen uniformly at random from the collection of all graphs which have N nodes and E edges. If pN 2 → ∞, then G(N, p) behaves fairly similarly to G(N, E) with E = N 2 p, as N increases [START_REF] Karyotis | Evolutionary Dynamics of Complex Communications Networks[END_REF][START_REF] Newman | The structure and function of complex networks[END_REF]. We choose E in the range of N log(N ) 2 < E < N (N -1)

2

. The lower bound ensures a connected topology, while the upper N (N -1) 2 signifies a massively dense network, i.e., a completely connected graph.

Scale-free (SF) is a type of network whose degree distribution follows a powerlaw. The fraction P (k) of nodes having k connections to other nodes scales as P (k) ∼ k -γ , where 2 < γ < 3 [START_REF] Karyotis | Evolutionary Dynamics of Complex Communications Networks[END_REF][START_REF] Newman | The structure and function of complex networks[END_REF]. Many real-world networks such as the WWW and the router network of the Internet belong in this category. We consider the Barabasi-Albert (BA) model of scale-free networks, which generates random scale-free networks using the preferential attachment mechanism. If a network begins with an initially connected and d-regular network of m 0 nodes (d < m 0 ) and new nodes are added to the network one at a time, each new node will connect to m ≤ m 0 existing nodes with a probability that is proportional to the degree of the latter. Such probability p i that the new node connects to the existing node i is p i = ki j kj , where k i is the degree of node i and the sum is over all pre-existing nodes j. Heavily linked nodes (hubs) tend to quickly accumulate even more links and usually have high betweenness and closeness centrality values.

Small-world (SW) is defined as a network where the typical distance L between two randomly chosen nodes grows proportionally to the logarithm of the number of nodes N , i.e., L ∝ log N [START_REF] Karyotis | Evolutionary Dynamics of Complex Communications Networks[END_REF][START_REF] Newman | The structure and function of complex networks[END_REF] and the clustering coefficient is high. Practically, nodes are linked with a small number of local neighbors. However, the average distance between nodes remains small, i.e., nodes are virtually close to each other. We consider the Watts-Strogatz (WS) model of SW. It starts from a regular lattice of degree d and randomly rewires each edge with a probability g p , connecting nodes that are otherwise far apart with shotcuts. The initial lattice ensures high clustering coefficient, while a suitable number of shortcuts can further reduce the average path length. For comparison purposes, one should consider that RGs exhibit a small average shortest path length (varying typically as the logarithm of the number of nodes) along with small clustering. Examples include some types of smartgrids, telephone networks, etc.

Case Study of the Application of the Proposed Framework: Impact of Edge Churn on Complex Networks

In this subsection we investigate via simulations the impact of edge churn on complex network evolution. We performed them in MATLAB, averaging over 50 topologies for each considered scenario. The feature vector consists of the following features, in this order: average degree centrality (normalized average node degree), average path length, average clustering coefficient, average betweenness centrality, average closeness centrality, where averaging is performed for all net-work nodes. We consider N = 100 nodes for all topologies, E = 2000 edges for ER-RGs, initial d-regular lattice with d = 10 for BA-SF and initial lattice with d = 10 and a rewire probability g p = 0.25 for WS-SW networks. Edge addition takes place with probability 0.60, and edge deletion with probability 0.4.

Impact of Edge Churn on Each Type of Network

In this subsection, we study how the use of the introduced network feature vector can aid at tracking and visualization of the evolution of several types of networks when random and preferential edge churn takes place. The results are shown in Figs. 1,2,3, where edge churn takes place in an initial RG, SF and SW topology correspondingly.

For visualization purposes, we divide the set of features in two subsets one including the degree, clustering coefficient and average path length and the other all the centrality metrics (degree, betweenness and closeness). The subfigure (a) of each figure corresponds to the first subset and the subfigure (b) to the second. The vectors with only a graph type indication in the legend correspond to the features' values in the initial topology, while the vectors having also an edge churn mechanism indication in the legend correspond to the final topology after the application of the edge churn. From Fig. 1, we observe that an RG is not significantly affected by edge churn, either PA-based or random, as it is intuitively expected. The final vectors remain very close to the initial ones for RG for all types of churn. The churn type (e.g., random, preferential) does not seem to affect the final vector, which differs from the initial one mainly due to the increase of the average node degree, as an outcome of network densification. This behavior is intuitively expected for RGs, since due to the random nature of the initial network, even preferential additions/deletions eventually behave similarly to random ones. Given the rest of random connections, the ones added/deleted do not alter considerably the initial structure of the RG. From Fig. 2, we observe that an evolved SF network under all types of edge churn examined, eventually moves closer to an RG, as the node degree increases due to network densification (nodes with low degree in the initial SF structure also obtain edges) and so does the clustering coefficient. Note that the SF topology has the lowest clustering coefficient compared to the RG and SW ones. The final topologies under PA edge churn based on degree and betweenness centrality distinguish themselves among the rest by staying closer to the initial SF topology, while the second also moves closer to the SW topology, as shown more clearly in Fig. 2(a).

For an initial SW topology under edge churn (Fig. 3), we can make similar observations as in Fig. 2, as the network feature vector moves towards the one corresponding to an RG. However, the final topology under preferential edge churn based on betweenness centrality distinguishes itself among the rest by staying closer to the SW feature vector. Thus, such illustrations of the network evolution reveal important observations for maintaining a non-random network structure, e.g., for SF and SW. For the first, in order to maintain a completely non-random structure, node degree and betweenness centrality should be prevented from changing significantly. For the SW network only betweenness centrality should be prevented from drastic modification. Comparative Analysis of Churn Types on Complex Relational Topologies In this subsection, we compare the evolved complex topologies with the initial ones, in terms of the distance and similarity metrics. In the following table pairs, the left-hand corresponds to the (Euclidean) distance metric between the pure topologies (not evolved) indicated in each row and the final topologies derived by the initial topologies indicated in columns. The right-hand table corresponds to the (cosine) similarity metric. As an example, the entry in row 1 and column 1 of Table 1 corresponds to the distance between the feature vector of a pure RG (row) and the feature vector of an evolved RG topology (column) under RC. The observations follow the lines of the ones made in Section 5.2.

Tables 1, 2 present the distance and similarity of the evolved topologies for random edge churn compared to the initial ones. Similarly, table pairs 3-4, 5-6, 7-8, present the distance and similarity of the evolved topologies compared to the initial ones for the PA edge churn based on degree, closeness and betweenness centralities, respectively.

For random edge churn, the most important observations are that it has almost no impact on RG topologies, lesser (and small) impact on SF and noteworthy impact on SW (which move closer to RG topologies). This is expected since random edge modification are not expected to affect an already random network. However, they are expected to have some effect on a SW, where the few shortcuts may be modified and thus affect its structure.

A similar but modified trend is emerging in preferential based edge churn. Evolved RGs are not affected notably by any type of preferential edge churn and always remain close enough to the features of the initial random graph. Evolved SFs and SWs after PA edge churn based on degree and closenness centralities become almost RGs as their distance to a pure RG network decreases and their similarity increases. This is counter-intuitive for SF graphs when considering PA edge churn based on degree centrality, however edges are added and deleted probabilistically, although preferentially, while the characterization of the network is now more holistic based on a set of features and not on specific features. Notably SF modified via PA edge churn based on betweenness centrality maintains its pure graph's properties as shown in Tables 7-8 and also observed in the previous subsection. The same observation is true for the evolved SW graph although to a lesser degree. Note that more specified observations targed to the extent of change of each topology can be made by comparing the absolute values of the distances. Another general observation is that with respect to distance, SF-SW networks are relatively close and RG is farther away from them in the 5-dimensional network feature space considered. This is also expected since several features of SF networks are also present in SW graphs [START_REF] Karyotis | Evolutionary Dynamics of Complex Communications Networks[END_REF][START_REF] Newman | The structure and function of complex networks[END_REF], which is also illustrated in the plots of the previous subsection. All the above observations can be exploited for advanced network design and optimization. By taking into account the anticipated network changes due to edge churn, network planning will be more realistic, and more accurate optimization can be achieved.

Conclusions

In this work, we demonstrated a framework for quantifying network evolution and similarity based on the mapping of a set of network parameters on points of a vector space. It allows to track network evolution by observing the trajectory of the network feature vector, which can include different graph features, and thus accommodate various applications. Consequently, it is a very flexible, yet descriptive framework for analysis and study of network evolution. In this work, we further demonstrated the potentials of the proposed framework by analyzing edge churn for various complex networks of relational type, namely random, scale-free and small-world. The outcomes are in accordance with observed trends and can be used for advanced network design/optimization.

In the future, we plan to demonstrate the full spectrum of potentials by extending the proposed framework to a topology control one. By proper definition of a cost function, optimal control theory can be used to drive the network feature vector to desired values, thus allowing modifying one type of network to another based on feasible trade-offs and precise specifications.

Acknowledgment

E. Stai gratefully acknowledges the Foundation for Education and European Culture (IPEP) for financial support.

Fig. 1 .

 1 Fig.1. Illustration of the proposed framework when edge churn with diverse attachment types is applied to an initial random graph.

Fig. 3 .

 3 Fig.3. Illustration of the proposed framework when edge churn with diverse attachment types is applied to an initial small-world graph.

Table 1 .

 1 Distance comparison for random churn.

	H H Pure H Fin. H H	RG	SF	SW
	RG	0.1195	0.2333	0.1643
	SF	0.3936	0.1068	0.1389
	SW	0.4133	0.4602	0.4262

Table 2 .

 2 Similarity comparison for random churn.

	H Pure H H Fin. H H	RG	SF	SW
	RG	0.9979	0.9816	0.9923
	SF	0.9541	0.9875	0.9855
	SW	0.9013	0.8378	0.8618

Table 3 .

 3 Distance comparison for preferential churn (degree-based).

	H H Pure H Fin. H H	RG	SF	SW
	RG	0.1882	0.0997	0.0581
	SF	0.4593	0.2055	0.2752
	SW	0.4353	0.3251	0.3424

Table 4 .

 4 Similarity comparison for preferential churn (degree-based).

	H Pure H H Fin. H H	RG	SF	SW
	RG	0.9954	0.9945	0.9977
	SF	0.9466	0.9837	0.9728
	SW	0.9067	0.9225	0.9183

Table 5 .

 5 Distance comparison for preferential churn (closeness-based).

	H H Pure H Fin. H H	RG	SF	SW
	RG	0.1881	0.0767	0.0521
	SF	0.4599	0.2041	0.2733
	SW	0.4377	0.3734	0.3475

Table 6 .

 6 Similarity comparison for preferential churn (closeness-based).

	H Pure H H Fin. H H	RG	SF	SW
	RG	0.9955	0.9983	0.9981
	SF	0.9463	0.9811	0.9728
	SW	0.9055	0.8967	0.8967

Table 7 .

 7 Distance comparison for preferential churn (betweenness-based).

	H H Pure H Fin. H H	RG	SF	SW
	RG	0.1873	0.2130	0.1875
	SF	0.4567	0.1833	0.3261
	SW	0.4292	0.2468	0.2372

Table 8 .

 8 Similarity comparison for preferential churn (betweenness-based).

	H Pure H H Fin. H H	RG	SF	SW
	RG	0.9953	0.9699	0.9783
	SF	0.9474	0.9840	0.9647
	SW	0.9094	0.9566	0.9646

Network churn is the process where users enter and leave a network modifying its topology in terms of nodes and/or links[START_REF] Holzer | Monitoring Churn in Wiireless Networks[END_REF].