Pierre Schneider

"Those round Arabia on the Persian Gulf are specially praised" (Pliny the

Keywords: Indian Ocean, Persian Gulf, Arabia, Strait of Hormuz, pearls, pearling, Rome, luxury

Westerners began to use pearls in the second century B.C. Their consumption dramatically increased from the first century B.C. onwards, due to huge Roman demand.

Most pearls were imported from the Persian Gulf and the Gulf of Mannar (India).

Oyster banks were constituted of hundreds of thousands of seashells scattered over level areas of coral rocks and sand. Most of them were at a distance ranging from about 200 meters to up to 100 km from the shore. Divers would rarely descend deeper than approximately 20 meters, although valuable pearls could be obtained at greater depths. 8 Most pearl banks were located between 11 and 27 m. J. G. Lorimer, who served as a British political resident in the Gulf in the early 20 th century, compiled a map listing hundreds of pearl banks between Ras Tannurah and Dubai with their local names, an area also known as the "Great Pearl Bank". While most of them lay on the western part of the Gulf, pearls were also fished along the Iranian side, mainly between Lingeh and Tahiri, and around Kharg Island. 9 However, they were in greater abundance along the Arabian side in modern times as well as in Antiquity, as will be seen below.

2) The background of the Greco-Roman body of knowledge: the Mediterranean taste for Indian Ocean pearls

In fact, the bulk of information about pearling in the ancient Persian Gulf is provided by Greek and Roman texts. This situation did not occur by chance, for the Greco-Roman knowledge of Indian Ocean pearls was compiled and developed upon concurrently with their growing consumption of pearls. Behaving as any customers would, Greek and above all Roman buyers were curious to know as much as possible about this fascinating ornament.

Had pearls been held in lower esteem by them, our written documentation would have been reduced to almost nothing. Thus, a short presentation of the historical, cultural and social backdrop of the Mediterranean taste for pearls seems necessary in order to better elucidate my inquiry. 10 All written and archaeological evidence that has been handed down to us conclude that Indian Ocean pearls were unknown in the Mediterranean world until Alexander the Great's expedition in Asia (335-323 BCE). Indeed, no piece of jewellery with pearls preceding the Hellenistic period -i. e. the period beginning with Alexander's rule -has been recovered so far. In addition, there is no pre-Hellenistic Greek text mentioning a precious item that would allude to the pearl produced by marine molluscs. In fact, it is very likely that the Greeks -or more precisely, the Greco-Macedonians -discovered this gorgeous object in Persia, where Alexander and his army stayed between 331 and 329 BCE. This fact is supported by pieces of written (see in particular Chares of Mitylene's fragment below) as well as archaeological evidence (the Persian taste for pearls is proven by several remarkable pieces of jewellery unearthed by archaeologists). The Musée du Louvre features among its works a necklace which was originally buried in a sarcophagus at Susa; it was made of approximately 400-500 pearls, all of which decayed except for 238 which have remained unspoiled. The excavations at Pasargadae have also yielded true pearls. 11 It is also likely that the Greco-Macedonians had the opportunity to observe pearls in north-west Indiathe country which marks the final stop of the king's adventure -, for, they were undoubtedly held in high esteem by local peoples. Back in Babylonia after his Indian expedition, Alexander the Great instructed several officials to reconnoitre the "Persian Sea" -viz. the Persian Gulf -. It is thanks to these explorers -in particular, Androsthenes of Thasos, Chares of Mitylene -and to Nearchus -who commanded the fleet for the voyage from the Indus to the Persian Gulf in 325 B.C. -that we have the first Greek reports pertaining to pearl-oysters, pearls and pearling. Two facts require our particular attention: firstly, these explorers introduced the Persian name meaning pearl (margarita) into the Greek language and to the Mediterranean world. Secondly, thanks to their reports, we know that pearls were a highly desirable decorative accessory for Indians, Persians and other Asian peoples at that time.

It appears that it was not until the second century BCE that the Greeks began to show some interest in pearls as ornaments, but how and why this phenomenon developed is yet unknown. A limited number of archaeological artefacts -necklaces and earrings -and a lone Alexandrian papyrus are evidence for the development of a taste for pearls in the Eastern Mediterranean, occurring as it did around the mid-second century. 12 This evolution may have something to do with the beginning of an established Greek presence in the Indian Ocean and the increasing presence of Mediterranean merchants in the southern Red Sea area, where they could purchase pearls from Arabian middlemen. From the sparse topographical details provided by the classical sources, we are as yet unable to identify the place where such exchanges took place, but Aden -called Eudaimôn in Greek -is a plausible candidate. 13 It can be said with more certainty that the late second century B.C. represents a turning point for the pearl trade. Firstly, following the discovery of the monsoon winds by the Greeks 14 , some enterprising Alexandrian merchants began to set out from Egypt and established direct access to Indian markets; the frequency with which they reached these places cannot be determined. Due to this shift in trade patterns, an official base of operation ("epistrategos of the Erythraean and Indian Seas"), clearly connected to Red Sea and wider Indian Ocean trade, was founded in the early-first century BCE and the growth of direct exchange with India was undoubtedly responsible for this administrative innovation. 15 Secondly, and more importantly, Roman consumption began to increase approximately around this time. This conclusion is supported by Pliny the Elder, who claims that the Latin word unio -referring to large and splendid Indian Ocean pearls -was coined in the late second century B.C., while the Greek word margarita would apply to smaller specimens (Pliny the Elder, Nat. Hist. 9, 123). Therefore, it would be reasonable to surmise that quality pearls were more and more in demand among Roman elites during this period, implying that Roman customers had now acquired a more refined taste. At any rate, Pliny's remark clearly stresses the fact that the Romans had overtaken the Greeks as the foremost lover of pearls and as potential customers. [START_REF]On Pliny's moralistic and literary treatment of pearls[END_REF] A set of Latin texts -including the passage of Pliny the Elder quoted above -provides corroborating evidence that Roman consumption continued to increase in the early first century BCE, allowing for the deduction that pearls of various qualities -from common to splendid -were available on the market. [START_REF]For further details[END_REF] In particular, Pliny the Elder mentions a date which is pivotal to this: the annexation of Egypt and the subsequent Roman control of the infrastructure of eastern trade, which allowed pearls to become ubiquitous in Rome (Pliny the Elder, Nat. Hist. 9, 123). The importance of this event is echoed by Strabo, who states that from this time on, seamen and merchants were less reluctant than before to cross the Bab al-Mandab and head to India. They were also more willing to invest capital, as shown by the expression "large fleets" employed by Strabo (Strabo, Geogr. 17, 1, 13). 18 It is indeed very likely that the pearl markets located in the Gulf area of the Gulf and South India had become much more accessible, and that, as a result, Roman demand could be satisfied in terms of quantity and quality. In fact, Roman demand, or more precisely the Roman craving for pearls, rose so rapidly that the word "madness" (insania) shows up in several documents. This word refers to tremendous demand, whose popularity was to last for at least a century. 19 Archaeological remains (even though, as R. A. Donkin rightly writes, "of the vast quantity of pearls in circulation at any one time, all but a minute fraction has disappeared"20 ), also attest to this phenomenon, for numerous items of jewellery made of pearls have been recovered (earrings, necklaces, rings, hair ornaments and so on), with remarkable objects unearthed from the devastated cities of Pompei and Herculaneum.21 Earrings, in particular, come in an incredible variety of forms: the most remarkable objects were made of two or three gorgeous pearls, while a very popular type consisted of several pearls made up into ear pendant so as to rattle, hence the name crotalia -from crotalium, "castanet" -given to this model. It is worth noticing here that an acceptable pair of crotalia could be made of a bunch of small, uneven pearls, meaning that wearing pearls was no longer the privilege of the Roman elite class.

3) The geography of pearl fishing according to classical sources

Almost all pearls destined for Roman customers were supplied by two regions: on the one hand South India and Sri Lanka; on the other, the Persian Gulf. With respect to the latter, despite the decent amount of evidence which has survived, we nonetheless lack accurate topographical details on the location of Gulf fisheries. In addition, modern historians are confronted with the vexing question of how the Greeks and Romans would call the Indian Ocean. The names Erythra thalassa (Greek) and mare Rubrum (Latin) -both literally meaning "Red Sea" -referred to the whole western Indian Ocean. As for the Persian Gulf and the modern Red Sea, they were respectively called Persikos kolpos -Persicus sinus (Persian Gulf) and Arabikos kolpos -Arabicus sinus (Arabian Gulf). However, both gulfs were interchangeably (and frequently) given the name of the whole body of water, viz. Erythra thalassa / mare Rubrum. Accordingly, when one comes across expressions such as:

Britain produces gold and silver and other metals: conquest is worthwhile. Their sea also produces pearls, but somewhat clouded and leaden-hued. Some people suppose that their pearl-fishers lack skill; in the "Red Sea" (Latin: mare Rubrum) we are to imagine them torn alive and still breathing from the shell, while in Britain they are gathered only when thrown up on shore. (Tacitus, Agricola, 12, 10-12) or:

A night to be marked with Thetis'22 Erythraean (Latin: Erythraeis) gems, long to be remembered, whose spirit will live forever. (Statius, Silvae, 4, 6, 18-19) there appears to be no way to decide if such pearls originated in the Gulf of Mannar (South India and Sri Lanka) or in the Gulf. It is only when in the same passage "Erythraean" pearls are contrasted with the "Indians" that it can be affirmed that the former adjective points to the Persian Gulf while the latter indicates Southern India and Sri Lanka (Gulf of Mannar), as is made clear by the following two instances23 : Their (= pearls) whole value lies in their brilliance, size, roundness, smoothness and weight, qualities of such rarity that no two pearls are found that are exactly alike: this is doubtless the reason why Roman luxury has given them the name of 'unique gems' (Latin: unio). (...) There is also a great variety in their actual brilliance; it is brighter with those found in the "Red Sea" (Latin: mare Rubrum = pearls from the Persian Gulf) whereas those found in the Indian Ocean (= pearls from South India and Sri Lanka) resemble flakes of mica, though they excel others in size. (Pliny the Elder, Nat. Hist. 9,112-113) The dimensions of the pearl are those of a fish's eye of large size, and it is produced off the coast of India and certain islands in the "Red Sea" (Erythra thalassa = the Persian Gulf). (Theophrastus, De lapidibus, 6, 36) Regrettably, only a handful of documents detailing the fishing areas in the Gulf have been handed down to us, and the small percentage of texts that have survived provides only fragmentary information. Be that as it may, some authors provide snippets of information that can partly fill in the blanks left by the lack of documentary evidence.

The first question that comes to mind is whether the two main fishing areas -the Iranian and the Arabian coasts -had been distinguished between by the Greeks and Romans.

By the time Alexander's officials and sailors were exploring the Gulf, no such distinction shows up in the extant evidence. Theophrastus, who borrowed his information from those who participated in the naval expeditions ordered by Alexander the Great, offers no clues.

His statement only proves that the Persian Gulf was identified as an important area of production as early as 325-324 BCE, and that most fisheries were located close to islands, which was a characteristic feature of pearling in the Gulf. 24 However, the growth of Mediterranean consumption, fuelled by Roman demand, may have caused this situation to change. Pliny the Elder, who witnessed Rome's ravenous craving for pearls, speaks of "Arabian pearls". By doing so, he explicitly points to pearls collected in the south-western part of the Gulf -between Bahrain and the Strait -:

The most highly valued of human possessions (...) is the adamas (= diamond), which for long was known only to kings, and to very few of them. (...) Next in value in our estimation come the pearls of India and Arabia, which we discussed in Book IX among the products of the sea. (Pliny the Elder, Hist. Nat. 37, 204) Elsewhere he writes:

The most productive (sc. island) is Ceylon, and also Stoidis (...) and the Indian promontory of Perimula; but those (= pearls) round Arabia on the Persian Gulf of the "Red Sea" (= found along the Arabian coast of the Gulf) are specially praised. (Pliny the Elder, Hist. Nat. 9, 106) By linking pearls to Eastern Arabia, instead of mentioning the whole Gulf, Pliny evokes the dominant trade patterns of his time; apart from the pearls imported from India, Roman demand seemed to be met primarily by Arabian producers. The reason why Pliny does not report on pearls of Persian origin remains obscure. In my opinion, his statement indicates the lower productivity of the Iranian coasts. Be that as it may, the same Pliny reports intensive pearling activity taking place in Bahrain (in Greek and Latin Tylos), and demonstrates that this island ranked higher in terms of pearling activity than any other fishing zone in the Gulf 25 : Fifty miles inland is the Attene district26 ; and opposite to it and the same number of miles (= 50 miles) distant from the coast is the island of Tylos, extremely famous for its numerous pearls, with a town of the same name, and next another smaller island 12 miles away from the cape of Tylos. (Pliny the Elder, Hist. Nat. 6, 148) This fact is echoed by another authority who wrote several decades before Pliny -Isidorus, a The Persian Sea (Greek: Persikos thalassa) must obviously be equated with the Persian Gulf.

Although the name of the island referred to by Isidorus has been lost, it can be identified with certainty as Bahrain. Firstly, both Isidorus and Pliny stress the abundance of pearls, which suggests that they are talking about the same place. 28 Secondly, considering that Bahrain / Tylos had been controlled by the Characene kingdom since the second century BCE, and was still part of it in the early first century CE, Isidorus is likely to have been reporting on pearling activity in a region belonging to his native country.

It is worth adding here that a document going back to the time of Alexander the Great indicates that Bahrain was a prominent production centre well before the Greek and Roman desire for pearls started to develop. As mentioned above, one year before he died (in 323 BCE), Alexander, who perhaps was considering an expedition into northern Arabia, ordered several men to reconnoitre the limits of the Arabian Peninsula. A certain Androsthenes of Thasos was sent down the Euphrates to skirt the coast of Arabia as far as contemporary author.

possible and explore the shores of the Persian Gulf. Sometime afterwards, Androsthenes wrote an account of his voyage which has been entirely lost to us except for a small collection of fragments, one of which details the features of Tylos / Bahrain. The descriptions of Bahrain's mangroves and freshwater springs, of the flourishing farming taking place there et cetera illustrate how accurate Androsthenes account was. In another excerpt, which is not explicitly linked with Tylos, he offers some pertinent observations on pearls and pearl oysters 29 :

There are many types of whelks, cowries, and other shellfish, which are very different from those we know; purple shellfish and large quantities of the other types of shellfish are also found. One unusual type, which they refer to as the berberi, 30 is the source of the margaritis-stone (pearl), which is very expensive throughout Asia and is sold in Persia and other inland regions for its weight in gold. The shellfish looks like a scallop, but is not striated and instead has a smooth, thick shell; and it does not have two "ears," like the scallop, but one. The stone is found in the creature's flesh, like a tubercle in a pig's flesh. Sometimes it looks very much like gold, to the extent that it can be difficult to tell them apart when they are set side by side, while at other times it looks like silver or is perfectly white and resembles fisheyes. (Androsthenes of Thasos, FGrH 711 F1 = Athenaeus of Naucratis, Deipn. 3, 45) Since Androsthenes was a member of the naval expedition led by Nearchus (see below), he could theoretically have collected these data while heading along the Iranian coast towards the Shatt al-Arab. I am not, however, inclined to favour this opinion and would rather theorize that an account of such good quality was part of the description of Bahrain, where Androsthenes undoubtedly stayed for some time in order to complete his mission. 31 Later, a Greek author active during the mid-first century CE, who wrote a treatise entitled the Periplus of the Erythrean Sea, reported on fisheries that must be located in the southern part of the Gulf. He no doubt refers to the cluster of pearl banks lying between the coast of Qatar and that of Sharjah, illustrating again the productivity of the southern Gulf.

29 For a discussion of the value of Androsthenes' description, see Schneider (a) 78-83. 30 Carter (b) 11 rightly dismisses any connection between the word berberi and the Bronze age site on Bahrain named Barbar. 31 Contra: Carter (b) 14:"It is surely significant that none of these sources, all based on the accounts of the Greek voyagers in the fourth century BC, associates pearls with Bahrain, despite the fact that Androsthenes had certainly visited the island."

This observation appears during his description of the sea route to Arabia and India:

Near the furthest headland of the Kalaiou Islands and the mountain called Kalon there comes, not far beyond, the mouth of the Persian Gulf (= the Strait of Hormuz), where there are many pearl fisheries. (Periplus of the Erythraean Sea, 35)

In fact, neither the Kalaiou Island nor the mountain Kalos have been positively identified, due to some discrepancies in the author's narrative. But L. Casson argued effectively that it deals with the Jazair Daymaniyat -west of Muscat -and an unidentified peak of the Ruus al Jibal chain, "near its end in the Musandam Peninsula". 32 Interpreted in a restrictive manner, the text would support the idea that pearl fishing was carried out outside the Gulf, that is to say in the Arabian Sea along the Omani coast, which is not implausible. L. Casson, one of the best commentators of the Periplus, nevertheless offers a more specific interpretation, claiming that the author of the Periplus is referencing fisheries inside the Gulf, despite the geographical inaccuracy of this passage. According to him, the expression "many pearl fisheries" refers to "the so-called Great Pearl Bank", which "stretches along the southern bight of the Persian Gulf roughly between Dubai and Qatar". 33 Without objecting to Casson's interpretation, I nonetheless would not exclude the possibility that the author of the Periplus also had in mind the fisheries scattered along the Iranian side of the Gulf. Their existence is attested to by several comprehensive accounts dating back to the time of Alexander the Great. For instance, according to Chares of Mitylene, an official who managed Alexander's household (a position equivalent to that of a court chamberlain) pearl fishing was practiced in four areas bordering the northern shore of the Gulf:

In the Indian Sea (= the Gulf of Mannar) as also along the coast of Armenia 34 , Persia, Susa and Babylon, a creature that resembles an oyster is caught. (...) They extract white bones, which they call margaritai, from it, and use them to produce necklaces, bracelets and ankle bracelets. The Persians, the Medes, and all the inhabitants of Asia are more interested in these than in jewellery made of gold. (Chares of Mitylene, 32 Casson 176-178. 33 Casson 178. 34 Armenia obviously cannot produce pearls. Athenaeus of Naucratis, who quotes Chares, is very likely to reproduce a corrupted reading -unless he was himself responsible of the present faulty version. "Carmania"a region which roughly corresponds to the modern province of Kerman -was suggested as a good emendation (Schneider (a) 82, n. 22).

Contrary to Chares, who did not accompany any exploratory missions, Nearchus, as mentioned above, had first-hand experience with the Eastern Gulf. He composed -maybe in the early third century BCE -an account of his voyage entirely lost except for extensive fragments preserved by later authorities. Strabo, drawing on Nearchus's description of the Persian side of the Gulf, mentions an anonymous island located north of the Strait of Hormuz:

Nearchus goes on to say that there is an island at the beginning of the Persian Gulf (= north of the strait) where quantities of valuable pearls are to be found. (Nearchus, FGrH 133 F28 = Strabo, Geogr. 16, 3, 7) This text is more or less geographically accurate, although we do get further details from Arrian, a second century CE Greek historian, who provides us with the best paraphrasing of Nearchus' narrative (Nearchus, FGrH 133 F1= Arrian, Ind. 37, 2-3). According to Arrian, Nearchus, after he crossed the Strait of Hormuz, sailed past six islandsmooring his ships at some of them -: Organa, a place devoid of inhabitants; Oarakta, a large island home to a resident group of farmers; an anonymous island sacred to Poseidon; another anonymous island; Pylora, a desert island; and finally, Kataiê, a desert island sacred to Hermes and Aphrodite marking the northern limit of Carmania. Whereas the identification of Oarakta as Qeshm is beyond question, the others are merely conjectural. It has been suggested that Kataiê could be Jazireh ye-Kish, but other islands surrounding Qeshm -Hormoz, Larak, Hengam, Farur and even Kish -could be good candidates. 35 At any rate, the pearl bank mentioned by Strabo was located around one of these islands.

There is also Pliny the Elder's testimony, which reports that pearl fishing was conducted in a place lying offshore on the eastern side of the Gulf: (...) an uninhabited island, Caecandrus, 50 miles out at, sea, and next to it, with a strait flowing between, Stoidis, with a valuable pearl-fishery. (Pliny the Elder, Nat. Hist., 6, 110) Regrettably, Pliny's geographical information is so incoherent that it is difficult to distinguish 35 Note, however, that only Farur and Kish (Qais) are listed in Lorimer's index of fisheries (Lorimer 2279-2280). Compare with the map given by Carter (b) 189. See also Carter (a) 153, quoting a XVIII th century account. substantiated facts from errors. Let us conclude that Pliny's text is a derivate -directly or indirectly -of Nearchus' text, as proven by the comparison with the following excerpt: Sailing along the Persian coast they put in at a place called Ilas, where a harbour is formed by a small desert island, called Kaikandros. The voyage [from Kataiê to Kaikandros] is 400 stades. At daybreak they sailed to another island, an inhabited one, and anchored there. Here, according to Nearchus, there is pearl fishing, as in the Indian Sea (i. e. as in South India). (Nearchus, FGrH 133 F28 = Arrian, Ind. 38, 2-3) Evidently, these two versions of Nearchus' report cannot be reconciled; not only does Arrian ignore the name Stoidis, but there is no correspondence between the two measurements, except for the fact that 50 Roman miles equates to 400 stades. This divergence notwithstanding, Pliny and Arrian admittedly to the same place. The question that now comes to mind is whether the island close to Cascandrus/Kaikandros is the same one referred to by Strabo (see above). This is apparently taken for granted by R.

Carter, but I am less confident, since Kaikandros, being located 400 stadia north-west of Kataiê, does not really lie "at the beginning of the Persian Gulf". 36 Given that Kaikandros has long been identified with Hindarabi, Nearchus' mysterious island may be Jazireh ye-Lavan. As for Pliny's Stoidis, there is no satisfactory answer given.

With respect to the pearls obtained off the Iranian coast of the Gulf, no detailed account later than the late fourth century and early third century BCE has been bequeathed to us. This fact may be consistent with my assumption (expressed above) that the growth of Mediterranean demand was mainly satisfied by the southern Gulf producers -hence the expression "Arabian pearls" occurring in Pliny's book. This statement may be corroborated by a neglected piece of evidence which is deserving of our attention. At some point in Pliny the Elder's description of the Arabian Peninsula, the following incidental remark appears:

Taken as a whole, they (= the Arabs, i. e. all tribes dwelling in the Arabian Peninsula) are the richest peoples in the world, because vast wealth from Rome and Parthia (= the kingdom ruling Persia at that time) accumulates in their hands, as they sell the produce they obtain from the sea (= pearls) or their forests (= frankincense and myrrh) and buy nothing in return. (Pliny the Elder, Nat. Hist. 6, 162) Although the Parthians undoubtedly could obtain pearls in the coastal areas that was under their control, they apparently did not accrue a surplus, as they seem to have been purchasing pearls from the Arabs. Whatever the reason, one can cautiously hypothesize that greater quantities were collected by Arab divers in the southern part of the Gulf which permitted them to meet Roman and Parthian demand.

4) The techniques of pearl gathering according to classical sources

Alexander the Great's companions, who produced the first accounts on pearls and pearling in the Gulf, may have described how oysters were collected and opened as well as other techniques for cultivating pearls. Unfortunately, all details on this subject have since been lost to time. We only have fragments referring to pearl catching in an altogether general manner, such as Chares' above quoted passage ("a creature that resembles an oyster is caught").

With the increasing Roman taste for pearls, this state of affairs was to change significantly. In the early first century CE, it was widely known that pearl-oysters were collected by divers in the Persian Gulf as well as in Southern India. For instance, in the Periplus of the Erythraean Sea the fisheries of the Gulf are referred to as "diving areas", or "places where people dive". As for Pliny the Elder, in a passage where he blames humans and in particular his Roman contemporaries for desiring luxuries, he says:

Man has come to go as far as China for silk, and to explore the depths of the Red Sea (Latin: mare Rubrum, i. e., the Indian Ocean) for the pearl and the bowels of the earth for the emerald. (Pliny the Elder, Nat. Hist. 12, 2) However, Greco-Roman knowledge about pearling was not limited to this general idea, but had reached a certain degree of complexity, as shown by Isidorus of Kharax and the latesecond century author Aelian: They (= probably the natives of the Persian Gulf) say that when there is constant thunder and downpours of rain, the oysters reproduce most, and the pearls are the most numerous and largest. During the winter the oysters tend to go down into their hiding-places deep in the sea. (Isidorus of Kharax, FGrH 781 F1 = Athenaeus of Naucratis, Deipn. 3, 46) So, then these shells which are the mothers of the aforesaid pearls are sought for when the weather is fine and the sea smooth. (Aelian, Nat. Anim. 10, 13) According to Isidorus, pearl-oysters were out of the reach of Gulf divers during winter. The same author also heard from the natives that oysters only produced pearls during the rainy season. One can conclude from such statements that pearl diving would start after winter.

The second document seems to confirm this by indicating that diving was carried out during the warm season -when the sea was "smooth" and water not too cold.

Isidorus of Kharax also briefly describes the boats used by divers, stating that the island of Bahrain (see above) is surrounded by rafts made of reeds at pearling season. The Greek word employed by Isidorus points to small crafts manned by a few individuals. That boats made of reeds were used in this area has been confirmed by research focusing on early seafaring in the Persian Gulf. Bronze Age glyptic and archaeological remains (viz., reedimpressed bitumen) prove that such vessels were in use as early as the third millennium BCE.

A 5-metre reconstitution has recently been made by T. Vosmer and his team based in eastern Oman. 37 Therefore, we can surmise from Isidorus' testimony that, at the time of pearling season, hundreds of small reed boats carrying one or more divers flocked all around Bahrain, apparently not far from the shore.

That pearls were also gathered from deep waters is indirectly implied by a rather odd text, showing that Greeks and Romans believed that the sun's rays could spoil pearls even though they were hidden inside the shellfish. 38 But, says Pliny the Elder, pearls from the deep sea preserve a special brilliance, being too deeply immersed for the rays to penetrate. (Pliny the Elder, Nat. Hist. 12, 2) This belief may echo a local tale transmitted to Mediterranean people from the Gulf, but this is beyond the scope of this paper. The information worth retaining is that Greeks and Romans were aware of pearls being collected at a greater distance from the coast, which is confirmed by Isidorus of Kharax: Oysters found deep in the sea produce pearls that are more translucent, purer, and larger, whereas those that move upward toward the surface, because they are It can be concluded from both documents that divers would venture far from the coast in order to gather oysters and collect pearls of superior quality. 39 Indeed, the same Isidorus claimed that the Gulf fishers dived 20 fathoms down into the sea to procure shellfish. This figure -which equates to 32 meters -contrasts sharply with contemporary data. For instance, R. Carter drawing on Lorimer's Gazetteer of the Gulf, writes that in modern times, divers used to descend to 8 fathoms (15 meters) or more. 40 Isidorus' unbelievable figure is even more surprising as he proved himself an excellent and reliable source. For instance, he is the only classical author who demonstrated knowledge that pearl-oysters prefer to live in reefs, where they can secure themselves with their byssus (a bundle of filaments secreted by many species of bivalve molluscs): They (i. e., pearl oysters) cling to rocks or reefs, put down roots (= filaments of byssus) and stay there and produce pearls. (Isidorus of Kharax, FGrH 781 F1 = Athenaeus of Naucratis, Deipn. 3, 46) Diving deep into the sea requires ropes and stone weights which are not mentioned in Isidorus' surviving fragment. However, these indispensable tools were not unknown to the Greco-Romans, as proven by a second century Christian document entitled Physiologus.

Most commentators agree that this anonymous text was written in Alexandria in Egypt, which was a major hub for Indian Ocean goods imported into the Roman world. It is likely that the merchants involved in the pearl business were well informed about pearling, so that Alexandria was probably one of the best places in the Mediterranean to get accurate information.

There is a stone whose name is achates (= agate): those who seek pearls as a trade find them by means of this achates. The divers attach an achates to a very strong rope and send it down into the sea; and the achates goes where the pearl (= the pearl oyster) stands and does not move more. Immediately the divers follow the rope and find the pearl. (Physiologus latinus, De achate lapide) 39 Carter (b) 185 doubts the ancient opinion that deeper water yielded pearls of better quality: "Pearls from oysters in shallow water close to the shoreline are said to be of perfectly reasonable quality." 40 Carter (a) 141.

The Physiologus consists of descriptions of animals, birds, and fantastic creatures, sometimes stones and plants, provided with moral content. Because of its symbolic value, the Physiologus often misrepresents facts. Nonetheless, a certain amount of robust information can be recovered from these otherwise distorted data. This is the case here with the story of the stone called achates which guides divers straight to oysters. In my opinion, this achates is nothing more than stone weights used by divers, which proves that some Greco-Romans were aware of this piece of equipment as early as the second century CE, if not before. That said, Arab and European authors of medieval and modern times were better informed about the divers' tools. Let us quote, for instance, Ibn Battuta, the famous Moroccan scholar who widely travelled the medieval world over a period of thirty years:

The pearl fisheries which are between Siraf, and Bahrain are situated in a quite gulf of the sea, not unlike a large valley. To this place comes a great number of boats, and in these are the divers, with the merchants of Fars and Bahrain. When one of the divers intend to down, he places something upon his face made out of tortoise shell, and in this a place for the nose is cut out; he then ties a rope round his middle and goes down. The time they will remain under water varies (...). When the diver gets to the bottom of the sea, he finds the shells firmly fixed in the sand among trees (of coral).

He then either tear off with his hands or cuts them away with an iron knife and puts them into a leather bag which hangs to his neck. (Travels in Asia and Africa, chapter 10 ["Persia"]) Such simple objects may well have been in use in Antiquity, but they do not show up in the extant evidence. For instance, Isidorus gives an excellent description of how the oyster must be dragged away from the rock where it is attached without referring to any kinds of tools:

Pearl-fishers run a risk when they stick their hand straight into an open shell. For then it closes, and their fingers are often sheared off; some of them even die then and there. But if they manage to get their hand sideways underneath the shells, they can easily pull them away from the rock. (Isidorus of Kharax, FGrH 781 F1 = Athenaeus of Naucratis, Deipn. 3, 46) The proposition that a pearl-oyster can grievously harm or even kill a fisherman raises some doubts. Nonetheless, pearl diving was genuinely considered to be a hazardous occupation by Greco-Roman authors, not because of illnesses as cold or ear infections, but because of sharks, which are repeatedly said to be of major concern to divers. For instance, Pliny the Elder writes:

If the hand (i. e., of a diver) is inserted first it (i.e., the mollusc) cuts it off with its sharp edge, the most just penalty possible -for it is armed with other penalties also, as for the most part it (=the pearl-oyster) is found among rocks, while even in deep water it has sea-dogs (=sharks) in attendance. (Pliny the Elder, Nat. Hist. 9, 110)

5) From the Gulf to the Mediterranean (i): marketplaces and trade routes

Analysing the financial organisation of pearl industry in its latest phase (19-20 th centuries), R. A. Carter distinguishes three levels of markets:

The lowest level of market comprised the boats themselves, which some merchants would visit to purchase pearls.

At the second level there were temporary markets and supply centres set up near the pearl banks.

Dalma is the best known of these, an island in the heart of the pearl banks which had a small permanent population, which was boosted during the pearling season, when it became an important centre.

Finally the next tier includes the permanent pearling centres of the Gulf, i.e., the coastal towns established before and during the 18 th century CE. These were dominated by Manama (Bahrain), with Dubai recognised as the Gulf's secondary pearling emporium by the early 20 th century (...). From the permanent centres in the Gulf, the bulk of the pearls went to Bombay (Mumbai), whence they were redistributed to the global market. 41 Unsurprisingly, Greco-Roman documents are not as prolific as modern sources with respect to market organisation, for which the loss of numerous classical texts is not the sole cause. Whereas European travellers could easily access the Persian Gulf from the 17 th century onwards (or at least they were not prevented to access it), it is likely that not many visitors from the ancient Mediterranean world -merchants, seamen, ambassadors, private individuals etc. -would go there on a regular basis. As a result, only a few prominent markets of the Gulf area are documented, and rather poorly at that, while the existence of first and second level markets has been omitted entirely from classical literature and other sources.

The places that correspond to Carter's "permanent markets" are briefly described by the Periplus of the Erythraean Sea and Pliny the Elder, two reliable sources about the Indian Ocean trade patterns in the early Imperial period, although the picture they provide is incomplete and somewhat biased; the local trade is generally ignored. However, before reviewing the extant documents, I would like to stress the fact that there is no written evidence showing that an important marketplace operated on the Iranian coast, which hints that Arab tribes played the main role in the redistribution of pearls on the regional scale (however, with regard to the sway that kings of Persis may have held over the opposite side, see below). If this imbalance does not result from that loss of documentation, it evades easy explanation. All hypotheses that can be made -the efficiency of eastern Arabian networks, the lower productivity of eastern Gulf fisheries etc -will remain speculative.

Let us begin with the Periplus of the Erythraean Sea:

The centre of the passage through the mouth (= the Strait of Hormuz) is about 60 stades (= around 10 km), beyond which the very great and very wide Persian Gulf extends into the innermost parts. At its extreme end is an established mart called Apologos, lying near Spasinou Kharax and the river Euphrates (...) Sailing along the coast through the mouth of the Gulf, after six courses there is another mart of Persis called Ommana. (...) From both marts (= Omana 42 and Apologos) are sent to Barygaza (= Broach, in north-west India) and Arabia much pearl, though inferior to the Indian.

(Periplus of the Erythraean Sea, 35-36) Although this passage is not completely clear -especially from the geographical point of view, as will be seen below -, it provides the names of two permanent markets in the Gulf area, Apologos and Omana. The former is given the status of nomimon emporion, translated by L. Casson into "legally limited port of trade". The denomination of Apologos as nomimon emporion is common to two other ports named by the anonymous author, Mouza and Adoulis. No doubt the existence of official regulations is implied by the adjective nomimos, but it remains difficult to grasp what kind of regulations this term covers. In contrast, Omana is a simple emporion ("port of trade").

The location of Omana is not easily determined, for the Periplus gives only vague indications. Its identification varies considerably according to how the expression "sailing along the coast through the mouth of the Gulf" (Greek: parapleusanti) is interpreted. If this verb means "to sail along the mouth without crossing it", Omana must then lie somewhere on the Iranian coast outside the Gulf. The measure provided by the author ("six runs") is anything but helpful in precisely identifying the location: Hormuz and Chāh Bahār (25°17' N -60°40'E) have been suggested by some commentators, while others have located Omana further east in the vicinity of Gwadar West Bay, approximately at latitude 25°10' N and longitude 62°20' E. 43 Some historians, however, object to these interpretations, arguing the parapleusanti may be understood as "sailing along the mouth to cross the strait". Omana has accordingly been located somewhere on the Arabian shore of the Gulf -in this case the port was under Persian control ("there is another mart of Persis called Ommana"). 44 In particular, D. T. Potts puts forth a location which seems to him archaeologically suitable, namely the site of ed-Dur. 45 Identifying Apologos is less problematic though its exact location cannot be determined. Apologos lay at the bottom of the Gulf and at the mouth of the Euphrates (Shatt el-Arab). Supposing that the toponym Ubulla / al-Ubulla derives from the ancient name Apologos, then this place was probably situated close to modern Basra. 46 Unfortunately, we have no hope of obtaining further information, for archaeological remains are very unlikely to be ever recovered in this area. Let us add that Apologos was not the only trading place in this area, for other names are known to us: Teredon, Forat, not to mention the capital city of the Mesene (Maisan), Kharax.

Be that as it may, the Periplus testifies that in the first century CE, if not earlier, two "permanent" markets lay at each end of the Gulf, allowing pearls to be exported abroad:

Apologos at its bottom and Omana, located close to the Strait of Hormuz. According to the author, a certain quantity of pearls was transported from both emporia to Barygaza, an important harbour in north-west India, which brings to mind the more recent routes linking the Gulf with north-west India (pearls from the Gulf area have been exported to Bombay / Mumbai for centuries). 47 Pearls were also exported to Arabia. Assuming that the author is referring to Arabian peoples who did not conduct pearling, this Arabia is not easily identifiable. In my opinion, some pearls obtained in the Gulf may have been shipped from Omana to south-west Arabia where several kingdoms had developed (e. g., Saba, Himyar, Hadramawt ...), to be either purchased by locals or re-exported from one of their ports of Khaṭṭ lies opposite to Bahrain, that is to say close to the main fisheries in the Gulf (see Pliny the Elder, Nat. Hist. 6, 148). This could attest to the role of Attene as a marketplace which redistributed the production coming from the nearby fisheries. In addition, Gerrha and Attene were probably linked with the northern Gulf. In the 4 th century BCE, Alexander's companions had already observed that aromatics were brought from Gerrha to Babylonia via the Euphrates River. In the late 1 st century BCE, according to D. T. Potts, Gerrha was in close connection with the Characene kingdom, which controlled the marketplace of Apologos:

The rise of Charax in the late 2 nd century and its growing commercial importance in the late 1 st century B.C. may explain why, as Juba (a Greek speaking author) says, the Gerrhaeans at some point made yet another reversal and began sending their wares to Charax and the Parthian empire. 49 Thus, despite the manifest lack of evidence, I would nevertheless assume that a local route connecting the fisheries of Bahrain to Apologos via Gerrha was in operation at that time.

With such paltry evidence, one can only draw a fragmentary picture of the supply chain. Considering the numerous fisheries off the Arabian coast, the pearls collected on this side were apparently transported to places located either at the mouth of the Gulf, or in the vicinity of the Strait. One may surmise that the production of southern fisheries would reach destinations less distant-i. e. Omana -while pearls from Tylos would preferably have been transported to Apologos, but such simple reasoning -mainly based on transport costsshould be avoided; the market structure and trade patterns of high value products such as pearls may have been much more sophisticated. As for the circuits starting from the Iranian fisheries, documentation concerning them is entirely absent. Several scholars have expressed a certain degree of scepticism about the theory of "Arab control" over Gulf trade. G. Young, for instance, while not denying that the Periplus is uninformed about the Gulf, rejects Salles' conclusion that "this was because the Gulf was closed to Roman shipping". No "ban" prohibited the merchants of Alexandria from sending ships into the Gulf, he says; they simply did not have any reason to go there. That said, Young seems to agree with Salles in that local agents (Palmyrenes, Parthians, Arabs, Persians, Characenians ...) took an active interest in local and interregional traffic. Conversely, the inner Persian Gulf was of little interest to Mediterranean shippers. 51 In other words, Apologos and Attene were primarily a destination for local agents, while Omana may have been reached by some Westerners, the number of which cannot be estimated.

6) From the Gulf to the Mediterranean (ii): cultural transfers

As mentioned above, the very first Mediterranean people who witnessed the commercialisation of pearls were Alexander's companions. It is common knowledge that the king and his friends managed to collect ample amounts of information about the countries they visited. It is also widely known that they did not hesitate to investigate many of the issues regarding the support of autopsia (direct observation), as did Alexander's companions when they discovered pearls and pearl fisheries in the Persian Gulf. They did their best to describe this marvel of Nature and to better understand its origin. In his account, Chares of Mytilene refers to the pearl as a psêphos ("stone"), which implies an implicit comparison with the stones of fruit or bones of animals. Androsthenes of Thasos, who was sent by Alexander to explore the western side of the Gulf, visited Tylos / Bahrein, where pearls were actively being fished. He put forward a different theory: the margaritês appeared to him to be the result of a physiological disorder. Both theories rest on a kat'analogian ("based on comparison") reasoning.

In the first century BCE, as I noted earlier, things were altogether transformed. As the Romans' desire grew more and more intense for this ornament, so did the importation of pearls increase in magnitude. As a result, pearls of varying quality were offered for sale in Rome, segregated in terms of colour, shape, size, lustre etc. This is proven by the many expressions referring to their quality and of their imperfections that appear in Pliny the Elder's Natural History. In addition, this increasing consumption must have solicited more and more curiosity on the part of the consumers. They certainly wished to understand how an Erythraean shellfish could create such a marvel. There is little doubt that the rational explanations provided by the first observers had to be abandoned. The analogy with a parasite or a bone could not explain the perfect roundness, whitish hue and incredible lustre of the uniones. In fact, as features in Pliny's account, a new set of explanations had been introduced and adopted, which I shall summarize below. 52 At a particular time of year, Indian Ocean oysters require fertilizing. Thus, they swim up to the surface of the sea, open their valves and receive a drop of dew of pure water. With this seminal liquid supplied by the morning sky, the oysters fall "pregnant". Now, instead of giving birth to oyster offspring, they instead produce a pearl which is in a liquidous state l until it is taken out of its sea habitat. In other words, the pearl is regarded as a drop of celestial water which has absorbed a certain quantity of pure light and then develops inside the shellfish. The qualities and imperfections of pearls are determined by the qualities of the celestial water and the light provided. A pure dewdrop produces a pearl possessing a perfect whiteness and a spherical form; if the water is turbid, the round pearl is opaquer in appearance. If the sky is clear when the oyster is fertilized, the pearl often displays a beautiful lustre (this explains the poor quality of British pearls: the western part of the Outer Ocean lacks the bright light of the eastern edge of the inhabited world). Some circumstances may develop which disturb or interrupt the development of pearls. Flashes of lightning make oysters close upon themselves and stop feeding. As a result, pearls stop developing; such is the origin of small pearls. Storms and thunder cause the oysters to close upon itself suddenly, which produce "blistered" pearls. Pliny also states that a pearl whose surface is not smooth is an old pearl etc. In conclusion, this explanatory system developed in tandem the evolution of Roman consumption. Each type of pearl was traced back to its developmental origin.

Let us further analyse this subject. The crux of this theory is that oysters are understood to absorb light and water. Thanks to Isidorus of Charax, we can ascertain that this idea originates from the Persian Gulf.

Isidorus of Charax, in his Description of Parthia, says that there is an island in the Persian Gulf where pearls are found in abundance; the island is surrounded with reed rafts from which the natives dive and bring up bivalves. They say that the mollusk is most apt to teem with pearls when thunderstorms and downpourings of rain are frequent, and the pearls found then are most numerous and of good size. (Isidorus of Kharax, FGrH 781 F1 = Athenaeus of Naucratis, Deipn. 3, 46) The idea that pearls are made of water and light turns out to be a story known to the 52 On this topic, see Schneider (a) 124-128. native divers; certainly, it is not a theory of Greco-Roman origin. As a matter of fact, this tale appears in later texts: al-Idrīsī reports the following anecdote from residents who lived around the shores of the Persian Gulf as follows:

They [= pearls] result principally of the rains of February; if no rains fall at time, the divers find nothing for the rest of the year. This is an incontestable fact, which no one in the area doubts. 53 This seems to be an incidence of cultural transfer. The first Greek ideas about how the pearls are formed were replaced by local tales as the opinion of the Arab divers proved more appropriate than the rational approach developed by Alexander's companions. This raises a couple of vexing questions. Firstly, how were such tales introduced into the Mediterranean world? At a time Mediterranean consumption had increased and an ever-growing number of pearls were being imported, I believe that merchants must have played an important role. As the Periplus of the Erythraean Sea proves, some of them ventured as far as the Strait of Hormuz and probably as far as Omana itself. In addition, archaeological fieldwork at Myos Hormos and Berenike has established the presence of Arab and Indian merchants in Egypt. Some even settled in Alexandria. This means that merchants did not carry only commodities; they also played a role in the cross-cultural contacts between the Mediterranean world and the Indian Ocean peoples, of which the story of the oysters being fertilized by a drop of water is an example.

As a matter of fact, the theory of pearls being made of water and light was not entirely contrary to Greco-Roman thinking. The idea that gemstones result from coagulation (pêxis) of water and /or light was expressed by the Greek scholar Posidonius (1 st century BCE). Accordingly, the idea that the pearl is a drop of light imprisoned in water which becomes solid could be readily accepted by the Greco-Romans. However, this oriental tale was probably adapted to the Greco-Roman mind (this is what we call interpretatio). Some traces of the Greco-Roman interpretation are present in Pliny's account. The sun, which was believed to be responsible for the Ethiopians' dark skin, was also said to be responsible for the colour of brownish pearls. More interestingly, according to the native divers, the abundance of pearls resulted from thunderstorms. However, we read in Aristotle that thunderstorms have undesirable effects on poultry eggs (Aristotle, Hist. Anim. 6, 2). It is not

  Greek speaking historian of oriental origin, born in Kharax -, who flourished around the early Christian era (late first century BCE -first century CE). 27 As a native of the Gulf area, Isidorus was obviously familiar with pearling. His account, entitled Journey through Parthia, has survived through citations. The following one relates to pearling: Isidorus of Charax claims in his Journey through Parthia that there is an island in the Persian Sea where large numbers of pearls are found. The island is therefore surrounded by rafts made of reeds. (Isidorus of Kharax, FGrH 781 F1 = Athenaeus of Naucratis,Deipn. 3, 46)

37

  Carter (b) 3-4. affected by the sun's rays, are of an inferior colour and smaller. (Isidorus of Kharax, FGrH 781 F1 = Athenaeus ofNaucratis, Deipn. 3, 46)

  trade, such as Kanê (Khor Rori), Eudaimôn (Aden) or Mouza (in the vicinity of modern Moka /al-Mukha). The Periplus may also refer to a circuit connecting Apologos to Palmyra, Gaza or Petra, since these places belonged to Arabia as defined in Greco-Roman geography.The author of the Periplus', whose knowledge of the Persian Gulf is very limited, does not refer to other major markets. This lack of information is hardly compensated by other sources. There exists only an elusive passage by Pliny the Elder who, at some point in his geographical presentation of the Persian Gulf, refers to two busy places named Attana and Homna: (...) and also Homna and Attana, towns said by our traders to be now the most frequented places in the Persian Gulf. (Pliny the Elder, Nat. Hist. 6, 149) Pliny affords us no clues as to the location of Homna. Most scholars consider this name aperhaps accidental -repetition of Omana, which Pliny had mentioned earlier. With respects to Attana (also spelled Attene), this name appears twice, first as a region (Pliny the Elder, Nat. Hist. 6, 148) and second as a place (oppidum) of trade, which, given the context, must be a port. Attene has been connected with the Chattenia of Polybius, the Atta of Ptolemy and finally with the Arabic name al-Haṭṭ. C. J. Robin argues that Attene / al-Haṭṭ, or Khaṭṭ, (al-ʿUqayr) is the port of the kingdom of Gerrha /Hagar (Hajar; al-Hufuf). 48

  We do not know to what extent the marketplaces of Apologos -and Attene -were frequented by the Greek and Roman merchants based in Alexandria. That Pliny refers several times to Roman (?) merchants of his time (nostri negotiators -e. g., Pliny the Elder, Nat. Hist. 6, 149) informing him about the Gulf, he says nothing about the actual level of commercial penetration by Mediterranean traders. According to J.-F. Salles, they did not venture further north than ed-Dur / Omana. (…) il n'est pas évident que des navigateurs romains en provenance de la mer Rouge aient eu librement accès à tous les ports de la mer Érythrée. L'hypothèse avancée est que le golfe Arabo-Persique constituait une branche autonome de la mer Érythrée sous contrôle arabe (Characéniens, Palmyréniens, Arabes du Golfe) et sans doute perse, inaccessible aux marchands occidentaux : leur ignorance de cette région en deviendrait alors parfaitement justifiées. 50

43

  For a good review, seePotts 306-310.

44

See Wiesehöfer.

45

Potts 306-310.

46

Casson 179-180.

47

Kunz & Stevenson 98.

See Schneider (c) 136-144 for references.

Donkin 90.

Schneider (a) 289-306.

In Greek mythology, the sea goddess Thetis was the wife of Peleus and the mother of Achilles.

For a full discussion of this question, seeSchneider (a). R. Carter is of different opinion: quoting a fragment by Chares of Mitylene (see below), he claims that the" Indian sea" referred to is a "vague statement" about pearling practiced "in the wider Indian Ocean"(Carter (b) 11).

On the importance of islands seeCarter (a) 197.

Pliny's source is unknown to us (see, however, Carter (a) 14). It cannot be excluded that Pliny refers to a

About this place, see below, n. 46.

Kharax was originally named Alexandria, after Alexander the Great, and was perhaps even personally founded by him. In 127 B.C., an independent state, Characene (alternatively known as Maisan), was established in this part of the Gulf, with Kharax as capital. It was annexed by Rome later.

This is the main reason why I would not agree with Carter (b) 15: "It is usually assumed that the 'island in the Persian Sea' alludes to Bahrain, but it could just as easily refer to Stoidis" on Stoidis, see below).


Hervé Baumann
email: herve.baumann@liafa.jussieu.fr

Pierluigi Crescenzi

Pierre Fraigniaud
email: pierre.fraigniaud@liafa.jussieu.fr

Inondation dans les r éseaux dynamiques † ‡

Keywords: Gossip protocol, epidemic protocol, evolving graphs, broadcasting

Cette note résume nos travaux sur l'inondation dans les réseaux dynamiques. Ces derniers sont définis à partir d'un processus Markovien de paramètres p et q générant des séquences de graphes (G 0 , G 1 , G 2 , . . .) sur un même ensemble [n] de sommets, et tels que G t est obtenu à partir de G t-1 comme suit : si e / ∈ E(G t-1 ) alors e ∈ E(G t ) avec probabilité p, et si e ∈ E(G t-1 ) alors e / ∈ E(G t ) avec probabilité q. Clementi et al. (PODC 2008) ont analysé différent processus de diffusion de l'information dans de tels réseaux, et ont en particulier établi un ensemble de bornes sur les performances de l'inondation. L'inondation consiste en un protocole élémentaire où chaque noeud apprenant une information à un temps t la retransmet à tous ses voisins à toutes les étapes suivantes. Evidemment, en dépit de ses avantages en terme de simplicité et de robustesse, le protocole d'inondation souffre d'une utilisation abusive des ressources en bande passante. Dans cette note, nous montrons que l'inondation dans les réseaux dynamiques peut être mis en oeuvre de fac ¸on à limiter le nombre de retransmissions d'une même information, tout en préservant les performances en termes du temps mis par une information pour atteindre tous les noeuds du réseau. La principale difficulté de notre étude réside dans les dépendances temporelles entre les connexions du réseaux à différentes étapes de temps.

Introduction

Gossip protocols have been identified as simple, efficient, and robust mechanisms for various network and system tasks, such as, e.g., multicast, resource location, and distributed databases management. In parallel, the epidemiology community has considered several models for the analysis of the spreading of an infection in a population. These models include the famous SIR (Susceptible-Infective-Removed) and SIS (Susceptible-Infective-Susceptible) models, aiming at capturing the way a virus disseminates in a population as a function of the reaction to the virus experienced by every people. Gossip (and epidemic) protocols tolerate a high degree of dynamism in their running environment. It is therefore of premier importance to evaluate the precise impact of this dynamism on the efficiency of the gossip protocols. It is indeed known that network dynamics can have a tremendous impact in certain circumstances. This impact can be quite positive whenever the network evolution is ergodic (e.g., when measuring the global bandwidth of an ad hoc radio network [START_REF] Grossglauser | Mobility increases the capacity of ad-hoc wireless networks[END_REF]), but also quite negative whenever the network evolution is arbitrary (e.g., when measuring the cover time of random walks [START_REF] Avin | How to Explore a Fast-Changing World[END_REF]).

In the framework of gossip protocols and epidemiology, one way to handle dynamism is to assume that the network evolves with time as a sequence (G 0 , G 1 , G 2 , . . .) of graphs on the same set of vertices, where the graph G t considered at time t is an Erdös-Renyi random graph drawn in G n,p . In this model, several investigations have been recently performed to measure the impact of the network evolution on the performances of algorithms. For instance, it was proved that radio broadcasting performs efficiently, even for p below the connectivity threshold of G n,p (see [START_REF] Clementi | Communication in Dynamic Radio Networks[END_REF]). In [START_REF] Avin | How to Explore a Fast-Changing World[END_REF] it is proved that the cover time of a random walk remains polynomial for any p > 0. Threshold phenomena have also been identified ; for † A part of this work was done during the stay of the second author at University Paris Diderot as Invited Professor. All authors were supported by COST Action 295 "DYNAMO". The first and third authors received additional supports from ANR project ALADDIN, and from INRIA project GANG. The second author received additional supports from the EU under the EU/IST Project 15964 AEOLUS.

‡ The full version of this paper will appear in Proc. 28th ACM Symp. on Principles of Distributed Computing (PODC 2009) instance, it was proved that SIR epidemic only contaminates a constant number of nodes if p = c/n with c < 1, but contaminate a constant fraction of the nodes if c > 1 (see [START_REF] Draief | Thresholds for virus spread on networks[END_REF]).

In term of modeling real world networks, the elementary random process

(G 0 , G 1 , G 2 , . . .) with G t ∈ G n,p
suffers from the absence of time dependencies. The network does evolve but the structure of the network at time t is independent from its structure at time t ′ < t. This does not precisely reflect what is observed in many contexts such as wireless networks (the connection between two nodes is highly correlated to the previous existence of this connection) and P2P networks (the occurrence of an information exchange between two participants is highly correlated to the existence of previous exchanges). A more evolved model capturing time-dependencies has been recently considered in [CMM + 08, CMPS09]. Similarly to the elementary random process, the model, called edge-Markovian process, and denoted M n,p,q , generates a random sequence of graphs

(G 0 , G 1 , G 2 , . . .) on the same node set [n].
This sequence is set based on a birth-rate p and death-rate q as follows § : G 0 is an Erdös-Renyi random graph in G n, p where p = p/(p + q), and, for any t > 0, a non-existing edge e / ∈ E(G t-1 ) exists in E(G t ) with probability p, while an existing edge e ∈ E(G t-1 ) disappears from E(G t ) with probability q.

In their companion papers [CMM + 08, CMPS09], Clementi et al. analyzed the flooding protocol in edge-Markovian dynamic graphs, i.e., in graph sequences generated by the edge-Markovian process. Flooding in dynamic graphs is the gossip mechanism in which every node becoming aware of an information at step t forwards this information to all its neighbors at all forthcoming steps t ′ > t. Flooding is a core mechanism for information dissemination in contexts in which the network topology is highly dynamic and unknown, such as P2P networks, mobile networks, or any networks susceptible to faults, and several variants of flooding designed to limit the bandwidth consumption have been proposed [CL07, LCC + 02]. Clementi et al. produced several bounds on the flooding time in edge-Markovian dynamic graphs. In particular they proved that there is a wide class of dynamic graphs in which the flooding time does not (asymptotically) depends on the edge death-rate q.

Despite the interests of flooding in term of simplicity and robustness, this protocol suffers from a severe drawback in dynamic graphs : it requires every node, upon reception of a source message, to forward this message during all forthcoming time steps. This results in a waste of resources in terms both of link bandwidth and node computation. Of course, if one knows that flooding completes in T time steps, then all nodes can be bounded to be active for only this amount of time. Nevertheless, T is typically growing with the size n of the network, hence been active T steps still results in a significant waste of resources. Our objective is to force flooding to perform more parsimoniously, by limiting the number of steps during which every node is active in forwarding a message to its neighbors, yet allowing the message to eventually reach all nodes in short time. Parsimonious flooding enables to save bandwidth and computational resources, and potentially energy as well, the latter parameter being known to be crucial for ad hoc and sensor networks.

Previous work.

There is a vast body of literature on broadcast and gossip protocols in static networks. Broadcasting in random graphs G n,p has been analyzed in [START_REF] Feige | Randomized broadcast in networks[END_REF][START_REF] Frieze | The shortest-path problem for graphs with random arc-lengths[END_REF][START_REF] Pittel | On spreading a rumour[END_REF], and randomized gossip protocols in specific metrics have been analyzed in [START_REF] Kempe | Spatial gossip and resource location protocols[END_REF][START_REF] Kempe | Protocols and impossibility results for gossip-based communication mechanisms[END_REF]. In all these cases, there are no time-dependencies induced by any evolution of the network structure.

Several papers tackle information spreading problems in the context of wireless networks (see, e.g., [CLF + 07, SD09]) but the time-dependencies are either ignored, or overcame by assuming sufficiently long time slots for these dependencies to become negligible. In fact, the only information spreading work we are aware of, dealing explicitly with time-dependencies regarding the presence of the links, is [CMM + 08, CMPS09]. In the former, the model is the one considered in this paper, i.e., edge-Markovian dynamic graphs. In the latter, the authors consider other types of Markovian dynamics, including the geometric Markovian evolving graphs, and give general bounds on the flooding time in Markovian dynamic graphs satisfying certain expansion properties.

More specifically, it is proved in [CMM + 08] that, for any initial graph G 0 (i.e., not necessarily G 0 ∈ G n, p ), and any birth-rate and death-rate 0 < p, q < 1, the flooding time in edge-Markovian graphs is at most § The general definition in [CMM + 08] assumes that G 0 can be arbitrary, and the definition of edge-Markovian evolving graphs given here is called stationary edge-Markovian evolving graphs in [START_REF] Clementi | Information Spreading in Stationary Markovian Evolving Graphs[END_REF].

O(log n/ log(1 + np)).

For the stable graph G 0 (i.e., E(G 0 ) = / 0), it is proved that, for any 0 < p, q < 1, the flooding time is at least Ω(log n/np)), and if p ≥ c log n/n for c > 1 then the flooding time is at least Ω(log n/ log(1 + np). The case G 0 ∈ G n, p is considered in [START_REF] Clementi | Information Spreading in Stationary Markovian Evolving Graphs[END_REF], where it is proved that for p ≥ c log n/n with c large enough, the flooding time is at most O( log n log n p + log log n p) and at least Ω( log n log n p ).

Parsimonious flooding

For a positive integer k, we say that a flooding protocol (in dynamic graphs) is k-active if each node forwards a source message only during the k time steps immediately following the step at which the node receives that message for the first time. For instance, the 1-active flooding protocol is the standard flooding protocol for static networks : a message is forwarded only once, at the step immediately following its reception. However, in dynamic networks, the flooding protocol may have to be active for k > 1 steps in order for the message to reach all nodes. The smaller the parameter k, the lesser the resource consumption by the protocol.

Our objective is to determine the minimum k for which the k-active flooding protocol completes correctly, i.e., the message eventually reaches all nodes. We define the reachability threshold for the flooding protocol in M n,p,q as the smallest integer k such that, for any source s ∈ [n], the k-active flooding protocol from s completes correctly almost surely ¶ . Clearly if flooding completes in T steps in M n,p,q , then the reachability threshold of flooding in M n,p,q is at most T . But in fact, we will show that, for a large spectrum of parameters p and q, the reachability threshold is just o(T ), and often even just constant. Moreover, we will also show that being active for a number of steps equal to the reachability threshold (up to a multiplicative constant) is sufficient for the flooding protocol to complete in optimal time, i.e., in asymptotically the same number of steps as when being perpetually active.

Our results

In this paper, we first revisit the results in [CMM + 08, CMPS09], and we give tight bounds on the flooding time (without any bound on the activity constraints) for all possible values of p, q ∈ (0, 1) whenever G 0 ∈ G n, p . These bounds are summarized in Table 1, where p = p p+q . For p ≥ c log n n with c > 1, flooding performs, a.s., in Θ( log n log(n p) ) rounds. For 0 < p ≤ c n with c > 0, flooding a.s. performs in Θ( log n np ) rounds. In between, the situation is more complex, and depends on the relative values of p and p (see Table 1). If np ≥ log n p then the flooding time is Θ( log n log(n p) ), whereas if np ≤ log n p then the flooding time is Θ( log n np ). In parallel to the computation of the bounds on the flooding time, we have established tight bounds on the reachability threshold. If p ≥ c log n n with c > 1, then this parameter is equal to 1. That is, just one step of activity is enough to make sure that, a.s., the message reaches all nodes. If p ≤ c log n n with c < 1, then the reachability threshold is Θ( log n np ). (Note that the condition is on p while the value for the reachability threshold depends on p).

Interestingly enough, for any p, q ∈ (0, 1), we also prove that if k is the activity threshold, then, a.s., an O(k)-active flooding protocol completes in the same time as flooding without constraints on the activity, up to a multiplicative constant. In other words, the reachability threshold for the flooding protocol is essentially of the same order of magnitude as the activity threshold for this protocol to complete in optimal time, up to multiplicative constants. In particular, for p ≥ c log n n with c > 1, one step of activity is sufficient for flooding to complete in optimal time. Similarly, for 1 n ≪ p ≤ c log n n with c < 1, and np ≥ log n p, the reachability threshold Θ( log n np ) is significantly smaller than the optimal flooding time Θ( log n log(n p) ), yet being active for O( log n np ) steps is sufficient for flooding to complete in asymptotically optimal time O( log n log(n p) ). For all the remaining cases, the thresholds for completion and for optimality coincide, up to multiplicative constants. ¶ A series of events E n holds almost surely (a.s.) if Pr[E n ] → 1 when n → ∞, i.e., Pr[E n ] = 1 -o(1).

R éf érences

  These events holds with high probability (w.h.p.) if Pr[E n ] ≥ 1 -O( 1 n α ) for some α > 0. 0 < p ≤ c n , c > 0 1 n ≪ p ≤ c log n n , c < 1 p ≥ c log n n , c > 1 np ≤ log n p np ≥ log n p

	Flooding time	Θ( log n np )	Θ( log n np )	Θ( log n log(n p) )	Θ( log n log(n p) )
	Reachability threshold	Θ( log n np )	Θ( log n np )	Θ( log n np )	1
		TAB. 1: Summary of results


Antoine Lima

Véronique Cherfaoui

Philippe Bonnifait

Evidential Trustworthiness Estimation for Cooperative Perception

Keywords: Cooperative Perception, Trust, Multi-Robot System, Belief Functions

Intelligent Vehicles can exchange their perception information using wireless technology in a cooperative and decentralized manner. This has the potential to extend the range of perception and thus improve anticipation for complex driving maneuvers and decision making. However, information received from other peers can be erroneous and has to be used carefully. In this paper, we present a method that allows each peer to assign a trust in the information received from other peers based on comparisons with its current knowledge of the world. We describe how this process is managed using the Dempster-Shafer theory. We also present how positive and negative evidence cues can be developed in this problem, in particular by using detectability grids. An experimental evaluation, carried out with real vehicles, is reported to show that this formalism behaves correctly.

Introduction

In order to navigate safely, intelligent vehicles need to perceive their environment. Their on-board sensors, such as cameras or LiDARs, are generally sufficient for local navigation tasks but not for more complex maneuvers because of the limited range of the sensors and because there are occlusions in their Field Of Views (FOVs). For example, in Fig. 1, v 1 cannot see if a vehicle is coming from behind the building on its right and will thus have to either be cautious and engage slowly or break strongly once the vehicle becomes visible. Cooperative Perception (CP) aims at improving the navigation performance in such situations by taking advantage of perceptual information captured by others. Indeed, using upcoming wireless technologies, it is possible for vehicles and the infrastructure to exchange perceptual information with each other. By integrating this information to its own, one's knowledge of its surroundings can be extended further and behind obstructions. For example, in Fig. 1, v 3 could warn v 1 of its presence and v 2 could warn v 1 that a vehicle is present in front of it. However, although the authenticity of peers can be cryptographically guaranteed using public-private key pairs [START_REF] Chowdhury | Secure Information Sharing among Autonomous Vehicles in NDN[END_REF], security vulnerabilities or perception malfunctions (e.g. sensors failures) can still generate erroneous information that should not be incorporated to one's own. Fig. 1: Three cooperative vehicles at an intersection. v 1 cannot see v 3 because it is hidden by a building but v 2 and v 3 can see each other.

To prevent this, we propose an information processing and data fusion system that confronts the information received and the information from the embedded system to estimate the trustworthiness of the peers. This information can then be used in a cooperative tracker to attenuate or ignore information from untrustworthy peers. This process is done locally by each peer, without communicating its trust, by verifying that the received information matches with its knowledge of the world. For example, detecting objects at the same location creates trustworthiness while mismatching or illogical information creates untrustworthiness. In Fig. 1, because v 1 partially shares objects and FOV with v 2 and v 3 , it will trust them and thus anticipate v 3 earlier. After a review of related works in Section 2, we will introduce the problem at hand using a dense representation of the detectable or undetectable space from the point of view of different peers in Section 3. In Section 4, trustworthiness estimation will be formulated. Finally in Section 5, a simulation study and experimental results based on real data will be given and analyzed.

Related Works

The field of Cooperative Perception began with [START_REF] Günther | Realizing collective perception in a vehicle[END_REF] demonstrating its potential for safety in intelligent transportation systems. Since then, the European Telecommunications Standards Institute (ETSI) standardized the Cooperative Perception Message (CPM) [START_REF]Analysis of the Collective Perception Service[END_REF], composed of the sender position, sensor descriptions and a list of objects. It is used in many cooperative approaches, as studied in [START_REF] Caillot | Survey on Cooperative Perception in an Automotive Context[END_REF]. A part of the research effort is focused on preventing attacks as CP works on a public network. The most common form of attack prevention is misbehavior detection, as reviewed in [START_REF] Van Der Heijden | Survey on Misbehavior Detection in Cooperative Intelligent Transportation Systems[END_REF]. In this paper, the authors list and classify numerous approaches as being standalone or shared, distributed or centralized and node or data centric. For example, in [START_REF] Obst | Multi-sensor data fusion for checking plausibility of V2V communications by vision-based multiple-object tracking[END_REF], errors are detected by comparing received positions with detections from embedded sensors. In [START_REF] Ambrosin | Design of a Misbehavior Detection System for Objects Based Shared Perception V2X Applications[END_REF], four levels of checks ranging from simple bound check to object comparison are used to emit reports on misbehaving peers. When enough reports are received about a faulty peer, their certificate is revoked, excluding them from communication. In [START_REF] Allig | Trustworthiness Estimation of Entities within Collective Perception[END_REF], a probability of trustworthiness is estimated for each pair of peers, by checking the consistency of their object lists and their respective detection probabilities across space. More recently, [START_REF] Liu | MISO-V: Misbehavior Detection for Collective Perception Services in Vehicular Communications[END_REF] compared occupied or free space in the form of grids and verified that detected objects matched with these grids. In [START_REF] Aeberhard | Object existence probability fusion using dempster-shafer theory in a high-level sensor data fusion architecture[END_REF], sensors estimate a probability of existence for each object. When fusing object from multiple sensors, they switch to an evidential representation and use a persistence probability to model the field of view of fused sensors. A trust parameter representing the sensor's information reliability is fixed for each sensors. Our method can be seen as an unification of [START_REF] Ambrosin | Design of a Misbehavior Detection System for Objects Based Shared Perception V2X Applications[END_REF][START_REF] Allig | Trustworthiness Estimation of Entities within Collective Perception[END_REF][START_REF] Liu | MISO-V: Misbehavior Detection for Collective Perception Services in Vehicular Communications[END_REF] where fault detection generates untrustworthiness and confirmation creates trustworthiness.

Problem Statement With Object Detectability

Consider a driving situation composed of N vehicles v 1 , v 2 . . . . Every cooperative vehicle perceives surrounding objects o ∈ O and Free Space F S. Objects can be any kind of road user (e.g: pedestrians, cooperative or non-cooperative vehicles) or static features (e.g: traffic signs) whereas F S are areas explicitly characterized as being free. Objects and F S are broadcast to peers in the communication range, supposed to be further than the perception range. In addition, the Fields Of View (FOV) of the sensors are shared as per the CPM such that all exchanged information is vectorial in order to reduce the amount of data exchanged. Upon reception, each peer assesses the trustworthiness of the sender, as illustrated in Fig. 2.

As the problem at hand combines multiple point of views, a method to define which area was seen by the cooperating peers is needed. For this, we extend the detection probability of [START_REF] Allig | Trustworthiness Estimation of Entities within Collective Perception[END_REF] with the capacity to state that there cannot be objects in the measured F S by using an evidential representation. In this representation, the ground plane is divided into cells of fixed size that contain a mass function This grid (called detectability grid) is built with the process illustrated in Fig. 3.

Outside the FOVs and behind buildings, the state is unknown. The F S characterized by local sensors (e.g. LiDAR points that hit the ground) is used to express the impossibility to detect objects within it. In space neither free nor unknown, an object is likely to be detected. Object detectability is used in two different ways when receiving information. First, the sender's detectability grid is reconstructed to assess its objects coherency (e.g. there is no object in the FS or out of the FOV). Then, the receiver's detectability grid is constructed to verify the coherency of received objects and to only compare objects that are detectable by it and the sender.

Evidential Trustworthiness Estimation

Trustworthiness in the information sent by other peers is estimated by every peer individually as a mass function noted m T j about peer j. It is designed to be used in a subsequent cooperative fusion to ignore or discount objects originating from untrustworthy peers. As such, it is defined on 2 Ω T with Ω T = {T , T } to express that information from j is trustworthy and can be integrated without hesitation or conversely not integrated at all. In the rest of this paper, trust mass functions are normalized and will be given in the following order: T , T , Ω T . Mass functions are particularly adapted to the problem at hand for several reasons. Firstly, similar to humans, trust evolves over time and can be forgotten when peers are not interacting anymore, which can be managed with discounting. Secondly, as new peers have an unknown degree of trustworthiness, choosing a wrong prior could lead to ignoring good information or including misleading one during transient phases. Finally, this provides more information for a subsequent cooperative tracker to make more or less cautious decisions when peers are only partly trustworthy (because their information contains both valid and invalid values). Trustworthiness is sequentially estimated using an evidential network. Similar to state filtering, the current estimate at time t m T j (t|t) is derived as the combination of the previous estimate m T j (t -1|t -1) and new evidence about "Co- , defined on 2 Ω T and group leaves of the network as described later on. Those leaves express simple and non-dogmatic constraints on either trustworthiness or non-trustworthiness based on different aspects 'of the received information. In the combination process, every term is discounted by an associated factor that is not be explicitly noted here for the sake of clarity. Therefore, leaves always express some degree of belief on m(Ω T ). Dempster's rule + is well adapted in this case and is used for combination:

m T j (t|t) = Λ ∆t m T j (t -1|t -1) + m cohe j + m cons j + m conf j (1)
where Λ ∆t is a discounting factor that depends on the elapsed time ∆t, moving an Λ ∆t -proportion of every focal set to the unknown [START_REF] Kurdej | Conservative, Proportional and Optimistic Contextual Discounting in the Belief Functions Theory[END_REF].

Coherency

m cohe models that the information contained in a message has to be coherent within itself. Multiple constraints are combined using Dempster's rule, three of them are given here as an example:

m cohe j = m obd j + m atc j + m spc j (2)
m obd j expresses that objects cannot exist inside the FS or outside the perception range of the peer. For this, the detectability measure j m D of the sending peer j is used. For example, an object that is in the FS is by definition undetectable and its detectability will be low. Similarly objects outside the FOV are unknown and will have a low detectability. We use a constant D min threshold to assign a mass on the untrustworthiness parametrized with a constant β pen for such objects:

m obd j = + o∈Oj j m D o (D)<D min 0 β pen 1 -β pen (3)
m atc j expresses that object attributes have to be likely. For example, the speed v o of the object o has to be coherent with a normal behaviour which is handled by a scalar sigmoid function Φ illustrated in Fig. 5 with parameters µ v , σ v and δ v chosen to reflect speed limits:

m atc j = + o∈Oj Φ(v o , µ v , σ v , δ v ) (4)
with

Φ(x, µ, σ, δ) = m1 = 1-CDF x-µ-2σ σ √ 2 2 m2 = 1+CDF x-µ-2σ-δ σ √ 2 2 1 -m1 -m2 (5)
m spc j expresses that objects have to be spatially coherent. For example, cars should be close to the road network. Again a scalar metric is computed and used as input to a sigmoid function Φ. For instance, let C j be the subset of O j classified as cars and d o the distance between a car o and the road. With adapted parameters µ d , σ d and δ d , the mass is:

m spc j = + o∈Cj Φ(d o , µ d , σ d , δ d ) (6)
Please note that other constraints can be added following the same formalism, such as modeling that object sizes or covariances should be of reasonable values.

Consistency

m cons models that objects must follow coherent trajectories in time and not change their dynamics in an unpredictable way (by making improbable position jumps between two messages for instance). For this, previously received objects are predicted O j (t|t -1) and associated with newly received objects O j using an assignment function noted A. The similarity function described in [START_REF] Zoghby | Optimal object association from pairwise evidential mass functions[END_REF] is used to compare objects and is noted m S a,b . It compares different characteristics of the objects and yields a mass function defined on Ω S = {S, S} to express that a and b can correspond to the same physical object or to two different ones:

m S Oj = + o(t-1),o∈A(Oj (t|t-1),Oj ) m S o(t-1),o m cons j = 0 m S Oj ({ S}) 1 -m S Oj ({ S}) (7)
Thus m cons j expresses untrustworthiness when objects mismatch with their past but is vacuous otherwise.

As we have seen, the coherency and consistency constraints (m cohe j and m cons j ) can only express untrustworthiness. Therefore, other criteria have to be used to allow trustworthiness to increase.

Confirmation through free space and objects

m conf models that received objects and F S should match with the current knowledge of the world of the receiving peer. For this, the detectability m D of the receiving and sending peers are used to represent that peers have different FOVs and that comparisons cannot be made on non-overlapping areas.

m conf j = m osi j + m odi j + m ofi j + m fsi j (8)
m osi j and m fsi j model that trustworthy information should match with the local one. The received F S is compared using the method of [START_REF] Liu | MISO-V: Misbehavior Detection for Collective Perception Services in Vehicular Communications[END_REF] in m fsi j . Received objects O j are matched using an assignment function noted A and compared using the similarity function m S defined in [START_REF] Zoghby | Optimal object association from pairwise evidential mass functions[END_REF]. The local object detectability grid m D is used as a discounting factor to only compare objects that are locally detectable:

m osi Oj = + o,oj ∈A(O,Oj ) (1-m D oj (D)) m S o,oj m osi j = m osi Oj (S) 0 1 -m osi Oj (S) (9)
Conversely, m odi j models that received objects must not mis-match local ones O. For this, the j-detectability of objects not matched with the assignment function A is used:

m odi Oj = + o∈ & A(O,Oj ) (1-m D o (D)) j m D o m odi j = 0 m odi Oj (D) 1 -m odi Oj (D) (10)
Similarly m ofi j models that the received objects O j must not be inconsistent with the free space F S estimated locally:

m ofi j = + oj ∈Oj 0 m D oj (& & D) 1 -m D oj (& & D) (11)

Results

In order to illustrate and validate our approach, we implemented the equations detailed in Section 4, first in a simple situation in Section 5.1 then on real data in Section 5.2.

Simulation Study

In this section, trustworthiness estimates are obtained by running simulations implementing Fig. 2 on the situation of Fig. 1 for 2 seconds with varying parameters. Curves of Fig. 6 correspond to the trustworthiness a vehicle attributed to another one under different conditions. Simulation results about a particular parameter are plotted on top of each other to compare its impact, while others remain unchanged and optimal.

One can see in Fig. 6a that the object detectability value plays a major role. When it is low, trustworthiness converges more slowly, which is a desired behavior. In Fig. 6b, the presence of objects that do not exist creates untrustworthiness while matched objects creates trustworthiness. The same can be seen in Fig. 6c, where erroneous sizes generate untrustworthiness.

Experimental Results

To validate our approach, it has been applied to real-world data using the same dataset as in [START_REF] Lima | Data Fusion with Split Covariance Intersection for Cooperative Perception[END_REF]. In it, three vehicles v 1 , v 2 and v 3 were driven in an busy roundabout with v 1 stopped at one of the roundabout entrance while v 2 and v 3 followed each other inside of it. In post-processing, LiDAR point clouds and RTK GNSS receivers have been processed to generate object lists and FSs. The different parameters (e.g. discounting factors) have been tuned on some preliminaries tests to get smooth trust variations. Fig. 7 shows the trust estimated by the three vehicles in each other over the course of 22 seconds. At the beginning, trustworthiness in the others is completely unknown. Exchanged information is faithful up to time t = 12 s when v 3 starts sending erroneous information (incoherent sizes, omitted and ghosts objects) then stops at time t = 16 s. In this case, v 3 is voluntary lying to v 1 and v 2 but its internal information remains correct. Note that v 2 and v 3 always share perceived areas, but only do with v 1 from t = 6 to t = 12 s. As a result, v 1 is uncertain in the trustworthiness of v 2 and v 3 and reciprocally when they do not share objects, while v 2 and v 3 trust each other rapidly. A transient phase can be observed when v 1 starts perceiving common areas with v 2 and v 3 . At first, only untrustworthiness is expressed because small inconsistencies between objects at the boundary of their FOVs accumulate without enough positive information to counteract. Once enough F S and objects are shared, trustworthiness can increase.

Fig. 7: Real data: Estimated trustworthiness of v 1 , v 2 and v 3 in each other, where i m T j denotes the trust of i in j. The green curve is m(T ) and the red curve m(T ) + m( T ).

When v 3 sends erroneous information, one can see on 1 m T 3 and 2 m T 3 that it is detected by v 1 and v 2 , with untrustworthiness being rapidly estimated and maintained. Once v 3 stops sending erroneous information at t = 16 s, v 1 and v 2 increase their trustworthiness in it. These results illustrate well that to have trustworthiness in another peer, overlapping and coherent objects and free space are necessary. In our opinion, the laws of physics of the real world are the most reliable way to induce trust. Finally, by comparing 2 m T 1 with 3 m T 1 and 1 m T 3 with 2 m T 3 , one can see that the trustworthiness estimated about a particular peer differs from the other due to different points of view. This is another consequence of our choice to manage trust in a decentralized way. In terms of computation performance, the trustworthiness estimation can take up to 500 ms per iteration with our current implementation in Python/C++. As such, it cannot run in real time as communications are at 10 Hz. However, this is not necessarily an issue as this process can be run asynchronously at lower frequencies.

Conclusion

In this paper, we have proposed a method to estimate trustworthiness in other peers in the context of decentralized cooperative perception. This formulation combines misbehavior detection techniques and positive confirmation thanks to mass functions to express trustworthiness, untrustworthiness or lack of information about another peer. The trust information that peers create in others is personal and never shared. Thus, two peers will have different trusts in a third peer. The convergence of the method has been illustrated in a simple simulation, then confirmed on a real-world situation. It has shown to react quickly to erroneous information to prevent its propagation. In future work, the interaction trust and object estimation will be studied. In addition, a cooperative dataset with a ground truth on the existence of objects will be acquired to illustrate the effectiveness of this formulation and the impact of trustworthiness estimation on the non-propagation of erroneous information.

Fig. 2 :

 2 Fig. 2: Decentralized cooperative perception with trustworthiness estimation.

Fig. 3 :

 3 Fig. 3: Illustration of a detectability grid from the point of view of v 1 . (a) Hidden area in grey, FS in red and detectable in green. (b) Resulting grid: nondetectability in red, detectability in blue and unknown in grey.

Fig. 4 :

 4 Fig. 4: Evidential network for trustworthiness computation at time t. Red arrows only convey untrustworthiness, green trustworthiness, and orange both. herency", "Consistency" and "Confirmation" as illustrated in Fig. 4. They are respectively denoted m cohe j , m cons j and m conf j

Fig. 5 :

 5 Fig. 5: Sigmoid function Φ(x) with parameters µ = 2, σ = 0.25 and δ = 2 producing an arbitrary mass function m.

Fig. 6 :

 6 Fig. 6: Simulation results: Trustworthiness between two vehicles with different levels of object detectability in (a), number of ghost objects in (b) and number of objects with incoherent sizes in (c). Continuous lines are m(T ) and dashed lines are m(T ) + m( T ).

Acknowledgments This work was carried out within SIVALab, a shared laboratory between Renault and Heudiasyc (UTC/CNRS).