Stéphane Boyaval

Christophe Dumouchel

DECONVOLUTION TECHNIQUE TO DETERMINE LOCAL SPRAY DROP SIZE DISTRIBUTIONS -APPLICATION TO HIGH-PRESSURE SWIRL ATOMIZERS

This paper reports the last part of a study on the drop size distribution of sprays produced by high-pressure swirl atomizers dedicated to gasoline direct injection. This last part focuses on the deconvolution of line-of-sight forward diffraction measurements of the spray drop size distribution in order to understand the variation of the mean drop diameter D 43 in the head of the spray. First, the spatial resolution of the deconvolution technique is improved by developing a continuous deconvolution procedure. Second, this new procedure is applied on a series of four GDI injectors. The results show that the variation of the mean diameter D 43 in the head of the spray is related to the presence of the pre-spray. Furthermore, the deconvolution technique allowed the determination of the pre-spray drop size distribution. It is found that the pre-spray drops are still a problem in GDI application and that there are very much a function of the injector geometry.

Introduction

This paper reports an investigation on the volume-based drop size distribution of sprays produced by swirl atomizers dedicated to direct-injection spark-ignited engines. Because of the use of high injection pressures to reduce the atomization time, the spatial density of the spray is high. This prevents from classical measurements of spray drop size distribution. In a previous investigation [1], this problem was overcome by combining an experimental approach to the application of the Maximum Entropy Formalism (M.E.F.). The resulting procedure, briefly described hereafter, succeeded in obtaining drop size distribution for a series of four swirl atomizers used at different injection pressures. The behavior of the four injectors was examined under both steady and transient working conditions.

The experimental part of the procedure made use of the line-of-sight forwarddiffraction technique. The effects of multiple light scattering caused by the high spray density were evaluated according to the Obscuration parameter Obs given at each measurement. In accordance with previous investigation, it was found that the multiple light scattering effect is non-negligible when Obs > 0.6. Furthermore, characteristics of the drop size distributions resulting from the mathematical inversion procedure were then corrected from any multiple light scattering effects. For each working condition, we focused on the determination of the mean drop diameter D 43 and of the relative span factor ∆ v of the volume-based drop size distribution. A result is presented in Fig. 1 that reports the temporal evolution of Obs, D 43 and ∆ v during one injection, for one injector (Inj.3) used at an injection pressure ∆P i = 50-bar and an injection time t i = 3-ms.

The measurements were conducted at a distance d = 50-mm from the injector.

The experimental procedure is restricted to the determination of drop size distribution characteristics as presented in Fig. 1 was overcome by the application of the Maximum Entropy Formalism (M.E.F.) following a procedure finalized in a previous investigation [2]. The M.E.F. application leads to an analytical expression for the volume-based drop size distribution f v , namely:

( ) ( )                 -         = - q 0 q 4 0 q 3 q 4 q v D D q 1 exp D D q 4 q D f Γ (1)
where D represents the drop diameter, Γ, the gamma function, and, q and D q0 are two parameters. It was found that the use of the experimental couple of information (D 43 ; ∆ v ) to determine the parameters q and D q0 ensures an acceptable description of the volume-based drop size distribution [1,2].

The reliability of the application of the M.E.F. in the present context was checked for injection pressures up to 70-bar. In each situation the main peak of the distribution was satisfactorily predicted and a good description of the big drop population was observed. Therefore, the M.E.F. distribution ensures a good estimation of the maximum drop diameter. This latter information is of paramount importance in G.D.I. application since the drops should not exceed 50-µm in diameter [START_REF] Zhao | SAE Technical Paper[END_REF].

The investigation summarized above reports drop size distribution spatially integrated along a diameter of the spray. The purpose of the present study is to determine local drop size distributions to have a better information on the position of the big particles in the spray. This is achieved by applying a deconvolution technique allowing the determination of local information from space integrated measurements.

The deconvolution procedure

The deconvolution technique allows the conversion of line-of-sight averaged drop size distribution into spatially resolved information. As a first step, the classical procedure described in the literature for similar situations is applied [START_REF] Gomi | [END_REF][START_REF] Yule | Symposium (International) on Combustion[END_REF][START_REF] Dodge | [END_REF]. This technique requires that the drop size distribution of the spray is axisymmetric. It was shown that sprays produced by highpressure swirl atomizers satisfy this assumption [1]. The spray can be divided into annular rings as shown in Fig. 2, each ring being characterized by a local drop size distribution that we want to determine. To achieve this, a series of N measurements is performed, each measurement corresponding to a given position y i of the laser beam (see Fig. 2): y 1 is the external position and y N is the central position where the measurement is conducted along a diameter of the spray. The distance between each consecutive position (y i -y i+1 ) is equal to the width δ of the laser beam. The series of 31 intensities I j (y i ) collected on the 31 diodes at each measurement are corrected from multiple light scattering effects using correction factor series established for each diode according to the obscuration [1]. For 5 diodes among the 31, Fig. 3 presents the correction factor series versus Obs (diode 31 is the external diode). As said in the introduction, it can be seen in this figure that the multiple light scattering has a reduced effect when the Obscuration parameter is less than 0.6.

The deconvolution procedure is applied on the corrected intensity series I j (y i ) to determine the intensity per unit length series I j (r k ). Using the classical formulation, the two intensity series are related by: ( )

∑ = = i 1 k k j ik i j ) r ( I g 2 y I i = 1 to N, j = 1 to 31 (2)
where the coefficients g ik , defined in Fig. 4, are given by:

∑ + = - -       + = i 1 k m im 2 i 2 k ik g r 2 r g δ (3)
For each ring, defined by a radial position r k , a series of 31 intensities I j (r k ) is available and can be sent to the inversion procedure of the instrument to get a local volume-based drop size distribution and its characteristics. The spatial resolution of the deconvolution procedure is equal to the diameter of the laser beam used to perform the measurement. In our case this diameter is equal to 8-mm. Reducing the laser beam diameter can increase the accuracy of the deconvolution procedure. We did not have the opportunity to do so and we decided to reconsider the previous procedure in order to develop a continuous deconvolution technique.

The continuous deconvolution technique is identical to the classical method except that the number of measurements N is increased and the radial shift between each measurement is reduced. This radial shift is now equal to δ/p where the step p is an even number. Figure 5 shows the position of the measurements for a step p = 2. It can be shown that the equation system (2) takes now the form:

( ) ( ) ( ) ( ) ( )          ≤ ≤ + + = ≤ ≤ = ∑ ∑ ∑ - + - = - - = - + = N i 1 2 p N- o r G 2 r G y I 2 p N- i o r G y I 1 2 p N i N 2 k k j 1 i N 2 i k k j i j 1 p i i k k j i j r f 1 r f (4)
where the functions G j are given by: ( ) ( )

∑ = = i 1 k k j ik i j r I g 2 r G (5)
As in the previous situation, the resolution of the equation system (4) allows to determine series of local intensities I j (r k ) that are introduced in the mathematical inversion procedure to calculate the volume-based drop size distribution. The equation system (4) leaves us with a problem. Indeed, according to the last term of the second equation of system (4), the number of radial positions involved in the procedure is equal to N+p/2-1. When the step p is greater than 2, this number is higher than the number of available measurements, i.e., N. Thus a number of p/2-1 supplementary equations is required to close the system. This problem can be avoided if the step p is not greater than 2. This is why we decided to limit the application of the continuous deconvolution procedure with a step p = 2. In the present context, the highest accuracy of the deconvolution procedure is equal to 4-mm (half of the laser beam diameter). Finally, despite the fact that the deconvolution procedure allows the determination of local volume-based drop size distribution, it was preferred in the following to concentrate on the experimental determination of the mean drop diameter D 43 and of the relative span factor ∆ v . The reason for this is that the correction performed on the intensities may lead to unrealistic representation of the small drop population. An example is presented in Fig. 6 that shows the volume-based drop size distribution obtained for Inj. 2 used at 65bar. It can be observed that the small drop population is not physical. The determination of the two parameters D 43 and ∆ v is believed to be of much better quality and allows the reconstruction of the volume-based drop size distributions from the application of the mathematical distribution issued from the application of the M.E.F. (Eq. [1]). Figure 6 shows the M.E.F. distribution. It can be seen that the representation of the big drop population is in accordance to the experimental result.

The deconvolution procedure presented in this paper was applied on a series of four highpressure swirl atomizers dedicated to GDI. These injectors are those studied in a previous investigation [1]. They differ from their internal geometrical characteristics as indicated in table 1 that presents the shape parameters ∆' and ∆'' introduced by Dombrowski and Hasson [7] to characterize swirl injector behavior. All the measurements presented in this paper were performed 50-mm far from the injector.

Results

One of the main objectives of the present work is to understand the maximum D 43 observed in each situation in the head of the spray. An example of this maximum can be seen in Fig. 1 where D 43 reaches a value greater than 100µm 2.7-ms after the injection command. It was suspected that this maximum is due to the presence of the pre-spray. Injected at the very beginning of the injection, the prespray is composed of liquid having no rotational momentum and penetrates the gas environment with a great axial velocity compared to the fully developed flow. Pre-sprays are known to mainly contain the biggest drops of the spray. Furthermore it has often been reported that pre-sprays drastically increase the penetration length of the spray. For the four injectors studied here, it can be seen in Fig. 7 that the pre-spray does not induce extended penetration length. The pre-spray drops have a small velocity and are rapidly overtaken by smaller drops produced slightly later and belonging to the main spray body. The pre-spray drops take more time than the drops of the front edge of the spray to reach the location where the drop size distribution measurement is performed.

The application of the deconvolution procedure should confirm this result. This is why we decided to apply the deconvolution procedure for the sprays reporting the maximum D 43 in the spray head. For each injector used at an injection pressure of 50-bar, the series of measurements required for the procedure was conducted at the time after the injection command where the maximum D 43 was observed [1]. For the step p = 1, the number of measurements N was equal to 6 to cover the all spray. With p = 2, this number becomes equal to 11. All the measurements were performed at 50-mm Injector ∆' ∆'' from the injector. Figure 8 presents the results of the deconvolution procedure performed with Inj. 3. This figure shows the local characteristics D 43 and ∆ v obtained either with a step equal to 1 and 2. It must be first observed that globally speaking, the results obtained for the two steps are in good agreement. For both distribution characteristics, a unique radial evolution is found. However, one should note that the results reported in the outskirts of the spray show some differences. Indeed, with a step equal to 2, an increase of the mean diameter D 43 is observed when the outskirts of the spray are approached. This behavior is accompanied by a peak in ∆ v at the position 3.8-mm. Such results are not obtained when the deconvolution procedure is conducted with a step p = 1. It is difficult here to claim that the benefit of accuracy due to an increase of the step parameter allows the observation of behavior hidden when the analysis is conducted with a smaller step. However, one should add here that many different investigations found in the literature [START_REF] Parrish | SAE technical Paper n°970629[END_REF][START_REF] Xu | SAE Technical Paper n°980493[END_REF][START_REF] Yoo | Eighth International Conference on Liquid Atomization and Spray Systems[END_REF] reported the presence of big drops in the outskirts of the spray and mainly during the first stage of the injection. Figure 8 indicates also that the mean drop diameter D 43 increases as the axis of the spray is approached and reaches a maximum at r = 0-cm. This confirms that the maximum D 43 reported in the head of the spray is due to the presence of big particles located on the axis. These particles are of course those that constitute the pre-spray.

As explained in the previous section, the mathematical distribution issued from the application of the M.E.F. and given by Eq. ( 1) can be used to represent the global volume-based drop size distribution. The relevance of the use of this mathematical function was discussed in detail in [1] and can be appreciated in Fig. 6. Although they are some discrepancies, it must be kept in mind that the big drop population, i.e. the tail of the distribution, is usually well reproduced by the mathematical distribution. The application of the mathematical distribution was fully detailed in [2]. It requires the knowledge of the characteristics D 43 and ∆ v for the determination of the parameters q and D q0 . The results presented in Fig. 8 were used to determine the local drop size distributions of the spray produced by Inj. 3, 2.7-ms after the injection command and at 50-mm from the injector. These distributions, presented in Fig. 9, were calculated for the deconvolution step equal to 2 which offers the determination of a volume-based drop size distribution each 4-mm. The series of drop size distribution presented in Fig. 9 reports a clear dependence between the size of the particles and their position. Considering the fact that the M.E.F. distribution offers a rather good description of the big drop population, we concentrate on this very population. It can be seen in Fig. 9 that the large drops are located either in the outskirts of the spray or in the center, confirming what was observed in Fig. 8. As said above, the presence of large drops in the outskirts of the spray was reported by experimental investigations found in the literature. These drops are probably produced right at the beginning of the atomization process and keep a ballistic trajectory because of their size.

As suspected, the biggest drops of the spray evolve in the center. This location confirms that these drops belong to the pre-spray and that the maximum D 43 reported during the first injection stage is well related to the presence of a low velocity pre-spray. The pre-spray contains very large drop. Indeed, the distribution obtained at 0.2-cm spread above 600-µm. Studying GDI swirl injector showing an extended pre-spray, Parrish and Farrel [START_REF] Parrish | SAE technical Paper n°970629[END_REF], reported in comparable working condition (∆P i = 48.3-bar) a pre-spray with a mean diameter D 32 not greater than 60-µm. This difference can be explained by considering the velocity of the prespray. In Parrish and Farrell study, the pre-spray velocity was higher than the main spray velocity. In the present case, this is the opposite. As far as the penetration length is concerned, the present injectors are better than those studied by Parrish and Farrell since they induce no extended penetration length (see Fig. 7). However, considering the degree of atomization of the pre-spray, the injectors studied by Parrish and Farrell showed a better efficiency: it is known that the higher the velocity, the higher the degree of atomization. In consequence, the pre-spray drop size distribution is still an important problem in GDI application. The prespray drop size distribution is a function of the internal geometry of the injector. This is presented in Fig. 10 that shows the pre-spray drop size distribution for the four injectors. The injection pressure is ∆P i =50-bar. Generally speaking we can see that the injectors with the largest shape parameter ∆' (Inj. 1 and 2) produce a better atomized pre-spray. It is important to add here that these very injector are those that produce less atomized sprays in the fully open conditions, i.e., in the main spray body [1]. This last remark shows the trickiness of the pre-spray problem.

Conclusion

The work reported in this paper presents a method to improve the spatial resolution of the deconvolution procedure applied on space integrated spray drop size distributions. These distributions are obtained from a line-of-sight forward diffraction technique. The deconvolution procedure allows the transformation of the measurements into local information. In classical deconvolution procedure, the spatial resolution is equal to the laser beam diameter. To improve the accuracy, one has to reduce the diameter of the laser beam. In the present study we suggested another method to increase the accuracy of the procedure without any modification of the instrument. Theoretically speaking, this new method allows reaching a high degree of spatial resolution. Indeed the spatial resolution of the procedure is now given by the radial shift between two consecutive measurements, shift that can be reduced without any limitation. This is why this technique was called the continuous deconvolution technique. This new deconvolution technique was applied on a series of four GDI high-pressure swirl injectors. This allowed understanding that the variation of the mean diameter D 43 in the first injection stage is due to the passage of the prespray in the measuring volume. Thus, it was possible to determine the pre-spray drop size distribution and to show the influence of the injector geometry on this characteristic. It was found that the pre-spray drops are much bigger when the pre-spray is very slow. In GDI application this constitutes an advantage as far as the penetration length is concerned. However, the poor atomization quality might be an important problem. It was found also that the prespray drop size distribution is a function of the injector geometry: the injectors with the best atomization efficiency in fully open condition are those producing bigger pre-spray drops. This result shows that the pre-spray is still an important and very tricky problem in GDI application

Figure 1 .

 1 Figure 1. Temporal evolution of Obs, D 43 and ∆ v during one injection.

Figure 2 .

 2 Figure 2. Description of the deconvolution procedure.

Figure 3 .

 3 Figure 3. Correction factors for the collected intensity versus the Obscuration. Influence of the diode [1].

Figure 4 .Figure 5 .

 45 Figure 4. Definition of the coefficients g ik .

Figure 6 .Figure 7 .Figure 8 .

 678 Figure 6. Comparison between a measured and a calculated drop size distribution (Inj. 2, ∆P i = 65-bar, Obs = 0.97).

Figure 9 .

 9 Figure 9. Local volume-based drop size distribution. . Inj. 3, ∆P i = 50-bar, 2.7-ms after the injection command. Deconvolution step: 2.

Figure 10 .

 10 Figure 10. Influence of the injector geometry on the prespray drop size distribution.

Table 1 .

 1 Shape parameters of the four injectors.

	Inj. 1	1.92	1.0
	Inj. 2	1.92	0.5
	Inj. 3	1.18	1.0
	Inj. 4	1.18	0.5


Laurent Gauthier
email: laurent.o.gauthier@gmail.com.

A Cliometric Perspective on Cultural Spread: Roman and Christian Names in Ancient Greece *

): economics can bring new perspectives in epidemiological modeling by endogenizing certain parameters, which can have a significant impact on how contagion dynamics are understood and projected.

The research into the endogeneization of important parameters in epidemiological models, however, has been purely concentrated on medical or biological applications. We propose a utility-based model for cultural contagion, which extends the class of so-called SIR models in epidemiology, and apply it first to the spread of Romanity in the ancient Greek world, through the dynamics of Roman names acquisition, and then to the spread of Christianism through Christian names acquisition. The dynamics of the transition from a pure Greek world to a Romanized world, explosive at the outset, appear to have been fundamentally driven by an intense adoption of Romanity, but combined with an equally intense return to traditional Greek traits. The transition from pagan to Christian names, on the other hand, saw less of a reversal effect.

The "spectre of the natural sciences of the social," as [START_REF] Sperber | Explaining Culture: A Naturalistic Approach[END_REF] put it in his seminal work in anthropology on explaining culture through contagion mechanisms, can be seen as an intellectual goal, maybe stemming from Auguste Comte's positivist perspective on sociology [START_REF] Ritzer | Sociological Theory[END_REF] 2014). Beyond these theoretical connections one may establish between natural sciences and social relations, studying the spread of culture through the lens of epidemiology has a kind of natural appeal: it seems to intuitively make sense. The dynamics of epidemics, with their sometimes explosive behavior in a pandemic, combined with oscillations or endemic patterns, seem to capture the phase transitions of cultural spread. The notion of contagion, as an analogy or a heuristic, can be applied to a very wide range of historical or social questions, as the fifty case-studies in [START_REF] Delaurenti | Cultures of Contagion[END_REF] illustrate. In the context of this link between biology and the social sciences, the analogy of contagion and cultural transfer further extends to the evolution, and improvement in fitness over time, of infections. For example, Sperber (1997) applied the mechanics of evolution, in particular through selection and improved fitness to the study of how culture disseminates. In trying to understand "how things change," in the most general way, this particular angle of enquiry appears promising.

While anthropological approaches tend not to rely on formal models, but rather on Weberian Idealtypus, the mathematical models of epidemiology have been directly applied to social contagion in many occasions. The most ubiquitous modeling approach in epidemiology relies on compartments, representing buckets of population, such as susceptible, infected and recovered individuals. Under the acronym SIR, this class of models is the workhorse of the field [START_REF] Brauer | Mathematical Models in Population Biology and Epidemiology[END_REF]. These kinds of models belong to the realm of complex systems analysis, due to their typical non-linear nature [START_REF] Boccara | Modeling Complex Systems[END_REF]. Such formal epidemiological models have been applied, some with adhoc adjustments, to the social sciences. [START_REF] Bettencourt | The Power of a Good Idea: Quantitative Modeling of the Spread of Ideas from Epidemiological Models[END_REF], for example, applied SIR modeling to the spread of Feynman diagrams amongst physicists. [START_REF] Hill | Infectious Disease Modeling of Social Contagion in Networks[END_REF] adapted the SIR model to account for "spontaneous" social infection, and applied it to the obesity epidemic. [START_REF] Sooknanan | When Behaviour Turns Contagious: The Use of Deterministic Epidemiological Models in Modeling Social Contagion Phenomena[END_REF] offers a survey of recent research in the field. One particular application has been the analysis of church growth through contagion effects in religious following. Following [START_REF] Hayward | Mathematical Modeling of Church Growth[END_REF] and Hayward (2012), [START_REF] Jo | Sustainability of Religious Communities[END_REF] accounted for the evolution of certain South Korean church communities with an adjusted SIR model. This approach into the modeling of religious preferences is clearly distinct from economists' perspectives on the subject, such as the ones surveyed in [START_REF] Iannaccone | Introduction to the Economics of Religion[END_REF]. In fact, the epistemological framework in which the application of epidemiology models to social contagion has been carried out does not seem to strongly recoup with economic research. These approaches are closer to quantitative sociology, and, to some extent, to a mean-field approach to agent-based simulation [START_REF] Conte | Simulating Social Phenomena[END_REF]Takadama, Cioffi-Revilla, and Deffuant 2010;[START_REF] Acerbi | Individual-Based Models of Cultural Evolution: A Step-By-Step Guide Using R[END_REF]. If one considers historical phenomena, then these approaches are somewhat similar in philosophy to cliodynammics [START_REF] Turchin | Historical Dynamics: Why States Rise and Fall[END_REF] 2018), because they do not address the optimality of the underlying individual choices. In the particular case of Roman history, [START_REF] Brughmans | Formal Modelling Approaches to Complexity Science in Roman Studies: A Manifesto[END_REF] have argued for a more generalized application of complex systems analysis and the associated formal models to historical material. The inconsiderate use of dynamic models to study history has been criticized, for example by the review of Turchin ([2003] 2018) in [START_REF] Bonneuil | History and Dynamics: Marriage or Mésalliance?[END_REF].

The interaction of networks and epidemiological models has been widely studied, both from a pure epidemiological perspective and from the standpoint of social sciences applications. SIR models operate on an assumption that the population is well mixed; reflecting the granularity of the social networks that connect people can significantly affect the spread of epidemics (social or biological) in models [START_REF] Keeling | Networks and Epidemic Models[END_REF][START_REF] Bernardes | Inadequacy of SIR Model to Reproduce Key Properties of Real-World Spreading Cascades: Experiments on a Large-Scale P2p System[END_REF]. Some approaches to the modeling of information spread rely only on network modeling, such as, for example, [START_REF] Ruan | Kinetics of Social Contagion[END_REF], [START_REF] Bikhchandani | Information Cascades and Social Learning[END_REF], [START_REF] Draief | Epidemics and Rumours in Complex Networks[END_REF] or [START_REF] Golub | Naïve Learning in Social Networks and the Wisdom of Crowds[END_REF], but this type of modeling requires the detailed knowledge of the underlying network structure. SIR-type models remain more common, since they only make very general assumptions about the underlying network.

While the integration of epidemiology into anthropological or sociological questions relates to social sciences in general, the links to economics have indeed not been numerous. Only recently did [START_REF] Shiller | Narrative Economics: How Stories Go Viral and Drive Major Economic Events[END_REF] examine the importance of information spread for economics in general. The way in which information travels on social networks, and the kind of information that travels in this way, have a significant impact on economic decisions at many levels. The particular links between economics and epidemiology were surveyed in [START_REF] Avery | An Economist's Guide to Epidemiology Models of Infectious Disease[END_REF]: economics can bring new perspectives in epidemiological modeling by endogenizing certain parameters, which can have a significant impact on how contagion dynamics are understood and projected. The research into the endogeneization of important parameters in epidemiological models, so far, has been purely concentrated on medical or biological applications, even if they may factor in economics-related concepts or metrics [START_REF] Morin | Dynamics with Economically Driven Contact Rates[END_REF][START_REF] Manfredi | Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases[END_REF][START_REF] Arthur | Adaptive Social Contact Rates Induce Complex Dynamics During Epidemics[END_REF][START_REF] Alutto | On SIR Epidemic Models with Feedback-Controlled Interactions and Network Effects[END_REF][START_REF] Engle | The Behavioral SIR Model, with Applications to the Swine Flu and COVID-19 Pandemics[END_REF].

Factoring in more complex spread mechanisms needs to rely on behavioral economics, and game theory in particular, such as in [START_REF] Reluga | Game Theory of Social Distancing in Response to an Epidemic[END_REF].

The use of complexity sciences in economics has grown, and it is expected to provide a better understanding of the details of many social mechanisms [START_REF] Foster | From Simplistic to Complex Systems in Economics[END_REF], and in the particular case of cliometrics, complexity sciences have expanded the range of methods that one could consider in empirical or theoretical modeling [START_REF] Abry | The Contours of a Cliophysics: How Can Econophysics Enrich Cliometrics? Case Studies in Debt Issues and Global Capital Markets[END_REF][START_REF] Bastidon | Graph-Based Era Segmentation of International Financial Integration[END_REF]. Hence, developing an economic perspective into the endogeneization of parameters in an SIR-like model does not have to be related to epidemiology, but can also allow us to form a more precise understanding of social phenomena. We will concentrate here on two particular historical examples of diffusion of a cultural trait: the spread of Roman names, and then the spread of Christian names in ancient Greece. The uses of epidemiology in history or historical anthropology have typically been more concerned with past infections [START_REF] Raoult | Paleomicrobiology: Past Human Infections[END_REF], and there has not been many applications of epidemiological models to past social dynamics. Nevertheless, many studies in historical research on Antiquity have relied on the notion of network, in particular among city-states, in order to appreciate the diffusion of numerous cultural features, such as [START_REF] Malkin | A Small Greek World: Networks in the Ancient Mediterranean. Greeks Overseas[END_REF] or [START_REF] Müller | Les réseaux des cités grecques : archéologie d'un concept[END_REF] for example. We have limited information about the structure of these social networks in ancient cities, but we do possess large amounts of data on certain particular aspects of the Ancients' social life. In particular, there is a large volume of data on the names that individuals carried, thanks to very numerous inscriptions. Without detailed network information, the use of a general approach based on epidemiological models appears to be sensible.

In the first section, we will first discuss the context of the Romans' effective conquest of the Greek world and its subsequent Christianization. We will examine the nature of the data available that may allow us to measure these cultural transitions quantitatively. Then, in the second section, we will focus on the formal modeling of the adoption of a cultural trait, starting with the basic SIR model. We will extend this model with an economic perspective by having individuals maximize a certain utility function, which result in particular SIR-like dynamics. Finally, we will discuss the available data on ancient Greek names, and fit the dynamic model to that data. We will also examine some potential econometric drivers of the intensity of Roman and Christian name spread; in particular, we find that the close proximity to a major battle site was a driver of Romanity.

The dynamics of the transition from a pure Greek world to a somewhat Romanized world appear to have been fundamentally driven by an intense adoption of Romanity, but combined with an equally intense return to traditional Greek traits. The dynamics of Christianization shared some common dynamics but it appears that there were much more limited benefits to switching back to prior names in this case.

Romanization and Christianization

In this section, we will begin with an overview of the historical relationships between the Romans and the Greeks in Antiquity, and look into the available data on the use of Roman names by the ancient Greeks. We will then examine the history of chritianization and its spread through the ancient Greek world.

Greeks and Romans: Onomastics as Cultural Spread

Graecia capta ferum victorem cepit et artis intulit agresti Latio. [. . . ] Greece, the captive, made her savage victor captive, and brought the arts into rustic Latium1 .

The great 1st century BCE poet Horace expressed what was then considered rather obvious, the fact that a form of Greek culture had seized the Romans, affecting their aesthetics in significant ways. The Romans had an ambivalent relationship to Greek culture: on one hand they aspired to the refinement of Greek art, on the other, they resented how it pushed them away from their revered and idealized rusticity. In any case, they recognized the importance and influence the Greeks had on them [START_REF] Henrichs | Graecia Capta: Roman Views of Greek Culture[END_REF], and the topos of the Greek conquest of the Roman mind is one of the fixtures of ancient history. We may wonder, however, to what extent the Greek culture was affected by the Romans, which, one would expect, should be the first order effect.

The Romanization process started in areas of ancient Greece that were directly conquered by Rome, such as Macedonia and Achaia. Indeed, the drivers of the Roman expansion into the Greek world, a series of upheavals, retaliations and wars, spanned from the 3rd century to the 1st century BCE, spotted with frequent battles. Table 1, relying on the chronologies in [START_REF] Bugh | The Cambridge Companion to the Hellenistic World[END_REF], shows the locations of the major conflicts in this process. The battles shown in the table did not necessarily involve Romans fighting against Greeks, but due to the various alliances, typically there were Greeks fighting on both sides. Apart from Pyrrhus's attack in Southern Italy at the beginning of the 3rd century, the combats were concentrated, using today's geopolitical definitions, in Continental and Northern Greece, and in Western and Northern Turkey.

Roman governors introduced Roman laws, institutions, and practices to these regions, and encouraged citizens to adopt Roman culture. Citizens of Greek poleis were enticed to become Roman citizens, which gave them additional privileges and responsibilities. The Latin language became an important language in administration and communication, alongside with Greek, and Greek cities adopted new forms of architecture, sculpture or painting inspired by Roman art.

Religious beliefs and practices were also somewhat influenced by Roman religions, for example, during the imperial period, through imperial cults. Over the following centuries, Romanization expanded in areas of Greece that were not directly conquered by Rome, such as Attica and Boeotia. Greek city-states continued to adopt Roman customs and institutions, and local elites sought Roman citizenship. Thus Romanization was a continuous and gradual process that had a profound impact on Greek societies and led them to become an integral part of the Roman Empire.

The Roman conquest of the Greek world had for a long time been considered as a part of Roman history, but a change in this historiographical perspective, around the 1980s, led to the study of the Roman expansion from the perspective of the Greeks [START_REF] Veyne | Les Grecs : de la paix d'Apamée à la bataille d'Actium[END_REF][START_REF] Chamoux | La civilisation hellénistique[END_REF][START_REF] Préaux | Le monde hellénistique : la Grèce et l'Orient de la mort d'Alexandre à la conquête romaine de la Grèce[END_REF]. There are many ways through which one could study the expansion of Roman cultural traits through the Greek world. For example, once may consider the use of certain facilities or the manner in which they are constructed: [START_REF] Kelly | Roman Bathhouses on Crete as Indicators of Cultural Transition: The Dynamics of Roman Influence[END_REF] One particular approach benefits from a sizable volume of data supporting it: onomastics, the study of names. The study of the names of the ancient Greek or Romans typically relies on inscriptions, as the epigraphic sources are where one finds the most numerous mentions of personal names [START_REF] Karila-Cohen | Prosopographia Attica 2.0 : base de données et raisonnement prosopographique[END_REF][START_REF] Turchin | Historical Dynamics: Why States Rise and Fall[END_REF]. The evolution of Greek names in general has been recently tackled in Parker (2019). The analysis of names, including their etymology or their distribution, can be an efficient way to drill into social relationships among the Ancients. For example, [START_REF] Parker | Theophoric Names and the History of Greek Religion[END_REF] examined theophoric names, names derived from that of a god, and related their occurrences to the expansion of particular cults. [START_REF] Oulhen | Les noms théophores composés féminins à Athènes[END_REF] looked into feminine theophores and stressed how much scarcer they were than for men, pointing out a form of isolation of women from cultual practice. [START_REF] Broux | Would You Name Your Child After a Celebrity? Arsinoe, Berenike, Kleopatra, Laodike and Stratonike in the Greco-Roman East[END_REF] explored the use of Ptolemaic queens' names in the population in the Hellenistic and Roman periods, and showed that it expanded as a form of fashion. [START_REF] Vlassopoulos | Athenian Slave Names and Athenian Social History[END_REF] focused on Athenian slave names, and studied their etymology and distribution in order to better understand how these names were formed.

More specifically focusing on Roman influence, [START_REF] Lewis | Notes on Slave Names, Ethnicity, and Identity in Classical and Hellenistic Greece[END_REF] examined how slaves received Roman names, and [START_REF] Kantola | Social Standing and Latin Names in Greek: Case Studies on Name Catalogues of the Early Imperial Period[END_REF] specifically focused on the use of Roman names by people in prominent social positions, in the Imperial period. The survey of Roman onomastic practices by [START_REF] Salway | What's in a Name? A Survey of Roman Onomastic Practice from c . 700 B.C. To A.D. 700[END_REF] shows that after Roman citizenship was granted to all residents by Caracalla in 212 CE, many in the Greek world used the first name (praenomen) Aurelius, which led some Roman elites to choose other names, such as Flavius, in order to signal a higher status. Dogaer and Depauw (2017) examined intercultural contacts and cultural spread in Graeco-Roman Egypt using onomastics by focusing on names that combined etymologies from Egyptian divinities with Greek suffixes, and showed these hybrid names increased over time. The mechanics through which different cultural influences could be combined into names in Galatia (corresponding to modern Northern Turkey) during the Hellenistic and Roman periods has been studied by [START_REF] Coşkun | Intercultural Anthroponomy in Hellenistic and Roman Galatia[END_REF].

In order to carry out a large-scale study of the evolution of Greek names, we rely on the Lexikon of Greek Personal Names (LGPN), an electronic database that systematically records Greek personal names [START_REF] Parker | Lexikon of Greek Personal Names[END_REF], and our method for extracting this data is detailed in the Appendix. The LGPN attempts to collect all attested occurrences of Greek names, along with categorization information, such as the date estimate, based on the epigraphic source, geographic location information, and parental or child information. In many instances, the names of individuals, anthroponyms, appear with the name of their father, the (2002) and from [START_REF] Kantola | Social Standing and Latin Names in Greek: Case Studies on Name Catalogues of the Early Imperial Period[END_REF]. Then, for each Roman name in that list, we applied every possible combination of phonetic variations in the transliteration, also according to the main ones identified in the references. Finally, if any Greek name matched one of the Roman name phonetic combinations, we categorized it as Roman.

Figure 1 displays the LGPN onomastic data that could be linked to specific poleis, where the local share of Roman name occurrences is represented by the size of the dots. We can see that, as one could expect, Roman names occurred more frequently in Magna Graecia (Southern Italy and Sicily), as one would expect, and also in Northern Greece and Western Turkey. The Continental Greek world and the Aegean do not seem to show high occurrences of Roman names. It does appear as though the higher concentration of Roman names loosely matches the geographic distribution of major battles. We can also look at the occurrences of Roman names through time. The transition from Greek to Roman names follows a very similar pattern, both in shape and in absolute value. In fact, these transitions average around 80% of the occurrence of Roman names.

Although we observed, earlier, a potential geographic link between battle sites and occurrences of Roman names, this evolution data tells us there is a significant time difference: the battles listed in Table 1 generally took place one or two centuries before the ramp up in Roman names took place, according to Figure 2. Year (date)

Pct Share

Variable

Given the relatively low overall rate of occurrence of Roman names, the sparsity of the data prevents us from carrying out a detailed dynamic and geographic analysis. Nevertheless, if we examine a few specific poleis for which there are enough observations, we can see that for each one, the overall dynamics resemble the ones for the aggregate, as shown in Figure 4. The absolute level varies from one polis to the next, consistently with what we noted in Figure 1, but, with few exceptions, the initial increase and the plateauing are similar enough. The most apparent exception is Delos, which was an important commercial hub during the Hellenistic period. Over the following centuries, Christianity became the dominant religion of the Greek world, and the cultural heritage from the Ancients was significantly influenced by Christian teachings. There were still pagans in late Antiquity, in Greece and elsewhere, but the archaeological evidence shows that temples and sanctuaries were transformed into churches, destroyed, and sometimes kept for aesthetic reasons [START_REF] Saradi | Late Paganism and Christianisation in Greece[END_REF].

The Christianization of the Roman Empire has been extensively studied over more than two hundred years, and has been animated by a series of historiographical debates. Among them, [START_REF] Macmullen | Christianizing the Roman Empire: A.D[END_REF] considered that paganism stayed strong until the 4th Century, and was able to also evolve, but was in opposition with the Christian principles and was effectively bankrupted by the Christian emperors. [START_REF] Brown | Authority and the Sacred: Aspects of the Christianisation of the Roman World[END_REF], on the contrary, saw an evolution from a religious system centered on places, the temples, to one centered on the persons, holy men. For [START_REF] Markus | The End of Ancient Christianity[END_REF], between then end of the 4th century and the end of the 6th century, it is not so much dogma or the Church's organization that changed, but rather the cult of martyrs, the sacralization of time and space, and the development of asceticism. There was a shift towards religious culture solely based on the Bible. [START_REF] Veyne | Les Grecs : de la paix d'Apamée à la bataille d'Actium[END_REF], to the contrary of the more systemic understanding of Christianization in these works, analyzed the transformation as a kind of accident, a bifurcation due to Constantine that was not necessary in a historical perspective.

One of the advantages of becoming a Christian was the sense of community and belonging that came with being part of the faith. Christianity also offered a sense of hope and salvation, particularly for individuals who were marginalized or oppressed in society, and the promise of eternal life in heaven could be an incentive incentive for many people to convert. The early Christian communities also provided support for the poor, the sick, and the elderly. In addition to the signs that people could wear showing their faith, such as the cross, their names could also constitute an important signal. The onomastic study of early Christianity is therefore an important angle into the understanding of this cultural spread. [START_REF] Bagnall | Religious Conversion and Onomastic Change in Early Byzantine Egypt[END_REF] was the first to use onomastics in order to assess the spread of Christianity, in the case of ancient Egypt. Also concentrating on Egypt in the Late Antiquity, Depauw and Clarysse (2013), using a dedicated dataset, defined various possible sets of Christian names, and analyzed their frequency from the 4th to the 8th century. Having built their data in a way that allowed them to identify Christian individuals, they showed that "the very high specificity of Christian names for Christians practically rules out 'false positives' " (p. 413). Revisiting some of results of [START_REF] Bagnall | Religious Conversion and Onomastic Change in Early Byzantine Egypt[END_REF], they show that there was a strong acceleration in the early 4th century in the share of Christians. Staying focused on Late Antiquity, Destephen (2019) analyzed Christian names in Asia Minor through epigraphic evidence, and focused in particular on clerics and monks, including bishops, priests and deacons. He showed their names were strongly concentrated, with a few divine, biblical or pious names accounting for a large percentage of the total.

In order to study the occurrence of Christian names, we rely on the LGPN in the same manner as we did for Roman names. We created a list of less than 30 names categorized as Christian, based on the names used by [START_REF] Destephen | Christianisation and Local Names in Asia Minor: Fall and Rise in Late Antiquity[END_REF] and [START_REF] Depauw | How Christian Was Fourth Century Egypt? Onomastic Perspectives on Conversion[END_REF], and on the Latin or ancient Greek names of Christian saints from late Antiquity. These names (in Latin form)

include Petrus, Paulus, Ioannes, Domnus, or Simon, for example. Any Greek name corresponding to the transcription of one of these names or of one of its derivatives, such as a feminine, was flagged as Christian. Since, as was pointed out by [START_REF] Destephen | Christianisation and Local Names in Asia Minor: Fall and Rise in Late Antiquity[END_REF], naming practices evolved in the late Christian Antiquity towards a heavy concentration, and a few names became very common, this short list of names appears to be sufficiently representative. We restrict the time period for the analysis to the Common Era.

Figure 5 shows a synchronic geographic distribution of Christian names, where the size of the points is related to the share of Christian names in all the names observed. If the larger concentrations appear to be exclusively in the northern part of the map, it is a reflection of the fact that there are very few Greek inscriptions over the period in North Africa. There does not appear to be any clear pattern in the geographic distribution of Christian Greek names, a priori. We can also examine the prevalence of Christian names in a diachronic perspective. Figure 6 shows the total rate of occurrence of Christian names, along with the rate of transition of non-Christian to Christian names. We can see that there was a very clear acceleration at the end of the 4th century. Emperor Theodosius's Edict of Thessalonika in 380 CE declared the Christian religion as the state religion for the empire, excluding various alternative currents, and further promoted laws preventing paganism.

We can see in Figure 7 that, while there were always transitions from Christian names to non-Christian names, at an overall rate of about 75%, this dropped significantly in the late 4th and early 5th century, during the time when paganism began being outlawed. At the same time, the transition of non-Christian names to non-Christian names started progressively to decline in the 5th century. Since epigraphic sources dried up in the Late Antiquity, as the habit of inscribing religious or administrative information in stone disappeared, the data becomes in fact sparse after the 4th century. As a result, detailed geographic differences cannot truly be distinguished, as Figure 8 illustrates.

Modeling Cultural Spread

Cultural evolution has been extensively analyzed with the tools of economics or game theory:

the seminal works in [START_REF] Boyd | Culture and the Evolutionary Process[END_REF] and Boyd and Richerson (2005), in particular, established a general approach into the theoretical modeling of the existence of culture, as well as many of its specifically human features. In this stream of research, concentrating on human evolution, the core focus is on the conditions that make the emergence of culture, in parallel with genetics, optimal and evolutionarily stable. The study of how information or technique may spread, within groups where some may innovate and others immitate, also recoups with the mechanics of the evolution of culture [START_REF] Chamley | Rational Herds: Economic Models of Social Learning[END_REF]). These approaches, however, do not focus on specific cultural traits, but rather on the general mechanism that favor, or hinder, the spread of general forms of culture. The analysis of how particular cultural traits, types of information, or technical knowledge percolate through a population typically requires that one reflects detailed and specific assumptions and constraints, and may not easily fall within a very general economic framework. In consequence, this type of research work appears to have been mostly tackled through the use of agent-based simulations. For example, [START_REF] Conte | Simulating Social Phenomena[END_REF], and more recently, Takadama, Cioffi-Revilla, and Deffuant (2010) covered a wide variety of different social situations, many of them involving diffusion through interactions.

For the purpose of analyzing how a particular cultural aspect may spread, relying on relatively sparse data, we cannot specify a very complex model; on the other hand, it is important to understand how the dynamics may arise from simple assumptions concerning individual behavior.

Combining the techniques used in epidemiological modeling with some simple economic setup, we will attempt to determine a reasonable framework applicable to the use of Roman names in the Greek world. In this section, we will first describe the most basic SIR epidemiological model and discuss some aspects of its dynamics. Stressing that, while it may account for a diffusion phenomenon in a population, it does not fundamentally account for how a cultural trait could spread, we then turn to a more economics-based approach to this model. We will explicitly model the choice by people to adopt a new cultural trait, and to later shed it, and examine the dynamics that can be derived from this set up.

The Basic SIR Model

The susceptible-infected-recovered (SIR) model consists in a system of non-linear ordinary differential equations, each one representing the population in these categories. [START_REF] Brauer | Mathematical Models in Population Biology and Epidemiology[END_REF] offers an overview of the most basic model, along with some more complex models adapted for specific viral or bacterial behaviors observed in natural history. As can be seen in [START_REF] Frauenthal | Mathematical Modeling in Epidemiology[END_REF], these models have existed for some time, although they were not designated by this particular acronym.

In our approach of this SIR model, we will express the susceptible S, infected I and recovered population R as fractions of the total population, and will not specifically model the total population size, for simplicity. At each time step, the susceptible population is reduced, and the infected population is increased, by an amount proportional to the product of the susceptible and infected populations. The parameter β represents the probability, over a short period of time, that a susceptible person gets infected through random interactions with the infected population.

The parameter γ represents the probability that an infected person recovers: the expected time during which a person is infectious is

1 γ .                  dS t = -βI t S t dt dI t = βI t S t dt -γI t dt dR t = γI t dt
At the beginning of a potential epidemic, we consider that S 0 is very close to 1, and I 0 very close to zero. For an epidemic to occur, it is necessary that the numbers of infected people initially increases, so it is necessary that dI 0 > 0, or equivalently, approximating S 0 by 1, β γ > 1. The ratio r 0 = β γ effectively expresses the number of people one infected person will infect, and conditions whether an epidemic will occur, and how fast it will spread.

Closed form expressions of SIR dynamics are difficult to determine, so one can only formally derive certain specific properties [START_REF] Turkyilmazoglu | Explicit Formulae for the Peak Time of an Epidemic from the SIR Model[END_REF]. In particular, by setting dIt dt = 0, we can see that the number of infected people peaks when the number of susceptibles reaches 1 r 0 . Also, we have dSt dRt = -r 0 S t , so that ln (S t ) -ln (S 0 ) = -r 0 (R t -R 0 ). However, S 0 is very close to 1, and R 0 is very close to zero, so we can approximate as S 0 = 1 and R 0 = 0. Also, since the total population is scaled to 1, we have R t = 1 -I t -S t . However, as t goes to infinity, the number of infected people vanishes, so that R ∞ = 1 -S ∞ . Hence, at the limit, the total number of susceptibles remaining as a fraction of the total population, S ∞ verifies the equation ln (S ∞ ) = r 0 (S ∞ -1).

Such a model can be resolved numerically, Figure 9 displays two examples of SIR dynamics, one with β = 0.7 and γ = 0.35, so that r 0 = 2, and the other with β = 0.7 and γ = 0.1, so that r 0 = 7.

While the modeling approach used in epidemiology generates dynamics that resemble what we may observe in the cultural spread of Roman names or of Christian names, there is no fundamental reason why such cultural features should spread by contact in the same way as a disease.

An Economic SIR Model

In order to better account for the adoption of a cultural trait, we will develop here a model where the "infected" compartment reflects an actual choice by the susceptible population, rather than a mechanistic diffusion of the trait by contact, as in the epidemiological SIR model we just reviewed. The attractiveness of a particular cultural trait may be driven by the prestige of those who exhibit it, for example [START_REF] Jiménez | Prestige-Biased Social Learning: Current Evidence and Outstanding Questions[END_REF], and while there is a notion of contact involved in the fact that potential adopters of the trait could randomly be exposed to it, there is also a notion of choice in adopting the trait.

We express the gain that an individual would obtain from a transition from the susceptible population S to the infected I as a function of the number of people who are currently infected as well as the number of people who will be transitioning at the same time. Given b > 0 and c > 0, we have, for an individual i who is transitioning from S to I:

G i (p) = bI -cp,
where p is the number of new entrants into I, expressed as a proportion of the population. The parameter b expresses a measure of attractiveness of the new cultural trait, while c is a cost of adoption in the form of a sharing cost of the new trait. The term bI captures a network effect, whereby the more people there are in this I community, the more beneficial it is for new entrants.

However, the more new entrants there are, the worse-off they are, individually, because the newly acquired benefits need to be shared individually to some extent, which is captured by the term cp.

Once individuals are in the I compartment, infected with the new cultural trait, we consider that they will over time "forget" the benefits of being in that category. Indeed, as we model families transitioning into one compartment to another, it is actual individuals who make the decisions, from their own limited perspective. Given g > 0 and e > 0, an individual i in I, when going back to the initial cultural trait, is assumed to receive an additional gain of:

F i (q) = g -eq,
where q is the proportion of people transitioning at the same time. The term g represents an absolute gain in reverting to the prior cultural trait, while e is the cost of doing so. The form of F i expresses a baseline attractiveness, consistent with the notion of "forgetting" the benefits of the new cultural trait, combined with an effect that reduces this attractiveness if there are many people transitioning at the same time, a cost similarly expressed to that in G i .

Proposition 2.1 (Cultural Dynamics). When individuals behave optimally at equilibrium, the dynamics of the population compartments S, I and R are as follows:

                 dS t = -βI t S 2 t dt dI t = βI t S 2 t dt -γI 2 t dt dR t = γI 2 t dt with β = b 2c and γ = g 2e .
Proof. The gain is the same for all individuals in the same situation. Each potential new entrant from S chooses a probability p i of transitioning, so that the expected gain for the individuals in S writes:

E [G i (p, p i )] = p i (bI -cp) .
However, at the optimum, the mixed-strategy equilibrium calls for setting p i to the same value p 0 that maximizes G i for all. Besides, p, expressed relative to the entire population, is then simply equal to p 0 S . Hence, we have:

E [G i (p 0 )] = p 0 bI -c p o S .
Maximizing over p 0 , we obtain p * 0 = bIS 2c . Expressed as a continuous evolution, the variation in S t can therefore be written as a function of the probability p * 0 taken as a proportion of S t :

dS t = -p * 0 S t dt = - bI t S t 2c S t dt = -βI t S 2 t dt, with β = b 2c .
Now we express the gain from a transition from I to R. For an individual with a probability q i of transitioning, the expected gain is:

E [F i (q, q i )] = q i (g -eq) .
However, assuming at the equilibrium all select the same probability q 0 , we have q = q 0 I . Then, the optimal probability is q * 0 = gI 2e . Expressed as a continuous rate of transition, we hence have:

dR t = q * 0 I t dt = gI t 2e I t dt = γI 2 t dt.
The dynamics for I can be derived from those for S and for R, and we obtain the result from the proposition.

These dynamics are not exactly the same as those from a standard SIR model, because of the quadratic terms in I and in S. The share of infected I solves a non-linear differential equation:

Proposition 2.2 (Infected Dynamics). In the cultural spread SIR model, the share of infected verifies I t = F ′ t where F is the solution of the following differential equation:

F ′′ t + γF ′2 t -β F ′ t (1 + βF t ) 2 = 0.
Proof. From dS t = -βI t S 2 t dt, and considering as an approximation that S 0 = 1, we get:

dS t S 2 t = -βI t t 0 S ′ s S 2 s ds = -β t 0 I s ds 1 - 1 S t = -β t 0 I s ds S t = 1 1 + β t 0 I s ds .
Replacing this expression for S into the dynamics for I, we obtain:

dI t = βI t S 2 t dt -γI 2 t dt I ′ t = βI t 1 + β t 0 I s ds 2 -γI 2 t .
We now define F t = t 0 I s ds, so that F ′ t = I t and F ′′ t = I ′ t . The equation can hence be written:

F ′′ t = βF ′ t (1 + βF t ) 2 -γF ′2 t F ′′ t + γF ′2 t -β F ′ t (1 + βF t ) 2 = 0.
The condition for an epidemic to occur in this model writes dI 0 > 0, that is βI 0 S 2 0 -γI 2 0 > 0.

Approximating S 0 by 1 at the beginning of the epidemic, this is equivalent to β γ > I 0 . With these dynamics, and epidemic is sure to occur: r 0 only needs to be greater than some ε > 0. The share of the population who becomes infected cannot be written with a closed-form expression, but we have the following condition: Proof. Setting dI = 0, we find that the maximum infection rate is reached when S = I r 0 . Setting t m = {t : I t = max u I u }, we can hence write:

S 2 tm = 1 1 + β tm 0 I s ds 2 = I tm r 0 ,
and hence at the maximum, I verifies the following:

I tm = r 0 1 + β tm 0 I s ds 2 .
Based on the expression for S as a function of I, S t = 1 1+β t 0

Isds

, we have:

S ∞ = 1 1 + β ∞ 0 I s ds .
We can see that the entire population is never converted as far as ∞ 0 I s ds < ∞, and this also holds reciprocally.

Numerical simulations with various parameters show that S t comes near zero, but at a slow rate, so that in practice the entire population is not converted in a reasonable amount of time.

The dynamics we have expressed above do not explicitly reflect the rates of transition from one population to another, but we can compute them explicitly. We write T I the rate of transition of S to I, and T R t the rate of transition from I to R.

Proposition 2.4 (Population Transition Rates). The rates of transition T I and T R t verify:

T I t T R t = r 0 S t .
Proof. We are interested in the actual rates of transitions, so we need to express the rate at which transitions appear within a family. Indeed, the decision to give a name only occurs when there is a birth, which we will assume takes place at some rate ρ. T I is the rate of transition of S to I, relative to the population who are in a position to make such a transition, and we express it over a period of time dt:

T I t = -dS t ρSdt = βI t S 2 t ρS t = β ρ I t S t ,
The transition from I to R, expressed equivalently, is:

T R t = dR t ρI t dt = γI 2 t ρI t = γ ρ I t .
Hence, we have:

T I t T R t = β γ S t = r 0 S t .
Therefore, within the context of the model, the ratio of transitions into the converted category relative to transitions out of the converted category is proportional to the susceptible population.

3 Applying the Cultural Spread Model to the Ancient Greek's

Use of Roman and Christian Names

In this section, we fit the cultural SIR model we developed earlier, and see to what extent it can account for the empirical dynamics of the use of Roman names and christian names, in aggregate

or at the polis level, when enough data is available. Then, we will focus on the explanatory factors that may account for the absolute levels of Romanity and Christianity that can be observed through the use of names.

In order to parameterize the dynamic model to the data, we follow the usual method in epidemiology, and carry out ordinary least-squares (OLS) estimations [START_REF] Brauer | Mathematical Models in Population Biology and Epidemiology[END_REF]. The minimization is non-linear, and the model's projected "infection" rate is computed as the numerical solution to the system of differential equations that describe the model. Given the scarcity of the data, and the need to aggregate observations across buckets of one or two generations (25 or 50 years), there are only a few observations over time on which we can apply the model. This situation is somewhat comparable to epidemiology studies considering the very early stages of a new infection wave, when very few observations are available because only a fraction of the infected may be tested for the new disease.

SIR Cultural Model Fit Application to Roman Names

We consider a time span ranging from 500 BCE to 500 CE, further filtered so that there are more than 50 observations of Roman names, and aggregated into buckets of 50 years. For the polis-level observations, we also required that there be at least 4000 observations. The OLS minimization is carried out without applying weights. Table 2 displays the parameters that were obtained for the fits, for a selection of poleis as well as for the entire dataset. Figure 10 shows the actual and projected data for the same poleis.

There are not many poleis for which we had substantial amounts of data, and the parameters are somewhat variable at the polis level. In fact, Delos appears to stand out, with a very high γ, while for the other poleis, its value only varied by a multiple of 1 to 2. The β seems particularly low in Athens, on the other hand. Delos, in the Hellenistic period, was a particular case, as it was a city mostly populated by transient people, and many merchants. It was considered as a market hub, in particular for the trade of slaves, and it suffered heavy destruction, by the Athenians in the 2nd Century BCE, and during the Mithridatic wars. In these conditions, it may seem logical that both the rate of adoption of the new rulers' culture, but also the rate of renewal and forgetting, would be high in Delos. Athens, on the other hand, as a Greek cultural center, may have been slower than other cities to adopt new cultural norms We can observe that the parameters can lead to a prolonged peak, or rather a plateau as in the case of Athens, for the "infected" dynamics, unlike the patterns we observed on the basic SIR model, in Figure 9.

Figure 11 displays the actual and projected data across the entire dataset. The parameters for the whole data, in Table 2, are somewhat lower than the specific parameters for each polis.

The parameter γ, which drives the return to the original culture, is quite high relative to the parameter β, which is the rate of adoption of the new culture by traditional Greek people. If we run the model forward with the fitted parameters for an extra 500 years, until 1000, then the remaining susceptible population compartment is still about 67% of the entire population. Hence, according to Romanization dynamics as they were until the 3rd century CE, only a minority of the Greek families would have been affected by this trend. The fit shown above were only carried out based on the size of the I compartment, from which the parameters were derived. However, we have seen earlier that determining the implicit ρ, or generation length, was not necessary if we looked at the ratio of transitions

T I t T R t = r 0 S t .
Hence one may also use the empirical rates of transitions in order to derive the value of r 0 = β γ . Table 3 shows a regression of T I t over T R t S t . The coefficients obtained in the regression are significant, and the R-squared is fairly high. In the empirical relationship, the ratio

T I t T R t = r o S t comes out at
approximately 0.042S t , while the fitted dynamics gave us an r 0 of about 0.069. While these do not match exactly, they have a comparable order of magnitude. The observed transitions into Roman names are less than the transitions out of Roman names would imply, according to the model. The onomastic data for which transitions are available is quite smaller than the entire dataset, since in the case of many inscriptions, one cannot not determine the person's father name, and this data inconsistency may account for the divergence in the estimation of the r 0 parameter.

SIR Cultural Model Fit Application to Chritianism

In order to apply the model to Christian names, we restrict the time period to the Current Era, up to 700 CE, but again apply the same filters for the number of observations. As we have pointed out earlier, the number of observations drops quickly after the end of the Antiquity, as the epigraphic habit disappeared. The data is unfortunately too sparse to carry out polis-level analyzes, and in any case the very notion of a city-state tended to disappear under the Roman Empire and, later, the Byzantine Empire.

Parameters shown in Table 4 are hence only applicable to the entire dataset. Figure 12 also plots the actual and fitted evolution in the share of Christian names over time. The available data shows a steep acceleration, continuing into the early Middle Ages, and the model matches that pattern. The parameter β corresponding to this fit is very close to the one we observed for Romanization. This is interesting, as it could imply that the rate of adoption of Romanity and Christianity, as per our measures, were comparable. Nevertheless, the short list of names we categorized as Christian is arbitrary, and could be expanded, or reduced, which would likely change the fitted parameters. The parameter γ, which represents the rate of reversion to the priori cultural traits, however, is very low: about 50 times smaller than what we observed in Table 2. Hence, the rapid spread of Christianity does not appear to be due to a particularly high rate of adoption, but rather to a high rate of continuous adherence to the new cultural trait, compared to Romanization. This could reflect a particularly high cost e in reverting to a non-Christian name, or a very low attractiveness g in doing so, in the economic model. We can also run the model forward, as we did in the case of Roman names, for an extra 500 years, until 1200. The remaining susceptible population reaches about 28% of the total. Based on the spread of Christian names in late Antiquity, continuing along the same dynamics, a majority of the families would have been touched. Naturally, the actual evolution of this trend in the Greek world was affected by other significant dynamics, such the spread of Islam. The transition data fit, shown in Table 5, tells us that one cannot find a good relationship between the rates, as would be expected according to the model. As we pointed out earlier, the transition data is much more reduced than the occurrence data, and this is all the more the case in the Late Antiquity.

Cross-Sectional Analysis

In this section, we explore some cross-sectional factors that may account for the level of Romanization. While we do have a sizable amount of data on Roman names after the initial ramp up in occurrences, this is not the case for Christian names. Indeed, their rate of occurrence was still steeply increasing in the early Middle Ages, while epigraphic evidence disappeared, so that it is not possible to isolate a stable rate; there is too little data.

The so-called POLIS data [START_REF] Johnson | POLIS[END_REF], derived from the encyclopedic work of [START_REF] Hansen | An Inventory of Archaic and Classical Poleis[END_REF] on ancient Greek poleis, gathers aggregate information on over a thousand ancient Greek cities. This data includes mainly categorization information, such as, for example, whether there was a democratic regime at a point in time, how much the city was discussed in ancient texts, or an approximate measure of the city's geographic extent. We linked this POLIS data with the LGPN data, in order to be able to categorize each polis according to these metrics. Further, we also calculated, for each polis to which name observations could be attached, the distance to the closest battle recorded in Table 1. The data shown in the chart is filtered so that only the rates of occurrence corresponding to the plateau we observed in the dynamics are factored in, approximately, in order to avoid periods of significant variations of occurrence rates over time.

Figure 13 displays the relationship between the distance to the closest battle and the rate of occurrence of Greek names, filtering out locations where there was less than a total of 500 observations. We can see that there is a clear relationship, and the intuition we derived from Table 6 shows the results from three logistic regressions relating the occurrence of Roman names to several potential explanatory variables. The first regression uses all the variables that were available in the data, excluding those that are very sparsely populated and would significantly reduce the number of observations. We can see that, in that full regression, the coefficient for the distance to the closest battle is negative, and hence the more distant it is, the less likely are people to give their children Roman names. The second regression contains the full list of polis-level variables, but excludes the distance measure. We can observe that, while the remaining coefficients vary to some extent, their generally keep the same sign and same level of significance, which would suggest that this battle proximity metric is independent enough from the other effects. The size of a city has a positive impact on the spread of Romanity, and so does the fact that the city belonged to the League of Delos, that is a pact among many Ionian cities and islands, corresponding to the Athenian empire of the 5th century BCE. Democracy, in these regressions, appear to have implied a lesser interest in Romanity, but it is also strongly correlated with being in the League.

The third model shows the coefficients obtained if we only keep a limited number of explanatory variables. We can see that in this case, the distance to a battle site plays a stronger effect, and city size also still plays a role. Democracy, now has a strong positive coefficient, so that the fact that a polis was a democracy increased the use of Roman names. In a democratic city-state, as compared to an oligarchic one for example, citizens were more involved in the actual running of the city's affairs, and hence more likely to be in close proximity to the Roman rulers or delegates.

As a result, it is likely that a greater share of the population, in a democracy, would be dealing with Romans in an official capacity, leading to more absorption of their culture, and names. The "Fame" metric, established by [START_REF] Johnson | POLIS[END_REF], captures a city's importance through the amount of texts that were written about it. It reflects, in a way, the cultural importance of that city in ancient Greek. It may be that the poleis with the most established cultural aura, all else being equal, were less likely to adopt Roman names.

Conclusion

Although epidemiological models, with their full formal complexity, have been used in the social sciences to some extent, effectively developing some general ideas that were introduced by anthropologists such as Dan Sperber, they did not truly percolate to economics. In economics, according to the survey by [START_REF] Avery | An Economist's Guide to Epidemiology Models of Infectious Disease[END_REF], epidemiological modeling has been largely taken as face value, and economic tools were mostly used at the intersection between the economy and the spread of diseases.

It would seem that the epistemological framework of infectious disease tracking, understanding and modeling could be fructuous if applied to economics in a broader sense. For [START_REF] Lowe | Why Social Scientists Should Engage with Natural Scientists[END_REF], it is important that social scientists engage with natural scientists; this relationship, however, is not simply to be viewed as the transfer of formal or quantitative approaches from the hard sciences to the broader humanities. It should go in both directions, as the social sciences can raise epistemological questions of import to other fields. In the context of historical sciences, one can argue that it is a good objective to try and put more history in cliometrics [START_REF] Gauthier | Putting Clio Back in Cliometrics[END_REF]. By joining history, some economic modeling, and a cliometric perspective on ancient Greek epigraphic data, we have tried to do that precisely.

The simple economic model we proposed, combined with dynamics inspired from epidemiology gave us a reasonable way of accounting for historical onomastic data, and provided us with a narrative for the limited expansion of a particular aspect of Roman culture into the Greek world.

The same model, applied to the expansion of Christianity in ancient Greece through the use of particular names, showed us that this particular cultural reaction followed fundamentally different dynamics.

Appendix

The electronic interface to the LGPN is not designed for its data to be processed and analyzed in bulk [START_REF] Parker | Lexikon of Greek Personal Names[END_REF]. It is designed as a tool to query a name or a name root, and observe its occurrences. Figure 14 shows the results of query for a name. This feminine name, Αβα, appears 20 times and for each one of those there is a separate identification, the ID, because each entry may be considered a separate individual. Information is provided on the volume and publication from which this observation is pulled. Some chronological information is given (on the first line, for example, "i BC-i AD" means "within a century before or after CE"), itself typically derived from the epigraphic analysis of the inscription where the name was observed.

The reference gives an inscription and a line number. On the first line, "SEG XLV 1499, 2" refers to the Supplementum Epigraphicum Graecum volume 45:1499. Finally, the reference field shows the name of the mother and the daughter of this person, based on prosopographic analysis.

One can also see from Figure 14 that although it is possible to obtain the data in various formats, this data only pertains to the multiple occurrences of a single name. It is hence impossible to simply download the entire data in a structured form in one batch; it has to be reconstructed from the data pertaining to each possible name. In addition, the various file formats that are illustrated in the figure do not all contain the same information. In particular, the file in CSV format contains the core individual and name data, while the XML file contains data on relationships, locations, and bibliographical references. Both data files hence need to be extracted each time.

In order to programmatically flow through the steps of a manual search, we use the package RSelenium [START_REF] Harrison | RSelenium: R Bindings for 'Selenium WebDriver[END_REF]. First, the function "extract_name_list" runs searches for the names starting with every possible letter, in order to generate an up-to-date and comprehensive list of names accessible in the database. Then the functions "extract_all_names_csv" and "extract_all_names_xml" loop through all these names and download the corresponding files (which takes about a day on a fast server).

The specialized function "extract_xml_spec," given a file name, parses and flattens the XML tree and extracts the information on family relationships, name aliases, bibliographical references, and geographical details. The CSV data is easier to process and does not require specialized treatment. The function "create_single_dt" treats all the files in parallel processing (this still takes another day on a fast server). # Get list of names srcnm <-remDr$getPageSource() linklist <-gsub("[?]$", "", gsub("<", "", gsub(">", "", gsub("\"", "", gsub("'", "", str_extract_all(srcnm, "\"http The resulting data is structured, in the sense that, for example, the entirety of the names in the LGPN is in the "namesdata" dataframe, and the entirety of all family relationships are in the "relvarsdata" dataframe. However, it still requires further processing in order to be more precisely categorized, and also in order to be made compatible with other data.

[:][/][/].*\"[>][<]")[[1]], fixed = T), NA
In particular, the geographic information obtained from the LGPN comes both in the form of coordinates and as place names. We had to address some issues raised with geographic location with the PHI, about inscriptions, but in the case of the LGPN it is more complex, as there are several layers of interpretation. First, there is a distinction between where an inscription is found and where the people that it refers to used to live, a piece of information that may not be explicit in the source. Further, when ancient locations are referred to in a source, it is not necessarily clear where exactly that location is.

Geographic coordinates provided in an electronic source may hence give an artificial sense of precision, although their semantics are imprecise: on one extreme they could point to the exact location where an inscription was found (and the meaning of that location actually remains to be understood), on the other they could point to the center of a modern city according to Google Maps, because that city is presumably the same as the location referenced in a source.

Given these considerations, our pre-processing maps the location information from the LGPN to that of the POLIS catalogue, first based on names and then based on coordinates (using a distance threshold in terms of degrees of latitude and longitude). The names are manually identified, as the code below illustrates. c("Thessalonike", "Thessalonica"), c("Athens", "Athenai"), c("Delphi", "Delphoi"), c("Taras Tarentum", "Tarentum"), c("Chytroi", "Chyton"), c("Gorgippia", "Gorgippeia"), c("Poseidonia Paestum", "Poseidonia"), c("Makyneia", "Makynea"), c("Rhithymna", "Rhithymnos"), c("Syracuse", "Syrakousai"), c("Chersonesos", "Chersonasos"), c("Metropolis", "Matropolis"), c("Polurrhenia", "Polyrhen"), c("Aigai", "Aigeai"), c("Korinth", "Korinthos"), c("Thisbe", "Thisbai"), c("Alalkomenia", "Alalkomenai"), c("Mantineia Antigoneia", "Mantinea"), c("Hermione", "Hermion"), c("Sybaris Thourioi Copiae", "Thourioi"), c("Myrmekion", "Myrmekeion"), c("Bouthrotos", "Bouthroton"), c("Nysa", "Nyssa"), c("Tralles Seleukeia", "Tralleis"), c("Stratonikeia", "Stratonicea"), c("Knossos", "Knosos"), c("Hierapytna", "Hierpytna"), c("Hyele Velia", "Hyele Elea"), c("Besbykos", "Bysbikos"), c("Hierapolis Kastabala", "Hierapolis"), c("Telmessos", "Telemessos"), c("Hipponion Vibo Valentia", "Hipponion"), c("Gonnoi", "Gonnos"), c("Salymbria", "Selymbria"), c("Pargasa Bargasa", "Bargasa"), c("Taucheira Arsinoe", "Taucheira"), c("Angeiai", "Angeia"), c("Bisanthe Panion", "Bisanthe"), c("Myrleia Apameia", "Myrleia"), c("Phanagoreia", "Phanagoria"), c("Brentesion Brundisium", "Brentesion"), c("Nikaia", "Nicaea"), c("Halaisa", "Alaisa"), c("Koliorgeis", "Koliyrgeis"), c("Kasthanaia", "Kasthanaie"), c("Methana Arsinoe", "Methana"), c("Kytenion", "Kytinion"), c("Megalopolis", "Megale polis"), c("Pythion", "Pythoion"), c("Olympe", "Olympa"), c("Kassope", "Kassopa"), c("Trichonion", "Trichoneion"), c("Euesperides Berenike", "Euhesperides"), c("Thebai", "Thebe"), c("Pyrgoi", "Pyrgos"), c("Phellos", "Phelloe"), c("Pithekoussai Aenaria", "Pithekoussai"), c("Alyzia", "Alyzeia"), c("Bylliones", "Byllis"), c("Isioi", "Issioi"), c("Laodikeia", "Laodicea"), c("Charadros", "Charadrous"), c("Adramyteion", "Adramyttion") names(polis_coord) <-c("geo_polis_id_map", "geo_PolisNameMap", "LongPolis", "LatPolis") matdist <-rdist(namesdata[, c("lat", "long")], polis_coord[, c("LatPolis", "LongPolis")])

  examined the spread of Roman bath houses in Crete. With a different and more quantitative perspective, Glomb et al. (2018) looked at the spread of certain Egyptian cults as a function of the distribution of Ptolemaic military facilities.

Figure 1 :

 1 Figure 1: Geographic Distribution of Roman Name Concentrations in the Greek World, and Location of Battles Involving Greeks and Romans

  Figure 2 displays the evolution in the share of Roman names, across all geographic locations, through time, using the base LGPN data. The Figure also displays the "Greek to Roman" transition: the percentage ofparents with a Greek name who elect to give a Roman name to a child, relative to all entries in the data mentioning a Greek-named parent at the time. We can observe that the epigraphic material only starts showing a noticeable number of Roman names in Greek after the 1st century BCE, and these occurrences plateau roughly between the 2nd and 5th centuries CE. The spread of Roman names in the Greek world appear to have taken place only after the series of wars and battles that set the Romans in a dominant position relative to the Greek poleis.

Figure 2 :

 2 Figure 2: Evolution in the Aggregate Occurrences of Roman Names, and in the Transition of Greek to Roman Names

Figure 3 :

 3 Figure 3: Evolution in the Transitions of Greek to Greek Names and Roman to greek Names

Figure 4 :

 4 Figure 4: Evolution of the Rate of Occurrence of Roman Names in a Selection of Poleis

Figure 5 :

 5 Figure 5: Geographic Distribution of Christian Name Concentrations in the Greek World

Figure 6 :Figure 7 :

 67 Figure 6: Evolution in the Aggregate Occurrences of Christian Names, and in the Transition of Non-Christian to Christian Names

Figure 8 :

 8 Figure 8: Evolution of the Rate of Occurrence of Christian Names in a Selection of Poleis

Figure

  Figure 9: Typical SIR-Model Dynamics r0 = 2 r0 = 7

Proposition 2. 3 (

 3 Condition for Maximum and Full Infection). In the cultural spread SIR model, the maximum infection rate at time t m verifies: I tm = zero, and equivalently, everyone becomes converted at some point in time, if and only if ∞ 0 I s ds = ∞.

Figure 10 :

 10 Figure 10: Projected and Actual Rates of Occurrence of Roman Names for Selected Poleis

Figure 11 :

 11 Figure 11: Projected and Actual Rates of Occurrence of Roman Names for All Data

Figure 12 :

 12 Figure 12: Projected and Actual Rates of Occurrence of Christian Names for All Data

Figure 1

 1 Figure 1 can be confirmed. Even though the pick up in Roman names took place at the end of the Hellenistic period and during the early Empire, and the rates we are observing are measured after 100 CE, it seems its intensity may have been related with the Roman's show of force in the

Figure 13 :

 13 Figure 13: Relationship Between the Rate of Occurrence of Roman Names at the Polis Level and the Distance to the Closest Major Battle

Figure 14 :

 14 Figure 14: Query for a Name in the LGPN

  )] dta[, Biblio := ifelse(elem. == 'title' & level6 == 'bibl' & level7 == 'title', shift(value., type = 'lag'), NA)] biblivars <-dta[!is.na(Title), list(PersonId, Title, Biblio)] return(biblivars) } if (tbtype == 'placevars') { # Extract place references dta[, PlaceName := ifelse(elem. == 'placeName' & is.na(attr.) & level6 == 'birth', valu e., NA)] dta[, PlaceCode := ifelse(elem. == 'placeName' & is.na(attr.) & level6 == 'birth', ifelse(shift(attr., n = 1, type = 'lead') == 'key', shift(value., n = 1, type = 'lead'), ifelse(shift(attr., n = 2, type = 'lead') == 'key', shift(value., n = 2, type = 'lead'), NA)), NA)] placevars <-dta[!is.na(PlaceName), list(PersonId, PlaceName, PlaceCode)] return(placevars) } if (tbtype == 'relvars') { # Extract relationship data dta[, Relationship := ifelse(elem. == 'persName' & attr. == 'type' & value.=='relationship' & level7 == 'label', shift(value., n = 2, type = 'lag'), NA)] dta[, RelName := ifelse(elem. == 'persName' & attr. == 'type' & value.=='relationship' & level7 == 'label', shift(value., type = 'lag'), NA)] relvars <-dta[!is.na(Relationship), list(PersonId, Relationship, RelName)] in flas) { print(paste("Doing directory", fla)) dirpath <-paste0(csv_path, "/", fla, "/") csv_nms <-list.files(dirpath) alldt[[fla]] <-paste0(dirpath, csv_nms) %>% future_map_dfr(read_csv, col_types = cols("volume" = col_character(flb]] <-paste0(dirpath, xml_nms) %>% future_map_dfr(extract_xml_spec, tbtype = 'placevars') print("done placevars") alldt_biblivars[[flb]] <-paste0(dirpath, xml_nms) %>% future_map_dfr(extract_xml_spec, tbtype = 'biblivars') print("done biblivars") } relvarsdata <-bind_rows(alldt_relvars) namevarsdata <-bind_rows(alldt_namevars) biblivarsdata <-bind_rows(alldt_biblivars) placevarsdata <-bind_rows(alldt_placevars) save(relvarsdata, file = relvarsdata_path) save(namevarsdata, file = namevarsdata_path) save(biblivarsdata, file = biblivarsdata_path) save(placevarsdata, file = placevarsdata_path) }

)

  manual_map_df <-as.data.frame(do.call(rbind, manual_map), stringsAsFactors = F) names(manual_map_df) <-c("CleanSettlementLGPN", "PolisNameMap") lgpn_locs <-unique(namesdata$CleanSettlement) lgpn_locs <-lgpn_locs[!is.na(lgpn_locs)] polis_locs <-all_polis[, c("polis_id", "PolisName")] multi_names <-polis_locs %>% group_by(PolisName) %>% summarise(Nb = n()) %>% filter(Nb > 1) %>% as.data.frame() polis_locs <-polis_locs[!(polis_locs$PolisName %in% multi_names$PolisName), ] names(polis_locs) <-c("polis_id_map", "PolisNameMap") lgpn_locs <-lgpn_locs[gsub(" ", "", lgpn_locs) != ""] polis_locs <-polis_locs[gsub(" ", "", polis_locs$PolisNameMap) != "", ] # Compute name string distance matdist <-adist(lgpn_locs, polis_locs$PolisNameMap) min_dist <-apply(matdist, 1, min) pot_maps_nm <-list() for (bdi in 1:length(lgpn_locs)) { pots <-polis_locs[(1:dim(polis_locs)[1])[matdist[bdi, ] == min_dist[bdi]], ] pottmp <-data.frame(CleanSettlementLGPN = lgpn_locs[bdi], MinDistMap = min_dist[bdi], pots) pot_maps_nm[[bdi]] <-pottmp } name_map <-bind_rows(pot_maps_nm) zero_match <-name_map %>% filter(MinDistMap == 0) %>% select(polis_id_map, PolisNameMap, CleanSettlementLGPN) manual_match <-manual_map_df %>% left_join(polis_locs) lay1_match <-bind_rows(zero_match, manual_match) names(lay1_match) <-c("name_polis_id_map", "name_PolisNameMap", "CleanSettlement") namesdata <-namesdata %>% left_join(lay1_match) # Mapping by geo coordinates and verification ## Map by geo coordinates polis_coord <-all_polis[, c("polis_id", "PolisName", "xcoord", "ycoord")] polis_coord <-polis_coord[!is.na(polis_coord$xcoord), ]

Table 1 :

 1 Most Important Wars and Battles Involving Greeks and Romans

	War Name	War Period Battle Location	Battle Year
	Pyrrhic War	280-275 Battle of Heraclea
		Battle of Asculum
		Battle of Beneventum
	First Macedonian War	214-205 No significant event
	Second Macedonian War	200-197 Battle of the Aous
		Battle of Cynoscephalae
	Roman-Seleucid War	192-188 Battle of Thermopylae
		Battle of the Eurymedon
		Battle of Myonessus
		Battle of Magnesia
	Aetolian War	191-189 No significant event
	Galatian War	189 Battle of Mount Olympus
		Battle of Ancyra
	Third Macedonian War	171-168 Battle of Callicinus
		Battle of Pydna
	Fourth Macedonian War	150-148 Second Battle of Pydna
	Achaean War	146 Battle of Corinth
	First Mithridatic War	90-85 Siege of Athens and Piraeus	87-86
		Battle of Chaeronea
		Battle of Orchomenus
	Second Mithridatic War	83-82 Battle of Halys
	Third Mithridatic War	73-63 Battle of Cyzicus
		Battle of Cabira
		Battle of Tigranocerta
		Battle of Artaxata
		Battle of the Lycus
	War with Pontus	47 Battle of Zela

Note:

All the dates are BCE.

Table 2 :

 2 Roman Names Spread: Fitted Parameters for Selected Poleis

	Place	Beta Gamma
	Athens	0.009	0.214
	Delos	0.030	1.003
	Ephesos	0.028	0.168
	Miletos	0.019	0.310
	All	0.011	0.160

Table 3 :

 3 Scaled Empirical Relationship Between the Transition of Roman Names to Greek and the Transition of Greek Names to Roman

			Dependent variable:
			GreekToRoman
			OLS
		Base-case	Weighed by Number of Observations
		(1)	(2)
	I(RomanToGreek * S)	0.042 * * *	0.041 * * *
		(0.006)	(0.006)
	Observations	12	12
	R 2	0.803	0.794
	Adjusted R 2	0.785	0.775
	Residual Std. Error (df = 11)	0.015	1.774
	F Statistic (df = 1; 11)	44.759 * * *	42.281 * * *
	Note:		* p<0.1; * * p<0.05; * * * p<0.01

Table 4 :

 4 Christian Names Spread: Fitted Parameters Across Poleis

	Place	Beta Gamma
	All	0.011	0.008

Table 5 :

 5 Scaled Empirical Relationship Between the Transition of Christian Names to Non-Christian and the Transition of Non-Christian Names to Christian

			Dependent variable:
			NonChristianToChristian
			OLS
		Base-case	Weighed by Number of Observations
		(1)	(2)
	I(ChristianToNonChristian * S)	0.057	0.014
		(0.043)	(0.011)
	Observations	13	13
	R 2	0.129	0.114
	Adjusted R 2	0.057	0.040
	Residual Std. Error (df = 12)	0.106	3.381
	F Statistic (df = 1; 12)	1.780	1.548
	Note:		* p<0.1; * * p<0.05; * * * p<0.01

Table 6 :

 6 Logistic Regression of the Rate of Occurrence of Roman Names

		Dependent variable:
			IsRoman
			logistic
		Full Variables and Distance	Full Variables	Selected Variables
		(1)	(2)	(3)
	Greek	0.426	0.527 *
		(0.300)	(0.294)
	DelianLeague	0.444 * * *	0.492 * * *
		(0.095)	(0.088)
	HasWalls	1.722 * * *	1.793 * * *
		(0.247)	(0.240)
	SizeProxy	0.001 * * *	0.001 * * *	0.001 * * *
		(0.0002)	(0.0002)	(0.0002)
	GaveProxeny	0.078	0.031
		(0.104)	(0.099)
	Fame	-0.005	0.004	-0.057 * * *
		(0.017)	(0.016)	(0.004)
	Democracy	-0.271 *	-0.322 * *	0.511 * * *
		(0.149)	(0.140)	(0.089)
	HasVictories	0.017	0.049
		(0.101)	(0.096)
	Colonies	-0.011 *	-0.013 * *
		(0.006)	(0.006)
	BattleDistance	-0.379		-1.169 * * *
		(0.258)		(0.231)
	Constant	-5.414 * * *	-5.624 * * *	-3.242 * * *
		(0.423)	(0.395)	(0.099)
	Observations	37,400	37,508	37,400
	Log Likelihood	-5,973.272	-5,996.915	-6,013.342
	Akaike Inf. Crit.	11,968.540	12,013.830	12,036.680
	Note:		* p<0.1; * * p<0.05; * * * p<0.01

# General data on Greek names extract_name_list

	/Data/ExtractLGPN/XML_downloads/"
	allst_path <-"../../Data/ExtractLGPN/allst_names.Rdata"
	namesdata_path <-"../../Data/ExtractLGPN/namesdata.Rdata"
	relvarsdata_path <-"../../Data/ExtractLGPN/relvarsdata.Rdata"
	namevarsdata_path <-"../../Data/ExtractLGPN/namevarsdata.Rdata"
	placevarsdata_path <-"../../Data/ExtractLGPN/placevarsdata.Rdata"
	biblivarsdata_path <-"../../Data/ExtractLGPN/biblivarsdata.Rdata"
	dwlink_csv <-";style=csv"
	dwlink_xml <-";style=xml"
	## Overall setup
	lgpn_start_link <-"http://clas-lgpn5.classics.ox.ac.uk:8080/exist/apps/lgpn1-search/index.ht
	ml"
	partial_link <-"http://clas-lgpn2.classics.ox.ac.uk/cgi-bin/lgpn_search.cgi?name="
	# Start browser phantom
	pJS <-phantomjs()
	pJS$process
	remDr <-remoteDriver(browserName = 'phantomjs', port = 4567)
	remDr$open()
	remDr$navigate(lgpn_start_link)
	alllt <-paste0(letters, "*")
	allnms <-list()
	for (ix in 1:length(alllt)) {
	try({
	Sys.sleep(0.5)
	# Find entry for search
	myel <-remDr$findElement("id", "namequery")
	# Send letter wildcard
	myel$clearElement()

<function() { myel$sendKeysToElement

(list(alllt[ix]

, key = "enter"))

Horace, Epistles,

2, 156-157, Horace (1929).