Introduction

There has recently been a great deal of interest in the effects of the sizes of communities on the social and economic conditions of settlements, drawing on theoretical and empirical work on complex systems (Bettencourt, 2021; Lobo et al., 2020; Ortman et al., 2020; Smith, 2018). As Lobo et al. (2020) have shown in an important article, there is now widespread evidence for systematic relationships between the sizes of sites and some of their most important attributes in a range of ancient and modern contexts, and these seem to be surprisingly uniform from context to context.Footnote 1 This body of work, known as settlement scaling theory, has not only shown that various social and economic measures tend to increase faster than the populations of settlements (i.e. in a super-linear fashion), but also that different kinds of infrastructure tend to increase slower than the populations of settlements (i.e. in a sublinear fashion) (Lobo et al., 2020; Ortman et al., 2020; Smith, 2018). One of the most important aspects of these findings is that the relationships in different contexts tend to increase or decrease at particular rates, with specific exponents, which are usually a multiple of either a 1/3 or 1/6. For example, infrastructure tends to increase at the rate of 2/3 or 5/6, while social and economic measures increase at the rate of 4/3 or 7/6. These relationships have now been found across a wide range of contexts, from pre-historic villages and medieval cities to modern metropolitan areas in the USA and Europe, and India, China, and Brazil. These regularities have been explained by the roles of settlements as ‘social reactors’, which concentrate people in space and time and therefore increase the opportunities for them to interact, share resources, and exchange knowledge, skills, and ideas (Bettencourt, 2013). This has allowed scholars to propose a series of formal, mathematical, models for these relationships, based on the central observation that settlements can be regarded as social networks embedded in the built environment (Bettencourt, 2013; Lobo et al., 2020).

Although these regularities were first observed in contemporary cities, they were quickly identified in both non-urban and non-modern contexts, beginning with the new world and then moving on to the old world (Bettencourt & West, 2010; Bettencourt et al., 2007; Lobo et al., 2020; Ortman et al., 2014; West, 2017). More recent work, for example, has shown that they can be applied to various aspects of the cities of the Greek and Roman world, including the sizes of public spaces, such as fora and agorae, the dimensions of street networks, the numbers and widths of city gates, the seating capacities of theatres and amphitheatres, and numbers of inscriptions (Hanson, 2020, 2021; Hanson & Ortman, 2020; Hanson et al., 2017, 2019). Having said this, although it is clear that there is a series of relationships between the sizes of sites and their various attributes, it is not yet known whether these relationships extend to life within settlements, such as between different neighbourhoods, blocks, or individual households.

There are three reasons for suspecting that the same, or similar, relationships might exist both within and across settlements. The first is that, since these regularities are essentially the result of the fact that all social networks are embedded in, and mediated by, the built environment, they should occur at any scale where (or when) social groups or communities interact with each other in a physical context, including at the household level. The second is that, since these models are very general, and are not tailored to specific conditions, such as specific forms of transportation infrastructure, they should not only be applicable to a range of contexts (such as the urban and non-urban, modern and non-modern referred to above), but also to a range of scales, from groups of houses to entire urban systems. Finally, these relationships are not simply a product of the numbers of people who live within settlements, but also of the number of people who regularly interact with each other within them, which means that we would not necessarily expect the social networks of a given group to be contained within a certain boundary, whether that is a specific house or settlement.

This therefore raises the questions of (1) whether the same, or similar, social network effects exist at different social, spatial, and temporal scales, including both individual houses and entire cities and (2) whether these relationships are common to settlements from a wide range of regions and periods, revealing another important commonality in how settlements operate, or whether they are restricted to larger, and more socially and economically developed, agglomerations, reflecting an important limit on the roles of social networks within the built environment. Uncovering (or failing to uncover) these relationships could have significant implications for our understanding of not only the Roman Empire, but also ancient and modern societies in general.

Aims and Objectives

In this article, I will investigate these ideas by focusing on one site, namely, Pompeii. This is the best candidate, given both the amount of the site that has been uncovered (even if the quality and quantity of these excavations have varied over time) and the amount of research that has been done on its domestic architecture (for example, Flohr, 2017; Wallace-Hadrill, 1994). I will begin by drawing on recent work, which provides detailed evidence for the sizes of structures and the numbers and different kinds of rooms that are associated with them, which can be broken down into two main categories: individual houses and shops or workshops with living quarters (the latter are generally referred to as tabernae) (Flohr, 2017). Although this material has been used in the context of settlement scaling theory before, it has only been used as one data point as part of a wider attempt to examine the relationships between the inhabited areas, population densities, and estimated populations of a selection of sites; hence, its use is quite different here (Hanson & Ortman, 2017). I will combine this material with evidence for both the numbers and diversities of small finds, derived from two of the most recent excavations on the site, in Insula 1, 10 and VI, 1, respectively. This information can then be used to investigate two different issues. The first is whether there are any relationships between the numbers and different kinds of rooms and the sizes of the residential units in Pompeii, focusing on the entire excavated area. The second is whether there are any relationships between the numbers and different kinds of small finds and the sizes of residential units, using two case studies.

The results show that there is indeed a series of relationships between (1) the numbers and different kinds of rooms and the sizes of structures, which are almost identical for houses and tabernae, and (2) the numbers and different kinds of artefacts and the sizes of structures. Importantly, the slopes, or exponents, of these relationships are similar to those in other contexts, including settlements in both the Greek and Roman world and medieval Europe and beyond, despite the fact that the former describe the pattern within a single site, while the latter describe the pattern across sites. This suggests that they might be the result of the same forces, with significant implications for our understanding of how settlements function.

Data and Methods

The evidence for the numbers, sizes, and different kinds of rooms within structures has been derived from Flohr’s (2017, 2018) database of buildings in Pompeii. As Flohr (2017) has noted, this database was specifically created not only to estimate the population of the settlement, but also to (a) investigate the overall patterns of investment and consumption, (b) measure the overall level of inequality within the settlement, and (c) examine its relationship with its hinterland. This was achieved by using a combination of the areas, the numbers (and numbers of different kinds of) rooms, and the presence and absence of different kinds of interior decoration within structures, along with measures of inequality such as Gini coefficients and Lorentz curves, to explore the distributions of different features across residential units within the excavated sections of the settlement (for an example, see Fig. 1). The complete database that was used to do this has been made available online, which means that it can also be put to other uses, as here (Flohr, 2018).

Fig. 1
figure 1

The rank-size distribution of residential units in Pompeii (using data from Flohr, 2018). Note that, although atria and peristyles are much more concentrated in the largest structures, they can be found in structures of almost all sizes

Full size image

As Flohr (2017) has noted, this material reveals that there is a relatively smooth transition from the largest to the small structures. This suggests that, although the distribution of wealth within Pompeii was by no means equal, it was not as neatly stratified as has been assumed.Footnote 2 In fact, there was a wide spectrum from relatively wealthy to relatively poor. Meanwhile, although certain kinds of rooms, such as atria and peristyles, are much more concentrated in the largest structures, they can be found in structures of almost all sizes. This suggests that there was a substantial market for ‘elite’ commodities, such as these rooms. One of the other interesting features of the results is that the rank-size distribution of both the sizes of structures and the sizes of certain rooms approximately follows a power law. Although this kind of distribution can come about through a number of means, it is often the result of social network effects. This suggests that it might be possible to understand the numbers and sizes of structures in terms of the theoretical and empirical models discussed above.

Since structures are often divided into smaller residential units, scholars normally distinguish between freestanding buildings and the individual units within them. In what follows, I have focused on individual units, rather than freestanding buildings, since these are the smallest units and are therefore the most likely to have been occupied by individual households. These units have been identified by following the standard practice of locating all the entrances from the street and regarding all the rooms that can be accessed from each one as part of the same unit (Flohr, 2017; Hanson & Ortman, 2017). As Flohr (2017) has noted, although this approach is not entirely unproblematic, given that it is possible for some entrances to have led to more than one unit, it should be more than sufficient for most general research, including for the purposes of this article. The structures can then be divided into five types, namely, houses, tabernae, apartments, (unspecified) units, and walled areas (Eschebach & Müller-Trollius, 1993; Flohr, 2017, 2018).Footnote 3 The first group includes atrium-peristyle houses, while the second includes both shops and workshops, the third the apartments above the previous two groups, and the fourth and fifth are essentially those that defy categorization. The first group is identified through narrow entrances, atria or peristyles, and suites of other small rooms. The second are identified through wide thresholds, fixed counters (or other mercantile equipment), and limited numbers of rooms. The third group is the most difficult to identify, given that there is very little or no evidence for upper floors throughout most of the site. These structures can therefore only be identified through the existence of external staircases, which means that it is not possible to estimate the areas or the numbers of rooms within them. I have therefore focused on houses and tabernae, rather than apartments (or the other structures described above), given that there is no evidence for the numbers and types of rooms in these structures. As noted elsewhere, since the terms that are used to describe these structures can be confusing, I have used the specific terms ‘houses’ and ‘tabernae’ to refer to the first and second groups, respectively, and the general term ‘residential unit’ to refer to both of them, as in earlier studies (Hanson & Ortman, 2017).

The rooms within these structures can then be divided into 16 types, i.e. cubicula, shops, staircases, corridors, atria, fauces, gardens, kitchens, tablina, alae, halls, courtyards, oeci, peristyles, porticos, and unspecified rooms (Bragantini et al., 1981; Flohr, 2017, 2018; Hodske, 2007, 2010).Footnote 4 These types are based on a combination of the Pitture e Pavimenti di Pompei and Pompeii Pitture e Mosaici, with various other additions which are too varied to list here (for details see Flohr, 2017, 2018). As Flohr (2017) has noted, although there has been a great deal of discussion about the domestic architecture of Pompeii, these types of structures have all been sufficiently well defined and described by scholars to be recognized and identified from the standing remains without too many problems and without too much loss of consistency.

In total, the database includes 1168 units, including 497 houses, 468 tabernae, 133 apartments, 52 (unspecified) units, and 18 walled areas (Tables 1 and 2) (Flohr, 2018). These range from about 4 to 2908 m2, have an average area of about 200 m2, and contain a total of 10,294 rooms, giving an average of about 9 rooms per structure. The houses range from 55 to 2908 m2 (Casa del Fauno), with an average area of 394 m2, and contain a total of 8589 rooms, suggesting an average of 17 rooms per structure (Flohr, 2018). The tabernae range from 7 to 208 m2, with an average area of 42 m2, and contain a total of 1293 rooms, indicating an average of 3 rooms per structure (Flohr, 2018). Both kinds of structures can be regarded as being part of the same continuum of residential units, given that there is a certain amount of overlap between them. The latter tend to be much more modest, however, so they are almost always interpreted as accommodating those of a lower, or at least more varied, social and economic status (Flohr, 2018: 72). These structures are derived from an area of about 44.32 hectares (Flohr, 2018). The area that has been studied therefore equates to about three-quarters or two-thirds of the settlement, depending on whether extra-mural areas are included or not (Hanson, 2016; Hanson & Ortman, 2017).

Table 1 The numbers, average areas (in m2), total numbers of rooms, and average numbers of rooms in structures in Pompeii (using data from Flohr, 2018)
Full size table
Table 2 The numbers of different kinds of rooms in structures in Pompeii (using data from Flohr, 2018)
Full size table

There are two issues with these structures that need to be briefly addressed before proceeding. The first is whether the tabernae that were often attached to the houses should be treated as part of the same architectural unit or not (Hanson & Ortman, 2017). Unfortunately, it can be difficult to decide whether these were used by the residents of the houses or rented out to others, given the lack of evidence for who owned and occupied them. It is also possible that some of these structures were completely unoccupied and that their owners (or renters) lived elsewhere. The second is how much extra space or additional rooms would have been provided by upper floors and whether these would have been used to provide more living space for the same household or separate living space for a different household (Hanson & Ortman, 2017). Although it is relatively straightforward to identify these upper floors from the presence of footings for staircases or ladders, it is more difficult to determine whether these upper floors would have covered the entire ground floor of each structure or whether they would have also extended over the ground floors of a neighbouring structure. It is also possible that a few adjacent units were connected through these upper floors, although this is unlikely. As Flohr (2018: 58) has noted, although this means that both the areas and numbers of rooms for some structures are likely to have been either under- or over-estimated, it is unlikely to ‘fundamentally alter’ the results.

It is also important to point out that the database is confined to the structures that have been fully excavated and buildings within the city walls (Flohr, 2018). It is also restricted to the types of rooms that could be found at least 100 times. Finally, as Flohr (2018) has noted, the database simply shows the situation in AD 79, ‘without revealing how this situation came into being’. This means that any evidence that it provides for the domestic architecture of Pompeii should be used with the upheavals of AD 60 s and 70 s in mind.Footnote 5

An important omission from this database, however, is coordinates for each unit. I have therefore derived the evidence for the locations of these structures from the Pompeii Bibliography and Mapping Project (Poehler, 2021). This provides a shapefile of the 1164 buildings that have been identified in the excavated part of the site. I then linked the two datasets using the standard IDs for each structure, which are based on their doorways, although it should be noted that there are some differences of identification and interpretation, which means that they do not align perfectly (and therefore that some structures have not been mapped). I used the online version of this database to create a centroid for each feature in ArcGIS and assessed the relative densities of structures using the kernel density estimation tool in the same program, giving the results in Fig. 3. I have then digitized the street network of Pompeii, using standard maps and plans, in ArcGIS. This has been partially reconstructed, following some of the suggestions made by Poehler (2017a, b) in recent work. I then calculated the betweenness centrality of each intersection, using Gephi. This measure essentially reflects the likelihood that traffic on the network will pass a given node, offering a simple index of how well connected the node was (Fig. 4) (Brughmans, 2013).

I have also collected a limited amount of evidence for the numbers and diversities of small finds that are associated with these residential units. To do this, I combined the information from two different sections of the site: Insula I, 10 and VI, 1 (Fig. 2) (Allison, 2008; Cool, 2016). The database associated with the former has also been made available online (Allison, 2009). Although both were excavated relatively early on, the former was re-evaluated during the late 1990s and early 2000s, while the latter was re-excavated by the Anglo-American Project in Pompeii between 1995 and 2006, meaning that they offer the most complete and up-to-date impression of life in Pompeii available. These sources provide us with evidence for 10 residential units in Insula 10, 1, including both the Casa del Menandro and Casa degli Amanti, and 6 in Insula VI, 1, including both the Casa delle Vestali and the Casa del Chirurgo (Table 3). Although the size of the sample is small, the sizes of these structures cover a wide range, from as little as 24 m2 to as much as 1632 m2 (their average area is 304 m2). This means that this sample is representative of the distribution of the sizes of structures within Pompeii. Although these insulae have not been randomly selected, they also include a good range of different kinds of buildings, as well as offering glimpses into two very different sections of the settlement (one near a gate, the other near the monumental centre).

Fig. 2
figure 2

The locations of the two case studies, Insula I, 10 and VI, 1, within Pompeii (adapted from Dobbins & Foss, 2005)

Full size image
Table 3 The areas (in m2) and the numbers and diversities of small finds associated with residential units in Insula I, 10 and VI, 1 (using data from Allison, 2008, and Cool, 2016)
Full size table

It is important, however, to make two points about the material. The first is that, although both sources include basic information about the forms, materials, and functions of each artefact, this material has been identified, interpreted, and categorized in slightly different ways. The second is that different amounts of material were retrieved from each set of excavations. As has been noted elsewhere, this means that it is important to adjust the evidence for the numbers of different kinds of small finds in each structure by the sizes of the samples that have been used, given that we are more likely to find a larger number of categories if we look at a larger number of observations. This can be achieved by dividing the numbers of different kinds of objects by the total numbers of objects and then multiplying the result by the size of the settlement (this is to convert it from what is essentially a per capita measure back into an aggregate measure) (Hanson et al., 2017). This means that, although it is possible to make comparisons between the relative numbers of different kinds of artefacts, it is not possible to compare their absolute numbers. It is also worth noting that the material from Insula VI, 1 contains a mixture of simple counts and estimated vessel equivalents (EVE). This is why there are a few fractions in the results below.

Another important question is whether we would expect the sizes of households to have increased or decreased as the sizes of structures increased or decreased. It is much more difficult to estimate the number of inhabitants in these structures, however, since we do not have any independent evidence for the numbers of residents. There are various reasons, though, for believing that the sizes of households were not related to the sizes of structures; the most important of which is that, although we would expect many wealthier families to have owned slaves or had freedmen, we would also expect many poorer families to have shared what space they had with others, such as lodgers. There are also strong reasons for believing that the sizes of houses are a proxy for the wealth of their inhabitants, rather than their numbers of inhabitants (Kohler & Smith, 2018; Ortman & Davis, 2019). This is because we would not only expect larger structures to cost more to build in terms of land, material, and labour (and therefore to have cost more to buy or rent (although we have to be careful about assuming that the land values were the same across each settlement)), but also expect larger structures to be associated with a larger amount of moveable wealth, such as tableware, furniture, and textiles.

This line of reasoning does seem to be borne out by the evidence from Pompeii, given that larger structures often seem to be associated with both higher quality wall paintings and higher quality tableware, furniture, everyday utensils, and even foodstuffs, although a systematic survey of the relationship between them has not been done. While the numbers of rooms have been used as a proxy for the numbers of inhabitants in certain contexts, there are reasons for believing that both the numbers and diversities of rooms can also be seen as a reflection of rising standards of living or an increasing quality of life. This is because we would expect the construction of new, and new kinds of, rooms to be associated with the development of increasingly elaborate spaces for reception, entertainment, and sanitation. This seems to be true of Pompeii, since the construction of new and new kinds of rooms, such as peristyles, is clearly associated with a luxury lifestyle (as witnessed by both textual and epigraphic material and their interior decoration and known small finds). For the moment, I have therefore assumed that each structure represents a single household and that they had a more or less constant household size, meaning that their sizes and numbers and diversities of rooms can be taken as a reflection of the disposable income of their inhabitants, in keeping with Flohr’s work. The possibility that this was not the case, however, is discussed in more detail below.

As has been noted elsewhere, although we would expect there to be a great deal of variation in the sizes of households, most scholars have assumed that the sizes of households in non-modern contexts, before the demographic transition associated with the industrial revolution, usually ranged between 3 and 7 individuals, with an average of about 5 (Chamberlain, 2006; Hansen, 2006, 2008; Hanson, 2016; Hanson & Ortman, 2017; Hassan, 1981; Russell, 1958; Storey, 1997; Wilson, 2011). This is based on various kinds of evidence, including not only detailed analysis of birth and death rates, life expectancy, and household composition in non-modern contexts, but also comparison with better documented periods and places (Chamberlain, 2006). These figures are also supported by evidence from census returns from Egypt, which mostly date to the Hellenistic and Roman periods (Kennedy, 2006: 111). These censuses contain information about a total of 167 families, which had an average size of 4.3 (Bagnall & Frier, 1994; Hansen, 2006: 58; 2008: 278). The average size of households in urban areas, however, was about 5.3, while those in rural areas was about 4.8 (Bagnall & Frier, 1994: 67–9). As Huebner (2013: 39) has noted, this difference is accounted for by the fact that the former were more likely to have owned slaves. As a result, although the average sizes of these households initially seem small, they are perfectly in line with other societies where both birth and death rates were high (Huebner, 2013: 201). These figures also make sense, given what we know about the domestic architecture of the ancient world, such as the average sizes of structures and the average numbers of rooms within them (Hanson & Ortman, 2017). According to Hansen (2006: Table 2.1), although the average sizes of houses in the Greek world ranged from 100 to 294 m2, they had an overall average of 199 m2. Meanwhile, as Wallace-Hadrill (1994: 64–7; 76; 100) has shown, using a sample of houses from two parts of Pompeii (Regio I.vi-xii and Regio VI.ix.-xvi) and one part of Herculaneum (Regio III, IV, V, and VI), although the sizes of houses in these sites range from 25 to 2500 m2, they have an average size of 266, 289, and 241 m2, respectively. This is also in keeping with the figures discussed above (a general average size of 200 m2 and a specific average size of 394 m2 and 42 m2 for houses and tabernae, respectively).

As Flohr (2017: 53–4) has noted, although the estimates for the population of Pompeii have ranged from as little as 8000 to as much as 20,000, the overall consensus is that it was about 10,000. Although scholars have been skeptical of our ability to estimate the populations of ancient settlements, as has been argued elsewhere, it is possible to come up with new estimates for the populations of sites by investigating the overall relationship between the numbers of residential units per unit area and the inhabited areas of sites, using a sample of over fifty sites from throughout the Greek and Roman world (Hanson & Ortman, 2017). The evidence for Pompeii, which is based on the same database as this article, suggests a population density of about 130 people per hectare (this is based on an earlier sample of over a thousand residential units, with an average of about 5 people per household, from an excavated area of about 44 hectares). This, in turn, suggests an estimated population of about 8000 people, based on a conservative total inhabited area of about 60 hectares. There are reasons, however, for believing that the overall relationship between the estimated population densities and inhabited areas of the entire sample of sites might provide a stronger basis for estimating the populations of sites than individual cases, given that it helps to overcome any errors associated with the representativeness of the sample used. This suggests an estimated population for Pompeii of about 9938, based on a predicted population density of about 166 people per hectare (Hanson & Ortman, 2017). This is almost identical to the figure of about 10,000 that has been proposed by most scholars (Flohr, 2017).

As noted above, a final important point is that the scaling relationships that were discussed in the introduction to this article are not simply a product of the numbers of people who live within settlements, but also of the number of people who regularly interact with each other within them (Lobo et al., 2020).Footnote 6 This has important implications for this article, since it suggests that any relationships that can be found between the sizes of structures and their other attributes would be a product of both the numbers of people that lived within them and the numbers of individuals who regularly interacted in them. In other words, we would expect them to have been designed to accommodate both their residents and their social networks.

I have then assessed the relationships between the sizes of the structures and the numbers and different types of rooms using a common form of regression known as ordinary least squares regression (Tables 4, 5, 6, and 7). As has been noted elsewhere, this is feasible because \(y=b{x}^{m}\) and \(\mathrm{log}y=m\mathrm{log}x+\mathrm{log}b\) are equivalent expressions (Bettencourt, 2013; Hanson & Ortman, 2017, 2020; Hanson et al., 2019; Ortman et al., 2014). This means that we can estimate the exponent and pre-factor of the power function by estimating the y-intercept and slope of the best fit linear function, so long as it is based on log-transformed values (this is because the effects in question are multiplicative, rather than additive). Although there are some issues associated with performing regressions on logarithmically transformed count data, these should not apply here, given that the variation in the sizes and numbers of rooms per structure is large enough to be regarded as nearly continuous data.

Table 4 The results of regressing the numbers and diversities of rooms in tabernae against their sizes (in m2). All regressions are generated using ordinary least squares regression on log-transformed values
Full size table
Table 5 The results of regressing the numbers and diversities of rooms in both houses and tabernae against their sizes (in m2). All regressions are generated using ordinary least squares regression on log-transformed values
Full size table
Table 6 The results of regressing the numbers and diversities of rooms in houses against their sizes (in m2). All regressions are generated using ordinary least squares regression on log-transformed values
Full size table
Table 7 The results of regressing the diversities of small finds associated with residential units in Insula 1, 10 and VI, 1 against their sizes (in m2). The regression is also generated using ordinary least squares regression on log-transformed values
Full size table

Results and Discussion

In total, this material provides us with evidence for 965 residential units, including 497 houses and 468 tabernae. These structures contain 9882 rooms, of which 8589 can be found in houses and 1293 in tabernae. This data also provides us with evidence for 5735 small finds (including EVAs), of which 5185 come from Insula I, 10 and 550 from Insula VI, 1.

Although both kinds of structures are distributed throughout Pompeii, it is striking that the densities of both of them are highest in the centre of the settlement, including not only immediately to the north and west of the forum, but also along the central sections of the north–south running Via Stabiana and east–west running Via dell’ Abbondanza and Via di Nola (Fig. 3). This is not only consistent with well-known theoretical frameworks, such as the Alonso model (also known as bid-rent theory), which suggests that rents are highest in the centres of settlements, but also reminiscent of the densities and heights of buildings in various contemporary cities in both the USA and elsewhere (Alonso, 1964; Mandich, 2019; McCann, 2013). Although there is not a perfect correlation between them, it is also interesting to note that the densities of these structures coincide with the intersections with the highest betweenness centrality, which are usually interpreted as the nodes that are most likely to have been passed by traffic on the network (Fig. 4). This suggests that the densities of houses could be taken as a proxy for the densities of interactions within settlements (Hanson & Ortman, 2017). These results also reflect the common observation that smaller structures, such as shops and workshops, were more likely to be concentrated on the parts of the street network with the greatest (potential) traffic, given their at least partial reliance on passing trade (Poehler, 2017a, b). Finally, it is also worth noting that, although the Gini coefficient for both sets of structures is relatively high, 0.61, the coefficient for either just houses or just tabernae is much lower, at 0.43 and 0.33, respectively.

Fig. 3
figure 3

The densities of buildings in Pompeii. The densities were calculated by using the centroids of the shapefile created by the Pompeii Bibliography and Mapping Project (Poehler, 2021) and the kernel density tool in ArcGIS 10

Full size image
Fig. 4
figure 4

The street network of Pompeii. The street network was digitized from standard maps and plans and has been partially reconstructed, following some of the suggestions made by Poehler in recent work (Poehler, 2017a, b). The network analysis was performed using the statistics extension in Gephi. The symbols are proportional to the betweenness centrality of each intersection

Full size image

As shown in Tables 4, 5, and 6 and Figs. 5, 6, and 7, it is then possible to examine the fit of the relationship between the sizes of structures and the numbers and different kinds of rooms within them. The results reveal that there is a relationship between these variables, which is virtually identical for both houses and tabernae. This relationship is slightly sublinear, meaning that the numbers of rooms increase more slowly than the sizes of structures (in other words, there were fewer rooms per unit area as they increased in size). This means that the average size of these rooms increases. These results are interesting in their own right, since they suggest that the desire of the inhabitants to create larger rooms was generally greater than their desire for a larger number of rooms, potentially suggesting that the need for new social spaces was greater than the need for more private spaces, such as rooms for sleeping or storage. The slopes of these relationships are relatively stable from regio to regio, although the size of each sample varies. Next, these results also show that there is a relationship between the sizes of these structures and the diversities of rooms within them, which is also similar for both houses and tabernae. Although this is also sublinear, it increases much more slowly than the previous relationship, meaning that the diversities of rooms increase even slower as structures increased in size. These results therefore have significant implications for our understanding of the domestic architecture of the site, given that they suggest that neither the numbers nor the diversities of rooms are random and that both are linked, in some sense, to the sizes of the structures involved. Although it is also possible to detect a relationship between the sizes of structures and the numbers of rooms when the evidence for both houses and tabernae are combined, this relationship does not conform as well to our expectations, and the R2 is also a little lower. This supports the idea that these structures had different functions and should be treated as separate entities. Finally, it is interesting to note that the y-intercepts, or pre-factors, of these results, which can be used to get a sense of the baseline amount of each variable, also suggest that houses tended to have not only more rooms, but also a greater diversity of rooms, than tabernae.

Fig. 5
figure 5

The relationship between the numbers of rooms within structures and the sizes of houses (in m.2)

Full size image
Fig. 6
figure 6

The relationship between the numbers of rooms within structures and the sizes of tabernae (in m.2)

Full size image
Fig. 7
figure 7

The relationship between the numbers of rooms within structures and the sizes of both houses and tabernae (in m.2)

Full size image

It is also notable that the confidence intervals of the results for houses tend to be wider than those for tabernae, while the R2 of the relationships for the former tend to be higher than those for the latter (although note that both results are significant at the < 0.0001 level). This applies to the results for both the numbers and diversities of rooms. This suggests that the former conform to our expectations a little more closely than the latter and that the variation in the sizes of the former accounts for more of the variation in the numbers and diversities of rooms than the latter. This makes sense, given that we would expect the inhabitants of houses to have had a little more control over their living conditions than the occupants of tabernae. Finally, note that the R2 of the relationship between the sizes of structures and the numbers of rooms also tends to be lower than those for the diversities of rooms (this also applies to both houses and tabernae).

As discussed above, although it is more difficult to relate these results to population, there are some grounds for believing that, while there was a great deal of variation in the sizes of households, these sizes were not related to the sizes of the structures themselves, for the simple reason that they are more likely to be a reflection of households’ overall wealth and rising standards of living or quality of life. It is worth noting that, if this was not the case, and if larger structures were occupied by a larger number of people, we would expect the slope of both relationships to be shallower, meaning that both the numbers and diversities of rooms would increase more slowly with population.

One of the most striking aspects of these results is that both of these sets of relationships increase at a rate that is completely consistent with previous theoretical and empirical work, despite the fact that this previous work focused on the relationships between various aspects of the built environment and the estimated populations of a range of cities, towns, or villages, rather than the relationships between the various features and sizes of houses within a single settlement (Lobo et al., 2020; Ortman et al., 2020; Smith, 2018). The relationship between the sizes of structures and the numbers of rooms is consistent with our theoretical and empirical expectations for different measures of infrastructure, which tend to increase at the rate of about 2/3 (Lobo et al., 2020). This is illustrated, for example, by both the sizes of fora and agorae and the dimensions of street networks (Hanson et al., 2019). Meanwhile, the relationship between the sizes of structures and the diversities of the rooms within them is consistent with the average wealth of their inhabitants, which tends to increase at the rate of about 1/3. These results make sense, given that we would expect most rooms to have been used by more than one person, while we would expect a greater diversity of types of rooms to be connected to an increased need to accommodate visitors, perhaps meaning that it is also a symptom of increased wealth and status, greater socio-economic influence, and larger social networks. This suggests that these results are not just a result of a general, and therefore trivial, relationship between the sizes of structures and the numbers and different kinds of rooms within them and that it might be a result of the specific functions of these structures, including how individuals moved around them and interacted with each other. This, in turn, means that they might be subject to the same forces as those that generate patterns across settlements.

Although the size of the sample is small, these results show that there is also a relationship between the diversities of the small finds associated with houses and the sizes of these houses, which is completely consistent with our wider theoretical and empirical expectations (Table 7 and Fig. 8). The diversities of small finds also increase more slowly than the sizes of houses, at the rate of about 2/3. This means that, although the range of objects that were owned and used by the inhabitants of these households did increase as they increased in size, it did so at a slightly slower rate. In this respect, these results are reminiscent of recent work on the amount of specialization and diversification in cities, which suggests that the total numbers of activities that occur in cities increase slower than their populations (Hanson et al., 2017).Footnote 7

Fig. 8
figure 8

The relationship between the diversities of small finds and the sizes of structures (in m.2)

Full size image

Finally, as we will see below, although it is not possible to make direct comparisons between Pompeii and other ancient settlements, given the relatively limited amount of information that is readily accessible about the footprints of, and numbers and diversities of the rooms within, houses in other cities, it is also interesting to note that similar patterns can be observed in certain rural contexts, including within villas in north-west Gaul (Maréchal, 2020: Table 2). Although the results are not directly comparable, a preliminary analysis reveals that there is an almost identical relationship between the total footprints of these villas and the sizes of the hot rooms, or caldaria, of the bathing complexes found within them (Maréchal, 2020: Table 2). These bathing complexes also increase slower than the footprints of the villas in question, with a slope of almost exactly two-thirds (the exponent is 0.66, the pre-factor 1.41, and the R2 0.52), suggesting that these rooms also acted as a form of infrastructure. Although more work will have to be done to interpret these results and extend them to other sites, they are nonetheless important in being suggestive of the existence of similar relationships in other contexts, hinting at the existence of a wider phenomenon.

While both sets of results are somewhat unexpected, as noted above, it is important to remember that the results from other contexts are based on the numbers of people who regularly interact in settlements. This means that we would expect the results to be a product of both the numbers of people who lived within each household and the sizes of their social networks. In other words, we would expect these houses to be a product of, and designed to accommodate, both their residents and their wider family, friends, competitors, and dependents, supporting the earlier suggestion that the domestic architecture of Pompeii is a reflection of networks of patronage within the site (Robinson, 1997, 1999). This has important implications, since it not only suggests that the incomes of individual households are proportional to their numbers of social contacts, but also that the sizes of these incomes and the amount of specialization and diversification are all linked via these social contacts, very much in line with wider settlement scaling theory.

In this context, it is also important to reiterate that the systematic relationships that have been found between the various attributes of modern cities and their populations have been explained as being a product of embedding social networks within the built environment, meaning that individuals must balance the benefits of interacting with others with the cost of movement in space and time (Lobo et al., 2020). It is possible to create a series of general models for the overall consequences of increasing the sizes of cities by considering the overall social and economic benefits of interaction between the individuals in a given settlement, after accounting for the energetic costs of navigating the physical space that it consisted of over a set unit of time (in other words, these relationships are not simply the result of an incidental correlation between these variables, but a direct result of the changes that happen within social networks with increasing demographic scale). This allows for a situation in which the overall rate of social interaction increases a little faster than the populations of cities, resulting in increasing population density and increasing rates of various social and economic outputs, exhibiting increasing returns to scale, while the infrastructure that is required to support them generally increases a little slower, exhibiting economies of scale. The similarities between the earlier results published elsewhere, at the urban system level, and the new results published here, at the city level, might therefore have significant implications for our understanding of the structure of ancient societies, since they suggest that the inhabitants of individual houses also made choices about the relative costs and benefits of social interaction, informing not only the scope of the social networks that they operated within, but also the extent to which they were able to invest in different amounts, and different kinds, of domestic space to do so. Finally, it is also worth pointing out that this line of reasoning is not only perfectly consistent with the evidence for a relatively high level of social and economic inequality within Pompeii, with a broad spectrum between the most wealthy and most poor, but could also be used to offer an initial explanation for its emergence and development by locating it within the context of the long-term structural effects of the flow of resources among social networks embedded within the built environment—another matter for elaboration in future work.

Another important question is whether we would expect to find the same relationship across sites or whether we would expect the slope of the relationship to increase or decrease as the sizes of sites change. In other words, do the relationships between the numbers and diversities of rooms and the sizes of structures vary from site to site or do they always have the same slopes, with an exponent of either 2/3 or 1/3? This raises the question of whether it is possible to apply the same approach to a wider selection of sites, which could be used to assess how stable these patterns are from context to context. The most obvious candidates for such a comparison are some of the other comparatively well-documented sites, such as Herculaneum and possibly Ostia, Volubilis, Timgad, and Sabratha, although a considerable amount of effort will have to be invested in digitizing existing maps and plans, correcting mistakes using satellite imagery, and creating new databases of the housing stock of each site (Andrews, 2006; Etienne, 1960; Hanson & Ortman, 2017; Packer, 1971; Wilson, 2011).Footnote 8 It is also tempting to reevaluate the Forma Urbis to see if it can provide similar evidence for the ratio between the numbers of structures and the numbers of rooms in the capital, although care would have to be taken about how these were measured and interpreted (Rodriguez-Almeida, 1981). There is no reason, however, why this approach could not be extended to medieval and early modern or contemporary cities. It would also be interesting to investigate whether there is any geographical or chronological variation in the pre-factors of these relationships, in both ancient and modern contexts, which could then be taken as a reflection of different levels of social and economic productivity, especially given the possibility of geographically and chronologically specific scaling relationships influenced by subsistence organization (Fletcher et al., 2022). This could have important implications for understanding why these measures seem to be so stable over time. Finally, these results also call for a more detailed investigation of the relationship between the numbers and diversities of small finds and the sizes of residential structures in Pompeii and elsewhere, using a larger sample of buildings. This exercise could then also be extended to other sites in both ancient and modern contexts.

These results raise a number of other interesting questions for future research, given that the variation in the sizes of houses has previously been explained as being the result of scale-free networks (for more discussion of this issue, see Brughmans, 2010 and Brughmans, 2013). Here, new nodes tend to become attached to the most well-connected nodes, resulting in a ‘rich get richer’ effect (Brughmans, 2010: 278; Barabási & Albert 1999: 79–92). These results seem to be more in keeping with small world networks. In this case, although most nodes are only connected to a small number of other, local, nodes, some nodes are also connected to a handful of further-flung nodes (and therefore other small world networks) through a limited number of long-distance connections (Brughmans, 2010: 278; 2013: 645; Barabasi, 2002: 79–92). Small world networks have been associated with various kinds of real-world networks, including social networks. It is therefore tempting to draw parallels between the sizes of houses, their capacity for accommodating individuals, the comparative sizes of elite and non-elite networks, the development of small world networks made up of a relatively small number of super-connected nodes, and systems of patronage.

Finally, it is worth noting that it is possible to calculate the extent to which each observation deviates from the overall relationship: a value that is known as a residual. These residuals have been used as a reflection of the specific social and economic conditions of cities, once their sizes have been taken into account, suggesting that the residuals of houses and tabernae could be used in a similar manner (Bettencourt et al., 2010). It will take a little care, however, to determine how these values should be interpreted, given that they document the variation from the sizes of structures, and possibly from the numbers of individuals that regularly interacted with each other within them, rather than from their specific numbers of inhabitants. This could be an important subject for future research.

Conclusions

In this article, I have shown that there is not only a relationship between the sizes of houses and the numbers of rooms within them, but also between the sizes of houses and the numbers of different kinds of rooms within them, which are identical for both houses and tabernae. These both increase more slowly than the sizes of structures, but at slightly different rates (about 2/3 and about 1/3). Although larger houses had both more rooms and a greater range of rooms, there was not a linear relationship between them. One of the most remarkable aspects of these results is that both sets of relationships increase at a rate that is consistent with both the theoretical expectations of settlement scaling theory and empirical results from other contexts, despite the fact that they reflect the variation within settlements, rather than across them. Similar regularities seem to occur at both the settlement and household level.

Although we have to be careful about how we interpret these results, they do nonetheless strongly suggest that these regularities are caused by the same, or at any rate similar, underlying social and economic processes. This makes sense, given that they can both be seen as a product of how social networks, and the social and economic effects of these networks, change as they increase or decrease in size, once embedded within the built environment. The only difference is that one is a product of the social networks of individual households and the other is the product of the social networks of the inhabitants of a given settlement. In other words, both individual houses and entire cities can be seen as containers for social interactions, meaning that there is no conflict between these results and wider settlement scaling theory. This has vital implications for our understanding of urban life, since it suggests that the same, or similar, social network effects exist at multiple social, spatial, and temporal scales and that both individual houses and entire cities can be regarded as being part of the same continuum of settlement, running from individual houses and neighbourhoods to entire cities and urban systems. It might be possible to unite these different scales into the same framework in the future, enabling a more effective collaboration between urban scholars and the classical archaeologists who have devoted their careers to trying to understand the social and economic fabrics of individual ancient settlements.

Finally, although the size of the sample is small, these results show that there is also a relationship between the diversities of the small finds associated with houses and the sizes of these houses, which is also completely consistent with our wider theoretical and empirical expectations. The diversities of small finds increase more slowly than the sizes of houses, at the rate of about 2/3. Although the range of objects that were owned and used by the inhabitants of these households did increase as they increased in size, it did so at a slightly slower rate. An important object of future research will therefore be to extend these results to not only other cities within the Roman Empire, but also other settlements in other contexts, opening up the possibility of building a series of much more ambitious theoretical and empirical models of urbanism in future, which trace the variation both within and across settlements back to the net socio-economic and infrastructural consequences of embedding social networks within built environments.