Models of settlement hierarchy based on partial evidence Andrew Bevan and Alan Wilson Postprint of a paper to be published in 2013 in Journal of Archaeological Science. 1. Introduction This paper considers how we might construct useful understandings of past human settlement hierarchy in situations where our surviving evidence is patchy and incomplete. It builds on the continuing attractiveness of a family of spatial interaction and settlement evolution models that were first developed in the 1960s and 1970s in urban geography to capture, amongst other things, the growth of modern retail outlets (see Wilson 2012 for a recent overview; Wilson, 1967, 1970, Harris and Wilson, 1978), and then first applied to archaeological settlement data sets by Rihll and Wilson in a series of papers some twenty-­‐five years ago (1987a, 1987b, 1991; also Wilson 2012). Despite the promise of these earlier efforts, there has been little further work on these models, with only some exploratory consideration of how they might be recast as agent-­‐based simulations (Graham and Steiner 2006) or compared with alternative forms of dynamic network (Evans et al. 2012). These models have sometimes been dismissed as ‘gravity models’ but the approach here is fundamentally different. The spatial interaction elements are based on entropy-­‐maximising methods and the settlement dynamics on a development of Lotka-­‐Volterra equations first used in ecology (see Wilson, 2008, for a full account of these ideas which are also exolored more fully in section 2 below and in the Appendices). Here, we further extend their analytical scope (a) by demonstrating how the combination of point process models with spatial interaction models offers a viable approach in cases where there is only a very partial array of settlement evidence, (b) by grounding the model in more physically realistic routes of interaction, and (c) by considering whether it can also shed light on the co-­‐evolution of path and place hierarchies. Bronze Age settlement and political geography on the Greek island of Crete provides a useful, well-­‐known test case where all of these methodological developments can be explored. We start by outlining a common research problem in archaeology, and one that has further relevance for historians as well. Given a partially observed pattern of human settlement from a particular period of the past, there are many different questions that we might wish to ask, with any eye to understanding the causal logics behind the settlement structure, exploring long-­‐term dynamics of this structure or comparing it with a larger cross-­‐cultural sample. One key goal is usually to reconstruct a wider political and economic hierarchy, as reflected in and promoted by a certain spatial configuration of settlement. We are typically also interested in related questions such as what aspects of this hierarchy are a consistent feature across most periods in a study region, and hence possibly induced by the local environment, and which ones are predicated on a particular cultural logic and/or a set of historical circumstances. Of further interest is also the role of formal trail or road networks between settlements, the agricultural and political territories around them, and ultimately the manner in which whole landscapes become laden with cultural significance. 1 As David Clarke noted some time ago, archaeology is a discipline that seeks to understand ”unobservable hominid behaviour patterns from indirect traces in bad samples” (1973: 17). In line with this maxim, most archaeological settlement distributions are incomplete and of uneven detail, often dramatically so. Even if we were happy, for example, only to consider the largest 100 settlements across a given study region and to ignore smaller hamlets and farmsteads as of minor consequence to the overall whole (itself a potentially problematic assumption), then in most cases, we would still only be able to identify some but not all of these settlements on the ground archaeologically, and would probably be able to assess other variables such as the settlements’ relative importance only for a further subset of these. The reasons for missing data are various: in many parts of the world, archaeologists construct large-­‐scale gazetteers of settlement from regional inventories that have often accumulated in a haphazard fashion over many years (e.g. most sites and monuments records) and as the result of many different types of investigation (rescue excavations, rare research-­‐driven projects, chance finds, casual surface reconnaissance, literature searches, etc.). In some regions, more systematic archaeological surface surveys can claim greater data quality (e.g. Fish and Kowalewski 1990; Banning 2002; Alcock and Cherry 2004; Peterson and Drennan 2005), but such intensive reconnaissance techniques can usually only be implemented for comparatively small areas. Moreover, the vast majority of survey practitioners would agree that even these intensive methods still only provide an imperfect sample due to the fact that only certain kinds of evidence have survived into the present day in certain taphonomically-­‐favourable places, and even where this evidence exists, settlements from different periods are often superimposed as awkward palimpsests (also Groube 1981). Very occasionally, there are exceptions where larger scale, more complete coverage of one period is achievable for unusual reasons (for example, the 3rd millennium BC tell landscapes of north-­‐eastern Syria: Menze and Ur 2012), but it would be very undesirable to limit our analysis solely to this evidentially privileged minority of cases. Even where we have a near complete record of the distribution of human settlement and a wealth of ancillary detail to draw upon, it is still important to construct models that allow us (a) to make sense of patterns in the data in a formal way, and (b) to use these insights comparatively and longitudinally. How we deploy existing knowledge about settlement patterns is also a strategic issue, as we might wish to withhold some of the evidence from the model-­‐building exercise as a way of testing the plausibility of the results. There are thus several reasons for wishing to develop methods that are robust to missing information. Similarly, even if ultimately, we are left with several competing models that are underdetermined by existing archaeological observations, (Hodder 1977; van der Leeuw 2004) the exercise still serves to formalise our thoughts, narrow down the range of possibilities and indicate where further research would be useful. The following discussion explores some ways in which this might be possible, building upon three existing modelling traditions in geography, spatial statistics and network science: (a) interaction models based on the principle of maximum entropy together with models of the dynamics of settlements (b) point process models that can capture both the first and second order properties of an observed point pattern, and (c) shortest path calculations and models of network 2 evolution. The methods section below begins by outlining the potential of each of these in light of our research problem and thereafter, we consider how they might work together in an applied case from Bronze Age Crete. 2. Methods 2.1 Transport and Settlement Dynamics' Models In Appendix 1, we present the key ideas in the development of spatial interaction models – that have been used in a contemporary context, for example, to estimate flows of money from consumers to retail centres. These retail models have been extensively tested and can replicate the data representing a current situation very accurately. The method is based on entropy maximisation (Wilson 1970; Wilson 2000) that, in part, is designed to make the best use of partial data and hence potentially lends itself well to archaeology. In urban retail analysis, the modelled flows can be summed at each centre and this revenue compared to the retailers’ costs to provide an estimate of ‘profit’ at each location. This in turn drives the dynamics of the evolution of the system of centres (Harris and Wilson, 1978) and enables the ‘most likely’ structure of the system of centres to be articulated (what is sometimes referred to as its equilibrium structure). These methods can be transferred into archaeology if the ‘centres’ become ‘settlements’ and the 'flows’ between settlements are understood as some composite of the relative intensity of trade along different routes, daily or seasonal journeys to work and/or the permanent migration of people. The model can then be used to estimate the spatial distribution of settlement sizes. 2.2 Inhomogeneous Point Process Models While the above framework provides a useful way to predict and explain the relative importance or size of different settlements without building in this information from the outset, by contrast, it still assumes fairly complete knowledge of settlement locations across a study area. As we argued above, such complete knowledge is rare in archaeology, even in situations where we are only interested in larger sites. We therefore need a method for proposing the position of possible missing settlements, based on the locational properties of the known sample, and then a way of exploring the sensitivity of model results in the presence of this hypothetical component. Recent approaches to inhomogeneous point process modelling provide just such a framework. We can think of a settlement distribution as one historical realisation of a settlement process played out over time and across a larger area, but observed for a particular time-­‐
slice within a particular study region. In many cases, we can make a simplifying assumption and represent the settlements by their centroids (i.e. ignore their spatial extent) and consider their configuration as a point pattern marked by a particular set of size attributes, as we do in this paper. A useful general distinction to make from the outset is between the first-­‐ and second-­‐order properties of a point pattern (O’Sullivan and Unwin 2003: 51-­‐75; Illian et al. 2008; Gelfand et al. 2010: 263-­‐423). The first-­‐order properties of a 3 point pattern are those that describe the average intensity (per unit area) of points across a given study region, while its second-­‐order properties are those that describe the influence of neighbouring points (i.e. the covariance structure). For example, we might think of the availability of better or worse farmland as a variable that prompts a first-­‐order unevenness (inhomogeneity) in the intensity of human settlement. Beyond this however, there are also often second-­‐order patterns in human settlement, for example regular-­‐spacing due to competition over resources, or the reverse, clustering together of human habitation (for various reasons including protection, due to colonisation history etc.), or indeed some multi-­‐scalar combination of these (for further useful ways to model the economics of these attractive and inhibitory factors, see Fujita et al. 1999). In archaeology, there is already an established tradition of exploring the first-­‐order properties of archaeological site locations via logistic regression, albeit in terms of predicting site probabilities for heritage management purposes (e.g. Mehrer and Wescott 2006; Verhagen and Whitley 2011). There is also a fairly long archaeological tradition of considering second-­‐order properties of settlement via nearest neighbour distances, quadrat counts and various multi-­‐scalar methods (Hodder and Orton 1976; Orton 2004; Bevan and Conolly 2006), but as yet there has been little attempt to model these two characteristics together. However, recent advances in spatial statistics have involved the development of techniques for assessing the first and second order properties of a point pattern in a more coherent way and then parameterising them via an inhomogenenous point process model (Baddeley and Turner 2006; Illian et al. 2008). An inhomogeneous point process model is an equation which combines a prediction of (a) the varying density of observed points across a study area (the first order trend), calculated either solely from the observed point distribution or, in our case, as correlated with one or more predictor variables (in a manner similar to a traditional predictive model of site location using logistic regression), and (b) the degree to which the location of one point inhibits or attracts other points (and over what distances, i.e. second order properties). For our purposes below, we can thus model the varying first order intensity of observed settlement as it is relates to different covariate variables such as various aspects of the local environment, at the same time as imposing further second order constraints visible in the observed data (such as the typical spacing between settlements). We can then simulate additional points whose locations are conditional on this model (i.e. they are not wholly random, but random within certain spatial constraints). Due caution must be taken to avoid including recovery biases present in the incomplete observed data, but even so, this capacity to simulate plausible candidates for missing settlements, and to do so repeatedly, opens up important opportunities to apply the above interaction models in the context of a Monte Carlo simulation and we explore this further below (for MC methods in general, see Robert and Casella 2004) 2.3 Interaction Distances and Landscape Assignment These models demand that we model the impact of geographical distance (i.e. the frictions it introduces) on the way different regions of interest (e.g. settlements in our case) interact with one another. The simplest approach is to consider isotropic (i.e. ignoring travel direction), pair-­‐wise (i.e. assuming direct and 4 bilateral links rather than indirect and multilateral ones) and straight-­‐line (i.e. Euclidean, as-­‐the-­‐crow-­‐flies) distances between settlements. However, in many situations, it is desirable to incorporate greater realism in the form of optimal or average travel times, as suggested by real world data, computed for a discrete network, or estimated over a lattice-­‐based representation of a landscape (i.e. across a digital elevation model). In this paper, we adopt the well-­‐established idea of calculating shortest paths (‘least cost’, ‘optimal’; see de Smith et al. 2009: section 4.4) and employ a specific anisotropic method that produces reassuringly valid results when compared to actual walking times documented on Crete in the earlier 20th century (Bevan 2010: 30-­‐31).1 While it is certainly possible to explore the network created by linking each pair of settlements across a potentially large study area via direct, shortest, direction-­‐
dependent routes, we suspect that this often just trades off one kind of simplistic assumption (Euclidean distances) for another (direct bilateral links between all sites even over long distances), whilst making the model both more complicated and more computationally costly. In the case study below, we prefer to simplify things by generating a local network of anisotropic shortest paths between each settlement and its 10 closest neighbours across a landscape grid. In our case, we used a well-­‐known hiking function (Naismith’s rule) and a 90m digital elevation model (using GRASS’ r.walk; see Fontenari et al. 2005; also Bevan 2011). For similar efficiency reasons, we then take an average of the costs along the route to and from each settlement pair and treated the result as a weighted, undirected graph. These decisions about what preliminary network among settlements to use prove not to be crucial as long as there are sufficient path segments to allow at least one or two alternative routes among all nodes (as ensured above). However, a relatively simple weighted network of this kind provides many analytical advantages not least of which is the possibility to consider not just one shortest route between two sites, but multiple shorter routes. Here we use a method for solving for ‘k-­‐shortest paths’ between two points on a network that was proposed by Eppstein (1998) and which has been shown to be both efficient and reliable. Figure 1a offers an example with the overall network shown in white and the shortest path between A and B shown in black, while figure 1b-­d show the next three shortest routes though this network. 1 There are several promising methods for exploring partially or wholly random walks across landscapes grids that might also be relevant either in situations where we are dealing with long-­‐
term settlement aggregations and time-­‐averaged human interaction (e.g. McRae et al. 2008; Saerens et al. 2009) or at smaller, shorter scales (e.g. Helbing et al. 1997), but there are practical and theoretical reasons for using more discretely-­‐routed and goal-­‐oriented flows in the examples below (see also Tero et al 2010; de Martinis et al. 2012). We hope to return to these issues in a subsequent paper. 5 Figure 1. An example of the four shortest paths between two settlements in a wider landscape of eight settlements. In this case, all settlements exist on a rugged terrain modelled as a raster grid and an initial network of shortest paths has been created across this grid. This then supports the calculation of multiple possible routes where a direct link between A and B is the merely shortest, if it exists at all. Pedestrian travel time is shown in hours. 2.4 Evolving Places and Paths In section 2.1 and in Appendix 1, we outline a model of the evolution of settlements, but we can now also explore the more challenging task of modelling the evolution of networks – in this case, trails and roads. We make an assumption that on links where flows increase, the link capacity also grows as certain paths become more widely known and/or better maintained (e.g. via word-­‐of-­‐mouth, published itineraries, road surfacing, way-­‐stations, signage, etc.) – facilitating interaction between larger centres. This is a network analogue of the evolution of centres and the model is presented in detail in Appendix 2. In effect, of course, this becomes a component of a model of the co-­‐evolution of settlements and networks. 6 3. A Case Study: The Towns of Bronze Age Crete As a case study of the above methods, we consider Bronze Age settlement on the Greek island of Crete during the 2nd millennium BC. This is a period in which we have evidence for a palace-­‐centred culture commonly known as Minoan civilisation (see Bevan 2010 for wider references). Figure 2a shows a distribution map of better-­‐known Cretan settlement sites from this period, with an indication of their relative size and whether or not they have so far produced palaces or palatial-­‐style buildings that might imply greater political and economic importance. Several further settlements have as yet not produced palaces but are clearly quite large or important, while there are also many others that are lower order villages and towns. Beyond these, archaeological survey has also documented many even smaller sites (e.g. hamlets and farmsteads) but here we focus here on the upper end of the settlement hierarchy (very roughly those settlements over a couple of hectares in area). 3.1 Handling Missing Settlement Evidence Crete is one of the most densely surveyed and archaeologically explored regions of the world and the Bronze Age settlement evidence here therefore offers a strong starting point for thinking about settlement modelling in the context of emerging political hierarchy (see Driessen 2001; Whitelaw 2001; Bevan 2010). Even so, when the whole island is considered over its full c.8,600 sq.km extent, it represents a common archaeological situation in which the evidence nevertheless remains patchy and incomplete. We have clearly discovered only some, not all, of the medium-­‐ to large-­‐sized Cretan settlements from this period and can only estimate their relative size with very varying degrees of confidence. Moreover, we know little about the wider political and economic hierarchies associated with these settlements, but rather would like to infer all of these things from the partial sample of information that we already have. The spatial interaction and settlement dynamics' model outlined in section 2.1 is a good start given that it can be tailored to make few, if any, initial assumptions about site size. However, in its traditional form, it still (a) requires a comparatively comprehensive mapping of known settlement within a chosen study area to work effectively, and (b) suffers further problems with regard to missing interactions across and beyond the borders of the study region. There is a variety of ways in which we could address these two different kinds of missing data. The method advocated here as a potential solution to the first of these problems is to characterise the location of known settlements via an inhomogeneous point process model, use this to simulate the location of missing but similarly-­‐placed settlements and then run a multiple interaction models o the result. By repeating these steps many times (as we do in section 3.4), we can then determine if there are features of the settlement hierarchy that remain robust over multiple simulations and which can therefore be considered as reliable results despite the missing settlement evidence. 7 Figure 2. (a) A simplified distribution of centres with major palatial buildings and other larger settlements from Middle-­‐Late Bronze Age Crete, and (b) an example simulating additional settlements (open circles) conditioned on the positive relationship between observed settlements (red crosses) and access to flatter, more agriculturally favourable, land (underlying raster surface, where yellow-­‐green-­‐blue-­‐purple indicates from more to less access respectively). The 35 settlements shown in figure 2a may of course reflect investigative biases resulting from different levels of modern archaeological research across different parts of Crete, but for the purposes of this paper, we assume that they can still be used in an approximate way to model the underlying logic of Bronze Age settlement with respect to the wider landscape. For example, the background raster surface in figure 2b expresses the amount of flat land within 2.5km of any given raster cell (this distance being about half an hour’s walk or so, which is a commonly observed outbound journey time in cross-­‐cultural studies of commuting behaviour: Zahavi 1979; Marchetti 1994) and this surface makes for a significant predictor of the intensity of known sites (those shown as red crosses). We can then use this relationship to add a further set of similarly-­‐
placed points (for instance) in order to reflect a more complete possible distribution of larger settlements across the island. For our analysis below, we 8 have chosen to add 100 extra settlements in each case, based on our sense from survey evidence of the overall likely quantity of medium to large sites in this period (e.g. Driessen 2001; Whitelaw 2001), but we have also experimented with greater or fewer additional sites to establish that the results remain fairly consistent. If we choose, we can also impose one or more second-­‐order properties on the simulated point pattern, for example, a typical separation distance between the newly-­‐added sites that might reflect separate hinterlands. For the examples below, we set to minimum settlement spacing of 5 km that is close to a typical spacing among the observed sites and to the travel times often observed cross-­‐
culturally as a daily commute (see above). 3.2 Visualising Hierarchy There are a variety of ways in which outputs from an interaction and settlement dynamics' model might be visualised, and our focus below is initially on identifying possible settlements that may have acted as central places. Figure 3a shows an example of the equilibrium proposed by one example run of the model at given set of parameter values, where the size of the circles indicates the predicted relative size or importance of the settlements. We also adopt a method first proposed by Nystuen and Dacey (1961) and later modified by Rihll and Wilson (e.g. 1991: 66-­‐69) and circle in blue those ‘terminal’ sites whose own inflow total is larger than that of any single site receiving flow from them. Finally, for each settlement, we can map those outward links from a given site that are above a certain percentage of the maximum flow out for that site. In this particular example, we have done this for the maximum outflow links only and the resulting network exhibits a characteristic star-­‐shaped, tributary structure around each central place and some secondary clustering of secondary large settlements next to terminals. 9 Figure 3. Two examples of the output from interaction model runs using Euclidean distances between settlements: in both cases, α=1.05, β=0.15 (with the map scale conveying the distance decay implied by the latter parameter setting), but in (a) no weighting is used for off-­‐island interactions (G=0), while for (b) G=0.1 (see equation 14 below). Settlement sizes are shown by graduated open circles and terminal settlements are circled in blue. Each non-­‐terminal settlement has been given a link to its maximum creditor (shown as a black line). An impression of the territories around central places are shown in white outline (by selectively merging a Voronoi tessellation of all settlements). 3.3 Accounting for Edge Effects The results from figure 3a need to be modified in a Cretan context for at least two reasons: (a) they have not yet taken into account the impact of Crete’s rugged topography on patterns of interaction, and (b) they have not yet addressed the fact that settlement hierarchy on island will often be substantially affected by interactions with communities in neighbouring regions overseas (e.g. in this case, the Aegean islands, the Peloponnese, mainland Turkey and beyond). We return to the first of these problems below. The second is a special (maritime) case of a wider modelling issue to do with how to handle an absence of knowledge about interactions beyond the boundaries of a study area. In fact, 10 there are several more or less complicated ways in which we could account for such edge effects. One interesting possibility is to add in a series of large, external zones (e.g. the southern Peloponnese, the central Cyclades, the Anatolia coast, the Levantine coast) explicitly, but this requires specific information about the relative contributions of each of these. We adopt a simpler method here: to calculate an adjusted inflow for those settlements next to the coast (in this case, all points <1km from the coastline) such that they received extra interactions from beyond the island that are proportional to their on-­‐island importance (see Appendix 1 for a formal description). Figure 3b shows the result when the same parameter settings and settlement dataset are used as in figure 3a, but with some off-­‐island interactions via Crete’s harbour sites. The net result in this second example is that the settlements that become central places are now more likely to be those closer to the coast. This tension between terrestrial centrality and coastal accessibility is an important and dynamic feature of past settlement patterns (Taaffe et al 1963) and one that has been particularly important in Mediterranean history (Braudel 1972: 103-­‐62). It is worth noting this example run already shows some interesting similarities with the observed hierarchical patterns observed on Bronze Age Crete (compare with figure 2a), with major places in locations at or close to known Minoan palaces such as Knossos, Phaistos, Malia and Zakros, with further predictions for the importance of centres in the Chania and Gournia-­‐Mochlos areas. 3.4 Sensitivity Analysis The results from a single model run should be treated with caution given the fact that three quarters of the input settlements are simulated, rather than archaeologically-­‐observed. One of the strengths of the above approach however is that we can perform a sensitivity analysis by repeatedly generating new point simulations. More precisely, we keep the same 35 known larger sites in each run, but add a further 100 extra points each time based on the predicted intensities and inter-­‐settlement spacings set up by the point process model. For each of these simulated datasets, we can then run an interaction model (with consistent parameter values throughout) and assess how robust the results might be. For example, figure 4a shows the density surface produced from running a bivariate kernel (σ=5km) over all those settlements that are chosen as terminal sites in any of 100 model runs (where a settlement can count more than once if it is chosen more than once). Again, making a comparison with figure 2a, we can note some clear associations: central places repeatedly emerge in locations where we might expect them given what we know about the relative size and monumental elaboration of these sites. Some areas where we know very little archaeologically but suspect a missing large centre (such as in west-­‐central Crete, half way between Knossos and Chania) also show signs of probably having a regional centre. Figure 4b offers an alternative visualisation of the territorial integrity of these predictions: if we take the modelled hinterlands of control around each terminal settlement (i.e. those shown in white outline in figure 3b) and allow the polygon areas from each model run to clip those from the others, we end up with very small slivers in areas that are regularly border zones in most model runs, and larger swathes of consistently contiguous territory elsewhere. Thus the red areas in figure 4b are those that are likely to have been consistently on the frontier of Cretan regional spheres. 11 Figure 4. Spatial summaries, for 100 runs of the point process and transport models, of (a) the location of terminal sites (a kernel density surface with a Gaussian bandwidth of σ=5km), and (b) more (light blue) and less stable (green-­‐yellow-­‐red), parts of the landscape in terms of their allocation to territories. The interaction model consistently uses Euclidean distances, and parameters α=1.05, β=0.15 (with the map scale conveying the distance decay implied by the latter parameter setting). The larger Cretan sites and palatial buildings from figure 2 are also overlaid for comparison. Continuing with this emphasis on identifying terminal sites for the moment, a further way we can explore the sensitivity of model results is to vary not only the additional simulated settlements, but also the interaction model parameters. For example figure 5a shows the average number of terminal settlements over 20 point simulations that are identified for different model parameter values, while figure 5b expresses the variability of this result over the same set of model runs. In general, there is a tendency towards a few, more dominant settlements as we decrease the β parameter (that controls distance decay, see Appendix 1) and/or increase α (that models the importance of concentrated resources). However, there is also a region at α < 1.05, 0.03 < β < 0.07 where a large number of independent centres emerges with a wider envelope of outcomes around this 12 that vary depending on the placement of the additional simulated settlements. By virtue of the model set-­‐up, we would expect to find a bifurcation point like this around α=1, and it is tempting to interpret these results by invoking historical evidence for two distinct settlement regimes on the island: one in which there is only one or a few dominant centres (e.g. in the Venetian and Ottoman periods) and another where there are very many (the Classical tradition of “one hundred-­‐citied Crete”; see Perlman 1992). We argue below that part of the reason for these distinct regimes has to do with changing networks of wider Mediterranean interaction, but it is also conceivable that they relate to endogenous changes (e.g. in social norms, political structures and/or resource extraction technologies) that vary the degree (α in our models) to which local communities might concentrate their efforts. Figure 5. (a) Summary of the mean number of terminal settlements identified at different settings for different α and β settings over 20 different point simulations, and (b) variability in the number of terminals identified for different α and β settings over 20 point simulations, expressed as a coefficient of variation. 3.5 Incorporating Physical Paths While versions of these models that employ shortest paths are more computationally expensive, they allow us (a) to judge whether the results based 13 on simpler distance measures are altered when we pay greater attention to landscape realism, as well as (b) to assign these flows to suggested physical routes through the landscape. Figures 6a-­b adopts the same parameter settings and settlement simulation as used in figure 3, but builds a model where the distances between settlements are calculated across a network of more physically realistic paths. More precisely, the example network used below involves connecting each settlement to its nearest ten neighbours via shortest paths across a digital elevation model of Crete (in the manner described in the Methods section). Because the network is not fully-­‐connected between all nodes, certain path segments are now used as part of several different interactions. There are therefore at least two useful ways in which this can be visualised: (a) in the form of those inflows above a certain proportion of maximum debit following the modified Nystuen–Dacey method used above or (b) as summed total flow across each path segment. The first way of mapping the model results serves to emphasise patterns of greater and lesser dependence on the regional terminal sites and the routes via which this control is likely to be exercised. It also shows that different regions can have quite different internal hierarchical organisation. The second method offers a broader impression of interaction across the entire system, highlighting path segments that can be popular both because of the interactions between a single pair of settlements and/or because lots of small pairs of interactions use that route. Of additional interest is the character of the flow loads in perceived frontier areas between sub-­‐regions. Note finally that the model results are largely consistent with those from the equivalent Euclidean model in figure 3 and archaeologically plausible in terms of where they predict regional centres should emerge. However, there are a few minor differences with, for example, the larger sites in the central sub-­‐region moving to a more coastal location, and the shape of the east-­‐central sub-­‐regions (around Malia and Gournia) changing due to the impact of local mountainous terrain. 14 Figure 6. Two different ways to visualise flows between settlements based on a single example distribution and an example model with α=1.05, β=0.57 (roughly equivalent for travel time in hours to the β=0.15 used for Euclidean distance in km in fig.3, and with the map scale conveying the implied decay), G=0.1. In (a) only the summation for each path segment of the largest outflows from each settlement are mapped (i.e. to its maximum creditor), whereas in (b) the summation is for all flows. The light blue to orange to dark red colour ramp depicts lower to higher flow. Settlements sizes are shown by graduated open circles and terminal settlements are circled in blue. An impression of the territories around central places are shown in white outline in both cases (by selectively merging a Voronoi tessellation of all settlements). 3.6 Joint Path-­Place Hierarchies and Evolutionary Trajectories The traditional approach to interaction and dynamics’ models such as those used above is that they are left to run until their zone sizes (the settlements in this case) achieve some approximate equilibrium state where no major changes occur from iteration to iteration and the model stopping conditions are met. In this scenario, the links between zones (paths across the landscape in this case) are treated as a static feature with different flows along them having no feedback into model results (for ways that this has tried in the past via extra balancing parameters, see Alonso 1978; de Vries et al. 2001). However, we have already noted above that that there are at least two ways to think about more evolutionary trajectories: (a) to run several successive models, each individually to some equilibrium state, and alter one or more exogenous variables in each 15 distinct model to represent changes over time in scale economies, in transport technologies or in the importance of longer-­‐distance flows from outside the study region and/or (b) to think of the iterations in a single model run not so much as a search method for finding an equilibrium state, but as a series of evolutionary time-­‐steps. We briefly explore the opportunities that the latter option provides below, not least because it allows us to integrate the hierarchy of physical paths into this dynamic. More precisely, we can consider the impact of having not just one but multiple possible paths between each pair of sites (in the example below, we consider the three shortest paths rather than just the shortest), and by allowing the intensity of interaction along a path segment to feed back into the cost of interaction along that segment in the next time-­‐step according to equations 12-­‐13 in the Methods section above. As an example of the possibilities offered by this approach, figures 7a-­c present three possible time-­‐steps in an evolutionary model of Cretan Bronze Age paths and places. We use the same example distribution of observed and simulated settlements and the same α, β and G parameters as in figures 3b and 6, but also allow the costs along certain path segments to be modified by the amount of flow along them (equations 12 and 13), leading to a potentially rich interaction with changing settlement size. The hierarchy of paths that is visible even after only a few iterations emphasises the fact that allowing multiple routes between each pair of sites automatically imposes a hierarchy of activity on the landscape, as certain path segments get used multiple times for interactions among multiple settlement pairs (figure 7a). The initial starting assumption of equally-­‐sized sites may be rather naïve, but it does largely reflect the more egalitarian conditions observed archaeologically during much of the 3rd millennium BC, with few if any sites visible dominating beyond their immediate hinterlands. Thereafter we slowly see the emergence of more important regional centres and most of these are where we might roughly expect them to be developing based on archaeological evidence (figure 7b). The feedback between settlement size, network flow and travel cost also leads to the emergence of some very important short-­‐distance thoroughfares (emphasised as thicker red-­‐brown lines). Letting the model run further brings out very distinct regional city-­‐states, primarily at or near the places we might expect them, and with a route network that is highly regionalised and focused on a few major arteries. The details of these arteries is often informative, with for example, the connectivity of the northern and southern parts of central Crete reflecting an interaction that has long been thought to have been enhanced by formalised roads and bureaucratically-­‐
managed flows of palace commodities (e.g. Evans 1928: 60-­‐81; see also Bevan 2010: 32), while the backbone of interaction in the far eastern and far western parts of the island is in each case a major coastal route and only a limited number of others inland. 16 Figure 7. Three example iterations of a model in which both settlement sizes and path weights are both free to vary: a-­‐c are iterations 10, 250 and 500. Throughout, the following parameters have been used: α=1.05, β=0.57, G=0.1, μ=0.5, a=0.2, b=0.2. The paths are coloured according to the quantity of flow assigned to them (blue-­‐yellow-­‐red is lower to higher flow respectively, and white is no flow). Settlements sizes are shown by graduated open circles and terminal settlements are circled in blue. The same colour and symbol size scales have been used for each of the three examples. 5. Discussion The results from the above case study and example runs cannot be an exact mirror of the actual historical trajectory, and it would be missing the point of the modelling exercise to ever anticipate that they would be. Nonetheless they often capture important aspects of an observed reality and, moreover, serve to articulate for us some important static and dynamic features that underpin the political structure of Bronze Age Crete during the 2nd millennium BC. While the results do vary with different parameter settings, they do so within some very 17 specific behavioural regimes and with some striking consistencies. For example, they capture the emergence of regional ‘peer polities’ of comparable size and extent from an initially more fragment landscape of villages. They also point to a tension between the proximity of central places to the coast and their centrality on the island itself, depending on the relative importance of off-­‐island links and the configuration of neighbouring settlement. In archaeological terms, the different site histories of places such as Archanes, Galatas, Knossos and Poros-­‐
Katsamba arguably corroborate and nuance this observation (and at a small scale, so to do north-­‐south shifts in funerary practice within the Knossos volley during this period: see Preston 2004: fig.2). In many ways, this is also an example of a wider tension between maritime access (or its reverse, coastal vulnerability to piracy) and inland agricultural control that is a defining feature of settlement dynamics in all periods of Mediterranean history. Raising the α and/or lowering the β parameters also makes it possible to explore how power might further consolidate over time at a single site in the middle of the island (e.g. figure 5a bottom right corner), as it may well have done at Knossos. However, our current impression is that while we can arrive at either one of the two equilibrium points within a single model with different parameter choices, we cannot achieve a trajectory from one to the other within a single set of model iterations (e.g. as different notional time-­‐steps, in the manner of figure 7). Hence, although the issue clearly warrants further attention, we suspect that any final level of concentrated power at a single centre on the island is better construed as an exogenous effect rather than an endogenous one. For example, it may reflect a step-­‐change in the ambition and efficiency of political control (e.g. via the construction of formal road networks, enhanced shipping infrastructure or improved administrative practice) and hence better modelled as a shift in α or β over time. More importantly in our view, it may also reflect the changing shape of wider Aegean eastern Mediterranean interactions, that would best be incorporated via a coarser model, within which the current intra-­‐Cretan one might be nested. Stepping back from this particular study, there are many ways in which it would be interesting to extend further some of the ideas discussed above. For example, one future opportunity is to tune such models more closely by considering their goodness-­‐of-­‐fit to the observed size distribution of archaeological settlement in one time period, or the observable dynamics of their relative size over time (e.g. rank-­‐size plots and rank clocks: Batty 2006). A second is to incorporate a more subtle treatment of the uncertainty associated with archaeological estimates of settlement date, size and location via further Monte Carlo simulation (e.g. Crema et al. 2010). A third and related prospect would be to add a more formal treatment of possible biases in archaeological recovery and investigation into the modelling of unknown settlements. The current construction of a point process model from demonstrably patchy observed data is potentially problematic as it risks incorporating recovery bias into the simulation of additional sites (one reason why we have only used a single, simple covariate above). It would thus be useful to model this bias explicitly, either via a first-­‐order map of the intensity of archaeological investigation or as a ‘thinning’ point process). A fourth opportunity lies in incorporating different local access to resources in the 18 hinterlands around each settlement into the interaction model itself and there are good examples of how this might be incorporated as a simple additional parameter (e.g. as often included for modern retail models in terms of shop ‘floor space’; Harris and Wilson 1978) or as an cost-­‐benefit trade-­‐off between local exploitation of resources and external linkage (Knappett et al. 2008, 2011). A fifth and final opportunity we anticipate is a tighter coupling between the size-­‐
interaction component of the model and the simulation of settlement locations itself. The current approach treats a settlement configuration as already established in the landscape (and uses the point process model merely as a convenient way to fill in the gaps), with no further acts of wholesale abandonment or colonisation to occur (even if sites can dwindle in size to effectively zero, they still notionally continue to exist). A close link may be possible via treatment of the data as a marked point process where point birth, death and change in mark size are all treated together (for some relevant discussion, see Renshaw and Särkkä 2001). Part of the challenge however would be to retain some sense of the intervening physical networks of interaction as one of the factors potentially affecting new colonisation ventures, such that, for example, entirely new settlements can potentially emerge at the crossroads of existing routes. Another possible area of future research to tackle this issue would be to shift to an agent-­‐based form of model in which there is no prior specification of the original settlement configuration and, instead, is simply allowed to emerge from a grid of possible locations (Dearden and Wilson 2011). Bearing in mind these further possibilities, the overall modelling approach adopted above already highlights ways in which we can explore settlement locations, settlement hierarchies and wider settlement and transport landscapes despite our information about them being incomplete. It has sought to open up the bulk of the archaeological record to formal attention of this kind, rather than only privileging a few evidence-­‐rich examples. It has also stressed that physical routes both structure and are structured by settlement interactions, and that we gain much from treating them as co-­‐evolutionary phenomena. Archaeological settlement datasets offer one of our best, truly long-­‐term records of past social, economic and political organisation, and it is high time we returned to the established agenda of comparative settlement archaeology with these formal modelling perspectives in mind. Acknowledgements Our thanks to Mark Altaweel, Enrico Crema, Joel Dearden and Sean Downey for facilitating access to computing hardware, discussing approaches to software development, and/or commenting on drafts. The spatial interaction models developed here were written in the R statistical software environment by the corresponding author, with further functionality contributed by the graph, kBestShortestPaths, maptools, raster, rgdal, rgeos and spatstat packages. References 19 Alcock, S.E. and Cherry, J. (eds.) 2004. Side-­by-­Side Survey. Comparative Regional Studies in the Mediterranean World, Oxford: Oxbow. Alonso, W. 1978. A theory of movements, in Hansen, N.M. (ed.) Human Settlement Systems: International Perspectives on Structure, Change and Public Policy: 197-­‐
211. Cambridge, MA: Ballinger. Baddeley, A. J. and Turner, R. 2006. Case Studies in Spatial Point Process Modelling, in Baddeley, A.J., Gregori, P., Mahiques, M., Stoica, R. and Stoyan, D. (eds.) Modelling Spatial Point Patterns in R: 23-­‐74. Berlin: Springer. Banning, E.B. 2002. Archaeological Survey, New York: Kluwer. Batty, M. 2006. Rank Clocks, Nature 444: 592-­‐596. Bevan, A. and Conolly, J. 2006. Multiscalar approaches to settlement pattern analysis, in Lock, G. and Molyneaux, B. (eds.) Confronting Scale in Archaeology: Issues of Theory and Practice; 217-­‐234, New York: Springer. Bevan, A. 2010. Political geography and palatial Crete, Journal of Mediterranean Archaeology 23.1: 27-­‐54. Bevan, A. 2011. Computational models for understanding movement and territory, in Mayoral Herrera, V. and S. Celestino Pérez (eds.), Tecnologías de Información Geográfica y Análisis Arqueológico del Territorio: 383-­‐394. Mérida: Anejos de Archivo Español de Arqueología Braudel, F. 1972. 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Alonso's theory of movements: developments in spatial interaction modeling, Journal of Geographical Systems 3: 233-­‐256. 20 Driessen, J. 2001. History and Hierarchy: Preliminary Observations on the Settlement Pattern of Minoan Crete, in Branigan, K. (ed.) Urbanism in the Bronze Age: 51-­‐71. Sheffield: Sheffield Academic Press. Eppstein, D. 1998. Finding the k shortest paths, SIAM Journal of Computing 28.2: 652-­‐673. Evans, A.J. 1928. Palace of Minos at Knossos (Volume 2), London: MacMillan. Evans, T., Rivers, R. and C. Knappett 2012. Interactions in space for archaeological models,’ Advances in Complex Systems 15: 1150009. Fish, S.K. and Kowalewski, S.A. (eds.) 1990. The Archaeology of Regions. A Case for Full-­Coverage Survey, Washington: Smithsonian Institution Press. Fontenari, S., S. Franceschetti, D. Sorrentino, F. Mussi, M. Pasolli, M. Napolitano, and R. Flor (2005). r.walk. GRASS GIS. Fujita, M., Krugman, P. and Venables, A. J. 1999. The Spatial Economy. Cities, Regions, and International Trade, Cambridge, MA: The MIT Press. Gelfand, A.E., Diggle, P.J., Fuentes, M. and Guttorp, P. 2010. Handbook of Spatial Statistics, London: CRC/Taylor and Francis Graham, S. and J. Steiner 2006. Travellersim: Growing Settlement Structures and Territories with Agent-­‐Based Modelling, in Clark, J.T. and E.M. Hagemeister (eds) Digital Discovery: Exploring New Frontiers in Human Heritage. Computer Applications and Quantitative Methods in Archaeology. Proceedings of the 34th Conference, Fargo, United States, April 2006. Budapest: Archaeolingua. Groube, L. 1981. Black holes in British prehistory: the analysis of settlement distributions, in Hodder, I., Isaac, G. and Hammond, N. (eds.) Pattern of the Past: Studies in Honour of David Clarke: 185-­‐209. Cambridge: Cambridge University Press. Harris, B. and Wilson, A.G., 1978. Equilibrium values and dynamics of attractiveness terms in production-­‐constrained spatial-­‐interaction models. Environment and Planning, A 10: 371-­‐88. Helbing, D.J. Keltsch, and P. Molnár 1997. Modelling the evolution of human trail systems, Nature 388: 47–50. Hodder, I. and Orton, C. 1976. Spatial Analysis in Archaeology, Cambridge: Cambridge University Press. Hodder, I. 1977. Spatial studies in archaeology, Progress in Human Geography 1: 33-­‐64. Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. 2008. Statistical Analysis and Modelling of Spatial Point Patterns, New York: Wiley. 21 Knappett, C., Evans, T. and Rivers, R. 2008. Modelling maritime interaction in the Aegean Bronze Age, Antiquity 82: 1009-­‐1024. Knappett, C., Rivers, R. and Evans, T. 2011. The Theran eruption and Minoan palatial collapse: new interpretations gained from modelling the maritime network, Antiquity 85: 1008-­‐1023. Marchetti, C. 1994. Anthropological invariants in travel behaviour, Technological Forecasting and Social Change 47: 75-­‐88. McRae, B.H., Dickson, B.R., Keitt, T.H. and Shah, V.B. 2008. 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2724. Mehrer, M. W. and Wescott, K. (eds.) 2006. GIS and Archaeological Predictive Modeling, London: CRC/Taylor and Francis Menze, B.H. and Ur, J.A. 2012. Mapping patterns of long-­‐term settlement in Northern Mesopotamia at a large scale, Proceedings of the National Academy of Sciences, USA. Nystuen, J.D. and Dacey, M.F. 1961. A graph theory interpretation of nodal regions, Papers and Proceedings of the Regional Science Association 7: 29-­‐42. Orton, C. 2004. Point pattern analysis revisited, Archeologia e Calcolatori 15: 299-­‐315. O'Sullivan, D. and D. Unwin 2003. Geographic Information Analysis, Hoboken: Wiley. Peterson, C.E. and R.D. Drennan 2005. Communities, settlements, sites, and surveys: regional-­‐scale analysis of prehistoric human interaction. American Antiquity 70: 5-­‐30. Perlman, P. 1992 One hundred-­‐citied Crete and the “Cretan Politeia”, Classical Philology 87.3: 193-­‐205. Preston, L. 2004. A mortuary perspective on political changes in Late Minoan II-­‐
IIIB Crete, American Journal of Archaeology 108.3: 321-­‐348. Renshaw, E. and Särkkä, A. 2001. Gibbs point processes for studying the development of spatial-­‐temporal stochastic processes, Computational Statistics and Data Analysis 36: 85-­‐105. Rihll, T.E. and A.G. Wilson 1987a. Spatial interaction and structural models in historical analysis: some possibilities and an example, Histoire et Mesure 2.1: 5-­‐
32. 22 Rihll, T.E. and Wilson, A.G. 1987b. Model based approaches to the analysis of regional settlement structures: the case of ancient Greece, in Denley, P. and Hopkin, D. (eds.) History and Computing: 10-­‐20. Manchester: Manchester University Press. Rihll, T.E. and A.G. Wilson 1991. Modelling settlement structures in ancient Greece: new approaches to the polis,’ in Rich, J. and A. Wallace-­‐Hadrill, A. (eds.) City and Country in the Ancient World: 59-­‐95. London: Routledge. Robert, C.P. and Casella, G., 2004. Monte Carlo Statistical Methods, New York: Springer. Saerens, M., Achbany, Y., Fouss, F. and Yen, L. 2009. Randomized shortest-­‐path problems: two related models, Neural Computation 21: 2363-­‐2404. Taaffe, E. J. and Morrill, R. L. and Gould, P. R. 1963. Transport expansion in underdeveloped countries: A comparative analysis, Geographical Review 53.4: 503-­‐529. Tero, A., Takagi, S., Saigusa, T., Ito, K., Bebber, D.P., Fricker, M.D., Yumiki, K., Kobayashi, R. and Nakagaki, T. 2010. Rules for biologically inspired adaptive network design, Science 327: 439-­‐442. van der Leeuw, S.E. 2004. Why Model? Cybernetics and Systems 35.2: 117-­‐128. Verhagen, P. and Whitley, T.G. 2011. Integrating archaeological theory and predictive modeling: a live report from the scene, Journal of Archaeological Method and Theory 19.1: 49-­‐100. Whitelaw, T. 2001. From sites to communities: defining the human dimensions of Minoan urbanism in Branigan, K. (ed.) Urbanism in the Bronze Age: 15-­‐37. Sheffield: Sheffield Academic Press. Wilson, A.G., 1967. A statistical theory of spatial distribution models, Transportation Research 1: 253-­‐69. Wilson, A.G., 1970, 2011. Entropy in urban and regional modelling. London: Pion (re-­‐issued by Routledge, London, 2011). Wilson, A.G. 2000. Complex Spatial Systems. The Modelling Foundations of Urban and Regional Analysis, Harlow: Prentice-­‐Hall/Pearson Education. Wilson, A.G. 2008. Boltzmann, Lotka and Volterra and spatial structural evolution: an integrated methodology for some dynamical systems, Journal of the Royal Society. Interface 5: 865-­‐871. Wilson, A.G., 2010. Entropy in urban and regional modelling: retrospect and prospect. Geographical Analysis, 42, pp. 364-­‐394. Wilson, A.G., 2012. The Science of Cities and Regions: Lectures on Mathematical Model Design. Heidelberg: Springer. 23 Wilson, A.G. 2012. Geographical modeling for archaeology and history: two case studies, Advances in Complex Systems 15: 1150008. Zahavi, Y. 1979 The UMOT, Project, Technical Report for the U.S. Department of Transportation and the German Ministry of Transport (August 1979), Washington and Bonn. URL: http://www.surveyarchive.org/Zahavi/UMOT_79.pdf 24 Appendix 1. Spatial interaction and dynamics It was demonstrated in Wilson (1967) that spatial interaction flows could be accurately modelled using entropy-­‐maximising methods, and these methods have been extensively tested since (see Wilson, 2010, for a review). While, in the 'retail', singly-­‐constrained version of these models, the flows were predicted given known retail centre sizes, Harris and Wilson (1978) also showed how this distribution of sizes could in fact be modelled as endogenous variables. It was later observed that the dynamic equations were part of the Lotka-­‐Volterra family, originally developed in ecology in the 1930s. The core spatial interaction model can be represented as: Tij =
OiW jα e
∑ W αe
k
€
− βc ij
k
− βc ik
(1) where Tij is a matrix recording the quantity of resources -­‐ which could be goods, services, commuters or migrants, or some composite representation of these -­‐ flowing from each site i to each other site j; Oi is a measure of the size of flow originated at site i; Wj is the attractiveness or influence of site j (which for illustrative purposes can be measured by size); cij is some expression of the distance from i to j; α is a parameter used to represent the importance of attractiveness of a site -­‐ if greater than 1, representing positive returns to scale; and β is a parameter used to represent the decay of effective communication with increasing distance. In the first instance, the {Wj} terms are given exogenously and the model is used to calculate (in the retail case, total revenues, in our case, 'flows' between settlements). From the {Tij} matrix, the vector {Dj} can be calculated from Dj = ∑iTij (2) In this case, we interpret Dj as influencing the dynamics of settlement size via the an additional hypothesis (Harris and Wilson 1978): ΔWj = ε(Dj – KWj) (3) K can be interpreted as a parameter that converts 'size' to 'the sum of flows' and equation (3) is thereby asserting that if the actual sum of flows exceeds this starting point, then the settlement will grow in size; and vice versa. The speed of response is controlled by the ε parameter. This difference equation could be made more complex in a variety of interesting ways (see Wilson 2008). The model is run iteratively until it converges at a point where the sets of Wj and Dj values no longer change significantly.2 The simplest use of the set-­‐up proposed 2 For simplicity in the examples below, we assume exact parity between the units of flow and those of importance by setting K to 1. We also set ε to 0.01 throughout to ensure the dynamics of settlement growth or decline are comparatively slow in developing over different model iterations. There are certainly opportunities however to explore the degree to which faster or 25 by equation (3) for understanding settlement is to reproduce a most likely distribution of settlement sizes, given the observed spatial settlement configuration at one point in time. Such a model is attractive for many archaeological and historical situations, because the only necessary inputs are: (a) a set of site locations, (b) some measure of the distances between sites, and perhaps (c) a rough idea of conceivable outcomes that might suggest a suitable set of α and β parameter values to explore (Rihll and Wilson 1991). In other words, iterative solution of the equations above can be understood as a way of arriving at an equilibrium distribution of settlement sizes at a particular point in time. We note later however that if we either (a) can define timelines of exogenous variables (such as changes in scale economies, transport technologies or extra-­‐systemic flows, see below) and repeat the above solution for each step-­‐
change in these variables or (b) make the, perhaps more contentious, assumption that equation (3) represents raw growth and decay dynamics rather than simply a search method for an equilibrium state, then the model results can also be used to represent the evolution of the settlement structure over time. slower dynamics produce different results: our own initial impressions are that model results are reasonably robust across a range of ε, but that as ε → 1 or larger, the dynamics become far less liable to find equilibrium states. The exact point in the model at which changes in settlement size are considered insignificant and the model converges is an arbitrary stopping threshold which here has been set to a <0.1% aggregate change in the size values enduring for 100 consecutive iterations. 26 Appendix 2. A model of network evolution To make the model of network evolution explicit, we introduce some further notation. Let j, u and v be labels for the settlement centroids such that (u, v, h) may be a link on the hth shortest path from i to j. (u, v, h) ε Rijh is the set of links that make up the hth shortest route from i to j. Let ρuv be the ‘length’ of link (u, v). Let cijh be the generalised cost of travel from i to j on the hth shortest path, and let {γuv} be the set of link travel costs on link (u, v). So: h
c ijh = ∑ u.v.h ∈R h γ u,v
(
)
(4) ij
€
We then show how to load flows onto this network. If, as in Appendix 1, Oi is the total number of flows originating in zone i, Wj is the attractiveness of zone j, and Tij is the flow between i and j, then a standard spatial interaction model is, for some composite cij, Tij = AiOiW ka e
− βc ij
(5) Ai = 1/ ∑k W ja e − βc ik with €
(6) €
The composite impedance will be some kind of average over the k shortest paths. For example, e
€
− βc ij
= X ∑e
h
(7) for some normalising factor, X. (see Wilson 1970: chapter 2) and here we have simply taken a weighted mean cost across all paths from i to j, where the weights are proportional to the relative costs of the individual paths. However, these Tij will be divided across alternative routes, say according to Tijh = Tij e
€
− βc ijh
− µc ijh
/ ∑me
− µc ijm
(9) We now need to know the flows on each link. Let quv be the flow on link (u, v) and let Quvh be the set of origin-­‐destination pairs that use the (u, v) link on the hth best route. Then quv = ∑i, j∈Q Tijh (10) uvh
€
Adopting this approach not only allows us to consider the more likely possibility that multiple paths are in use between neighbouring settlements rather than just one, but also to consider the routes themselves as a dynamic system. For 27 example, we can adjust the travel cost of each path segment according to some function of its base cost and overall flow. Suppose, for simplicity of illustration, that a link cost is initially proportional to link length, γuv = ρuv. Suppose then that as more people travel along a link, certain path segments become more widely known and/or better maintained. Hence, as the trail becomes established, the perceived cost reduces, as γuv = f(quv)ρuv (11) where f(quv) decreases from an initial value of 1 when the model is initialised, say to some ‘capped’ number. One function we might adopt is f(x) = (1 -­‐ a) e-­‐bx + a (12) where a defines an asymptotic lower bound on the reduction in perceived cost, and b controls the shape of the decay curve (of course in some modelling situations, exceptionally high flow might discourage rather than encourage further travel, but we do not address this case here). This set of equations can then be integrated into the spatial interaction model outlined above such that it is not simply a hierarchy of settlements that emerges from the model of section 3.1 but also a hierarchy of paths between them, and where we can potentially consider not just how the result at a perceived equilibrium point, but also as it changes over all model iterations. This gives one possible sense of network evolution. Another, as noted earlier, is to consider a sequence of equilibria over time, with each step driven by changes in exogenous variables. In order to address edge effects from interactions beyond Crete in a simple way (see section 3.3), we calculate an adjusted inflow ( p) for those settlements next to the coast (in this case, all points <1km from the coastline) such that: D p = GDp + Dp
€
(13)
where Dp is the unadjusted inflow to a port from all other settlements and G is a parameter reflecting our sense of the relative importance of overseas flows relative to internal ones during the period in question. 28