Method update: beta-skeletons

This second update of the project’s method concerns the distance networks based on beta-skeletons described in an earlier blog post. We mentioned that the reconstruction of ancient trade routes is extremely complex as a number of variables should be taken into account, so our best bet is to focus on one parameter that might have been influential in determining trade routes. Using beta-skeletons and graph theory we will investigate whether the distance between centre of production and site of deposition is reflected in the ceramic evidence and whether it significantly influenced the selection of trade routes.

Although we mentioned in a previous post that the beta-skeleton would be compared with a reconstruction of trade routes based on the shortest path for every sherd from centre of production to site of deposition over this beta-skeleton, we now have to confess that this is nonsense as we would compare the beta-skeleton with a slightly altered version of itself that is based on a large number of assumptions concerning the intermediary sites. We realized that these shortest paths actually contain the hypothesis that we are testing, as they represent trade routes based on the ceramic evidence in which distance surpasses all other factors in importance.

To create such a network of trade routes we will make a beta-skeleton in which every site has at least one connection, so that all of them would be reachable. This will be done in ArcGIS with a beta-skeleton calculator programmed by dr. Graeme Earl, applied to all the sites in the database and their geographical coordinates. For every sherd the shortest path in geographical distance from centre of production to centre of deposition over this beta-skeleton will be calculated in pajek (although this can be done in ArcGIS, pajek is able to calculate geographical as well as graph theoretical shortest paths). Edge value will represent the number of sherds passing between two sites and edges with a value of zero will be discarded.

At this point we have a reconstruction of the trade routes over which the vessels would have been transported if the distance between start and ending point would have been the only factor taken into consideration by their transporters. This network embodies the hypothesis we want to test, which can be done by comparing it to another network visualisation of ceramic evidence. The networks of co-presence described in the previous post will provide this basis for comparison, as they do not contain any assumptions of their own (before their analysis that is).

Now, there is an obvious danger of comparing things with different meanings, so we need to be very clear of what aspects of both networks will be used for comparison. We will focus on a couple of phenomena that we think are represented in both types of networks: bridges and centrality.

A bridge is a line whose removal increases the number of components in the network (de Nooy et.al. 2005: 140). In our networks of co-presence a bridge is a site that forms the connection between two different groups of distribution networks. Such a site should play an important role in dispersing information on the pottery market as it is linked in with highly differing networks, but does not necessarily play a central role in the entire network. On the distance network these sites should play a similar role in connecting different distribution networks, in order for the hypothesis to be valid.

Sites belonging to the centre of a pottery distribution network can be easily reached by new pottery forms from diverse producing centres, they are central to the communications network of the pottery trade as it is represented in the ceramic evidence. This is true for both our shortest path network and our co-presence network, and can be measured using the closeness centrality method: sites are central in distribution networks if their graph theoretical distance to all other sites is minimal. In network terms: the closeness centrality of a vertex is the number of other vertices divided by the sum of all distances between the vertex and all others (de Nooy et.al. 2005: 127). Although this method will provide comparable numerical results (a score between 1 and 0), we will not compare these absolute values. Rather, we will focus on seeing whether sites that are central (or not) in our co-presence network are also central (or not) in our shortest path network.

Pairs of contemporary networks of both types will be compared using these methods in order to provide an answer to our hypothesis “was distance a significant factor in selecting trade routes?”

Relative Neighbourhood graphs and Beta-skeletons

Although our preliminary method indicates that a reconstruction of pottery trade flows involves a lot of complications, we cannot seem to let this research topic go. One reason for this is that most archaeological attempts to study the ancient economy make interpretations about trade routes based on ceramic evidence (e.g. Abadie-Reynal 1989 ; Fulford 1989), yet none have ever attempted a networks approach. In this post we will discuss a geographical network in which distance is a significant parameter, an assumption that is not without its complications.

We believe that relative neighbourhood graphs (RNG) and Beta-skeletons might prove to be useful tools for constructing distance-based networks. Unlike other types of cluster analysis (e.g. nearest neighbour) these methods take the position of all points in account. Jiménez and Chapman (2002); discussed the archaeological application of RNG, and summarize its construction as a graph in which “the link between two points is determined by taking into account not only the proximity between the two points, but also the relative distance of each pair to the remaining points ». Lines are drawn between two neighboring points that have no other points in a region around them. By varying the size (beta) of the region of influence for each pair of points, graphs (called Beta-skeletons) can be created with different levels of connectivity: if the region is small, more relationships will be drawn between the points; if the region is large, the network will start to fall apart in smaller networks (see Fig. 1).

Fig. 1 Beta-skeletons with varying regions of influence, indicating that for a higher value of beta the network will start to fall apart. Taken from Jiménez & Chapman 2002.

Of particular interest for our study is a Beta-skeleton of sites in the Eastern Mediterranean at the stage just before it starts to fall apart, so without any unconnected sub-networks (similar to the network for ‘Beta=2’ in Fig. 1). This Beta-skeleton can be analysed as a network, which will allow us to define the relative position of every site for the hypothesis “what if straight-line distance were a determining factor in the distribution of table wares?”

Such a network obviously avoids all complications but is invaluable in testing a distance-based hypothesis. For every ware in every period the number of sherds being transported from centre of production to centre of consumption can be plotted on such a Beta-skeleton (only including those sites in which the ceramics in question were found). We can easily compare the relative positions of sites in these transportation networks, as we know the influence of our basic ‘distance’ network.

To test our hypothesis that proximity is an important parameter in the distribution of table wares, we have to analyse our ceramic networks and compare them to our basic networks. If the relative position of sites weighted by the ceramic evidence is similar to sites in a ‘distance network’, we can conclude that distance played an important role in determining trade relations and thus trade routes. If there is a significant difference between ceramic and distance networks, we can conclude that distribution was influenced by other parameters, e.g. personal contacts of traders and land owners. Testing the hypothesis for 15-year periods will allow us to identify periods in which distance was more likely to be a determining factor than others.

Some of the numerous issues with this method should be listed:

• although RNG is a formidable method for cluster analysis, it still does not take into account any of the complexities that determine trade routes. Could this method be combined with a cost-surface analysis to paint a more accurate picture of regional overland trade?

• Will the ceramic evidence influence the distance network to such a degree that its basic connectivity can be altered?

• Using a Beta-skeleton as the basis for testing our hypothesis might lead us to find exactly what we were looking for (distance = significant) because it is inherent in the network. Should the Beta-skeleton be compared with a more neutral network of ceramic distribution through space?