In my madness series of posts published a few months ago I mentioned I was looking for a method to study processes of emerging intervisibilty patterns. I can finally reveal this fancy new approach to you Here it is: introducing exponential random graph modelling (ERGM) for visibility networks. In previous posts I showed that when archaeologists formulate assumptions about how lines of sight affected past human behaviour, these assumptions imply a sequence of events rather than a static state. Therefore, a method is needed that allows one to test these assumed processes. Just analysing the structure of static visibility networks is not enough, we need a method that can tackle changing networks. ERGM does the trick! I just published a paper in Journal of Archaeological Science with Simon Keay and Graeme Earl that sets out the archaeological use of the method in detail. You can download the full paper on ScienceDirect, my Academia page or via my bibliography page. But in this blog post I prefer to explain the method with LEGOs
Social network analysts often use an archaeological analogy to explain the concept of an ERGM (e.g., Lusher and Robins 2013, p. 18). Past material remains are like static snapshots of dynamic processes in the past. Archaeologists explore the structure of these material residues to understand past dynamic processes. Such snapshots made up of archaeological traces are like static fragmentary cross-sections of a social process taken at a given moment. If one were to observe multiple cross-sections in sequence, changes in the structure of these fragmentary snapshots would become clear. This is exactly what an ERGM aims to do: to explore hypothetical processes that could give rise to observed network structure through the dynamic emergence of small network fragments or subnetworks (called configurations). These configurations can be considered the building blocks of networks; indeed, LEGO blocks offer a good analogy for explaining ERGMs. To give an example, a network’s topology can be compared to a LEGO castle boxed set, where a list of particular building blocks can be used to re-assemble a castle. But a LEGO castle boxed set does not assemble itself through a random process. Instead, a step by step guide needs to be followed, detailing how each block should be placed on top of the other in what order. By doing this we make certain assumptions about building blocks and their relationship to each other. We assume that in order to achieve structural integrity in our LEGO castle, a certain configuration of blocks needs to appear, and in order to make it look like a castle other configurations will preferentially appear creating ramparts, turrets, etc. ERGMs are similar: they are models that represent our assumptions of how certain network configurations affect each other, of how the presence of some ties will bring about the creation or the demise of others. This is where the real strength of ERGMs lies: the formulation and testing of assumptions about what a connection between a pair of nodes means and how it affects the evolution of the network, explicitly addressing the dynamic nature of our archaeological assumptions.
More formally, exponential random graph models are a family of statistical models originally developed for social networks (Anderson et al. 1999; Wasserman and Pattison 1996) that aim to scrutinize the dependence assumptions underpinning hypotheses of network formation by comparing the frequency of particular configurations in observed networks with their frequency in stochastic models.
The figure below is a simplified representation of the creation process of an ERGM. (1a) an empirically observed network is considered; (1b) in a simulation we assume that every arc between every pair of nodes can be either present or absent; (2) dependence assumptions are formulated about how ties emerge relative to each other (e.g. the importance of inter-visibility for communication); (3) configurations or network building blocks are selected that best represent the dependence assumptions (e.g. reciprocity and 2-path); (4) different types of models are created (e.g. a model without dependence assumptions (Bernoulli random graph model) and one with the previously selected configurations) and the frequency of all configurations in the graphs simulated by these models is determined; (5) the number of configurations in the graphs simulated by the models are compared with those in the observed network and interpreted.
My madness series of posts and the recently published paper introduce a case study that illustrates this method. Iron Age sites in southern Spain are often located on hilltops, terraces or at the edges of plateaux, and at some of these sites there is evidence of defensive architecture. These combinations of features may indicate that settlement locations were purposefully selected for their defendable nature and the ability to visually control the surrounding landscape, or even for their inter-visibility with other urban settlements. Yet to state that these patterns might have been intentionally created, implies a sequential creation of lines of sight aimed at allowing for inter-visibility and visual control. An ERGM was created that simulates these hypotheses. The results suggest that the intentional establishment of a signalling network is unlikely, but that the purposeful creation of visually controlling settlements is better supported.
A more elaborate archaeological discussion of this case study will be published very soon in Journal of Archaeological Method and Theory, so stay tuned Don’t hesitate to try out ERGMs for your own hypotheses, and get in touch if you are interested in this. I am really curious to see other archaeological applications of this method.
Anderson, C. J., Wasserman, S., & Crouch, B. (1999). A p* primer: logit models for social networks. Social Networks, 21(1), 37–66. doi:10.1016/S0378-8733(98)00012-4
Lusher, D., Koskinen, J., & Robins, G. (2013). Exponential Random Graph Models for Social Networks. Cambridge: Cambridge university press.
Lusher, D., & Robins, G. (2013). Formation of social network structure. In D. Lusher, J. Koskinen, & G. Robins (Eds.), Exponential Random Graph Models for Social Networks (pp. 16–28). Cambridge: Cambridge University Press.
Wasserman, S., & Pattison, P. (1996). Logit models and logistic regressions for social networks: I. An introduction to Markov graphs and p*. Psychometrika, 61(3), 401–425.