Can I do cluster analysis of dyadic data? I have multilevel data that is dyadic in the unit of observation. The dyad is a unique pair of countries that sign a treaty, such that no dyad repeats itself. For example, the US-UK treaty, the US-Canada treaty, the US-Mexico treaty, and so forth (Mexico-Canada, Mexico-Panama, Mexico-Belgium and so forth are also unique dyads). 
Each treaty is coded on an interval variable Y that is the dependent variable of interest. The regression model has some variables that are specific to the dyad (e.g. difference in economic power), some variables are specific to only individual members of the dyad (e.g. level of democracy in a country).
Existing studies of this dataset pool all the observations together and run regression analyses. However, I am interested in whether there are significant differences in the mean of Y across countries. Thus, I would like to analyze differences across clusters, where the cluster is the country. 
My question is, given that dyadic nature of the observations, can I conduct cluster analysis for individual countries?
 A: Cluster analysis is a method of unsupervised learning - that is, you don't know the clusters in advance.  Here, you do know them, so cluster analysis is inappropriate. 
However, I don't think regular regression analysis is appropriate here because your independent variable is dyadic. Just pooling (somehow) across country seems wrong to me - and I'm not sure how it would work, anyway.
Instead, I'd suggest looking at social network methods.  David Kenny has written extensively about dyadic analysis, see e.g. this page from Columbia University's Mailman school. If that link rots, Googling "dyadic data analysis" finds a lot. Or see Dyadic Data Analysis by Kenny, Kashy and Cook, published by Guilford in 2006. I remember reading this and finding it useful, but I have forgotten the details. 
A: You don't need cluster analysis to aggregate per country...
Just select all data for that country, and average!
Your data is an edge-weighted graph, with the nodes being the countries. You can likely find some clustering algorithms to run on this, but I don't expect it to yield much insights. They will mostly ignore the edge weights, which is usually a distance, but your data isn't.
