On two mathematical representations for “semantic maps”

We describe two mathematical representations for what have come to be called “semantic maps”, that is, representations of typological universals of linguistic co-expression with the aim of inferring similarity relations between concepts from those universals. The two mathematical representations are a graph structure and Euclidean space, the latter as inferred through multidimensional scaling. Graph structure representations come in two types. In both types, meanings are represented as vertices (nodes) and relations between meanings as edges (links). One representation is a pairwise co-expression graph, which represents all pairwise co-expression relations as edges in the graph; an example is CLICS. The other is a minimally connected co-expression graph – the “classic semantic map”. This represents only the edges necessary to maintain connectivity, that is, the principle that all the meanings expressed by a single form make up a connected subgraph of the whole graph. The Euclidean space represents meanings as points, and relations as Euclidean distance between points, in a specified number of spatial dimensions. We focus on the proper interpretation of both types of representations, algorithms for constructing the representations, measuring the goodness of fit of the representations to the data, and balancing goodness of fit with informativeness of the representation.