Let me pose a puzzle. Mathematical sociologists like puzzles!
Jordi Bascompte recently published an article in Science (vol. 329, 13 August, 2010, p. 765-6) titled “Structure and Dynamics of Ecological Networks.” The piece is in the “Perspectives” section — which are commentaries and explanations of the “real” articles (which are often a bit too discipline-specific to be accessible to the average reader). Basecompte’s article is an explanation and appreciation of the piece by Elisa Thebault and Colin Fontaine (also vol. 329, 13 August 2020 p. 853-6) titled “Stability of Ecological Communities and the Architecture of Mutualistic and Trophic Networks.”
A very common use of graphs and networks in ecology is as a representation of web of relations among species. A “food web” for example, is composed of nodes (species) and directed relations (e.g. “eats”) as in Bear -> Salmon.
The food web is an “antagonistic” relation: one species feeds on the other (occasionally, such relations are reciprocated). The relation results in a “gain” for one node and a “loss” for the other.
Ecologists also focus on “mutualistic” relations where two species interact, but both benefit. Bees and flowers, for example are mutualistic — bees pollinate flowers, flowers provide food for bees. These kinds of networks can be thought of as undirected (though maybe they would better be regarded as multiplex and directed). Generally, mutualistic ecological relations form bi-partite networks.
Bear with me just a moment more…
The interesting results in the Science article are that networks of “antagonistic” relations in ecology tend to have a characteristic form of largely separated and rather densely connected clusters – something akin to “communities”. Mutualistic networks, on the other hand have a characteristic form composed of “generalists” (nodes with high degree) and “specialists” (nodes with low degree) of the species in each mode, in a nested “hierarchy”.
Networks of individual humans, groups, roles, organizations, or populations often have “antagonistic” relations — “A” dominates “B”. But, human networks are also connected by mutually beneficial ties (e.g. emotional support and identity confirmation).
In sociology, we tend to think of “antagonistic” networks as, perhaps, having an equilibrium tendency toward hierarchy. “Mutualistic” human relations, we tend to see as likely to be organized as communities — often largely separated from one another.
Are these patterns of findings really contradictory? They would seem to suggest that structures of inter-species networks follow rather different rules than the structures that emerge in relations among heterogeneous human agents (individuals, groups, organizations, communities, societies).