Introduction. Whether a directed network is a hierarchy is not a yes-or-no question. Network science has produced several quantitative measures, of which the most useful for the VSM is the morphospace of Corominas-Murtra, Goñi, Solé and Rodríguez-Caso. They condense a directed graph to its acyclic skeleton and compute three coordinates: treeness (how pyramidal and unambiguous the chain of command is), feedforwardness (how much of the flow avoids cycles) and orderability (what fraction of nodes can be placed in a strict order). A perfect tree sits at one corner of the cube, a fully cyclic network at the opposite corner. Placing ecological, cellular, technological and social networks in the space, they found four clusters separated by voids; two clusters were indistinguishable from random graphs of similar connectivity, and two—gene regulation and ecological networks—occupied regions they attributed to functional constraint. The essential move, common to all of network science, is the null model: a structural property is only evidence of design or selection if it differs from what degree-preserving random rewiring produces. Older measures from organisational sociology (Krackhardt’s connectedness, hierarchy, efficiency and least-upper-bound scores) address the same question with fewer distinctions.
Important authors. Ricard Solé, ICREA research professor at Universitat Pompeu Fabra and external faculty of the Santa Fe Institute, leads the Barcelona Complex Systems Lab from which the morphospace came; Bernat Corominas-Murtra was first author. Tamás Vicsek’s group in Budapest produced an alternative hierarchy measure. David Krackhardt at Carnegie Mellon introduced graph-theoretic hierarchy measures for informal organisations. Sergei Maslov and Kim Sneppen established degree-preserving rewiring as the standard null model.
Importance for cybernetics and the VSM. Beer described the VSM as neither hierarchy nor heterarchy and drew McCulloch’s heterarchy into the model’s ancestry (T6.1). The dispute has been conducted rhetorically for fifty years. It is now a computation: the VSM graph cannot sit at treeness 1 because it contains a reciprocal edge (the Three–Four homeostat), a shortcut (the algedonic channel) and a lateral coupling layer (S1–S1 through S2), so it has a location in the morphospace, and that location can be compared with a degree-matched null. Cybernetics has never subjected its canonical diagrams to this discipline; the VSM community has not either.
Importance for the article. Demonstration IV, claim IVb (§9.1–9.4), is built on this sub-theme: compute the specified graph’s morphospace coordinates and motif profile, compare against degree-preserving random graphs, and record whether the architecture carries information beyond its connectivity. §10.5 places it at stage two of the staged programme, and §9.3 costs it at weeks, laptop-scale, no fieldwork or ethics review—the paper’s model of a cheap discriminator. The import is from Variety and Channels Now §3 (C4); C4 §8 extends it to whether motif profiles are level-dependent, which is the fractality question behind recursion. What a reviewer will press on: the computation requires the formal specification IVa, which does not yet exist; a graph of eleven or so nodes is small for morphospace placement and the null distribution must be built carefully; and the claim that a “functionally constrained” location supports the necessity claim is an inference the morphospace authors themselves state cautiously.
Sources in the reading list.
- the three coordinates, the four clusters, and the null-model comparison that IVb copies.
Other important sources and authors.
- Corominas-Murtra, B., Rodríguez-Caso, C., Goñi, J., & Solé, R. (2011). Measuring the hierarchy of feedforward networks. Chaos, 21(1), 016108 — the earlier technical paper defining the measures.
- Mones, E., Vicsek, L., & Vicsek, T. (2012). Hierarchy measure for complex networks. PLoS ONE, 7(3), e33799 — an alternative global reaching centrality measure; useful as a second instrument.
- Krackhardt, D. (1994). Graph theoretical dimensions of informal organizations. In K. M. Carley & M. J. Prietula (Eds.), Computational Organization Theory (pp. 89–111). Lawrence Erlbaum — the organisational-sociology hierarchy measures; the bridge to management readers.
- Maslov, S., & Sneppen, K. (2002). Specificity and stability in topology of protein networks. Science, 296(5569), 910–913 — the degree-preserving rewiring null model in its standard form.
- Ravasz, E., & Barabási, A.-L. (2003). Hierarchical organization in complex networks. Physical Review E, 67(2), 026112 — hierarchical modularity via clustering-coefficient scaling; a different sense of “hierarchy” a reviewer may conflate with the VSM’s.
- Newman, M. E. J. (2018). Networks (2nd ed.). Oxford University Press — the reference text for graph measures and random-graph null models.