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This is a companion discussion topic for the original entry at https://skewed.de/lab/posts/higher-standards
This is a companion discussion topic for the original entry at https://skewed.de/lab/posts/higher-standards
Very cool reading and certainly needed. Thank you! Still, I’m worried that it may be conflating distinct aspects of network representations, e.g. model inference from model analysis. In the case of the mentioned Boolean networks, they certainly encode higher-order interactions, but not in their interaction graph. Rather in the automata logic which is not represented as a graph. Indeed, as Stefan Klamt and others have shown, that logic requires additional representations, e.g. for inhibition vs excitation, but more importantly for (nonlinear) logic, see, eg. doi:10.1371/journal.pcbi.1000385. In our work we have to use threshold networks, which are not graphs, to represent BN logic (doi:10.1371/journal.pone.0055946). In a sense, this debate takes me back to the Papert Minsky perceptron theorem. A single layer perceptron cannot represent an XOR, but with additional layers you can. Finally (space is short here) I believe Klir’s conceptualization of complex systems with relation theory is a better formalism to work this out. A graph is the simplest relation of a set of variables A (A^2), but many other types of relations can be defined on the cross product of variable sets A x B x C x ….. you can project a higher order relation to set A, but that does not mean that A^2 contains all the information that is in a higher order relation, right?
Dear Luis,
Thank you for the comment. Some answers below.
It is precisely our core argument that graphs encode very little about actual interactions: the mere existence of an edge only tells us that these two nodes interact, but not how they interact. This is true not only for Boolean networks, but for every other network model conceivable. It is always true that we need more information than the graph, typically in the form of functions that map the set of inputs of a node to its output.
More formally, a network model is defined by a graph G plus a set of functions f_i, conditioned on G, that map the values of the neighborhood of a node i to its output.
Your example using threshold logic is very good, and one we evoke in the paper: a network model with threshold functions is manifestly not modelling “pairwise” interactions, nor they have anything to do with hypergraphs.
I believe this point is moot given the issue you identified above, and which forms the basis of our argument: the graph does not completely define the “relations” (i.e. interactions) between entities, only their support; we always need functions on top.
What we also demonstrate in the paper is that the (G, f) setup above is maximally general, i.e. there’s no dynamical system representable by a hypergraph model (i.e. a hypergraph H plus some functions g_i) that cannot be represented by a projected graph with a suitably defined set of functions. This has to do with what you identify: since higher-order tensors encode more information, this information can only further constrain the interaction functions, to the extent it has any relevance to them. But since graphs contain the minimal amount of information—adjacencies and nothing more—they leave maximal freedom to the functions, and hence achieve maximal generality overall.
Besides this central point, even if we discard the set of interaction functions (and thus throw away the context that allows us to interpret what the edges actually represent) the combinatorial argument gets complicated once you realize that you can annotate the edges and nodes with arbitrary values. It turns out that every hypergraph can be recovered from a suitably annotated multilayer network, but the opposite is not true.
Not to mention, of course, that every hypergraph admits an equivalent bipartite graph representation.
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Thank you Tiago. This clarifies a lot. Obviously I need to read the paper 
So, the modeling “magic” happens with the functions one has to define in addition to variable interaction/association graphs. Would it be fair to say that such a parsing arguably kicks the pairwise/higher-order “can” to those functions? I get it, and deeply sympathize with the idea that the field needs to move on from the preaching of the advantages of hypergraphs. But from the viewpoint of estimation from data and downstream analysis of specific models, ultimately one needs to consider whether to allow only pairwise or higher order occurrences, be them conceptualized at the interaction graph and/or at the functions. If I’m optimizing airline operations between airports, pairwise interactions between airports are sufficient (flights are A to B). But if I’m studying gene regulation, in addition to an interaction graph, I need to define additional functions (e.g. the automata rules and ensuing dynamics in Boolean networks) to capture n-ary interactions that I know to occur in this phenomenon. So, it is useful to differentiate phenomena by the type of interactions they require for modeling. But I completely agree that it is a disservice to argue that these modeling considerations are somehow tied to a specific representation that is best™️ :).
BTW, only commenting because, as usual, your post is extremely interesting. Now I need to read the paper in detail. Thank you.
The HON literature hijacks the terminology in a rather nefarious way.
The distinctions between “pairwise” vs. “multivariate” interactions and “graph” vs. “hypergraph” representations are entirely orthogonal. The HON literature conflates these two axes in a way that is, at best, careless, and at worst, dishonest.
The examples you give concern exclusively the difference between pairwise and multivariate interactions, and have nothing to do with the distinction between graphs and hypergraphs. Yet the HON literature repeatedly insists that multivariate interactions require hypergraph representations, which is nonsense.
Moreover, it is often claimed that combining multivariate interactions with hypergraphs yields phenomena—such as abrupt transitions—that are supposedly “invisible” to graph-based models. However, their own mean-field calculations prove that the hypergraph structure itself plays no role at all in producing these behaviors.
The HON literature gets worse the more you read it, to the point of being, in my view, quite dishonest. I think it has moved the field backward, not forward.
As we discuss in the paper, it is certainly possible that for certain classes of interactions a hypergraph parametrization may offer genuine advantages. However, if such benefits exist, the HON literature has largely obscured them beneath a substantial amount of overstatement and confusion.
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