Graphs are maximally expressive for higher-order interactions

TL;DR

It is argued that clearly distinguishing between multivariate interactions, parametrized by graphs, and the functions that define them enables a more unified and flexible foundation for modeling interacting systems.

Abstract

We demonstrate that graph-based models are fully capable of representing higher-order interactions, and have a long history of being used for precisely this purpose. This stands in contrast to a common claim in the recent literature on "higher-order networks" that graph-based representations are fundamentally limited to "pairwise" interactions, requiring hypergraph formulations to capture richer dependencies. We clarify this issue by emphasizing two frequently overlooked facts. First, graph-based models are not restricted to pairwise interactions, as they naturally accommodate interactions that depend simultaneously on multiple adjacent nodes. Second, hypergraph formulations are strict special cases of more general graph-based representations, as they impose additional constraints on the allowable interactions between adjacent elements rather than expanding the space of possibilities. We show that key phenomenology commonly attributed to hypergraphs -- such as abrupt transitions -- can, in general, be recovered exactly using graph models, even locally tree-like ones, and thus do not constitute a class of phenomena that is inherently contingent on hypergraphs models. Finally, we argue that the broad relevance of hypergraphs for applications that is sometimes claimed in the literature is not supported by evidence. Instead it is likely grounded in misconceptions that network models cannot accommodate multibody interactions or that certain phenomena can only be captured with hypergraphs. We argue that clearly distinguishing between multivariate interactions, parametrized by graphs, and the functions that define them enables a more unified and flexible foundation for modeling interacting systems.

Figures (5)

• Figure 1: Graphs encode adjacencies (or neighborhoods), which define the domain of interactions, not the interactions themselves. The adjacency set constrains which variables can influence a node, but only when functions are defined on these adjacencies are the interactions specified. Since these functions are multivariate in general, they do not need to decompose into pairwise terms. This diagram shows a possible instance of the general proof-of-concept ODE system of Eq. \ref{['eq:netdyn']}, including the equations governing the dynamics of the nodes encircled. The adjacent nodes in blue (together with the red nodes) define the domain of each function.
• Figure 2: (a) Graph-based models are maximally general with respect to a particular set of adjacencies. (b) Hypergraph formulations include further constraints, amounting to specific choices of the node functions, whose structure honors the grouping of adjacent nodes in particular subsets that, if combined, form hyperedges (i.e. overlapping cliques), as identified by shaded regions in (b). A more concrete example is given in Fig. \ref{['fig:graph-comp']}.
• Figure 3: Hypergraph parametrizations are special cases of graph-based models, and thus offer no generalization. The node adjacencies (top) show the skeleton of each model, and the coupling functions (bottom) define the interactions. A hypergraph requires mutual, symmetric membership: if nodes $\{i,j,k\}$ form a hyperedge, a single shared coupling function of all three must appear in the equation of every node in that set. Panel (a) satisfies this constraint---the shaded regions mark groups of nodes that always appear together in function arguments, forming consistent hyperedges (Eq. \ref{['eq:group-wise']}). This can also be represented by the graph-based model of Eq. \ref{['eq:graph-redux']}. Panel (b) uses the same form (Eq. \ref{['eq:graph-redux']}), but no hypergraph can represent it, since changing the adjacencies and/or the couplings can break the mutual membership constraint. For example, $h_5^{(1)}$ groups $\{1,3,5,6\}$, but node 1 depends only on $\{1,4,5\}$ via $h_1$, node 3 on $\{3,5,6\}$ via $h_3$, and node 6 on $\{3,5,6,12\}$ via $h_6$---none share the same coupling function as node 5, so no hyperedge $\{1,3,5,6\}$ can exist. The colored edge endpoints in (b) indicate which coupling function each edge belongs to, for nodes with more than one. The existence of multivariate coupling functions is completely independent from any hypergraph structure. Therefore, simultaneously labeling the model of panel (a) "higher-order" and the one of panel (b) "pairwise" or "dyadic" would be arbitrary.
• Figure 4: Hypergraphs define groups of nodes that are not uniquely recovered from their graph projections due to the possible existence of cliques that are induced by the presence of other cliques. Panel (a) shows an example where the existence of hyperedges $(1,2,4)$, $(2,3,5)$, and $(4, 5, 6)$ means that the projected graph will also contain clique $(2,4,5)$, even though that is not included in the hypergraph. However, the bijection is fully restored via multilayer graphs, as shown in panels (b) and (c), where the requirement is that cliques must contain all edges with the same color.
• Figure 5: Abrupt transitions in homogeneous hypergraph models are identical to locally tree-like graph models with the same interaction functions, and belong to the same phenomenology class of graph-based abrupt transitions. The panels show the saddle point bifurcations (top) and corresponding abrupt transitions of the order parameter (bottom) for (a) bootstrap ($k$-core) percolation for $k=3$ on a Erdős--Rényi (ER) model with mean degree $z=5$, (b) interdependent percolation for an ER model and various mean degree values, $z$, as indicated in the legend, (c) higher-order synchronization with coupling parameters given by the legend, which corresponds to an ER model with mean degree $z$, such that $K_l \to K_l (N/z)^l$ in Eq. \ref{['eq:kuramoto']}, and (d) simplicial contagion, corresponding to an ER model with mean degree $z$, such that $\beta_k\to \beta_k(N-1)/[z {N-1\choose k-1}]$ in Eq. \ref{['eq:graph_inf']}, for $k\in\{1,2\}$, and $\mu=1$. The dashed lines in the bottom figures show the unstable fixed points.