# The Cost of Representing Higher-Order Dynamics on Graphs: A Theoretical Framework { :label: manuscript } :config: { :theme: default :typography: sans-serif :numbering: section } :: :author: { :name: Leo Torres :email: [email protected] } :: :abstract: Can graphs represent the dynamics of higher-order (multi-body) interactions, or are hypergraphs necessary? We decompose network dynamics into three components\: the combinatorial *structure* (graph or hypergraph), per-edge *coupling* functions, and per-node *aggregation* functions. Within this framework, we prove that graphs can always reproduce the trajectories of any hypergraph dynamics by absorbing the coupling logic into the aggregation. This generalizes a recent result by Peixoto et al., who showed the same under the assumption of flat dynamics. However, the graph representation pays a price in parsimony\: when the coupling and aggregation functions are polynomials of degree at most $D$, the graph aggregation requires $\Theta(k^D / D!)$ parameters relative to the hypergraph alternative, where $k$ is the interaction size. Without symmetry, the comparison can reverse. The framework also clarifies two other recent contributions by Moutuou and by Lacasa, and identifies overlooked modeling degrees of freedom including partial interaction symmetries and the role of non-additive aggregation. :: :toc: ## Introduction { :label: sec-intro } Peixoto et al. :cite:peixoto2026:: argued that graphs are "maximally expressive" for higher-order interactions, showing that any hypergraph dynamics can be equivalently represented on a graph. Moutuou :cite:moutuou2026:: responded that hypergraphs are strictly richer, and Lacasa :cite:lacasa2026:: showed that nonlinear dynamics on graphs can be lifted to linear dynamics on higher-order structures via Carleman linearization. We show that Peixoto et al. are correct, and that their result is more general than their framework reveals. Their proof relies on an assumption we call *flat dynamics*, where the decomposition into coupling and aggregation is bypassed entirely and a node's state depends only on the set of its neighbors. Under flat dynamics, graph sufficiency follows immediately\: if the dynamics cannot see the edge labels, the edge structure plays no role. The SCA framework proposed herein shows that graph sufficiency holds well beyond the flat regime, but it also reveals what the flat framework hides: the graph representation has a cost. When a $k$-body interaction is moved from a hypergraph to a graph, the complexity of the dynamic coupling migrates into the aggregation function, the per-node function that determines how each node combines contributions from its interactions. The SCA framework makes both responses precise. Moutuou :cite:moutuou2026:: shows that when a symmetric multi-body interaction is represented on a graph, the graph representation necessarily introduces asymmetry\: one node per interaction must play a distinguished role. In the SCA framework, this happens because the graph representation must absorb the symmetric coupling into the per-node aggregation, which breaks the shared structure. Lacasa :cite:lacasa2026:: shows that nonlinear dynamics on graphs can be lifted to linear dynamics on a richer combinatorial structure; in our framework, this is a different axis of the same tradeoff, trading coupling complexity for structural complexity. Beyond reconciling these perspectives, the SCA framework identifies modeling degrees of freedom that have been overlooked. The symmetry of the coupling function admits a range of intermediate possibilities. The aggregation function is an independent modeling choice with dynamical consequences that all three papers leave implicit. Common graph properties such as directedness, edge weights, and signs turn out to be constraints on the coupling and aggregation functions, not only on the combinatorial structure itself. The broader context is the growing literature on higher-order networks. Battiston et al. :cite:battiston2020:: and Majhi et al. :cite:majhi2022:: survey the field, cataloguing dynamical phenomena where higher-order interactions produce qualitatively different behavior than pairwise ones. Iacopini et al. :cite:iacopini2019:: showed that simplicial contagion models exhibit discontinuous phase transitions absent in pairwise models, launching a wave of work on higher-order dynamics. Pecora and Carroll :cite:pecora1998:: introduced the master stability function, the original decomposition separating network structure from local dynamics for synchronization; Gambuzza et al. :cite:gambuzza2021:: and Carletti et al. :cite:carletti2020:: extend this approach to simplicial complexes and hypergraphs respectively. Torres et al. :cite:torres2021:: examine when and why different mathematical representations lead to different conclusions about the same system, arguing that the choice of representation is itself a modeling decision. Bick et al. :cite:bick2023:: formalize coupled dynamical systems on hypergraphs and establish rigorous results on synchronization and pattern formation. Lucas et al. :cite:lucas2026:: propose an information-theoretic framework for measuring when higher-order structure can be reduced to pairwise interactions without loss of dynamical information. Our framework provides a language for comparing the representational choices that all of this work makes, often implicitly. The decomposition into structure, coupling, and aggregation has precedents in other communities. In dynamical systems, Golubitsky and Stewart's coupled cell networks :cite:golubitsky2023:: separate network structure from admissible dynamics, though they do not isolate aggregation as an independent component. In machine learning, message passing neural networks :cite:gilmer2017:: and graph network blocks :cite:battaglia2018:: explicitly decompose graph computations into per-edge message functions, per-node aggregation, and per-node update, the same three-way split. ## The SCA Framework { :label: sec-framework } We decompose network dynamics into three independent components\: structure, coupling, and aggregation. Fix a state space $S$ (say $\mathbb{R}$). We begin with the combinatorial structure, then layer on the dynamics. A *graph* is a pair $(V, E)$ with $E \subseteq \binom{V}{2}$ (simple, undirected). A *hypergraph* is a pair $(V, \mathcal{E})$ with $\mathcal{E} \subseteq \mathcal{P}(V) \setminus \{\emptyset\}$, where each hyperedge is a nonempty subset of $V$. We write $\mathbf{Graph}$ and $\mathbf{Hyp}$ for the respective classes. Three natural constructions relate graphs and hypergraphs. :definition: { :title: Structural constructions :label: def-structural-constructions } The *inclusion* $I$ sends each graph to the hypergraph with the same vertex set whose hyperedges are the size-2 edge set. The *clique expansion* $K$ sends a hypergraph $(V, \mathcal{E})$ to the graph $(V, E_K)$ where $\{u, v\} \in E_K$ iff $u$ and $v$ appear together in some hyperedge. The *neighborhood hypergraph* $N$ sends a graph $(V, E)$ to the hypergraph $(V, \{N[i] : i \in V\})$ where $N[i] = \{i\} \cup \{j : \{i, j\} \in E\}$ is the closed neighborhood of $i$. The *bipartite incidence graph* $B$ sends a hypergraph $(V, \mathcal{E})$ to the bipartite graph with node set $V \cup \mathcal{E}$ and an edge between $i$ and $e$ whenever $i \in e$. This is another graph representation of a hypergraph, but the coupling computation is relocated to auxiliary hyperedge-nodes rather than eliminated\: the parameter count is unchanged. :: Given a structure (graph or hypergraph), we layer on dynamics by specifying two families of functions: per-edge couplings and per-node aggregation. :definition: { :title: Dynamical graphs :label: def-dyngraph } A *dynamical graph* is a triple $(G, c, f)$ where :enumerate: :-: $G = (V, E)$ is a graph, :-: $c = \{c_e\}_{e \in E}$ with $c_e \colon S^2 \to S^2$ is a family of per-edge coupling functions, operating on the state of all nodes involved in $e$ and outputting one (possibly different) value to each such node, and :-: $f = \{f_i\}_{i \in V}$ with $f_i \colon S^{1+\deg(i)} \to S$ is a family of per-node aggregation functions combining the node's own state with contributions from all edges containing $i$ into a state update. :: We write $x_e = (x_i, x_j) \in S^2$ for the states of the nodes in edge $e = \{i, j\}$, and $c_{e,i} \colon S^2 \to S$ for the component of $c_e$ corresponding to node $i \in e$, that is, the scalar that edge $e$ delivers to node $i$ via the coupling $c$. With this notation, the update rule for the state $x_i$ of node $i$ is: $${ :label: eqn-update } \dot{x}_i = f_i\!\big(x_i,\; (c_{e,i})_{e \ni i}\big). $$ In words, the next state of node $i$ (namely $\dot{x}_i$) equals the aggregation (via $f_i$) of the node's current state ($x_i$) and the contribution of all the couplings it participates in ($c_{e,i})_{e \ni i}$). We assume all nodes' states update at the same time and instantly. :: :definition: { :title: Dynamical hypergraphs :label: def-dynhyp } A *dynamical hypergraph* is a triple $(H, c, f)$ where $H = (V, \mathcal{E})$ is a hypergraph, $c = \{c_e\}_{e \in \mathcal{E}}$ with $c_e \colon S^{|e|} \to S^{|e|}$ is a family of per-hyperedge coupling functions, and $f = \{f_i\}_{i \in V}$ is a family of per-node aggregation functions. The update rule for the state $x_i$ of node $i$ is the same as :ref:eqn-update::\: $$ \dot{x}_i = f_i\!\big(x_i,\; (c_{e,i})_{e \ni i}\big). $$ :: We write $\mathbf{DynGraph}$ and $\mathbf{DynHyp}$ for the classes of dynamical graphs and dynamical hypergraphs, respectively. The following examples show how standard dynamics decompose into $(G, c, f)$. :example: { :title: Graph dynamics :label: ex-graph-models } *SIS epidemic.* The update rule $\dot{x}_i = -\gamma x_i + \beta(1 - x_i)\sum_{j \sim i} x_j$ admits (at least) two decompositions\: $$ \begin{align} c_{\{i,j\},i} = x_j, \qquad f_i(x_i, (y_e)) = -\gamma x_i + \beta(1 - x_i)\sum_e y_e \\ c_{\{i,j\},i} = \beta(1 - x_i)\, x_j, \qquad f_i(x_i, (y_e)) = -\gamma x_i + \sum_e y_e. \end{align} $$ The first makes $c$ a trivial channel, simply transmitting a node's state to its neighbor through the edge; the second puts the infection logic in $c$ and leaves $f_i$ as a simple sum. The split between $c$ and $f$ is not unique. *Kuramoto oscillators.* Given the update rule $\dot{\theta}_i = \omega_i + \tfrac{K}{N}\sum_{j \sim i} \sin(\theta_j - \theta_i)$, define\: $$ c_{\{i,j\},i} = \sin(\theta_j - \theta_i), \qquad f_i(\theta_i, (y_e)) = \omega_i + \tfrac{K}{N}\sum_e y_e. $$ Here the coupling is antisymmetric ($c_{\{i,j\},i} = -c_{\{i,j\},j}$), a structural property visible in $c$ but invisible in a flat representation. :: :example: { :title: Hypergraph dynamics :label: ex-hyp-models } *Nonlinear eigenvector centrality.* On a hypergraph $H$, a natural generalization of eigenvector centrality uses the product over co-members defined as $$ \dot{x}_i = -x_i + \sum_{e \ni i} \prod_{j \in e \setminus \{i\}} x_j. $$ A *non-focal symmetric* decomposition (defined formally below) sets a shared coupling to $h_e(x_e) = \prod_{j \in e} x_j$, symmetric in all arguments, and lets each node's aggregation extract what it needs\: $$ c_{e,i} = h_e(x_e) = \prod_{j \in e} x_j, \qquad f_i(x_i, (y_e)) = -x_i + \sum_e y_e / x_i. $$ Every node in $e$ receives the same scalar $h_e$. The aggregation is additive (after dividing out $x_i$). *Higher-order Kuramoto.* On a 3-hyperedge $e = \{1, 2, 3\}$\: $$ c_{e,1} = \sin(\theta_2 + \theta_3 - 2\theta_1), \qquad f_i(\theta_i, (y_e)) = \omega_i + K\sum_e y_e. $$ The coupling $c_{e,1}$ is symmetric in $\theta_2, \theta_3$ but not in $\theta_1$. The coupling is neither fully symmetric across all nodes nor concentrated at a single distinguished node; both of these cases are defined formally below. :: ### Modeling assumptions { :label: sec-subclasses } The full generality of $\mathbf{DynHyp}$ allows arbitrary couplings and aggregation. The three papers under consideration :cite:peixoto2026,moutuou2026,lacasa2026:: each work in a restricted setting in our framework. Making these restrictions explicit is one of the main motivations of the SCA framework. :definition: { :title: Flat dynamics :label: def-flat } $\mathbf{DynHyp}_{\mathrm{flat}}$ is the subclass of $\mathbf{DynHyp}$ where, for every node $i$, the update $\dot{x}_i$ depends only on the states of the immediate neighbors $N[i] = \bigcup_{e \ni i} e$\: $$ \dot{x}_i = h_i(x_{N[i]}). $$ This is a constraint on the composition of $c$ and $f$, not on either component individually. Whatever $c$ and $f$ do internally, their combined effect sees only the union of neighbor states, not which hyperedge contributed which neighbor or how many times. :: This is the implicit assumption in much of the higher-order networks literature, and the explicit assumption in Peixoto et al. :cite:peixoto2026::. Under flat dynamics, a node's update depends only on the union of its neighbors, so the partition of neighbors into hyperedges is invisible. As we show in :ref:sec-result-flat::, this is precisely why graphs and hypergraphs become equivalent in their framework. :definition: { :title: Non-focal symmetric dynamics :label: def-nfs } $\mathbf{DynHyp}_{\mathrm{NFS}}$ is the subclass of $\mathbf{DynHyp}$ where for every hyperedge $e$, we have $c_{e,i} = h_e(x_e)$ for all $i \in e$ (every node in $e$ receives the same contribution) and $h_e$ is invariant under permutations of its arguments. The aggregation functions $f_i$ are unconstrained. :: The NFS and focal terminology and the distinction between them were introduced by Moutuou :cite:moutuou2026::. NFS requires that no node in a hyperedge plays a distinguished role in the coupling: every node receives the same symmetric value, and every node plays the same role. Differentiation between nodes happens only in the aggregation $f_i$. Moutuou shows that representing NFS dynamics on a graph necessarily introduces a distinguished node per edge (focal structure), breaking the symmetry that the hypergraph representation preserves. :definition: { :title: Focal dynamics :label: def-focal } $\mathbf{DynHyp}_{\mathrm{focal}}$ is the subclass of $\mathbf{DynHyp}$ where each hyperedge $e$ has a distinguished *focal node* (or *basepoint*) $b(e) \in e$. The coupling treats the focal node differently from the remaining nodes\: $c_{e,b(e)}$ may differ from $c_{e,i}$ for $i \neq b(e)$, but $c_{e,i} = c_{e,j}$ for all $i, j \in e \setminus \{b(e)\}$. The subclass $\mathbf{DynHyp}_{\mathsf{gf}}$ (*graph-focal*) is the further restriction where the hyperedges are exactly the closed neighborhoods of the nodes, $\{N[i] : i \in V\}$, each with $i$ as its focal node. :: Focal dynamics break the full symmetry of NFS, but minimally\: one node per hyperedge is singled out, but the remaining $|e| - 1$ nodes are still interchangeable. On the other extreme, we have fully asymmetric couplings, where every node plays a distinct role. We discuss the degree of symmetry of the couplings in :ref:sec-fibers, the Discussion::. :definition: { :title: Carleman linearization lift :label: def-carleman } Let $(G, c, f) \in \mathbf{DynGraph}$ such that every $c_e$ is a polynomial of degree at most $D$. Since each edge $e = \{i,j\}$ involves two nodes, the monomials of $c_e$ are products $x_i^a x_j^b$ with $a + b \leq D$. The *Carleman lift* introduces a new variable for each such monomial and produces a new SCA triple $(H', c', f')$ where $H'$ is an hb-graph (a hypergraph with multiset edges), $c'$ is linear, and $f'$ is additive. :: The Carleman lift trades coupling complexity for structural complexity\: the nonlinearity of $c$ is absorbed into the structure of $H'$, where each monomial variable becomes a node and the original nonlinear coupling becomes linear coupling between monomial nodes connected by hb-edges. This is Lacasa's construction :cite:lacasa2026::, recast in SCA terms. When $c_e$ is polynomial of degree $D$, the lift produces finitely many monomial variables (up to degree $D$ in $|V|$ original variables). For arbitrary smooth or analytic couplings, however, $H'$ is generically infinite (one node per monomial of every degree). One further connection\: lifted variables $z_{jk} = x_j x_k$ are symmetric in their indices, so the Carleman lift naturally produces NFS-like structures from graph dynamics. ## A worked example { :label: sec-example } We illustrate the framework with a single dynamical system viewed through each subclass. Let $V = \{1, 2, 3\}$ with $S = \mathbb{R}$ and global dynamics $$ \dot{x}_i = (1 - x_i) \prod_{j \neq i} x_j. $$ This models a symmetric group interaction\: each node is driven toward 1 at a rate proportional to the product of the other nodes' states. The 3-body symmetry (all nodes play identical roles) is the feature we track across representations. :html: { :path: fig_worked_example.html :static: fig_worked_example_static.svg :label: fig-worked-example } :caption: The same dynamics viewed through three SCA decompositions. Use the tabs to compare different dynamical systems. In each case, the NFS hypergraph keeps the coupling rich and the aggregation simple, the graph representation uses simpler couplings at the cost of more complex aggregation, and the flat representation collapses the decomposition entirely. The trajectories are identical across all three. :: :paragraph: {:label: sec-ex-nfs} **NFS on a hypergraph** Take $H$ with a single hyperedge $e = \{1, 2, 3\}$. Set the shared interaction $h_e(x_1, x_2, x_3) = x_1 x_2 x_3$ and the aggregation $f_i(x_i, y) = (1 - x_i) \cdot y / x_i$. Then $$ \dot{x}_i = f_i(x_i,\, h_e(x_e)) = (1 - x_i) \cdot \frac{x_1 x_2 x_3}{x_i} = (1 - x_i) \prod_{j \neq i} x_j. $$ The 3-body symmetry is explicit\: $h_e$ is a single permutation-invariant function shared by all three nodes. :paragraph: {:label: sec-ex-graph} **Graph representation** The clique expansion (:ref:def-structural-constructions::) gives $K_3$ with edges $\{1,2\}$, $\{1,3\}$, $\{2,3\}$. A valid decomposition\: set each edge function to act as a trivial channel, $c_{\{i,j\},i} = x_j$, and let node $i$ compute everything in aggregation: $$ f_i(x_i,\, x_j,\, x_k) = (1 - x_i) x_j x_k. $$ The trajectories of the resulting dynamical system are identical. But the 3-body interaction has been fragmented into three independently specified pairwise channels. Nothing in the graph representation records that these three edge functions were part of a single symmetric interaction. The same trajectories would result from a system where the three pairwise interactions happened to coincide by accident rather than by structural necessity. :paragraph: {:label: sec-ex-flat} **Flat dynamics** Under flat dynamics, structure drops out entirely. The update becomes $\dot{x}_i = h_i(x_1, x_2, x_3)$ with $h_i(x) = (1 - x_i) \prod_{j \neq i} x_j$. This is equally valid on $K_3$, on the hypergraph $\{1,2,3\}$, or on any structure whose flat neighborhood of every node is $\{1,2,3\}$. The dynamics cannot distinguish between them, which is precisely the content of :ref:prop-flat::. The three representations produce identical trajectories. They differ in what structural information is preserved. :enumerate: :-: The NFS hypergraph representation records that the interaction is a single 3-body symmetric function. A modeler reading this representation knows the interaction is genuinely triadic and permutation-invariant. :-: The graph representation distributes the interaction across pairwise channels. A modeler reading this representation cannot distinguish a genuine 3-body interaction from three coincidentally coordinated pairwise ones. :-: The flat representation discards all structural information. A modeler reading this representation cannot even tell how many interactions are involved. :: All three representations produce the same trajectories, confirming that graphs are equally expressive as hypergraphs in this toy example. But the representations differ in parsimony\: the NFS decomposition uses one coupling function and arity-2 aggregation, the graph decomposition uses three coupling functions and arity-3 aggregation, and the flat representation collapses the decomposition entirely. This difference matters for inference. A scientist trying to infer the interaction structure from trajectory data faces one symmetric function to estimate (hypergraph) versus three pairwise couplings and a more complex aggregation (graph). We quantify this gap in :ref:thm-complexity-gap::. ## Results { :label: sec-results } Beyond providing a common language, the SCA framework yields results that stand on their own. We begin with the special cases corresponding to each paper, then build to the general result. ### Flat dynamics\: graphs and hypergraphs are equivalent { :label: sec-result-flat } :proposition: { :title: Flat equivalence :label: prop-flat } Under the flat dynamics assumption, $\mathbf{DynHyp}_{\mathrm{flat}}$ is equivalent to $\mathbf{DynGraph}_{\mathrm{flat}}$ via clique expansion $K$. :: :proof: Let $(H, h) \in \mathbf{DynHyp}_{\mathrm{flat}}$ with update $\dot{x}_i = h_i(x_{N[i]})$. The clique expansion $K(H)$ is a graph where $\{u, v\}$ is an edge iff $u$ and $v$ share some hyperedge. Set $c_{\{i,j\},i} = x_j$ (trivial channels) and $f_i(x_i, (y_e)) = h_i\big(\{x_i\} \cup \{y_e\}\big)$. The flat neighborhood of $i$ in $K(H)$ equals $N[i]$ in $H$ by construction. Since each pairwise channel delivers one neighbor state, $f_i$ receives exactly the arguments of $h_i$, so the dynamics are identical. Conversely, any graph $G$ with flat dynamics embeds into $\mathbf{DynHyp}_{\mathrm{flat}}$ via $I$, which sends edges to size-2 hyperedges and preserves flat neighborhoods. The round-trip $K \circ I$ recovers $G$. The round-trip $I \circ K$ does not recover $H$ as a hypergraph, but preserves flat neighborhoods, and in $\mathbf{DynHyp}_{\mathrm{flat}}$ the dynamics depend only on flat neighborhoods. :: ### NFS dynamics\: what graphs forget { :label: sec-result-nfs } :proposition: { :title: NFS representability on graphs :label: prop-nfs } Any NFS dynamics on a hypergraph can be represented on a graph with identical trajectories, but the representation necessarily loses the NFS structure. :: :proof: Let $(H, c, f) \in \mathbf{DynHyp}_{\mathrm{NFS}}$. We construct a graph representation with identical trajectories, then show that NFS structure is lost. *Single hyperedge.* To fix ideas, let $e$ be a hyperedge of size $k$ with NFS interaction $h_e$. The clique expansion $K$ places an edge between every pair of nodes in $e$. For each node $i \in e$ and each $j \in e \setminus \{i\}$, set the pairwise coupling to act as a channel\: $c'_{\{i,j\},i} = x_j$. Node $i$ receives the states of all $k - 1$ other nodes in $e$ through these channels. Define the graph aggregation as $$ f'_i\big(x_i,\, (x_j)_{j \in e \setminus \{i\}}\big) = f_i\big(x_i,\, h_e(x_i, x_{e \setminus \{i\}})\big), $$ which reconstructs $h_e$ from the channeled states and applies the original aggregation. The trajectories are identical by construction. *Multiple hyperedges.* When $H$ has hyperedges $e_1, \ldots, e_m$, the clique expansion places an edge $\{i, j\}$ whenever $i$ and $j$ co-occur in some $e_\ell$. For non-overlapping hyperedges, the single-hyperedge construction applies independently. When hyperedges overlap, a single graph edge $\{i, j\}$ may arise from multiple hyperedges, but the channel $c'_{\{i,j\},i} = x_j$ delivers the same value regardless of origin. The aggregation $f'_i$ receives $x_j$ for every graph neighbor $j$ and must reconstruct the NFS contributions $h_{e_\ell}(x_{e_\ell})$ for each $e_\ell \ni i$. This is possible because $h_{e_\ell}$ depends only on states within $e_\ell$, all of which are available\: $$ f'_i\big(x_i,\, (x_j)_{j \sim i}\big) = f_i\big(x_i,\, \big(h_{e_\ell}(x_{e_\ell})\big)_{e_\ell \ni i}\big). $$ The right-hand side is well-defined because for each $e_\ell \ni i$, the set $e_\ell \setminus \{i\}$ is contained in the graph neighborhood of $i$. :: In this construction, since $h_e$ is symmetric in all its arguments, each $f'_i$ is symmetric in the arguments corresponding to $e \setminus \{i\}$. The original symmetry of $h_e$ has been split into $k$ independent symmetries, one per node. This is precisely what graphs forget. In the hypergraph representation, the shared symmetry is recorded *inside* the SCA triple\: a single coupling function $h_e$ that all nodes in $e$ receive. In the graph representation, the triple $(K(H), c', f')$ contains $k$ aggregation functions that happen to be individually symmetric, but nothing in the triple records that these symmetries have a common origin. The graph triple is complete for trajectory simulation but incomplete for representing the interaction structure. ### The landscape of dynamical network classes { :label: sec-result-hierarchy } The structural constructions of :ref:def-structural-constructions:: behave differently when extended to dynamical networks. The clique expansion $K$ does not extend cleanly\: a single hyperedge $e = \{1, 2, 3\}$ with coupling $c_e \colon S^3 \to S^3$ produces three graph edges under $K$, but decomposing $c_e$ into three pairwise functions requires arbitrary choices — different choices yield different $(K(H), c', f')$ with the same trajectories. The only way to make $K$ well-defined on dynamical hypergraphs is to absorb all hyperedge interactions into unconstrained per-node functions, which is precisely the flat dynamics collapse of :ref:def-flat:: (see :ref:prop-flat::). The neighborhood hypergraph $N$, by contrast, extends canonically, mapping $\mathbf{DynGraph}$ into $\mathbf{DynHyp}_{\mathrm{focal}}$. Given $(G, c, f) \in \mathbf{DynGraph}$, the construction $N$ produces $N(G) = (V, \{N[i] : i \in V\})$. The contribution $c_{\{i,j\},i}$ from edge $\{i, j\}$ goes to hyperedge $N[i]$, and $c_{\{i,j\},j}$ goes to $N[j]$. Each hyperedge $N[i]$ collects all contributions that node $i$ receives, with $f_i$ unchanged. The assignment is canonical\: each edge contribution has a unique destination determined by which node it targets, and the construction is focal by definition. These observations yield a hierarchy. :proposition: { :label: prop-hierarchy } The following hierarchy of classes holds, with each inclusion strict. $$ \mathbf{DynHyp}_{\mathrm{flat}} \hookrightarrow \mathbf{DynGraph} \hookrightarrow \mathbf{DynHyp}_{\mathsf{gf}} \subsetneq \mathbf{DynHyp}_{\mathrm{focal}} \subsetneq \mathbf{DynHyp} $$ Here $\hookrightarrow$ denotes a trajectory-preserving embedding\: every flat hypergraph dynamics can be realized on a graph (via $K$), and every dynamical graph can be realized as a graph-focal dynamical hypergraph (via $N$). :: :proof: As shown above, $N$ embeds $\mathbf{DynGraph}$ into $\mathbf{DynHyp}_{\mathrm{focal}}$. Its image is the class $\mathbf{DynHyp}_{\mathsf{gf}}$ where hyperedges are exactly closed neighborhoods, one per node. The first inclusion is strict because general focal hypergraphs allow a node to be focal in multiple hyperedges, or multiple hyperedges with the same focal node but different membership; in a graph, each node $i$ has exactly one closed neighborhood $N[i]$. The second inclusion is strict because NFS interactions (:ref:def-nfs::) have no focal node ($c_{e,i} = h_e(x_e)$ for all $i \in e$) and therefore lie outside $\mathbf{DynHyp}_{\mathrm{focal}}$. :: The flat class sits at the bottom of the hierarchy because flat dynamics depend only on the union of neighbor states, erasing the partition into hyperedges. Whether the underlying structure is focal, NFS, or fully general is invisible to flat dynamics — which is precisely why $\mathbf{DynHyp}_{\mathrm{flat}}$ embeds into $\mathbf{DynGraph}$. :html: { :path: fig_hierarchy.html :static: fig_hierarchy.svg :label: fig-hierarchy } :caption: Left\: structural containment of SCA triple classes. Right\: as sets of realizable trajectories, all classes collapse — graphs can simulate any hypergraph dynamics. The gap between the two panels is the paper's central tension\: structural richness versus trajectory equivalence. :: :remark: { :label: rem-structure-vs-trajectory } The structural hierarchy (:ref:fig-hierarchy::, left) and the trajectory-equivalence picture (right) are not in conflict. Propositions :ref:prop-flat:: and :ref:prop-nfs:: show that every class in the hierarchy can be trajectory-embedded into $\mathbf{DynGraph}$\: flat dynamics via $K$, NFS dynamics via $K$ with trivial channels. The hierarchy describes which SCA triples *exist as formal objects* — which decompositions are available — not which trajectories are realizable. All classes realize the same trajectories. What differs is the cost of the decomposition, which the next result quantifies. :: ### The coupling-aggregation tradeoff { :label: sec-result-cost } The previous results show that graphs can represent hypergraph dynamics under flat (:ref:prop-flat::) and NFS (:ref:prop-nfs::) assumptions. In fact, graphs can always reproduce hypergraph trajectories. :proposition: { :title: Graph expressiveness :label: prop-expressiveness } For any dynamical hypergraph $(H, c, f)$ with $|e| \geq 2$ for all $e$, there exists a dynamical graph $(G, c', f')$ on $G = K(H)$ with identical trajectories. :: :proof: Set $G = K(H)$, the clique expansion. For each edge $\{i,j\}$ in $G$, set $c'_{\{i,j\},i}(x_i, x_j) = x_j$ (trivial channel). Node $i$ then receives $x_j$ for every $j$ in its graph neighborhood, which equals $\bigcup_{e \ni i} e \setminus \{i\}$. For each hyperedge $e \ni i$, the states $x_e = (x_j)_{j \in e}$ are all available to $f'_i$ through these channels (plus $x_i$ itself). Define $$ f'_i\big(x_i,\, (x_j)_{j \sim i}\big) = f_i\big(x_i,\, \big(c_{e,i}(x_e)\big)_{e \ni i}\big). $$ The right-hand side is well-defined because $f'_i$ has access to all states needed to evaluate each $c_{e,i}$. The trajectories are identical by construction. :: This generalizes the result of :cite:peixoto2026:: from flat dynamics to the full SCA framework. The following result quantifies what this representation costs. :theorem: { :title: Aggregation Complexity Gap :label: thm-complexity-gap } Let $(H, c, f)$ be a dynamical hypergraph with $|e| \geq 2$ for all $e \in \mathcal{E}$. For each node $i$, let $r_i$ be the degree of $i$ in $H$ (number of hyperedges containing $i$) and $d_i$ the degree of $i$ in the clique expansion $K(H)$ (number of distinct neighbors). Then $f_i$ takes $1 + r_i$ inputs on $H$ and $1 + d_i$ inputs on $K(H)$. If the coupling and aggregation functions are polynomials of total degree at most $D$, the total parameter count of the two decompositions is\: :enumerate: :-: *Hypergraph.* Each coupling component $c_{e,i}$ is a polynomial in $|e|$ variables\: $\binom{D + |e|}{|e|}$ parameters. There are $|e|$ components per hyperedge, so the coupling cost per hyperedge is $|e| \cdot \binom{D + |e|}{|e|}$. Each aggregation $f_i$ is a polynomial in $1 + r_i$ variables\: $\binom{D + 1 + r_i}{1 + r_i}$ parameters. Total\: $\sum_{e} |e|\binom{D + |e|}{|e|} + \sum_{i} \binom{D + 1 + r_i}{1 + r_i}$. :-: *Graph (trivial channels).* Coupling has zero free parameters. Each aggregation $f_i$ is a polynomial in $1 + d_i$ variables\: $\binom{D + 1 + d_i}{1 + d_i}$ parameters. Total\: $\sum_{i} \binom{D + 1 + d_i}{1 + d_i}$. :: :: :proof: On $H$, each hyperedge $e \ni i$ delivers one scalar $c_{e,i}$ to node $i$, so $f_i$ takes $1 + r_i$ inputs. On $K(H)$, node $i$ has $d_i$ incident edges, so $f_i$ takes $1 + d_i$ inputs. A polynomial of degree at most $D$ in $m$ variables has $\binom{D+m}{m}$ monomials, giving the per-component counts. Each $c_e$ has $|e|$ components, each a polynomial in $|e|$ variables. In the graph case with trivial channels, $c$ contributes no parameters. Summing over all components gives the totals. :: We use *cost* to mean the total number of free parameters in an SCA decomposition. The theorem compares the cost of both decompositions. The graph has zero coupling parameters but more aggregation parameters; the hypergraph has coupling parameters (per hyperedge, per component) but fewer aggregation parameters. For the worked example of :ref:sec-example:: ($k = 3$, one hyperedge, $D = 2$)\: the hypergraph coupling has $3\binom{5}{3} = 30$ parameters (three components of $c_e$, each a polynomial in 3 variables), and the aggregation has $3\binom{4}{2} = 18$, totaling 48. The graph has $3\binom{5}{3} = 30$ aggregation parameters. Here the hypergraph has higher cost — the coupling parameters outweigh the aggregation savings. The balance shifts under NFS, where all $|e|$ components of $c_e$ collapse to one shared function $h_e$, reducing the coupling parameters from $|e|\binom{D+|e|}{|e|}$ to $\binom{D+|e|}{|e|}$. At $k = 10$, $D = 2$\: the NFS hypergraph total is $\binom{12}{10} + 10\binom{4}{2} = 66 + 60 = 126$, versus the graph's $10\binom{12}{10} = 660$. Symmetric interactions are where hypergraphs earn their keep. :corollary: { :title: NFS asymptotic gap :label: cor-asymptotic } For NFS dynamics on a single hyperedge of size $k$, the ratio of graph cost to hypergraph cost grows as $\Theta(k^D / D!)$ for fixed $D$ as $k \to \infty$. With multiple non-overlapping hyperedges the gap is at least as large. :: :proof: With one hyperedge, NFS aggregation at each node has $1 + 1 = 2$ inputs ($\binom{D+2}{2}$ parameters) while graph aggregation has $1 + (k-1) = k$ inputs ($\binom{D+k}{k}$ parameters). The ratio is $\binom{D+k}{k} / \binom{D+2}{2} = \Theta(k^D / D!)$ by Stirling. With $r$ non-overlapping size-$k$ hyperedges, graph aggregation at the central node has $1 + r(k-1)$ inputs, which grows even faster. :: :corollary: { :title: BIC penalty gap :label: cor-bic } The Bayesian Information Criterion (BIC) penalizes model complexity by adding $\tfrac{p}{2} \log n$ to the negative log-likelihood, where $p$ is the number of free parameters and $n$ is the number of observations. Under the assumptions of :ref:thm-complexity-gap::, the difference in BIC penalty between the graph and hypergraph decompositions is $$ \Delta_{\mathrm{BIC}} = \frac{\log n}{2}\left[\sum_i \binom{D+1+d_i}{1+d_i} - \sum_e |e|\binom{D+|e|}{|e|} - \sum_i \binom{D+1+r_i}{1+r_i}\right]. $$ When $\Delta_{\mathrm{BIC}} > 0$ the hypergraph decomposition is favored; when $\Delta_{\mathrm{BIC}} < 0$ the graph decomposition is favored. :: :html: { :path: fig_bic.html :static: fig_bic_static.svg :label: fig-bic-gap } :caption: Total parameter count for four SCA decomposition classes on a star hypergraph (node $i$ in $r$ non-overlapping $k$-hyperedges). Left\: $D = 2$ reference. Right\: adjustable $D$. NFS hypergraphs are consistently cheaper than graphs for large $k$, because the shared coupling $h_e$ is paid once per hyperedge. General hypergraphs, where each node receives a distinct coupling component, are the most expensive. Focal hypergraphs lie between NFS and graph. Use the sliders to explore how $D$ and $r$ affect the gap. :: :ref:fig-bic-gap:: reveals a striking pattern\: graphs are almost always more costly than hypergraphs with symmetric coupling (NFS or focal), often by orders of magnitude. The reason is that symmetry allows the coupling $c_e$ to be shared — paid once per hyperedge rather than once per node. Without symmetry (the general hypergraph), this sharing is lost and the hypergraph decomposition becomes nearly as costly as the graph. Symmetry in the interaction is what makes hypergraphs worth the overhead. :remark: { :label: rem-asymmetry } The coupling-aggregation tradeoff is not symmetric. Coupling can always be absorbed into aggregation (:ref:thm-complexity-gap::), because $f_i$ receives all coupling signals and can reconstruct any function of them. The reverse does not hold\: aggregation cannot in general be absorbed into coupling, because each $c_{e,i}$ sees only the states within edge $e$ and has no access to the signals produced by other edges. The operation of combining contributions from multiple edges is irreducibly a per-node computation. This is why aggregation is the component whose parameter count grows when coupling is simplified. :: ### Graph taxonomy as dynamical constraints { :label: sec-insight-taxonomy } Traditional graph taxonomy treats directedness, edge weights, signs, and multiplexity as distinct structural classes. In the SCA framework, all of these can be expressed as constraints on the coupling and aggregation functions, not on the combinatorial structure. An *undirected* interaction is one where the coupling is symmetric: $c_{e,i} = c_{e,j}$ for any edge $e = \{i, j\}$. Any coupling that violates this symmetry is effectively *directed*. The asymmetry lives in $c$, not in the edge set. A *weighted* interaction multiplies the coupling output by a scalar $w_{ij}$; *signed* interactions are the special case $w_{ij} \in \{-1, +1\}$. *Multilayer and multiplex* networks can be modeled with an SCA triple on an enlarged node set\: each (node, layer) pair is a node, intra-layer interactions are edges within a layer (or edge type), and inter-layer connections are edges between layers (if allowed). Directedness, weights, and signs constrain $c$ without changing the graph. Multiplex and multilayer networks change the graph (by enlarging the node set) but remain within $\mathbf{DynGraph}$. By contrast, moving from $\mathbf{Graph}$ to $\mathbf{Hyp}$ is a qualitatively different kind of change\: hyperedges group nodes in ways that edges cannot, and this grouping is visible to the dynamics through the couplings $c$. ## Discussion { :label: sec-discussion } Graphs are maximally expressive across the full range of network dynamics, not only under flat dynamics. But expressiveness has a cost\: the coupling-aggregation tradeoff (:ref:thm-complexity-gap::) quantifies how interaction complexity migrates from $c$ to $f$ when a hypergraph is replaced by a graph. The worked example (:ref:sec-example::) shows this concretely\: the same dynamics admits representations ranging from parsimonious (NFS, $f$ has arity 2) to costly (graph, $f$ absorbs everything). We note that the complexity gap compares the trivial-channel graph decomposition against the hypergraph; whether other graph decompositions with non-trivial coupling achieve lower total cost is open. Several further problems emerge from this tradeoff. Given a vector field $F \colon S^V \to S^V$ governing $\dot{x} = F(x)$, define its *interaction order* $\mathrm{ord}(F)$ as the smallest $k$ such that $F$ admits an NFS decomposition on some hypergraph with maximum hyperedge size $k$. If $\mathrm{ord}(F) = 2$, the dynamics can be decomposed into pairwise symmetric interactions and graphs suffice even under NFS. The question is whether natural dynamics exist with $\mathrm{ord}(F) > 2$ — dynamics that genuinely require higher-order interactions, not just for convenience but because no pairwise NFS decomposition exists. This is subtler than it appears. On a complete graph, pairwise NFS couplings can channel individual neighbor states to each node (e.g., $h_{\{i,j\}}(x_i, x_j) = x_i + x_j$ delivers $x_j = h - x_i$ to node $i$). The aggregation $f_i$ can then reconstruct any function of the neighborhood from these channeled values. So the obstruction to pairwise NFS is not the dynamics alone but the combination of dynamics, state space, and allowed graph structure. Characterizing when $\mathrm{ord}(F) > 2$ is a well-posed open problem that the SCA framework makes precise. The BIC penalty gap (:ref:cor-bic::) gives a first connection between the complexity gap and model selection\: given observational data, the decomposition with fewer free parameters is penalized less. But this is only a starting point. The BIC penalty compares parameter counts; a full model selection analysis would also account for the likelihood (how well each decomposition fits the data) and the geometry of the parameter space (how many parameter settings produce the same trajectories). Whether the parsimony advantage of NFS hypergraphs translates into better inference in practice — particularly under noise, finite data, and model misspecification — remains open. Most work on network dynamics implicitly assumes additive aggregation. The framework treats aggregation as an independent modeling choice, but little is known about its consequences. When a node belongs to multiple hyperedges that share members, the choice of $f_i$ determines whether the overlap is dynamically meaningful. :example: { :label: ex-overlap } Let $V = \{1, 2, 3, 4\}$ with two hyperedges $e_1 = \{1, 2, 3\}$ and $e_2 = \{2, 3, 4\}$. Nodes 2 and 3 belong to both. Suppose the NFS couplings $h_{e_1}$ and $h_{e_2}$ are both permutation-invariant. Then node 2's update is: $$ \dot{x}_2 = f_2\!\big(x_2,\; h_{e_1}(x_1, x_2, x_3),\; h_{e_2}(x_2, x_3, x_4)\big). $$ With additive aggregation $f_2(x_2, a, b) = a + b$, the two contributions are independent and the overlap is invisible. With multiplicative aggregation $f_2(x_2, a, b) = a \cdot b$, node 2's update is suppressed if either group interaction is near zero; a gating mechanism that may be desirable in this case of hyperedge overlap. :: Does the choice of $f_i$ affect the gap? The multiplicative gating example suggests that non-additive aggregation can produce qualitatively different behavior, but a systematic study is lacking. :paragraph:{:label: sec-fibers} The hierarchy of :cite:moutuou2026:: distinguishes focal from non-focal symmetric interactions, but the SCA framework reveals a finer spectrum between these extremes. Consider a hyperedge $e = \{1, 2, 3\}$ where nodes 2 and 3 are interchangeable but node 1 is distinguished\: $$ c_{e,2} = c_{e,3} \neq c_{e,1}. $$ This is neither focal nor NFS. The symmetry of $c_e$ is captured by a subgroup of $\Sigma_{|e|}$, the group of all permutations of the $|e|$ nodes in $e$. The trivial subgroup (no symmetry) gives fully asymmetric interactions, the full group gives NFS (complete symmetry), and any intermediate subgroup gives partial symmetry. :example: { :label: ex-partial-symmetry } Consider an enzyme-catalyzed reaction modeled as a hyperedge $e = \{\text{enz}, s_1, s_2\}$. Let $r(x_{\text{enz}}, x_{s_1}, x_{s_2})$ be a rate function that is symmetric in the two substrate concentrations but not in the enzyme. The coupling is\: $$ c_{e,\text{enz}} = r(x_{\text{enz}}, x_{s_1}, x_{s_2}), \qquad c_{e,s_1} = c_{e,s_2} = -r(x_{\text{enz}}, x_{s_1}, x_{s_2}). $$ The enzyme receives a saturation signal; the substrates are consumed at equal rates. The symmetry group is $\Sigma_2$ acting on $\{s_1, s_2\}$, a proper subgroup of $\Sigma_3$. The coupling $c_e$ is doing genuine computational work\: it computes a reaction rate and distributes it asymmetrically according to each node's role. This is neither representable as NFS (the enzyme is distinguished) nor as a trivial channel (the rate depends on all three states). :: This observation leads to a natural formalization. The map $\Phi$ that sends a triple $(G, c, f)$ to its induced global dynamics $F \colon S^V \to S^V$ forgets the decomposition. The preimage $\Phi^{-1}(F)$ is the collection of all triples that produce $F$. :definition: { :title: Local symmetry profile :label: def-local-symmetry } Let $(H, c, f) \in \Phi^{-1}(F)$. For each hyperedge $e \in \mathcal{E}$, the *local symmetry group* of $c_e$ is the subgroup $G_e \leq \Sigma_{|e|}$ under which $c_e$ is invariant\: $$ G_e = \{\sigma \in \Sigma_{|e|} : c_e(x_{\sigma}) = c_e(x) \text{ for all } x \in S^{|e|}\}. $$ The *local symmetry profile* of the triple is the collection $\{G_e\}_{e \in \mathcal{E}}$. :: In $\mathbf{DynGraph}$, every edge has $|e| = 2$, so $G_e \leq \Sigma_2$. No graph decomposition can express a $k$-body symmetry for $k > 2$ within a single edge function. In $\mathbf{DynHyp}$ with a $k$-hyperedge, $G_e$ can reach the full symmetric group $\Sigma_k$. Local symmetry profiles are partially ordered by componentwise inclusion. If $\Phi^{-1}(F)$ contains an NFS element, that element has $G_e = \Sigma_{|e|}$ for all $e$, which is the maximum of this partial order. The NFS decomposition, when it exists, is the maximally symmetric representative of $\Phi^{-1}(F)$. Whether this maximum is unique, and whether $\Phi^{-1}(F)$ has further structure (e.g., a lattice), is open. A complementary direction is to define equivalence relations on triples that identify dynamical systems with the "same behavior" and study the resulting quotient classes. Multiple notions of behavioral equivalence are natural\: identical ODEs, conjugate trajectories (related by coordinate change), or identical qualitative dynamics (same attractors and bifurcation structure). Under the coarsest equivalence (same ODE), the quotients of $\mathbf{DynGraph}$ and $\mathbf{DynHyp}$ coincide, recovering the observation of :cite:peixoto2026::. Under finer equivalences that account for the internal structure of decompositions, the quotients may differ. Studying how the relationship between $\mathbf{DynGraph}$ and $\mathbf{DynHyp}$ changes as the equivalence relation varies would characterize exactly where the graph-hypergraph distinction becomes dynamically meaningful. ## Conclusion { :label: sec-conclusion } Graphs can represent any network dynamics. The mechanism is simple\: absorb coupling into aggregation. The SCA framework makes this explicit and quantifies its cost. When interactions are symmetric, graphs pay a combinatorial price growing as $\Theta(k^D / D!)$. Without symmetry, the comparison reverses. Whether this matters depends on the goal\: for trajectory simulation, graphs suffice; for model selection, interpretability, and scientific understanding, hypergraphs may be decisive. If all you care about is trajectories, use graphs. If you care about structure, use hypergraphs. :references: @article{peixoto2026, title={Graphs are maximally expressive for higher-order interactions}, author={Peixoto, Tiago P. and Peel, Leto and Gross, Thilo and De Domenico, Manlio}, year={2026}, journal={arXiv preprint}, url={https://arxiv.org/abs/2602.16937}, } @article{lacasa2026, title={On the equivalence between nonlinear graph-based dynamics and linear dynamics on higher-order networks}, author={Lacasa, Lucas}, year={2026}, journal={arXiv preprint}, url={https://arxiv.org/abs/2602.21727}, } @article{moutuou2026, title={Graphs are focal hypergraphs: strict containment in higher-order interaction dynamics}, author={Moutuou, Elkaïoum M.}, year={2026}, journal={arXiv preprint}, url={https://arxiv.org/abs/2603.04215}, } @article{battiston2020, title={Networks beyond pairwise interactions\: structure and dynamics}, author={Battiston, Federico and Cencetti, Giulia and Iacopini, Iacopo and Latora, Vito and Lucas, Maxime and Patania, Alice and Young, Jean-Gabriel and Petri, Giovanni}, year={2020}, journal={Physics Reports}, doi={10.1016/j.physrep.2020.05.004}, } @article{torres2021, title={The why, how, and when of representations for complex systems}, author={Torres, Leo and Blevins, Ann S. and Bassett, Danielle S. and Eliassi-Rad, Tina}, year={2021}, journal={SIAM Review}, doi={10.1137/20M1355896}, } @article{bick2023, title={What are higher-order networks?}, author={Bick, Christian and Gross, Elizabeth and Harrington, Heather A. and Schaub, Michael T.}, year={2023}, journal={SIAM Review}, doi={10.1137/21M1414024}, } @book{golubitsky2023, title={Dynamics and Bifurcation in Networks: Theory and Applications of Coupled Differential Equations}, author={Golubitsky, Martin and Stewart, Ian}, year={2023}, publisher={SIAM}, doi={10.1137/1.9781611977332}, } @article{gilmer2017, title={Neural message passing for quantum chemistry}, author={Gilmer, Justin and Schoenholz, Samuel S. and Riley, Patrick F. and Vinyals, Oriol and Dahl, George E.}, year={2017}, journal={ICML}, url={https://arxiv.org/abs/1704.01212}, } @article{battaglia2018, title={Relational inductive biases, deep learning, and graph networks}, author={Battaglia, Peter W. and Hamrick, Jessica B. and Bapst, Victor and Sanchez-Gonzalez, Alvaro and others}, year={2018}, journal={arXiv preprint}, url={https://arxiv.org/abs/1806.01261}, } @article{iacopini2019, title={Simplicial models of social contagion}, author={Iacopini, Iacopo and Petri, Giovanni and Barrat, Alain and Latora, Vito}, year={2019}, journal={Nature Communications}, doi={10.1038/s41467-019-10431-6}, } @article{pecora1998, title={Master stability functions for synchronized coupled systems}, author={Pecora, Louis M. and Carroll, Thomas L.}, year={1998}, journal={Physical Review Letters}, doi={10.1103/PhysRevLett.80.2109}, } @article{gambuzza2021, title={Stability of synchronization in simplicial complexes}, author={Gambuzza, Lucia Valentina and Di Patti, Francesca and Gallo, Luca and others}, year={2021}, journal={Nature Communications}, doi={10.1038/s41467-021-21486-9}, } @article{carletti2020, title={Dynamical systems on hypergraphs}, author={Carletti, Timoteo and Fanelli, Duccio and Nicoletti, Sara}, year={2020}, journal={Journal of Physics: Complexity}, doi={10.1088/2632-072X/aba8e1}, } @article{majhi2022, title={Dynamics on higher-order networks: a review}, author={Majhi, Soumen and Perc, Matjaž and Ghosh, Dibakar}, year={2022}, journal={Journal of the Royal Society Interface}, doi={10.1098/rsif.2022.0043}, } @article{lucas2026, title={Reducibility of higher-order networks from dynamics}, author={Lucas, Maxime and Gallo, Luca and Ghavasieh, Arsham and Battiston, Federico and De Domenico, Manlio}, year={2026}, journal={Nature Communications}, doi={10.1038/s41467-025-68273-4}, } ::