Functional reducibility of higher-order networks (2024)

Introduction

Many complex systems exhibit an interconnected structure that can be encoded by pairwise interactions between their constituents. Such pairwise interactions have been used to model biological, social and technological systems, providing a powerful descriptive and predictive framework[1, 2, 3, 4, 5, 6].Recently, the analysis of higher-order structural patterns and dynamical behaviors attracted the attention of the research community as a powerful framework to model group interactions[7, 8, 9, 10], with applications ranging from neuroscience[11, 12] to ecology[13] and social sciences [14, 15], highlighting the emergence of novel phenomena and non-trivial collective behavior[16, 17, 18, 19, 20, 21].

Higher-order networks encode more information than pairwise interactions: for example, metabolic reactions are more realistically described by group interactions between any number of reagents and reactants, capturing information that would be lost by considering the union of pairwise interactions instead.However, this modeling flexibility comes at a cost: new data needs to be adequately recorded and stored as group interactions instead of pairwise ones,and new analytical[22, 23, 24, 25] and computational[26, 27, 28] tools need to be developed. Moreover, the complexity and computational cost of these tools increase exponentially as larger group interactions are considered.

It is therefore crucial to understand under which conditions higher-order representations need to be favored over classical pairwise ones, and whether it is possible to devise a grounded procedure to determine which representation provides the most parsimonious description of an empirical system.

A similar challenge was faced nearly a decade ago, when the advent of temporal and categorical data[29, 30, 31] allowed multilayer representations of complex networks[32]. For these representations, a principled approach was used to show that not all layers, or types of interaction, are equally informative: Some information can be discarded or aggregated to reduce the overall complexity of the model without sacrificing its descriptive power[33, 34].Although multilayer networks are different from higher-order networks, this approach, formally based on the density matrix formalism[35], provides a good candidate for the present case.Indeed, the idea is similar to the widely used information compression algorithms adopted in computer science:by exploiting the regularities in the data, one can build a compressed representation that optimizes the number of bits needed to describe the data with a model and those to encode the model itself.Similar approaches have also been used to coarse-grain complex and multiplex networks[36, 37, 38].

Here, we build on this long-standing research line and propose a principled approach to optimally compress systems with higher-order interactions—accounting for the complexity of the data and the complexity of the model.Specifically, we determine an optimal order of interactions up to which interactions need be considered to obtain a functionally optimal representation of the system—larger orders can be safely discarded.Formally, we do so by generalizing the concept of network density matrix[35, 39] to account for higher-order diffusive dynamics with the multiorder Laplacian[40, 17] and calculate a (modified) message length corresponding to each order. By minimizing this message length, in the spirit of the original minimum message length principle[41], we find the optimal compression of the data which, in turn, corresponds to the most parsimonious functional description of the system. In the following, we refer to this procedure as functional reduction. A higher-order network is fully reducible—to a pairwise network—if the optimal order is 1, while its reducibility decreases for increasing optimal orders. We demonstrate the validity of our method by performing an extensive analysis of synthetic networks and investigate the functional reducibility of a broad spectrum of real-world higher-order systems.

The advantage of this framework is that it provides a bridge between network analysis and information theory by means of a formalism that is largely inspired by quantum statistical physics, which has found a variety of applications from systems biology[42] to neuroscience[43], shedding light on fundamental mechanisms such as the emergence of network sparsity[44] and the renormalization group[45].

Results

.1 Density matrix for higher-order networks

The flow of information between nodes in a (pairwise) network can be modeled by different dynamical processes. Arguably, the simplest and most successful of these processes is diffusion, that can be described by means of the propagator $e^{-\tau\mathbf{L}}$, where $\mathbf{L}$ the combinatorial Laplacian and $\tau$ is the diffusion time. In particular, the information flow from node $i$ to node $j$ is described by the component $\left(e^{-\tau\mathbf{L}}\right)_{ij}$. Network states can then be encoded by a density matrix[35, 39] defined as

$$\bm{\rho}_{\tau}=\frac{e^{-\tau\mathbf{L}}}{Z},$$

(1)

where the partition function $Z=\text{Tr}\left(e^{-\tau\mathbf{L}}\right)$ ensures a unit trace for this operator.

Here, we generalize density matrices to higher-order networks. The most general formalism to encode higher-order networks is that of hypergraphs[8]. A hypergraph is defined by a set of nodes and a set of hyperedges that represent the interactions between any number of those nodes. A hyperedge is said to be of order $d$ if it involves $d+1$ nodes: a 0-hyperedge is a node, a 1-hyperedge is a 2-node interaction, a 2-hyperedge is a 3-node interaction, and so on. Simplicial complexes are a special case of hypergraph that is also commonly used: they additionally require that each subset of each hyperedge is included, too. In a hypergraph $H$ with maximum order $d_{\rm max}$, diffusion between nodes through hyperedges of order up to $D\leq d_{\rm max}$ can be described by the multiorder Laplacian[40, 46]

$$\mathbf{L}^{[D]}\equiv\mathbf{L}^{(D,\text{ mul})}=\sum_{d=1}^{D}\frac{\gamma_$$

(2)

which is a weighted sum of the Laplacians $\mathbf{L}^{(d)}$ at each order $d$ up to order $D$ ¹¹1In general, a hypergraph does not need to have hyperedges at every order below $D$, unless it is a simplicial complex. If there is no hyperedge of order $d$, both the Laplacian and the average degree in Eq.2 vanish and the result is undefined. In those cases, the sum thus needs to be taken over all orders below $D$ that exist: $\mathcal{D}=\{d\leq D:\langle K^{(d)}\rangle>0\}$. At each order, the weight is defined by a real coefficient $\gamma_{d}$ (which we set to 1 for simplicity) and the averaged generalized degrees $\left<K^{(d)}\right>$. Each $d$-order Laplacian is defined by$L^{(d)}_{ij}=K^{(d)}_{i}\delta_{ij}-\frac{1}{d}A^{(d)}_{ij}$,in terms of the generalized degrees $\bm{K}^{(d)}$ and adjacency matrix $\mathbf{A}^{(d)}$ of order $d$. The matrix $\mathbf{L}^{[D]}$ satisfies all the properties expected from a Laplacian: it is positive semidefinite and its rows (columns) sum to zero (see Methods for details).

Accordingly, the multiorder density matrix of hypergraph $H$, up to order $d$, is defined as

$$\bm{\rho}^{[d]}_{\tau}=\frac{e^{-\tau\mathbf{L}^{[d]}}}{Z},$$

(3)

with the partition function $Z=\text{Tr}\left(e^{-\tau\mathbf{L}^{[d]}}\right)$. Just like its pairwise analog in Eq.1, this operator satisfies all the expected properties of a density matrix: it is positive definite and its eigenvalues sum up to one. Importantly, the diffusion time $\tau$ plays the role of a topological scale parameter: small values of $\tau$ allow information to diffuse only to neighboring nodes, probing only short-scale structures. Larger values of $\tau$, instead, allow the diffusion to reach more remote parts of the hypergraph and describe large-scale structures. In this context, the meaning of “small” and “large” depends on the network structure, and can be estimated with respect to the magnitude of the largest ($1/\lambda_{\max}$) and smallest ($1/\lambda_{\min}$) eigenvalues of the Laplacians, respectively.

.4 Random structures

We now investigate the reducibility of two types of heterogeneous random structures: random hypergraphs and random simplicial complexes. To do so, we compute the optimal order as described above.

A random hypergraph is defined by a number of nodes $N$ and a set of wiring probability $p_{d}$ for each order required. At each order $d$, a hyperdedge is created for any combination of $d+1$ nodes with probability $p_{d}$, similarly to Erdős-Rényi networks. Random simplicial complexes are built in the same manner before adding the missing subfaces of all simplices, to respect the condition of inclusion. In both cases, we set $N=100$, $p_{d}=50/N^{d}$ and $d_{\rm max}=4$.

Figure3 shows the modified message length considering orders from 1 to 4, for 100 realizations of each type of random structure. In random hypergaphs (Fig.3a), the optimal order is the maximum, $d_{\rm opt}=4=d_{\rm max}$. This means that those random hypergraphs, at this diffusion scale $\tau=1/\lambda_{N}$, are not reducible, i.e., $\chi=0$. Instead, in the random simplicial complex case, $d_{\rm opt}=3$, reflecting a higher reducibility $\chi=1/3$.

The only difference between these two cases is that the hyperedges between different orders are correlated (nested) in random simplicial complexes but not in random hypergraphs. To test the effect of this feature, we start from random simplicial complexes and gradually change them to random hypergraphs by shuffling their hyperedges. Specifically, we change each hyperedge into another inexisting hyperedge with probability $p_{\rm shuffle}$. For $p_{\rm shuffle}=1$, the result is a random hypergraph, and Fig.S2 shows the results that confirm the above pattern.

.4.1 Effect of the diffusion time $\tau$ and density

As mentioned above, $\tau$ acts as a topological scale. Diffusion processes with different values of $\tau$ are thus expected to “see” different structures, which may result in different optimal order and reduced hypergraph. To illustrate this, we compute the modified message length curves for the random simplicial complex case, for five values of $\tau$ evenly spaced on a logarithmic scale, and where the second and fourth are $1/\lambda_{N}$ and $1/\lambda_{2}$, respectively. FigureS3 shows that as the topological scale increases (larger $\tau$), the modified message length (i) increases overall and (ii) the reducibility decreases to $\chi=0$. FigureS5 shows the same results for three values of the hyperedge density, where we can see that for higher densities, the modified message length curve becomes much more flat with no clear minimum (see FigureS5), suggesting a functional similarity between the large-scale structure at all orders.

We similarly tested the effect of the sole density on the reducibility (Figs.S4 andS5). In general, the overall density coefficient does not seem to significantly affect the reducibility.

.5 Real-world hypergraphs

We now investigate how reducible real-world hypergraphs are by considering 22 empirical hypergraph datasets from 4 categories: coauthorships, face-to-face contacts, biological systems, and online communications (“technological”).

First, we observe a great variety in the reducilibity values (Fig.4) and the associated modified message length curves (Fig.S6). Fig.4a-c show three examples with curves of very different shapes and with different reduciblity. Second, we note that datasets from different categories seem to be distinctively reducible: all coauthorship datasets are not reducible ($\chi=0$), half of the technological datasets are fully reducible (tag datasets, $\chi=1$) and the other half is not (email datasets, $\chi=0$), whereas most contact datasets have medium values of reducibility. The number of nodes, the maximum and optimal orders, and the hypergraph reducibility are reported for all datasets in Table1, and a description of the datasets is provided in Methods.

In Fig.S7, we also show the reducibility values against structural parameters of the hypergraphs: number of nodes, number of edges, density, and maximum order. We did not observe clear correlations with any of those parameters. In particular, the reducibility takes many values between 0 and 1 for any value of the structural parameters, which confirms the variety of reducibility values in the empirical datasets.

Dataset	Category	$N$	$d_{\rm max}$	$d_{\rm opt}$	$\chi$
coauth-mag-geology_1980	coauthorship	1350	17	17	0.00
coauth-mag-geology_1981	coauthorship	464	17	17	0.00
coauth-mag-geology_1982	coauthorship	1331	17	17	0.00
coauth-mag-geology_1983	coauthorship	535	14	14	0.00
congress-bills	other	1718	399	353	0.12
kaggle-whats-cooking	other	6714	64	64	0.00
contact-high-school	contact	327	4	3	0.33
contact-primary-school	contact	242	4	2	0.67
hospital-lyon	contact	75	4	2	0.67
hypertext-conference	contact	113	5	3	0.50
invs13	contact	92	3	2	0.50
invs15	contact	217	3	2	0.50
science-gallery	contact	410	4	4	0.00
sfhh-conference	contact	403	8	5	0.43
malawi-village	contact	84	3	2	0.50
dawn	bio	2290	15	1	1.00
ndc-classes	bio	628	38	11	0.73
ndc-substances	bio	3065	24	23	0.04
email-enron	technology	143	36	36	0.00
email-eu	technology	986	39	39	0.00
tags-ask-ubuntu	technology	3021	4	1	1.00
tags-math-sx	technology	1627	4	1	1.00

Discussion

All areas of natural and social science, as well as engineering ones, are undergoing a deluge of publicly available data with complex structure. This data allows for building more detailed and powerful models of systems from areas such as physics, biology, sociology, and technology. One class of such models is networks that encode group interactions rather than just pairwise ones: higher-order networks. Higher-order networks provide a natural framework for modeling systems where interactions between more than two units occur, such as chemical reactions of metabolic interest or social interactions. However, they are usually projected—by design—to pairwise interactions when data is gathered, and higher-order information is inevitably lost. By preserving that information, higher-order networks have the potential to yield a more reliable model of some empirical systems.Nevertheless, higher-order networks come with novel challenges: new theoretical and computational methods have to be developed, while algorithms become exponentially more complex for increasing order of the interactions.It is thus of paramount importance to understand under which conditions one should opt for higher-order modeling—or keep using traditional pairwise models.

Here, we have provided a principled method, at the edge between statistical physics and information theory, to guide researchers in identifying the most suitable representation for their data. Our method is based on minimizing a suitable modified message length, encoding both the number of bits required to describe the data given a model and the number of bits required to describe the model estimated by its entropy, to find the optimal order of group interactions. Orders above the optimal one can be safely discarded because they provide only redundant information about the system while increasing model complexity and the computational cost for its analysis.

The message length we defined is “modified” compared to the standard Bayesian definition. Indeed, the density matrices are operators—and not probability distributions—for which there is no direct generalization ofthe Bayes theorem. To partially overcome this, weused a Kullback-Leibler divergence to approximatethe model complexity and define an effective regularization term.

Remarkably, we show that not all systems require a higher-order model to be described and that even within the same class of systems, there is some level of variability. We started by testing the method on extremely regular cases, where the modified message length was flat for all orders, as computed analytically. We then applied the method to random structures, where results indicate that random hypergraphs are non-reducible: i.e., they are best represented by considering all possible orders of interaction. This result is expected because random hypergraphs are the most uncorrelated model possible: hyperedges are uncorrelated within each order, just like in Erdős-Rényi random graphs, but hyperedges of different orders are also uncorrelated, yielding incompressibility of the representation. In other words, all orders are relevant to describe the system’s structure and dynamics, since they encode genuine uncorrelated noise that is incompressible.Conversely, random simplicial complexes were reducible because hyperedges at different orders are correlated by design: the presence of correlation introduces some level of redundancy that is captured by our method.

Finally, we applied our method to empirical datasets that contain group interactions. The large variety of reducibility values obtained indicates that while the extra information encoded in higher-order networks may be optimal for some systems, others can be optimally represented with lower orders and in some cases even with only pairwise interactions. It is also important to note that the category of systems to which the datasets belong appears to be correlated with its reducibility, although more datasets are needed to perform a systematic and quantitative analysis of this phenomenon. Our results challenge the widespread assumption that complex network data must necessarily be investigated through the lens of higher-order dynamics. In fact, there are completely reducible and non-reducible systems, with a complete spectrum of cases between these two extremal cases, demonstrating that some orders might be irrelevant or uninformative to describe an empirical system. Moving forward, it will be crucial to collect new datasets from the biological sciences to determine whether and in what scenarios complex networks such as metabolomes and connectomes can be considered reducible or irreducible. This will help to understand under which conditions higher-order mechanisms and behaviors are essential for the function of these systems, as is often assumed nowadays.

Our work contributes to the increasing interest in reducing the dimensionality of higher-order network models[48, 49], and more generally of complex systems models[50, 51, 52, 45, 53]. In particular, phase reduction techniques for coupled oscillators have shown that higher-order interactions appear naturally when reducing the dynamics of (initially pairwise) coupled higher-dimensional oscillators to simple 1-dimensional phase oscillators [50]. This is consistent with recent results reducing complex to low-rank matrices, in which case higher-order interactions also appear naturally [53]. A renormalization group approach also identified important orders of interaction in empirical datasets which were not always pairwise [49]. Parallel but complementary to these approaches that look at “mechanisms”, other approaches such as maximum entropy models and higher-order information-theoretic measures look at “behaviors”, that is, time series of observables from those systems. Interestingly, several such studies have instead shown that pairwise models can be sufficient, under certain conditions, to describe the behaviors of systems that contain higher-order interactions, or mechanisms [54, 55, 56, 57]. Except for initial work [10, 58], little is known about the relationship between these results and approaches, suggesting a promising direction for future work.

Detecting redundancies and exploiting them with a principled approach to identify a compressed representation of the data,has the potential to enhance our understanding of empirical higher-order systems, and contributes to the increasing interest in dimensionality reduction of such networks[48, 49, 53]. The main advantage of our framework is that it builds on a consolidated formalism that is firmly grounded on the statistical physics of strongly correlated systems. As in the case of multilayer systems[33], we think that it is remarkable that it is possible to tackle such challenges by capitalizing on a formal analogy between quantum and higher-order systems, which can be further exploited to gain novel insights into the structural and functional organization of complex systems.

Methods

References

• Boccalettietal. [2006]S.Boccaletti, V.Latora,Y.Moreno, M.Chavez,andD.-U.Hwang,Complex networks: Structure and dynamics,Phys. Rep.424,175 (2006).
• Barabásietal. [2011]A.-L.Barabási, N.Gulbahce,andJ.Loscalzo,Network medicine: anetwork-based approach to human disease,Nat. Rev. Genet.12,56(2011).
• BullmoreandSp*rns [2012]E.BullmoreandO.Sp*rns,The economy of brainnetwork organization,Nat. Rev. Neurosci.13,336 (2012).
• Pastor-Satorrasetal. [2015]R.Pastor-Satorras, C.Castellano, P.VanMieghem,andA.Vespignani,Epidemic processes incomplex networks,Rev. Mod. Phys.87,925 (2015).
• Ciminietal. [2019]G.Cimini, T.Squartini,F.Saracco, D.Garlaschelli, A.Gabrielli,andG.Caldarelli,The statistical physics of real-world networks,Nat. Rev. Phys.1,58 (2019).
• DeDomenico [2023]M.DeDomenico,More is different inreal-world multilayer networks,Nat. Phys.19,1247 (2023).
• Lambiotteetal. [2019]R.Lambiotte, M.Rosvall,andI.Scholtes,From networks to optimal higher-ordermodels of complex systems,Nat. Phys.15,313 (2019).
• Battistonetal. [2020]F.Battiston, G.Cencetti,I.Iacopini, V.Latora, M.Lucas, A.Patania, J.-G.Young,andG.Petri,Networksbeyond pairwise interactions: Structure and dynamics,Phys. Rep.874,1 (2020).
• Battistonetal. [2021]F.Battiston, E.Amico,A.Barrat, G.Bianconi, G.Ferrazde Arruda, B.Franceschiello, I.Iacopini, S.Kéfi, V.Latora, Y.Moreno, etal.,The physics of higher-order interactions in complex systems,Nat. Phys.17,1093 (2021).
• Rosasetal. [2022]F.E.Rosas, P.A.Mediano,A.I.Luppi, T.F.Varley, J.T.Lizier, S.Stramaglia, H.J.Jensen,andD.Marinazzo,Disentangling high-order mechanisms and high-order behaviours incomplex systems,Nat. Phys.18,476 (2022).
• Petrietal. [2014]G.Petri, P.Expert,F.Turkheimer, R.Carhart-Harris, D.Nutt, P.J.Hellyer,andF.Vaccarino,hom*ological scaffolds of brain functional networks,J. R. Soc. Interface11,20140873 (2014).
• Santoroetal. [2023]A.Santoro, F.Battiston,M.Lucas, G.Petri,andE.Amico,Higher-order connectomics of human brain function reveals localtopological signatures of task decoding, individual identification, andbehavior,bioRxiv,2023.12.04.569913 (2023).
• Levineetal. [2017]J.M.Levine, J.Bascompte,P.B.Adler,andS.Allesina,Beyond pairwise mechanisms of species coexistencein complex communities,Nature546,56 (2017).
• Pataniaetal. [2017]A.Patania, G.Petri,andF.Vaccarino,The shape of collaborations,EPJ Data Science6,1 (2017).
• Cencettietal. [2021]G.Cencetti, F.Battiston,B.Lepri,andM.Karsai,Temporal properties of higher-order interactions in socialnetworks,Scientific Reports11,1 (2021).
• Iacopinietal. [2019]I.Iacopini, G.Petri,A.Barrat,andV.Latora,Simplicial models of social contagion,Nat. Commun.10,1 (2019).
• Zhangetal. [2023]Y.Zhang, M.Lucas,andF.Battiston,Higher-order interactions shape collectivedynamics differently in hypergraphs and simplicial complexes,Nat. Commun.14,1605 (2023).
• SkardalandArenas [2020]P.S.SkardalandA.Arenas,Higher order interactionsin complex networks of phase oscillators promote abrupt synchronizationswitching,Commun. Phys.3,1 (2020).
• Millánetal. [2020]A.P.Millán, J.J.Torres,andG.Bianconi,Explosive higher-orderkuramoto dynamics on simplicial complexes,Phys. Rev. Lett.124,218301 (2020).
• Ferraz de Arrudaetal. [2023]G.Ferraz de Arruda, G.Petri, P.M.Rodriguez,andY.Moreno,Multistability,intermittency, and hybrid transitions in social contagion models onhypergraphs,Nat. Commun.14,1 (2023).
• Alvarez-Rodriguezetal. [2021]U.Alvarez-Rodriguez, F.Battiston, G.F.deArruda, Y.Moreno,M.Perc,andV.Latora,Evolutionary dynamics of higher-order interactions insocial networks,Nat. Hum. Behav.5,586 (2021).
• Contiscianietal. [2022]M.Contisciani, F.Battiston,andC.DeBacco,Inference of hyperedgesand overlapping communities in hypergraphs,Nat. Commun.13,7229 (2022).
• Lotitoetal. [2022]Q.F.Lotito, F.Musciotto,A.Montresor,andF.Battiston,Higher-order motif analysis in hypergraphs,Commun. Phys.5,79 (2022).
• Bensonetal. [2018]A.R.Benson, R.Abebe,M.T.Schaub, A.Jadbabaie,andJ.Kleinberg,Simplicial closure and higher-order link prediction,Proc. Natl. Acad. Sci. U.S.A.115,E11221 (2018).
• Chodrow [2020]P.S.Chodrow,Configuration models ofrandom hypergraphs,J. Complex Netw.8,cnaa018 (2020).
• Landryetal. [2023]N.W.Landry, M.Lucas,I.Iacopini, G.Petri, A.Schwarze, A.Patania,andL.Torres,XGI: A Python package for higher-order interactionnetworks,J. Open Source Softw.8,5162 (2023).
• Lotitoetal. [2023]Q.F.Lotito, M.Contisciani,C.DeBacco, L.DiGaetano, L.Gallo, A.Montresor, F.Musciotto, N.Ruggeri,andF.Battiston,Hypergraphx: a library for higher-order network analysis,J. Complex Netw.11,cnad019 (2023).
• Praggastisetal. [2023]B.Praggastis, S.Aksoy,D.Arendt, M.Bonicillo, C.Joslyn, E.Purvine, M.Shapiro,andJ.Y.Yun,HyperNetX:A Python package for modeling complex network data as hypergraphs,arXiv10.48550/arXiv.2310.11626(2023).
• Muchaetal. [2010]P.J.Mucha, T.Richardson,K.Macon, M.A.Porter,andJ.-P.Onnela,Community structure in time-dependent, multiscale, andmultiplex networks,Science328,876 (2010).
• HolmeandSaramäki [2012]P.HolmeandJ.Saramäki,Temporal networks,Phys. Rep.519,97 (2012).
• Battistonetal. [2014]F.Battiston, V.Nicosia,andV.Latora,Structural measures for multiplexnetworks,Phys. Rev. E89,032804 (2014).
• DeDomenicoetal. [2013]M.DeDomenico, A.Solé-Ribalta, E.Cozzo, M.Kivelä,Y.Moreno, M.A.Porter, S.Gómez,andA.Arenas,Mathematical formulation of multilayer networks,Phys. Rev. X3,041022 (2013).
• DeDomenicoetal. [2015a]M.DeDomenico, V.Nicosia, A.Arenas,andV.Latora,Structural reducibility of multilayernetworks,Nat. Commun.6,6864 (2015a).
• GhavasiehandDeDomenico [2020]A.GhavasiehandM.DeDomenico,Enhancing transportproperties in interconnected systems without altering their structure,Phys. Rev. Res.2,013155 (2020).
• DeDomenicoandBiamonte [2016]M.DeDomenicoandJ.Biamonte,Spectral Entropiesas Information-Theoretic Tools for Complex Network Comparison,Phys. Rev. X6,041062 (2016).
• RosvallandBergstrom [2007]M.RosvallandC.T.Bergstrom,Aninformation-theoretic framework for resolving community structure in complexnetworks,Proc. Natl. Acad. Sci. U.S.A.104,7327 (2007).
• DeDomenicoetal. [2015b]M.DeDomenico, A.Lancichinetti, A.Arenas,andM.Rosvall,Identifying modular flowson multilayer networks reveals highly overlapping organization ininterconnected systems,Phys. Rev. X5,011027 (2015b).
• SantoroandNicosia [2020]A.SantoroandV.Nicosia,Algorithmic complexity ofmultiplex networks,Phys. Rev. X10,021069 (2020).
• Ghavasiehetal. [2020]A.Ghavasieh, C.Nicolini,andM.DeDomenico,Statistical physicsof complex information dynamics,Phys. Rev. E102,052304 (2020).
• Lucasetal. [2020]M.Lucas, G.Cencetti,andF.Battiston,Multiorder Laplacian forsynchronization in higher-order networks,Phys. Rev. Res.2,033410 (2020).
• WallaceandBoulton [1968]C.S.WallaceandD.M.Boulton,An information measurefor classification,The Computer Journal11,185 (1968).
• Ghavasiehetal. [2021]A.Ghavasieh, S.Bontorin,O.Artime, N.Verstraete,andM.DeDomenico,Multiscale statistical physics of the pan-viralinteractome unravels the systemic nature of sars-cov-2 infections,Commun. Phys.4,1 (2021).
• Nicolinietal. [2020]C.Nicolini, G.Forcellini, L.Minati,andA.Bifone,Scale-resolved analysis of brainfunctional connectivity networks with spectral entropy,NeuroImage211,116603 (2020).
• GhavasiehandDeDomenico [2024]A.GhavasiehandM.DeDomenico,Diversity ofinformation pathways drives sparsity in real-world networks,Nat. Phys.20,1 (2024).
• Villegasetal. [2023]P.Villegas, T.Gili,G.Caldarelli,andA.Gabrielli,Laplacian renormalization group for heterogeneousnetworks,Nat. Phys.19,445 (2023).
• Gambuzzaetal. [2021]L.V.Gambuzza, F.DiPatti,L.Gallo, S.Lepri, M.Romance, R.Criado, M.Frasca, V.Latora,andS.Boccaletti,Stability of synchronization in simplicial complexes,Nat. Commun.12,1255 (2021).
• Note [1]In general, a hypergraph does not need to have hyperedges atevery order below $D$, unless it is a simplicial complex. If there is nohyperedge of order $d$, both the Laplacian and the average degree in Eq.2 vanish and the result is undefined. In those cases,the sum thus needs to be taken over all orders below $D$ that exist:$\mathcal{D}=\{d\leq D:\langle K^{(d)}\rangle>0\}$.
• Neuhäuseretal. [2023]L.Neuhäuser, M.Scholkemper, F.Tudisco,andM.T.Schaub,Learning the effectiveorder of a hypergraph dynamical system,arXiv (2023),2306.01813 .
• Nurissoetal. [2024]M.Nurisso, M.Morandini,M.Lucas, F.Vaccarino, T.Gili,andG.Petri,Higher-order Laplacian Renormalization,arXiv10.48550/arXiv.2401.11298(2024),2401.11298 .
• Bicketal. [2016]C.Bick, P.Ashwin,andA.Rodrigues,Chaos in generically coupled phase oscillatornetworks with nonpairwise interactions,Chaos26,094814 (2016).
• Jiangetal. [2018]J.Jiang, Z.-G.Huang,T.P.Seager, W.Lin, C.Grebogi, A.Hastings,andY.-C.Lai,Predicting tipping points in mutualistic networks through dimensionreduction,Proc. Natl. Acad. Sci. U.S.A.115,E639 (2018).
• Laurenceetal. [2019]E.Laurence, N.Doyon,L.J.Dubé,andP.Desrosiers,Spectral Dimension Reduction of ComplexDynamical Networks,Phys. Rev. X9,011042 (2019).
• Thibeaultetal. [2024]V.Thibeault, A.Allard,andP.Desrosiers,The low-rank hypothesis of complexsystems,Nat. Phys.,1(2024).
• MerchanandNemenman [2016]L.MerchanandI.Nemenman,On the Sufficiencyof Pairwise Interactions in Maximum Entropy Models of Networks,J. Stat. Phys.162,1294 (2016).
• Shimazakietal. [2015]H.Shimazaki, K.Sadeghi,T.Ishikawa, Y.Ikegaya,andT.Toyoizumi,Simultaneous silence organizes structured higher-order interactionsin neural populations,Sci. Rep.5,9821 (2015).
• Schneidmanetal. [2003]E.Schneidman, S.Still,M.J.Berry,andW.Bialek,Network Information and ConnectedCorrelations,Phys. Rev. Lett.91,238701 (2003).
• Amari [2001]S.-I.Amari,Information geometry onhierarchy of probability distributions,IEEE Trans. Inf. Theory47,1701 (2001).
• Robiglioetal. [2024]T.Robiglio, M.Neri,D.Coppes, C.Agostinelli, F.Battiston, M.Lucas,andG.Petri,Synergistic signatures of group mechanisms in higher-ordersystems,arXiv:2401.1158810.48550/arXiv.2401.11588 (2024).
• GhavasiehandDeDomenico [2022]A.GhavasiehandM.DeDomenico,Generalized networkdensity matrices for analysis of multiscale functional diversity,arXiv preprint arXiv:2210.1670110.1103/PhysRevE.107.044304 (2022).
• Sinhaetal. [2015]A.Sinha, Z.Shen,Y.Song, H.Ma, D.Eide, B.-J.P.Hsu,andK.Wang,An overview of microsoftacademic service (MAS) and applications,inProceedings of the 24th International Conference on World Wide Web(ACM Press,2015).
• Barratetal. [2013]A.Barrat, C.Cattuto,V.Colizza, F.Gesualdo, L.Isella, E.Pandolfi, J.F.Pinton, L.Ravà, C.Rizzo,M.Romano, etal.,Empirical temporal networksof face-to-face human interactions,Eur. Phys. J. Spec. Top.222,1295 (2013).
• soc [2008]SocioPatterns: a collection of contacts datasets (2008),accessed: 2023-08-19.
• Kan [2015]W.Kan,What’s cooking? (2015).