SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Notice that this simplified Mixed Central Moment operator expression resembles the original Lancaster interaction in Equation (2) using the unique  $*$  notation'. Notice that the resulting terms after expanding Equation (2) are distinctive with coefficient  $(-1)^{|\pi_l| - 1}$  depending on the except we have multiple occurrences of  $\hat{0}$ . This is due to the fact that  $\mathbb{P}_i^*$  is the same as  $\mathbb{P}_i^*$ . For example, when  $d = 4$ , instead of direct product of the marginals  $\mathbb{P}_1\mathbb{P}_2\mathbb{P}_3\mathbb{P}_4$  that accounts for  $\hat{0}$ , there are also four other terms  $\mathbb{P}_1^*\mathbb{P}_2\mathbb{P}_3\mathbb{P}_4$ ,  $\mathbb{P}_1\mathbb{P}_2^*\mathbb{P}_3\mathbb{P}_4$ ,  $\mathbb{P}_1\mathbb{P}_2\mathbb{P}_3\mathbb{P}_4^*$ ,  $\mathbb{P}_1\mathbb{P}_2\mathbb{P}_3\mathbb{P}_4^*$  that are also equal to  $\hat{0}$ . Note that these terms have different signs comparing with  $\mathbb{P}_1\mathbb{P}_2\mathbb{P}_3\mathbb{P}_4$  because of  $\mathbb{P}_i^*$  and therefore result in  $(-1)^{d - 1}(d - 1)$  in the Möbius function.
Computed immediately from the estimated norm of Mixed Central Moment operator, the Lancaster interaction estimator is
$$
\begin{array}{l} | | \hat {\mathcal {M}} _ {d} | | _ {H S} ^ {2} = \left\langle \frac {1}{n} \sum_ {a = 1} ^ {n} \prod_ {i = 1} ^ {d} \tilde {\phi} ^ {i} (x _ {a} ^ {i}), \frac {1}{n} \sum_ {b = 1} ^ {n} \prod_ {i = 1} ^ {d} \tilde {\phi} ^ {i} (x _ {b} ^ {i}) \right\rangle \\ = \frac {1}{n ^ {2}} \sum_ {a = 1} \sum_ {b = 1} \prod_ {i = 1} ^ {d} \left\langle \tilde {\phi} ^ {i} \left(x _ {a} ^ {i}\right), \tilde {\phi} ^ {i} \left(x _ {b} ^ {i}\right) \right\rangle \\ = \frac {1}{n ^ {2}} \sum_ {a = 1} ^ {n} \sum_ {b = 1} ^ {n} \prod_ {i = 1} ^ {d} \tilde {K} _ {a b} ^ {i}, \\ \end{array}
$$
where  $\tilde{\phi} (\cdot) = \phi (\cdot) - \mathbb{E}[\phi (\cdot)]$
A.4 Proof of Proposition 4
This can be completed by manual expansion and then equating the terms. Alternatively we can prove it more easily by the relationship of Lancaster interaction and Streitberg interaction in Lemma 1.
$$
\begin{array}{l} \mathcal {M} _ {b} = \mathbb {E} \left\{\prod_ {i \in b} \left[ \phi^ {i} - \mathbb {E} [ \phi^ {i} ] \right] \right\} \\ \mathcal {K} _ {d} = \sum_ {\pi_ {s} \in \Pi (D)} \left(| \pi_ {s} | - 1\right)! (- 1) ^ {| \pi_ {s} | - 1} \prod_ {b \in \pi_ {s}} \mathcal {M} _ {b} \\ = \sum_ {\pi_ {s} \in \Pi (D)} \left(| \pi_ {s} | - 1\right)! (- 1) ^ {| \pi_ {s} | - 1} \prod_ {b \in \pi_ {s}} \mathbb {E} \left\{\prod_ {i \in b} \left[ \phi^ {i} - \mathbb {E} [ \phi^ {i} ] \right] \right\}. \\ \end{array}
$$
A.5 Proof of Proposition 5
Similar to the norm of the Mixed Central Moment operator, the norm of Mixed Cumulant operator can be obtained by simply squaring. Note that this can be computed directly using the product lattice discussed in Section 4.2.
$$
\begin{array}{l} | | \hat {\mathcal {K}} _ {d} | | _ {H S} ^ {2} \\ = \left\langle \sum_ {\pi_ {s}} (| \pi_ {s} | - 1)! (- 1) ^ {| \pi_ {s} | - 1} \frac {1}{n ^ {| \pi_ {s} |}} \sum_ {\{b _ {i} \}} \prod_ {i = 1} ^ {d} \tilde {\phi} ^ {i} (x _ {b _ {i}} ^ {i}), \sum_ {\pi_ {s} ^ {\prime}} (| \pi_ {s} ^ {\prime} | - 1)! (- 1) ^ {| \pi_ {s} ^ {\prime} | - 1} \frac {1}{n ^ {| \pi_ {s} ^ {\prime} |}} \sum_ {\{b _ {i} ^ {\prime} \}} \prod_ {i = 1} ^ {d} \tilde {\phi} ^ {i} (x _ {b _ {i} ^ {\prime}} ^ {i}) \right\rangle \\ = \sum_ {\pi_ {s}, \pi_ {s} ^ {\prime}} (| \pi_ {s} | - 1)! (| \pi_ {s} ^ {\prime} | - 1)! (- 1) ^ {(| \pi_ {s} | - 1) + (| \pi_ {s} | - 1)} \frac {1}{n ^ {| \pi_ {s} | + | \pi_ {s} ^ {\prime} |}} \left\langle \sum_ {\{b _ {i} \}} \prod_ {i = 1} ^ {d} \tilde {\phi} ^ {i} (x _ {b _ {i}} ^ {i}), \sum_ {\{b _ {i} ^ {\prime} \}} \prod_ {i = 1} ^ {d} \tilde {\phi} ^ {i} (x _ {b _ {i} ^ {\prime}} ^ {i}) \right\rangle \\ = \sum_ {\pi_ {s}, \pi_ {s} ^ {\prime}} (| \pi_ {s} | - 1)! (| \pi_ {s} ^ {\prime} | - 1)! (- 1) ^ {| \pi_ {s} | + | \pi_ {s} |} \frac {1}{n ^ {| \pi_ {s} | + | \pi_ {s} ^ {\prime} |}} \sum_ {\{b _ {i} \}} \sum_ {\{b _ {i} ^ {\prime} \}} \prod_ {i = 1} ^ {d} \tilde {K} _ {b _ {i}, b _ {i} ^ {\prime}} ^ {i} \\ \end{array}
$$
B Product Lattice
The partition lattice is reduced to an upper semilattice after centring. When  $d = 4$ , the reduced lattice only contains  $\mathbb{P}_{1234}$ ,  $\mathbb{P}_{12}\mathbb{P}_{34}$ ,  $\mathbb{P}_{13}\mathbb{P}_{24}$  and  $\mathbb{P}_{14}\mathbb{P}_{23}$  which are the 4 partitions with no singletons. Given the two arbitrary lattices  $\mathcal{L}$  and  $\mathcal{L}'$ , the product lattice formulated using the Cartesian product of  $S \times S'$  with product order can be defined. Given two elements  $(\sigma, \sigma')$ ,  $(\pi, \pi') \in \mathcal{L} \times \mathcal{L}'$ ,  $(\sigma, \sigma') \leq (\pi, \pi')$  if and only if  $\sigma \leq \pi$  and  $\sigma' \leq \pi'$ . In our case specifically,  $\mathcal{L}$  and  $\mathcal{L}'$  are the identical reduced partition lattices of  $d$  variables after centring, and the elements in the product lattice are analogous to the inner products when computing the kernel-based estimator. Below we show the product lattice at  $d = 4$ . Therefore, it can be utilised to directly compute the coefficients of the estimator in Proposition 5. Note that the order within each element is not particularly informative in this case as the kernel matrices are symmetric.
Product lattice
d = 4
Figure 4: Product lattice for  $d = 4$ . The two lines and squares represent the non-singleton factorisations with two blocks and one block. Dark green and dark red highlight the elements in  $\mathcal{L}$ , whilst light green and light red are used for the elements in  $\mathcal{L}'$ .
The resulting estimator using centred kernels when  $d = 4$  is
$$
\begin{array}{l} | | \hat {\mathcal {K}} _ {4} | | _ {\mathcal {H S}} ^ {2} = \frac {1}{n ^ {2}} \sum_ {i j} \tilde {K} _ {i j} ^ {1} \tilde {K} _ {i j} ^ {2} \tilde {K} _ {i j} ^ {3} \tilde {K} _ {i j} ^ {4} \\ - \frac {2}{n ^ {3}} \sum_ {i j k} \tilde {K} _ {i j} ^ {1} \tilde {K} _ {i j} ^ {2} \tilde {K} _ {i k} ^ {3} \tilde {K} _ {i k} ^ {4} - \frac {2}{n ^ {3}} \sum_ {i j k} \tilde {K} _ {i j} ^ {1} \tilde {K} _ {i k} ^ {2} \tilde {K} _ {i j} ^ {3} \tilde {K} _ {i k} ^ {4} - \frac {2}{n ^ {3}} \sum_ {i j k} \tilde {K} _ {i j} ^ {1} \tilde {K} _ {i k} ^ {2} \tilde {K} _ {i k} ^ {3} \tilde {K} _ {i j} ^ {4} \\ + \frac {1}{n ^ {4}} \sum_ {i j k l} \tilde {K} _ {i j} ^ {1} \tilde {K} _ {i j} ^ {2} \tilde {K} _ {k l} ^ {3} \tilde {K} _ {k l} ^ {4} + \frac {1}{n ^ {4}} \sum_ {i j k l} \tilde {K} _ {i j} ^ {1} \tilde {K} _ {k l} ^ {2} \tilde {K} _ {i j} ^ {3} \tilde {K} _ {k l} ^ {4} + \frac {1}{n ^ {4}} \sum_ {i j k l} \tilde {K} _ {i j} ^ {1} \tilde {K} _ {k l} ^ {2} \tilde {K} _ {k l} ^ {3} \tilde {K} _ {i j} ^ {4} \\ - \frac {2}{n ^ {4}} \sum_ {i j k l} \tilde {K} _ {i j} ^ {1} \tilde {K} _ {i l} ^ {2} \tilde {K} _ {k j} ^ {3} \tilde {K} _ {k l} ^ {4} - \frac {2}{n ^ {4}} \sum_ {i j k l} \tilde {K} _ {i j} ^ {1} \tilde {K} _ {i l} ^ {2} \tilde {K} _ {k l} ^ {3} \tilde {K} _ {k j} ^ {4} - \frac {2}{n ^ {4}} \sum_ {i j k l} \tilde {K} _ {i j} ^ {1} \tilde {K} _ {k l} ^ {2} \tilde {K} _ {i l} ^ {3} \tilde {K} _ {k j} ^ {4} \\ \end{array}
$$
Notice that a few terms are summed up twice. This is due to the symmetry in the product lattice we described above.
C Composite Null Hypothesis
The permutation test strategy we propose for Streitberg interaction and Lancaster interaction is carried out via multiple sub-hypotheses, and the rejection of the interaction is confirmed once all the sub-hypotheses are rejected. Instead of having to consider all possible factorisations  $(B_d)$  for the sub-hypotheses, by considering the lattice structure we only need to perform  $(2^{d-1} - 1)$  sub-tests of factorisations corresponding to the 2nd level of the partition lattice, since all other sub-tests are consequences of these.
In the case of rejecting the Streitberg interaction, it won't necessarily be the case that all sub-hypotheses are rejected, allowing us to derive a more detailed understanding of how the joint distribution is factorised. For example, in the 7 sub-hypotheses for the Streitberg interaction when  $d = 4$ , there is a clear difference between the number of rejections when the ground truth factorisations are  $\mathbb{P}_1\mathbb{P}_{234}$  or  $\mathbb{P}_1\mathbb{P}_2\mathbb{P}_{34}$ . The former results in 6 rejections while the latter results in only 4 rejections because  $\mathbb{P}_1\mathbb{P}_2\mathbb{P}_{34}$  is the mutual refinement of  $\mathbb{P}_{12}\mathbb{P}_{34}$ ,  $\mathbb{P}_1\mathbb{P}_{234}$  and  $\mathbb{P}_2\mathbb{P}_{134}$ . Hence, even though the vanishing conditions of Streitberg interaction are not 'if and only if', we can still narrow down the range of the possible ground truth factorisations using the rejected sub-hypotheses.
The operations can be easily explained using the lattice. Before performing any sub-test,  $\Pi(D)$  is the space of all possible ground truth factorisation. Whenever we reject a sub-hypothesis, the refinements of the associated partition  $\pi$  gets eliminated. For example, if we reject  $\mathbb{P}_1\mathbb{P}_{234}$ , then we also automatically reject  $\mathbb{P}_1\mathbb{P}_2\mathbb{P}_{34}$ ,  $\mathbb{P}_1\mathbb{P}_3\mathbb{P}_{24}$ ,  $\mathbb{P}_1\mathbb{P}_4\mathbb{P}_{23}$  and  $\mathbb{P}_1\mathbb{P}_2\mathbb{P}_3\mathbb{P}_4$ . In other words, partitions in the interval lattice  $[\hat{0}, \pi]$  are eliminated. The remaining partitions until there are no further rejections are the only possible choices and also form a lattice. Again we illustrate this idea using the partition lattice of 4 variables (see Figure 5).
Figure 5: Remaining factorisations with partial rejection of sub-hypotheses for  $d = 4$ . The greyed out partitions represent the eliminated factorisations due to the rejections in the sub-tests. A dashed line between two partitions indicates the refinement ordering (only shown for those not rejected). (a) In the case that 4 of the sub-hypotheses are rejected, all of their refinements are automatically eliminated. The remaining elements form an interval lattice and is isomorphic to the partition lattice of 3 variables. (b) If 6 of the sub-hypotheses can be rejected, the ground truth can only be either the factorisation remaining or non-factorisable. Similarly this is isomorphic to the partition lattice of two variables.
Remark: The simplification of the operators (computation of the test statistics) can be reflected in a collapsed partition lattice without singletons. However, one should not confuse this with the full lattice that is associated with the test procedures, i.e. one should always consider the full partition lattice to figure out the possible configurations of the ground truth factorisation based on what we outlined above. The two lattices have the same form however they have completely different implications. The lattice corresponding to test procedures can only be reduced as a result of the rejections. For example, 7 sub-tests are needed instead of just 3 tests that involve the factorisations without singletons when  $d = 4$ .
D Möbius Inversion
Instead of deriving the Möbius function as the inverse of the Zeta function, it can also be computed using the expression below [37],
$$
\mu (\sigma , \pi) = \left\{ \begin{array}{l l} 1 & \text {i f} \quad \sigma = \pi \\ - \sum_ {\sigma \leq \rho <   \pi} \mu (\sigma , \sigma) & \text {i f} \quad \sigma <   \pi \\ 0 & \text {o t h e r w i s e} \end{array} \right..
$$
We can check the validity of this formula by computing the Möbius on the trivial partition lattice associated with joint independence, which contains two elements,  $\hat{0}$  and  $\hat{1}$ ,
$$
\mu (\hat {1}, \hat {1}) = 1
$$
$$
\mu (\hat {0}, \hat {1}) = - \mu (\hat {1}, \hat {1}) = - 1,
$$
$$
\begin{array}{l} \Delta_ {I} ^ {d} \mathbb {P} = \mu (\hat {1}, \hat {1}) \mathbb {P} _ {1 \dots d} + \mu (\hat {0}, \hat {1}) \prod_ {i = 1} ^ {d} \mathbb {P} _ {i} \\ = \mathbb {P} _ {1 \dots d} - \prod_ {i = 1} ^ {d} \mathbb {P} _ {i}. \\ \end{array}
$$
E Links to Simplicial Complexes
A  $k$ -simplex  $\theta$  is a set of  $k + 1$  vertices  $\theta = [p_0, \dots, p_k]$ , i.e., a 1-simplex is a line, a 2-simplex is a triangle, 3-simplex is a tetrahedron, etc. A simplicial complex  $\Theta$  is a collection of simplices that satisfy two conditions: (i) if  $\theta \in \Theta$ , then all the sub-simplices  $v \subset \theta$  built from subsets of  $\theta$  are also contained in  $\Theta$ ; and (ii) the non-empty intersection of two simplices  $\theta_1, \theta_2 \in \Theta$  is a sub-simplex of both  $\theta_1$  and  $\theta_2$ . The first condition makes the inclusion ordering in the subset lattice natural for simplicial complexes.
Below we have the Zeta matrix in Figure 6 and Möbius matrix in Figure 7 to illustrate that the boundary operators (with no directionality) of  $(d - 1)$ -simplex can be founded as block matrices within them. The subset lattice doesn't include the non-simplicial terms and we have shown numerically that these terms are essentially in the synthetic experiment when  $\mathbb{P}_{1234} = \mathbb{P}_{12}\mathbb{P}_{34}$ .
Figure 6: Zeta matrix for the partition lattice of 4 variables. Singletons in the partial factorisations are omitted for simplicity. The matrix can be decomposed into small blocks which correspond to the boundary operators in 3-simplex. Here the two red blocks represent the Edge-to-Face operator and Face-to-Tetrahedron operator. The blue block corresponds to the non-simplicials, partitions with no singletons.
Figure 7: Möbius matrix for the partition lattice of 4 variables. Comparing with the Zeta matrix above we see that the block structure is preserved. Each block can be obtained using the Möbius inversion on the incidence matrix of its own partially ordered set, e.g.  $\{12|34, 13|24, 14|23, 12|3|4, 13|2|4, 14|2|3, 23|1|4, 24|1|3, 34|1|2\}$  forms a partially ordered set. The values are always  $-1$  since the partially order set only contains the elements from two neighbouring levels.
F Interplay of Test Power and Sample Size
To explore the effect of the different sample size on test power, we have performed four experiments, each with three interaction strengths, to show how a decrease in the sample size affects the null rejection rate (Fig 8). In all cases, the Lancaster and Streitberg tests are able to accurately detect higher order interactions with as low as 50-100 samples (depending on the interaction proportion/strength), whilst dHSIC requires substantially more samples. This provides further experimental evidence that the proposed Streitberg interaction test has superior sensitivity, as compared to dHSIC, in detecting higher order interactions.
Figure 8: Decay of accuracy rates as the sample size is decreased in the XOR example. Independence tests (first and third columns); interaction tests (second and fourth columns). In all cases, dHSIC degrades substantially faster than Lancaster and Streitberg.
G Computational Complexity and Practical Limitations
G.1 Computational Complexity
The original time complexity for computing the Streitberg interaction estimator is  $\mathcal{O}(B_d^2 n^{2d})$  where  $B_{d}$  is the Bell number, which represents the number of partitions in a set of cardinality  $d$ . Hence  $B_{d}^{2}$  is the number of terms in the Streitberg interaction estimator. For fixed number of samples  $n$ , our lattice formulation allows us to reduce the number of terms in the mixed cumulant operator from  $B_{d}$  to  $F_{d}$ . Table 1 illustrates this reduction in the number of terms before and after eliminating the partitions with singletons. After this reduction, the complexity becomes  $\mathcal{O}(F_d^2 n^{2d})$ .
Table 1: Number of terms in the test statistics before/after eliminating the partitions with singletons.
No. of variables	4	5	6	7	8
Streitberg⇒(centred)	15⇒(4)	52⇒(11)	203⇒(41)	877⇒(162)	4140⇒(715)
Lancaster⇒(centred)	12⇒(1)	27⇒(1)	58⇒(1)	121⇒(1)	248⇒(1)
For fixed  $d$ , we can further optimise the time complexity by optimal contraction ordering, using the fact that two index sets are disjoint, such that  $\mathcal{O}(n^{2d})$  becomes  $\mathcal{O}(n^{min(|\pi_s|,|\pi_s'|) + 1})$ . This reduction for different  $d$  can be found in Table 2.
Table 2: Time complexity for  $d$ -order interaction estimators.
d-order	dHSIC	Lancaster⇒(optimised)	Streitberg⇒(optimised)
2-way	O(n2)	O(n2)	O(n2)
3-way	O(n2)	O(n2)	O(n2)
4-way	O(n2)	O(n8)⇒O(n2)	O(n8)⇒O(n3)
5-way	O(n2)	O(n10)⇒O(n2)	O(n10)⇒O(n3)
Similarly, the time complexity for computing the  $d$ -order Lancaster interaction estimator naively is  $\mathcal{O}(2^{d + 1}n^{2d})$ . By centring, i.e. eliminating the partitions with singletons, the only element left is  $\hat{1}$ . Therefore the Lancaster interaction estimator can be computed in  $\mathcal{O}(dn^2)$ .
G.2 Practical Considerations
The problem of detecting  $d$ -order interactions among a group of  $e$  variables is combinatorial, as it entails checking groups of  $d$  variables chosen from the  $e$  variables. Clearly, such combinatorial problems become infeasible as  $e$  and/or  $d$  become large. Recent work on real data from various application areas has focused on revealing any interactions beyond pairwise, i.e.,  $d > 2$ . It has been shown that interactions with  $d = 3, 4$  already make a significant difference to the analysis of network structure and network dynamics [2, 21, 55], highlighting the potential benefits to be gained from tests that detect high order interactions.
Many well-recognised and widely used theoretical approaches only have closed forms for  $d = 3$  [56, 57], and cannot be generalised to arbitrary  $d$ , since the number of terms in those approaches is related to the Dedekind number which becomes rapidly intractable [58]. In contrast, here we show that the Streitberg interaction can be explicitly defined for any  $d$ , and we have devised theoretical and computational strategies to reduce its computational cost via the lattice theory formulation.
In the case where high order interactions for a range of  $d$  are of interest, it is possible to discount the computation of certain high order interactions when some lower-order interactions are present. If all the lower order interactions are absent, then testing the  $d$ -order interaction is the same as testing the joint independence for  $d$  variables. For example when  $d = 3$  and all pairwise interactions are absent, then testing the Streitberg interaction reduces to testing joint independence of 3 variables. More generally, if we test the interactions bottom-up, i.e., recursively from the lower orders upwards, the expression for Streitberg becomes simpler whenever there is a lower-order independence. The simplification is possible because the Streitberg interaction can be rewritten as the sum of the differences between a factorisation and the product of the marginals due to the fact that the sum of Möbius coefficients is zero. Hence the presence of a lower order independence allows us to simplify the  $d$ -order interaction formula.
Alternatively, there are also cost-reducing simplifications if we do the tests top-down (i.e. starting from order  $d$  downwards), although the problem still remains combinatorial. If there are partial rejections for some tests involved in the  $d$ -order Streitberg test, we can narrow down the possible choices of the factorisation (as discussed in Appendix C). This allows us to eliminate the lower order factorisations that fall in the lattice branches of the rejected second level factorisations, thus reducing the total tests needed for the factorisation of the  $d$ -variables. Further simplifications are achieved by accounting for overlaps of the lattice branches.
All these observations can be taken into account in the construction of hypergraph representations, as further discussed below in Appendix H.
H Neuroimaging Data
H.1 Preprocessing
Preprocessing of neuroimaging data was performed following Luppi et al. [54]. We summarise the key steps here, and refer interested readers to the original paper for further details.
We started from the 'minimally preprocessed' release of the fMRI data from the Human Connectome Project [50, 51]. This data was preprocessed with bias field correction, functional realignment, motion
correction and spatial normalization to Montreal Neurological Institute (MNI-152) standard space with  $2\mathrm{mm}$  of isotropic resampling resolution. We then removed the first ten points in the time series to avoid transient effects introduced by the scanner. Finally, we further denoised the data with the anatomical CompCor method, which involves regressing out potential noise confounds (specifically, five principal components of white matter activity, five principal components of cerebrospinal fluid activity, and 12 motion parameters including head translation, rotation, and their temporal derivatives). All preprocessing steps were performed using the CONN toolbox (https://www.nitrc.org/projects/conn/), version 17f58. The resulting volume was parcellated according to the Schaefer-100 atlas [52] by spatially averaging across all voxels in the same region for each timestep.
Table 3: Time taken for experiments in Figure 3.
SOM	VIS	SAL	DAN	DMN	FPN	LIM	Random
2-way	1s	2s	1s	2s	2s	2s	1s	12s
3-way	12s	18s	7	13s	8s	8s	1s	13s
4-way	5m36s	5m	2m44s	2m31s	2m	2m10s	1s	2m41s
5-way	2h18m24s	1h30m9s	1h3m	54m40s	1h12m4s	27m16s	2s	29m
We report the computational time for all fMRI experiments in Table 3. All experiments carried out on a 2015 iMac with 4 GHz Quad-Core Intel Core i7 processor and 32 GB 1867 MHz DDR3 memory.
H.2 Analysis of Simple Hypergraphs
As an illustration of future lines of work, we hierarchically constructed hypergraphs from the identified  $d$ -order interactions. Specifically, individual regions are nodes and the high-order interactions are used to define hyperedges, incrementally including the interactions of increasing order. The resulting hypergraphs integrate information across orders of interaction, and can be analysed by computing structural properties of hypergraphs. As an example, we show the degree assortativity of the different hypergraphs in Fig 9. We find that almost all RSNs display a negative degree assortativity (but much less so than random), and only FPN shows positive degree assortativity for higher order hypergraphs. Future analyses of these hypergraphs will study the links between high-order interactions in brain activity and different functional areas.
Figure 9: Hypergraph analysis of the neuroimaging data. Percentage of high-order interactions in RSNs (left) and degree assortativity of hypergraphs constructed from the high-order interactions detected (right).
H.3 Alternative Hypergraphs: Emergent and Redundant High-Order Interactions
To interpret the presence of lower order interactions in larger cliques, one can alternatively build a hypergraph as follows: i) If a  $d$ -order hyperedge is detected and none of the lower order  $(d - 1)$ -order hyperedges are present then we say that this  $d$ -order hyperedge reflects a purely synergistic (or emergent) interaction between the  $d$  variables; ii) if the  $d$ -order and all the lower order interactions are present, then we say that the  $d$ -order interaction is purely redundant (and corresponds to a simplicial complex construction); iii) if some, but not all, lower order interactions are present (e.g., for 4 variables, only the 4-way interaction and one 3-way interaction are present), then both synergy and redundancy are present in this group of variables. Such a construction could offer an alternative,
statistically-motivated approach to computing synergy and redundancy, an important current topic of research in computational neuroscience, and its representation through hypergraphs.
I Code
This code for performing the interaction tests and the synthetic experiments is provided in this anonymous Github repository https://github.com/barahona-research-group/streitberg-interaction.git.