Networked dynamic systems with higher-order interactions: stability versus complexity

ABSTRACT The stability of complex systems is profoundly affected by underlying structures, which are often modeled as networks where nodes indicate system components and edges indicate pairwise interactions between nodes. However, such networks cannot encode the overall complexity of networked systems with higher-order interactions among more than two nodes. Set structures provide a natural description of pairwise and higher-order interactions where nodes are grouped into multiple sets based on their shared traits. Here we derive the stability criteria for networked systems with higher-order interactions by employing set structures. In particular, we provide a simple rule showing that the higher-order interactions play a double-sided role in community stability—networked systems with set structures are stabilized if the expected number of common sets for any two nodes is less than one. Moreover, although previous knowledge suggests that more interactions (i.e. complexity) destabilize networked systems, we report that, with higher-order interactions, networked systems can be stabilized by forming more local sets. Our findings are robust with respect to degree heterogeneous structures, diverse equilibrium states and interaction types.


Constructing community matrices
As shown in the main text, we focus on the largest real part among all eigenvalues of the community matrix, M .Specific interaction types in set structured systems determine the corresponding community matrices.Here we consider four interaction types (random, exploitative, mutualistic, and competitive) and show how to construct corresponding community matrices in set structured systems where species within the same set interact with probability C.This construction process can also be extended to heterogeneous and local symmetric structure communities.Our main parameters are: number of species, S, number of sets, G, number of sets each species belongs to, K, connectivity in each set, C.

Random community matrices
We first discuss the construction of random community matrices: (i) Each species is randomly assigned to K sets.We denote that g i = 1 if species i belongs to set g, g = 1, 2, • • • , G. (ii) For different species i and j, if g i g j = 1, we draw a random value p from a uniform distribution U [0, 1].(iii) If p ≤ C, we sample M g ij and M g ji from a normal distribution N (0, σ 2 ), respectively.Otherwise M g ij = M g ji = 0, g = 1, 2, • • • , G. (iv) For the community matrix M , we have M ij = G g=1 M g ij if i ̸ = j and we set diagonal terms For the corresponding random community matrices in unstructured systems, we construct them following the classical way shown in [1].

Exploitative community matrices
For exploitative community matrices, step (i) is identical to constructing random community matrices in S1.1.Subsequent steps are as follows: (ii) If g i g j = 1, for pair of interactions M g ij , M g ji i>j , we draw a random value p 1 from a uniform distribution Otherwise, we do the opposite.(v) if p 1 > C, M g ij and M g ji are set to be 0, For the corresponding exploitative community matrices in unstructured systems, we construct them following the classical way shown in [1].

Mutualistic community matrices
For mutualistic community matrices, step (i) is identical to constructing random community matrices in S1.1.Subsequent steps are as follows: (ii) If g i g j = 1, for pair of interactions (M g ij , M g ji ) i>j , we draw a random value p from a uniform distribution For the corresponding mutualistic community matrices in unstructured systems, we construct them following the classical way shown in [1].

Competitive community matrices
For competitive community matrices, steps (i), (ii), and (iv) are identical to constructing mutualistic community matrices in S1.

Stability estimation for different interaction types
Here we derive the stability criteria for different interaction types (random, exploitative, mutualistic, and competitive) in set structured systems.While our estimation of stability criteria in set structured systems is approximate, it offers valuable insights into the general trends of stability.Precisely quantifying the stability of such systems can be challenging, but our theoretical framework adeptly captures the underlying dynamics, enabling us to predict the stability of set structured systems.

Expected number of common sets for two species
We begin with calculating the expected number of common sets, H, for two randomly chosen, distinct species [2].For any species i and j, the number of sets, H ij , that they have in common is Then we have Since each species belongs to K sets, the possible number of sets two species have in common is To determine the specific form of H, we sum up the product of k and the corresponding probability for all possible values of k: Simplifying Eq. (S1), we have According to Eq. (S2), we obtain that the expected number of common sets for any two species is given by K 2 G .Moreover, we calculate the expected interaction times, T , for any two species Next, we derive the corresponding stability criteria for different interaction types in set structured systems.

Random
For random communities in set structured systems, we set the diagonal terms of M to be 0 in the initial.Following our discussion in S1.1 and S2.1, for any two species i and j (i ̸ = j), we have , where X ∼ N 0, σ 2 .Consequently, we obtain We further estimate the largest real part of the eigenvalues of M .According to circular law [1, 3], the eigenvalue distribution of M /(σ SC K 2 G ) satisfies that: for any eigenvalue λ, as S → ∞, we have According to Eq. (S3), the largest real part of the eigenvalues of M approximates to σ SC K 2 G when S is sufficiently large.
Next, we take the diagonal strength into account.As shown in S1.1, we set M ii = −d for any species i, and the largest real part of the eigenvalues of M approximates to σ SC K 2 G − d as S → ∞.Then we get the stability criterion for random cases where G is the number of sets and K is the number of sets each species belongs to.
Compared with the stability criterion for unstructured systems in the random case We find that the set structure stabilizes the networked system if

Exploitative
For exploitative communities, we also set the diagonal terms of M to be 0 first.Following our discussion in S1.3 and S2.1, for any two species i and j (i ̸ = j), we have , where X ∼ N 0, σ 2 .Consequently, we obtain Here we consider the following theorem [4]: for a matrix A, if its elements A ij satisfy that then the eigenvalues of A are uniformly distributed on an ellipse if S is sufficiently large where a = 1 − τ and b = 1 + τ .
According to Eqs. (S4), (S5), the eigenvalues of M /(σ SC K 2 G ) are distributed on the ellipse shown in Eq. (S6), where thus the largest real part of the eigenvalues of M approximates to σ SC K 2 G (1 − 2/π) as S → ∞.Taking the diagonal strength into account, we have that the largest real part of the eigenvalues of Then we get the stability criterion for exploitative cases Similar to the random case, the set structure stabilizes the networked system in the exploitative community if

Mutualistic
For mutualistic communities, we follow our discussion in S1.2 and S2.1 and have With increasing S, the row sum of M approximates to S j=1 According to Gershgorin circle theorem [3, 5], for sufficiently large S, the largest real part of the π .Next, we get the stability criterion for mutualistic cases Compared with the stability criterion for unstructured systems in mutualistic cases we obtain that the set structure stabilizes the networked system if G > K 2 .

Competitive
For competitive interactions, we have According to Allesina and Tang [1], we first consider the eigenvalue distribution of the matrix N , which is defined as where Furthermore, according the theorem in Eqs.(S4)-(S6), N is now an elliptic matrix.Thus, the largest real part of the eigenvalues of N can be approximated as Then, according to Allesina and Tang [1], we obtain the largest real part of the eigenvalues of M , λ 1 , which approximates to when S is sufficiently large.
According to Eqs. (S10), (S11), we get the stability criterion for competitive cases where F (x) satisfies that The expected number of sets any two species have in common,

Log-normal distribution
Supplemental Figure S8: ∆Stability of heterogeneous set structured and corresponding unstructured systems with mixed interaction types under log-normal distribution equilibrium abundance.We sample X i * from the log-normal distribution with log-mean 0 and log-standard deviation 0.02.Other parameters are the same as those in Fig. S6.The expected number of sets any two species have in common,

Half-normal distribution
Supplemental Figure S9: ∆Stability of heterogeneous set structured and corresponding unstructured systems with mixed interaction types under half-normal distribution equilibrium abundance.We sample X i * from the half-normal distribution |X|, where X ∼ N (1, σ 2 ) and σ = 0.02.Other parameters are the same as those in Fig. S6.
3. And we have: (iii) If p ≤ C, M g ij and M g ji are drawn from a negative half-normal distribution −|N 0, σ 2 |.Otherwise both M g ij and M g ji are 0, g = 1, 2, • • • , G.For the corresponding competitive community matrices in unstructured systems, we construct them following the classical way shown in [1].
of sets any two species have in common, Supplemental FigureS2: Confirmation of the simple rule, K 2 /G < in set structured systems with various average connectivity C. (a-c) The ∆Stability of set structured and corresponding unstructured systems with increasing K 2 /G in dense (C = 0.3), middle (C = 0.5), and sparse (C = 0.7) communities with different interaction types, respectively.We set the number of species S = 500.Other parameters are the same as those in Fig.3.Supplemental FigureS3: Confirmation of the simple rule with diverse equilibrium abundances.The ∆Stability of set structured and corresponding unstructured systems with diverse equilibrium abundance (log-normal and half-normal).(a-d) We sample X i * from the log-normal distribution with log-mean 0 and log-standard deviation 0.05.(e-h) We sample X i * from the half-normal distribution |X|, where X ∼ N (1, σ 2 ) and σ = 0.05.Other parameters are the same as those Fig.6e-h.Supplemental Figure S4: Confirmation of the simple rule with diverse equilibrium abundances in heterogeneous set structured systems.The ∆Stability of set structured and corresponding unstructured systems with diverse equilibrium abundance distributions (uniform, log-normal and half-normal) in heterogeneous communities.(a) We sample X i * from the uniform distribution on [0.95, 1.05].(b) We sample X i * from the log-normal distribution with log-mean 0 and log-standard deviation 0.02.(c) We sample X i * from the half-normal distribution |X|, where X ∼ N (1, σ 2) and σ = 0.02.We set the average connectivity C = 0.2.Other parameters are the same as those in Fig.6e-h. of sets any two species have in common, Supplemental FigureS5: ∆Stability of set structured and corresponding unstructured systems with mixed interaction types.Interaction types in each set are randomly chosen from random, exploitative, mutualistic, and competitive interactions.(a-f) ∆Stability of six different random compositions involving two interaction types.Other parameters are the same as those in Fig.6eh. of sets any two species have in common, Supplemental FigureS6: ∆Stability of heterogeneous set structured and corresponding unstructured systems with mixed interaction types.We set the average connectivity C = 0.2 in both heterogeneous set structured and unstructured systems.Other parameters are the same as those in Fig.S5. of sets any two species have in common,Uniform distributionSupplemental FigureS7: ∆Stability of heterogeneous set structured and corresponding unstructured systems with mixed interaction types under uniform distribution equilibrium abundance.We sample X i * from the uniform distribution on [0.95, 1.05].Other parameters are the same as those in Fig.S6.