## Introduction

Percolation1,2,3,4 is one of the most fundamental critical phenomena defined on networks. As such, it has attracted large interest in the literature5,6,7,8,9,10,11,12,13,14. Indeed by predicting the size of the giant component (GC) of a network when links are randomly damaged, percolation can be used for the establishment of the minimal requirements that a structural network should satisfy in order to support any type of interactive process. Despite the great success of percolation, ordinary percolation is unsuitable to describe real-world situations that occur in neuronal and climate networks when the connectivity of these networks changes in time.

Typically, the dynamics associated to percolation is the one of a cascading process where an initial failure propagates within a network possibly affecting its macroscopic connectedness. In the last decade, large scientific activity has been addressed to generalized percolation problems that capture cascades of failure events5,15,16,17,18,19,20 on multilayer networks21,22,23 where the damage propagates back and forth among the layers reaching a steady state at the end of the cascading process. In duplex networks, period-two oscillations can be observed in presence of competitive or antagonistic interactions24,25,26,27,28 among the different layers of the multiplex networks. However, this phenomenon seems to be restricted to duplex networks. Finally in damage and recovery models on multilayer networks26,29,30 aimed at getting insight for the robustness of complex critical infrastructures and financial systems, also more than two coexisting stable configurations of percolation have been observed.

An important question that arises from these works is whether percolation can capture more general time-dependent variations in the connectivity of a network. Here, we give a positive answer to this question and we show that higher-order interactions, and specifically triadic interactions, can turn percolation into a fully fledged dynamical process in which the order parameter undergoes period doubling and a route to chaos.

Higher-order networks are ubiquitous in nature31,32,33,34,35,36. Paradigmatic examples are the networks that describe brain activity, chemical reactions networks, and climate37,38,39,40,41. Higher-order interactions may profoundly change the physical properties of a dynamical process compared to those displayed by the same process occurring on a classic network of pairwise interactions. Examples include synchronization42,43,44,45, random walk dynamics46, contagion dynamics47,48,49,50,51,52 and game theory53. However little is known so far about percolation in presence of higher-order interactions51,54,55,56,57,58.

In this paper, we focus on a paradigmatic type of higher-order interactions named triadic interactions which occur when a node regulates the interaction between two other nodes. Regulation can be either positive, in the sense that the node facilitates the interaction, or negative, meaning that the regulator inhibits the interaction. Triadic interactions occur in ecosystems, where the competition between two species can be affected by the presence of a third species59,60,61. In neuronal networks, the interactions between neurons/glia is known to be triadic with glias modulating the synaptic interaction between neurons62. In climate networks of extreme rainfall events, triadic interactions can be used to explain the situations in which the network links are modulated by large-scale patterns, such as Rossby waves, which have a regulatory activity on climate inducing long-range synchronization of rainfall between Europe, Central Asia and even East Asia40. Finally in chemical reaction networks, generalized triadic interactions could model the action of enzymes as biological catalysts for biochemical reactions. While triadic interactions have received large attention in ecology and neuroscience, theoretical analyses of triadic interactions have investigated exclusively small-scale ecological systems59,60,61.

Here, we change perspective and study the role of triadic interactions in shaping macroscopic network properties. Specifically, we investigate how triadic interactions can change the critical and the dynamical properties of percolation. We combine percolation theory1,3 with the theory of dynamical systems63,64,65,66 to define triadic percolation, i.e., percolation in presence of signed triadic interactions. We show that in triadic percolation the GC of the network displays a highly non-trivial dynamics characterized by period doubling and a route to chaos. We use a general theory to demonstrate that the phase diagram of triadic percolation has fundamental differences with the phase diagram of ordinary percolation. While ordinary percolation displays a second-order phase transition, the phase diagram of triadic percolation is much richer and can be interpreted as an orbit diagram for the order parameter. Our theory is validated with extensive simulations on synthetic and real-world networks. These results reveal that in triadic percolation the GC of the network becomes a dynamical entity whose dynamics changes radically our understanding of percolation.

## Results

Triadic interactions (see Fig. 1) are higher-order interactions between nodes and links. They occur when a node regulates the interaction between two other nodes. The regulation can be either positive, in the sense that the node facilitates the interaction, or negative, meaning that the regulator inhibits the interaction. For instance, the presence of a third species can enhance or inhibit the interaction between two species; also, the presence of a glia can favor or inhibit the synaptic interactions between two neurons. Triadic interactions can be added to a simple structural network. However, triadic interactions can also be introduced on top of an hypergraph, when one node regulates the strength of an hyperedge, or on top of multilayer networks, where triadic interactions represent inter-layer interactions between the nodes of one layer and the links of other layer. For instance, an enzyme is a node that can regulate an hyperedge (i.e., a reaction between chemicals); neural networks and networks of glias form instead two layers of a multiplex network interacting via triadic interactions.

Let us now formulate the simplest example of higher-order networks with triadic interactions. This higher-order network can be modeled as the composition of two networks: the structural network and the regulatory network which encodes triadic interactions. The structural network $${{{{{{{\mathcal{A}}}}}}}}=(V,\,E)$$ is formed by the set of nodes V connected by the structural links in the set E. The regulatory network $${{{{{{{\mathcal{B}}}}}}}}=(V,\,E,\,W)$$ is a bipartite, signed network between the set of nodes V of the structural network and the set of structural links E, with nodes in V regulating links in E on the basis of the regulatory interactions, either positive or negative, specified in the set W. Given a regulated link, a node at the end of the regulatory interaction is called positive regulator if the regulatory interaction is positive and negative regulator if the regulatory interaction is negative. Note that the sign is an attribute of the regulatory interaction and not of the node that acts as regulator.

In the following we will focus on percolation on this model of network with triadic interactions, however our results can be easily extended to hypergraphs and multiplex networks with triadic interactions as well.

We define triadic percolation as the model in which the activity of the structural links is regulated by the triadic interactions and the activity of their regulator nodes. Conversely, the activity of the nodes is dictated by the connectivity of the network resulting after considering only the active links. In particular, we assume that the activity of nodes and links is changing in time leading to the triadic percolation process defined as follows. At time t = 0, every structural link is active with probability p0. We then iterate the following algorithm for each time step t ≥ 1:

• Step 1. Given the configuration of activity of the structural links at time t − 1, we define each node active if the node belongs to the GC of the structural network in which we consider only active links. The node is considered inactive otherwise.

• Step 2. Given the set of all active nodes obtained in step 1, we deactivate all the links that are connected at least to one active negative regulator node and/or that are not connected to any active positive regulator node. All the other links are deactivated with probability q = 1 − p.

Note that for p = p0 = 1 the model is deterministic. However, for p < 1 (and p0 < 1) the model is stochastic, i.e., the activity of the nodes does not uniquely define the activity of the links.

In the proposed triadic percolation, links can be dynamically turned on and off by the regulatory interactions. The model only makes minimal and justifiable assumptions while remaining general. The assumption that only nodes within the GC of the network are considered functioning/active is well accepted in the literature concerning network robustness1,5. Also, the regulatory rule chosen for deactivating the links is the minimal rule for treating both positive/negative regulations in a symmetric way: given suitable conditions the activation of a single positive regulator or the deactivation of a single negative regulator can turn the activity of a link on. Finally, the introduction of annealed stochastic effects, present for p < 1 (and p0 < 1), represents a simple way to account for the unavoidable randomness that can affect the activation/deactivation of the structural links in real scenarios.

Triadic percolation can lead to a highly non-trivial dynamics of the network connectivity. For instance Fig. 2 illustrates the phenomenon of network “blinking” with nodes of the network turning on and off periodically to form GCs of different size. As we will see, this dynamics emerges at the bifurcation transition indicating the onset of the period-two oscillations of the order parameter, but oscillations of longer period and also chaos is observed depending on the model’s parameters.

Here we establish the theory for triadic percolation that is able to predict the phase diagram of the model on random networks with triadic interactions.

We assume that the structural network $${{{{{{{\mathcal{A}}}}}}}}$$ is given and contains N nodes and 〈kN/2 structural links, with 〈k〉 indicating the average degree of the network. We consider structural networks given by individual instances of the configuration model. To this end, we first generate degree sequences by selecting random variables from the degree distribution π(k). We denote with ki the structural degree of node i.

To generate the regulatory network $${{{{{{{\mathcal{B}}}}}}}}$$, we assume that every node i has associated two degree values, namely the number of positive regulatory interactions $${\kappa }_{i}^{+}$$, and the number of negative regulatory interactions $${\kappa }_{i}^{-}$$. For simplicity we consider the case in which both $${\kappa }_{i}^{+}$$ and $${\kappa }_{i}^{-}$$ are chosen independently of the structural degree ki (see the SI for the extension to the correlated case). Each structural link is assigned the degrees $${\hat{\kappa }}_{\ell }^{+}$$ and $${\hat{\kappa }}_{\ell }^{-}$$ indicating the number of positive regulators and the number of negative regulators, respectively. In particular, nodes’ degrees are extracted at random from the distribution P(κ+, κ), and links’ degrees are randomly extracted from the distribution $$\hat{P}({\hat{\kappa }}^{+},\,{\hat{\kappa }}^{-})$$ here taken to be uncorrelated so that $$\hat{P}({\hat{\kappa }}^{+},\,{\hat{\kappa }}^{-})={\hat{P}}_{+}({\hat{\kappa }}^{+}){\hat{P}}_{-}({\hat{\kappa }}^{-})$$. Once degrees have been assigned to nodes and links, we establish the existence of a positive (+) or negative (−) regulatory interaction between the structural link and the node i with probability:

$${p}_{\ell,i}^{\pm }=\frac{{\kappa }_{i}^{\pm }{\hat{\kappa }}_{\ell }^{\pm }}{\langle {\kappa }^{\pm }\rangle N},$$
(1)

where 〈κ±〉 denotes the average of κ over all the nodes of the network. In the creation of regulatory interactions, we allow any pair (, i) to be connected either by a positive of by a negative regulatory interaction but not by both. Note that as long as the network $${{{{{{{\mathcal{B}}}}}}}}$$ is large and sparse the latter condition is not inducing significant correlations.

Let us now combine the theory of percolation with the theory of dynamical systems to derive the phase diagram of the considered uncorrelated scenario. Let us define S(t) as the probability that a node at the endpoint of a random structural link of the network $${{{{{{{\mathcal{A}}}}}}}}$$ is in the GC at time t. Moreover, let us indicate by R(t) the fraction of nodes in the GC at time t (or equivalently the probability that a node at the end of a regulatory link is active). Finally, $${p}_{L}^{(t-1)}$$ is the probability that a random structural link is active at time t. By putting $${p}_{L}^{(0)}={p}_{0}$$ indicating the probability that structural links are active at time t = 0, we have that for t > 0, as long as the network is locally tree like, S(t), R(t) and $${p}_{L}^{(t)}$$ are updated as:

$${S}^{(t)} =1-{G}_{1}\left(1-{S}^{(t)}{p}_{L}^{(t-1)}\right),\\ {R}^{(t)} =1-{G}_{0}\left(1-{S}^{(t)}{p}_{L}^{(t-1)}\right),\\ {p}_{L}^{(t)} =p{G}_{0}^{-}(1-{R}^{(t)})\left[1-{G}_{0}^{+}\left(1-{R}^{(t)}\right)\right],$$
(2)

where the first two equations implement Step 1, i.e., a bond-percolation model1 where links are retained with probability $${p}_{L}^{(t-1)}$$, and the third equation implements Step 2, i.e., the regulation of the links. Here the generating functions G0(x), G1(x) and $${{G}_{0}}^{\pm }(x)$$ are given by:

$$\begin{array}{c}{G}_{0}(x)=\mathop{\sum}\limits_{k}\pi (k){x}^{k},\quad {G}_{1}(x)=\mathop{\sum}\limits_{k}\pi (k)\frac{k}{\langle k\rangle }{x}^{k-1},\\ {{G}_{0}}^{\pm }(x)=\mathop{\sum}\limits_{{\kappa }_{\pm }}{\hat{P}}_{\!\!\pm }({\hat{\kappa }}^{\pm }){x}^{{\hat{\kappa }}^{\pm }}.\end{array}$$
(3)

Equation (2) for the percolation model regulated by triadic interactions can be formally written as the map65:

$${R}^{(t)}=f\left({p}_{L}^{(t-1)}\right),\quad {p}_{L}^{(t)}={g}_{p}\left({R}^{(t)}\right),$$
(4)

which can be further reduced to a unidimensional map R(t) = h(R(t−1)). The previous set of equations lead to the theoretical prediction for triadic percolation defined on structural networks generated according to the configuration model. This solution are of mean-field nature: while triadic percolation dynamics has many interacting degrees of freedom given by the activity of each node and each link, and is characterized by a stochastic dynamics for p < 1, Eq. (2) [or equivalently the map Eq. (4)] involves only three/two variables and are deterministic. As we will see, despite this approximations, the proposed theoretical approach provides a very accurate prediction of the behavior of triadic percolation.

In presence of negative interactions, triadic percolation displays a time-dependent order parameter, given by the active fraction of nodes R(t). The order parameter R(t) undergoes a period doubling and a route to chaos in the universality class of the logistic map for structural networks with arbitrary degree distribution π(k) and regulatory connectivity generated by Poisson distributions $$P({\hat{\kappa }}^{\pm })$$ (see SI and Supplementary Figs. 15 for details). Triadic percolation has a very rich dynamical nature and displays the emergence of both “blinking” oscillations and chaotic patterns of the giant component (see Fig. 3). “Blinking” refers to the intermittent switching on and off of two or more sets of nodes which leads to periodic oscillations of the order parameter. Chaos implies that at each time a different set and number of nodes is activated. The map defined by Eq. (4) allows us to generate the cobweb of the dynamical process. Theoretical predictions display excellent agreement with extensive simulations of the model (see Fig. 3). The combination of negative and positive regulatory interactions present in triadic percolation leads to a much richer phase diagram than the one of ordinary percolation in absence of triadic interactions (see Fig. 4). The phase diagram of triadic percolation is found by monitoring the relative size R of the GC as a function of the parameter p indicating the probability that a link is active when all the regulatory conditions allowing the link to be active are satisfied. Clearly from Fig. 4, we see that while in absence of triadic interactions the transition is second-order; when signed positive and negative regulatory interactions are taken into account, the phase diagram of percolation becomes an orbit diagram. In particular, Eq. (2) predicts that the order parameter undergoes a period doubling and a route to chaos irrespective of the degree distribution of the structural network. Theoretical predictions are well matched by results of numerical simulations (see Fig. 4). Our theory allows to well approximate the dynamical behavior of triadic percolation for random Poisson and scale-free structural networks (see Supplementary Information (SI) and Supplementary Figs. 611 for a discussion about the effect of the structural degree distribution on the phase diagram of triadic percolation).

Results of numerical simulations denote a rich dynamical behavior of the model also if structural networks are taken from the real-world. In particular, we consider real-world structural networks constructed from empirical data collected in the repository of ref. 67, and we combine these real structural networks with synthetic regulatory networks capturing the triadic interactions. In Fig. 5, we show that also for these topologies the phase diagram reveals non-trivial dynamics with some regimes of (noisy) oscillations and some regimes of chaotic dynamics of the order parameter (for more information about these datasets see Supplementary Table 1 and Supplementary Fig. 12).

In absence of negative triadic interactions, when all regulatory interactions are positive, the dynamics always reaches a stationary point independent of time. In Fig. 6a we show a typical time-series for R(t) where it is apparent that R reaches a stationary limit R(t) = R, where R is independent of time. Moreover in Fig. 6b we also display the dependence of this stationary state with p, i.e., R = R(p). The agreement between theoretical predictions and results of numerical simulations is excellent. Interestingly, the order parameter R displays a discontinuous hybrid phase transition as a function of p showing that positive triadic interactions induce discontinuous hybrid percolation in higher-order networks (see Fig. 6 and the SI for the analytical derivation of this result).

In order to exclude that the observed chaotic behavior of triadic percolation is an artefact of the particular choice of the dynamics, we consider also a version of the model with time-delayed regulatory interactions, where each regulatory link is assigned a time delay τ and Step 2 of triadic percolation is replaced by:

• Step 2′. Given the set of all active nodes obtained in Step 1, each structural link is deactivated:

1. (a)

if none of its positive regulators is active at time at t − τ;

2. (b)

if at least one of its negative regulators is active at time t − τ;

3. (c)

if the structural link is not deactivated according the conditions (a) and (b), it can still be deactivated by stochastic events which occur with probability q = 1 − p.

We consider two models of triadic percolation with time delay which depend on the choice of the probability distribution for time delays of regulatory links (see the illustration of the models in Fig. 7):

• [Model 1] each structural link is regulated by regulatory links associated to the same time delay τ, with the time delay τ being drawn from the distribution $$\tilde{p}(\tau )$$;

• [Model 2] each regulatory link is associated to a time delay drawn independently from the distribution $$\tilde{p}(\tau )$$.

Note that both models reduce to triadic percolation without delays when $$\tilde{p}(\tau )={\delta }_{\tau,1}$$ where δx,y indicates the Kronecker delta. Interestingly both models lead to a route to chaos also in presence of a non-trivial distribution of time delays, although the universality class might be different from the one of the logistic map (see Fig. 7). This finding demonstrates that the route to chaos observed in triadic percolation is a robust feature of the triadic-percolation model. Finally we note that triadic percolation might be suitably generalized also to node percolation leading also in this case to a route to chaos for the order parameter R (see SI and Supplementary Fig. 13 for details about this generalization of triadic percolation).