Epidemic spreading on coupling network with higher-order information layer

Epidemic tends to break out with information spreading which occurs between pairwise individuals or in groups. In active social online platform, three or more individuals can share information or exchange opinion, which could be modeled as a clique beyond pairwise interaction. This work studies the influence of information with higher-order cliques whose closure probability is described by higher-order clustering coefficient on epidemic spreading. The coupled spreading process of disease and awareness follows an unaware-aware-unaware-susceptible-infected-susceptible model on multiplex networks. We deduce the epidemic threshold of multiplex network with higher-order cliques, and explore the effects of cliques on epidemic spreading process at slow, middle and fast pairwise information spreading rates. The results show that the epidemic threshold and higher-order clustering coefficient are positively correlated. Moreover, we find that the effect of higher-order cliques could be ignored when pairwise information spreading rate is large enough. Finally, we show that the steady infection ratio decreases with the increase of the higher-order clustering coefficient.


Introduction
The spreading epidemic in complex networks has been a major subject of research in the past few decades [1,2].Researchers have made significant developments in understanding the relationship between epidemic dynamics and human behaviors [3,4].The harm brought by epidemics drives humans to take actions, such as broadcast, isolation and vaccination, to improve prevention and control efficiency, and these measures would change the evolution mechanism of the epidemic in turn [5][6][7].After the outbreak of Covid-19, the public platform was full of massive information about the scale of disease [8], which would remind individuals to pay attention to the epidemic and adopt protective measures such as wearing a mask, washing hands frequently, staying home to reduce the risk of infection.A coupled, nonlinear system is formed to explore the interaction between diseases and information [9][10][11].
The study of epidemic on the multiplex network has developed more than ten years and many interesting results have been found in [12][13][14].The multiplex network related to awareness has attracted much attention in the study of epidemic spreading [4,[15][16][17][18][19]. Considering the interrelation between epidemic spreading and social awareness, Funk et al [20] added a sensitivity coefficient related to individual awareness to SIR model, and this method leads to a reduction in the outbreak scale but does not affect the threshold.Granell et al [21] provided a framework where information prevails in the upper virtual layer, and then the epidemic process is supported by the lower physical layer.They revealed that local awareness influence on epidemic threshold with two stages divided by a metacritical point where information shifts from disappearance to prevailing.They further analyzed the generalized model including both local and global awareness, and they found that the competition principle causes the consistent inhibition and the global awareness is highly efficient in reducing the steady infected proportion [22].Wang et al [23] investigated the structural correlation between two layers and showed an asymmetrically interacting, in which the epidemic threshold would be raised with an unchanged information threshold.da Silva et al [24] extended the information model to unaware-aware-stifler-unaware model under the Maki-Thompson rumor model, and pointed out that reducing the proportion of unaware individuals does not always decrease the prevalence of a disease, due to different timescales that control the relative rates of informaiton and disease spreading.Li et al [25] used a temporal multiplex network as the physical layer in Granell's framework and defined the tendency of aware nodes moving to a specified layer as the layer preference, to explore the relationship between the layer preference and spreading dynamics.The study concluded that the layer preference decreases the epidemic threshold.
Traditional works are based on the assumption that a healthy individual gets infected through either pairwise interaction or single exposure which are simply described by pairs of nodes [26].However, pairwise interaction alone is no longer sufficient to characterize the structures and dynamics on networks with higher-order interaction, which has been found widely in various systems, such as social systems [27,28], ecology [29,30] and neuroscience [31,32].Higher-order interaction is not a new concept but was often ignored within the structure of edges [33].To fully describe systems with interaction among groups, formulations with higher-order properties are required [34].Iacopini et al [35] formalized a social group as a simplex and adopted simplicial complexes as the underlying structure.They combined the transition route based on both pairwise and higher-order mechanisms, where the spreading through group interaction leads to first-order transitions and co-existence of endemic equilibrium and disease-free equilibrium.Li et al [36] captured the coexistence of interactions of different orders, and found that the change of dominant interaction could induce discontinuous transition to the endemic state and reveal a distinctive spreading phase diagram.Chowdhary et al [37] further introduced higher-order structures into temporal network, where simplices can be added or deleted over time, and found that temporality makes static infectious parameters less influential and reduces the inhibition higher-order structures add to epidemic.Matamalas et al [38] used the microscopic Markov chain approach (MMCA) and the epidemic link equations (ELE) to explore the epidemic process on networks with simplices.They presented an abrupt transition and showed that ELE can give a more accurate prediction than MMCA about the susceptible-infected-susceptible (SIS) model in simplicial complexes.Nie et al [39] and Li et al [40] focused on the influence of simplicial complex on the competing spreading.However, the influence of the higher-order properties on the epidemic spreading has not been fully understood.An adjustable higher-order property could help us better control the epidemic spreading.
In this work, we study the influence of information with higher-order properties on epidemic spreading, by bringing higher-order cliques into the structure of social awareness.We explore to find how the closer links among people in cliques, such as family groups, friend groups, work groups, and the density of such links, could have an effect on disease spreading.The coupled spreadings of epidemic and awareness on multiplex networks, which are made up of information and epidemic layers with one to one mapping pattern of corresponding nodes are studied.The Kronecker-clique Graph (KCG) [41] with adjustable higher-order clustering coefficient is utilized to characterize the information layer.Nodes in a clique represent families or friends, thus, they have an intimate relationship and higher willingness to accept information from clique members than that from ordinary neighbors.
The physical layer is constructed using different networks to study the impact of topology on the higher-order coupled spreading.We use MMCA to calculate the epidemic threshold of the multiplex network with higher-order cliques and analyze the influence of the third-order clustering coefficients.At slow, middle and fast pairwise information spreading rates, we further demonstrate that the inhibition of cliques is influenced by the pairwise information spreading process.Moreover, we verify the results through Monte Carlo (MC) methods and perform large numerical simulations to demonstrate the impact of higher-order cliques on the epidemic process.We find that higher-order structures are beneficial for disease inhibition, but not the more dense the higher-order structure, the better the inhibition effect.When network is too sparse or too dense, there is little difference in the inhibition between the higher-order network and the isomorphic lower-order network.A moderate distribution of higher-order structures promotes powerful diffusion of information and is effective in stopping epidemic outbreaks.
We briefly review previous related work and put forward our ideas in section 1.In section 2, we introduce higher-order clique and higher-order clustering coefficient, and then describe the mechanism of epidemic dynamic in detail.In section 3, we utilize MMCA to deduce the epidemic threshold with the influence of higher-order information layer.In section 4, to exclude the impact of pairwise interaction, we explore the effect of infectious parameters on the influence of higher-order structures.We confirm the theoretical results by MC simulations in section 5.In section 6, we conclude the whole work finally.

Higher-order clustering coefficient
The phenomenon that a friend of your friend is likely to be your friend, is usually reflected by the strength of triadic closure [42,43] in social networks.The clustering coefficient is proposed to measure the density of the triangle [44].The classical clustering coefficient of node i with the degree of k i is defined as [45] where E i denotes the existing edges among k i neighbours of node i and the denominator gives the number of possible edges.From a geometric perspective, the classical clustering coefficient can be replaced by the ratio of the triangles that contain node i to the connected triples centered on the node i.
In higher-order networks, the m-cliques and m-wedges are basic higher-order structures.As shown in figure 1, an m-clique is a fully connected group of m nodes.We regard an edge as a two-clique, a triangle as a three-clique, a tetrahedron as a four-clique and so on.An m-wedge is made up of an m-clique and an adjacent edge linking to a node in the m-clique, which is regarded as the wedge center.The higher-order clustering coefficient is used to measure the probability of multi-node closure in the networks [46].
Extending the classical clustering coefficient in the higher-order framework, the local higher-order clustering coefficient of node i is defined as [46] where K m+1 (i) is the set of (m+1)-cliques, W m (i) is the set of m-wedges with node i as the wedge center.
where K m+1 is the set of (m+1)-cliques, W m is the set of m-wedges in the whole network.When the wedge center is not appointed, an (m+1)-clique includes m 2 + m wedges, because there are m + 1 choice to select a node as the wedge center and choose another node in the remaining m nodes, to add the adjacent edge.The global clustering coefficient describes the clusters in the entire network, while the local clustering coefficient quantifies the characteristic of neighbors of a node that gather together.By the algorithm of constructing KCG, the surrounding information of each node is given in detail and the local higher-order clustering coefficient could be accurately obtained.Thus, we use the average local higher-order clustering coefficients to reflect the higher-order clustering property of the network in this work.A diagram of two-layered multiplex networks with higher-order cliques.Information is exchanged following UAU model, and epidemic spreads is described with the SIS model.Blue, orange, green and red dots represent nodes in unware, aware, suscepitible and infected compartments, respectively.Cliques are shown as orange triangles in the information layer.

KCG
Based on the topology of pairwise nodes, stochastic Kronecker graph (SKG) [47] could well describe the heavy-tailed distributions for in-degree, out-degree, eigenvalues, and eigenvectors [48], but lacks the higher-order structures that exist widely in reality [27][28][29][30][31][32].As a generalization of SKG, by relying on degree-dependent functions and embedding cliques, Li et al [41] put forward KCG, which is closer to the real world in terms of higher-order clustering coefficients, to depict the higher-order features of complex networks.
The KCG is generated with the following steps: Step 1: Set the initiator where a,b,c ∈ [0, 1] are decided by the probability of adding an edge between zero and one.Give the parameter h of Kronecker product, which determines the size of network and there are 2 h nodes in total.
Then we have for example Step 2: Get the SKG with parameters in step 1.The P h extended by P 1 after h − 1 Kronecker products provides the elements p ij ∈ P h , which is the probability of adding the edge (i, j) to the SKG.Get the maximun degree k max of the SKG.
Step 3: Give the dependent function f (k) to tag nodes.If the degree k of a node is greater than or equal to two, the node will be tagged with probability f (k).
Step 4: For each tagged node, fully connect all neighbors of one tagged node, and form a clique that includes the tagged node and its neighbors.
Step 5: Begin with k = 2 and embed cliques in three nodes, then increase k and repeat step 3 and step 4 until k reaches k max .

SIS-UAU model with higher-order cliques
The multiplex network G in figure 2 with two layers is modeled to study the interaction between information and epidemic, where the two layers have different topologies.The virus and information spreading processes follow SIS model and unaware-aware-unaware (UAU) model, which is a version of SIS model applying on information, respectively.Two processes are coupled through the self-protection and awareness when node is alerted or infected.
The information layer G A with adjacent matrix A, constructed by KCG, is considered as a virtual network to help information spreading across social media platforms such as WeChat, Tiktok, and Facebook.Nodes in the information layer are only in two states: unaware (U) and aware (A).The unaware individual will transfer to aware state when he is informed or infected.We reckon that the higher-order interaction depicts the intimate relationship, thus the information spread by an individual must be accepted by his or her clique members, but the individual may not be fully trusted by others not in the clique.The information spreading rate is used to denote the speed of information spreading, but in cliques we emphasize whether ones accept the news rather than receiving it.We here assume that receptive tendency and the information spreading rate both describe how quickly and successfully information spreads, thus not distinguishing between these two concepts in the following description.According to the information spreading process through higher-order cliques or pairwise edges, an unaware individual becomes alert to the epidemic from an aware neighbour not in cliques with probability λ where 0 < λ < 1.However, the individual accepts the information from an aware clique member with probability 1.After a while, the individual might forget the information and return to unaware state with probability δ, where 0 < δ < 1.
In the physical layer G B , we use adjacent matrix B to represent the connection between nodes, where b ij = 1 denotes there is physical contact which can make the virus exchange, b ij = 0 indicates no connection.Each node could be in susceptible (S) or infected (I) state.When the epidemic spreads, an unaware and susceptible node might be infected with probability β U .Cautious attitude can make susceptible individuals take measures to protect themselves from being infected, so the probability of those aware individuals getting infected decreases to β A = γβ U , γ ∈ [0, 1] is an attenuation coefficient associated with the coupling.The smaller the attenuation coefficient, the stronger the protection, information has no effect on inhibiting disease outbreak when γ = 1, on the contrary, the aware individuals are apart from disease completely when γ = 0. Infected nodes will recover with a probability µ.

MMCA
MMCA is used to describe the random transition from one state to another in a state space.We here study the coupling between information with higher-order properties and epidemic by deriving the MMCA dynamical equations.
We assume that an infected one knows epidemic information absolutely, thus there is no UI state in the spreading process, and each node transfers among three states, i.e.US, AS and AI, without the dynamical correlations [49].We have introduced the spreading processes in section 2.3.We denote r i (t) as the probability that node i is informed, q U i (t) and q A i (t) as the probability that node i in the unaware and aware state is infected at time t.The probabilities can be written as where p A j (t) = p AS j (t) + p AI j (t) and p I j (t) = p AI j (t).For embedding cliques, the links in information layer can be divided into two matrices A H and A L .Two nodes in one clique mean that there is higher-order interaction and a H ij = 1, and when there is pairwise interaction, a L ij = 1.We assume that if an edge belongs to higher-order structures, we will no longer consider its characteristic as a pairwise edge, thus A H + A L = A. We focus on the role of absolute information spreading where the information spreading rate is 1 in higher-order cliques.In this case, the order of cliques has few contributions to the spreading process.Therefore, we do not need to discuss the influence of various orders.To emphasize the higher-order structures, cliques will be reduced to ordinary pairwise edges to eliminate the higher-order contribution.
We call the information network with the same topology but no higher-order properties the isomorphic lower-order network.The influence of higher-order cliques is reflected in the information spreading dynamics, i.e. the information spreading rate.The isomorphic lower-order network has the same topology with corresponding higher-order network.In a higher-order network, the adjacent matrix A H ̸ = 0, but in the isomorphic lower-order network, the adjacent matrix A H = 0 and A L = A.
According to the transition probability trees in figure 3, considering the transitions between the three states and the probabilities given in equations ( 7)-( 9), the MMCA transition equations of node i from time t to time (t+1) are Figure 3. Transition probability trees for the states (a) US, (b) AS, (c) AI per time step.r i denotes the probability that an individual will be informed by any neighbour in A state.q U i (q A i ) denotes the probability that an unaware (aware) and susceptible individual will be infected by any infective neighbour.δ gives the forgetting probability from aware to unaware and µ indicates the recovery probability from aware to susceptible.
As t → ∞, the system will reach a steady state where the infection ratio no longer changes with time passing by, p AI i (t + 1) = p AI i (t).We obtain steady state probabilities p US i , p AS i and p AI i , thus equations ( 10)-( 12) can be written as At the critical point τ c , the system will transform from disease-free equilibrium to endemic equilibrium.When τ is near τ c , we assume that p AI i = ε i ≪ 1, and ignore the higher-order terms of equations ( 8) and ( 9) to obtain the probabilities approximations Near the critical point τ c , there are few infected nodes in the physical network leading to little information, so the interaction between two layers is quite weak or even decoupled.The information spreading process can be modeled as an independent SIS model.Associated with transitions between states U and A, equations ( 13) and ( 14) can be transformed into According to equations ( 15), ( 18) and ( 19) can be simplify as For the attenuation coefficient and β A = γβ U , we get Substitute equation ( 21) and (20), we obtain the steady equation Generalize the steady equation of one node to the whole network, and use ε = ε 1 , ε 2 , . . ., ε N T representing the steady-infected probability of all nodes.equation ( 22) is rewritten as where P = I n − (1 − γ) diag p A and p A = p A 1 , p A 2 , . . ., p A N , equivalently, P i i = 1 − (1 − γ)p A i and P i j = 0, i ̸ = j, depending on the solution in the information network.If there is a fixed point besides zero solution, the solution of equation ( 23) should satisfy Λ max ( β U µ PB) = 1, thus the epidemic threshold is

Critical point and phase transition
In traditional works, networks tend to consist only of edges with pairwise interaction.Although we explore the higher-order properties now, we cannot ignore the pairwise interaction, because pairwise edges and higher-order cliques coexist in networks.We denote the τ H c as the higher-order threshold derived from the multiplex network with higher-order information layer, which still contains pairwise interaction.We remove the higher-order structures, i.e. reducing cliques to edges, and then calculate the lower-order threshold τ L c in the isomorphic lower-order network.To exclude the inhibition caused by information spreading through pairwise edges, we let τ H c − τ L c depict the influence of higher-order cliques.We refer to the coefficient κ as the information spreading rate among fully connected nodes.Here, κ = 1 denotes the presence of a higher-order clique in which an individual fully believe the information from the clique members.Conversely, κ = λ means that the higher-order property is removed and information spread with pairwise interaction totally in the isomorphic lower-order network.Therefore, the probability of individual i in the A state at time t + 1 is calculated by We will discuss the equation ( 25) when κ = 1 and λ, and get the steady probability vector of aware individuals p and q to calculate the corresponding thresholds.
In the case of the information layer with higher-order cliques, κ = 1, we denote the state transition by the mapping Φ : , where 0 n is all-zero vector with dimension n.The p A (t + 1) = Φ p A (t) means for each node, we have The probability vector that nodes are aware in steady state is denoted by satisfying Φ(p) = p, when κ = 1.We obtain and then Once edges in cliques lose higher-order property, they degenerate to pairwise interaction and κ reduce to λ.In this condition, we use the mapping When κ = λ, the probability vector q = p A 1 , p A 2 , . . ., p A N T is the nontrivial fixed point of Ψ, and we derive We use Rayleigh entropy to establish the relationship between τ H c and τ L c .The minimum value of Rayleigh entropy corresponds to the minimum value of the matrix eigenvalues.Hence, the two eigenvalues can be written as a function with the inverse matrix of matrices P and Q, which are related to vectors p and q.The coefficient of each term in the polynomial of τ H c is larger than that of τ L c .The τ H c and τ L c are the minimum value of corresponding polynomials.When the two polynomials each reach their minimum value, τ H c must be larger than τ L c .We give a detailed proof process to show τ H c > τ L c in appendix A. When information spreads quickly or an individual receiving news strengthens his or her protection against disease, the harm of virus will decease.Here, in order to highlight the effect of cliques, we study the relationship between the difference ∆ = τ H c − τ L c and parameters, i.e. attenuation coefficient γ and information spreading rate λ.In appendix B, the theoretically explanation is given to show that the difference ∆ is negatively related to attenuation coefficient γ and information spreading rate λ.

Numerical simulations
We verify the theoretical results using both MMCA and MC simulations.We use KCG to construct the information layer, with the initiator P 1 = 0.05 0.9 0.9 c which could generate the network with a large range of third-order clustering coefficient C 3 .The c is the independent variable used to adjust the third-order clustering coefficient C 3 .From the simulations on KCG, we set the parameters a, b, and c to change from 0 and 1 with intervals of 0.05, respectively.The parameters a and c are actually equivalent, and we only need to adjust one of the parameters.We found that a maximum range of C 3 could be obtained, when a = 0.05 , b = 0.9.The C 3 is hard to reach 1, because embedded network sparsity is not high and the maximum C 3 is about 0.6 in KCG [41].We adjust c to change C 3 from 0.1 to 0.6 with an interval of 0.05.We set h = 10 and the network includes 1024 nodes.The degree-dependent function controls the selection probability of higher-order cliques when generating KCG networks.We follow the function f(k) = 1 k−1 , which could generate the widest range of C 3 [41].
According to the physical layer, we adopt two classical artificial networks, (i) the Erdos-Rényi random (ER) network [50] G ER with average degree < k >= 10, (ii) the Barabasi-Albert (BA) scale-free network [51] G BA with adding edges m = 5 to be consistent with average degree of G ER .
In the MMCA, we assume that initially each node has an infected and alert probability of p I i (0) = 0.001 and p A i (0) = 0.001, resulting from approximately one infected node in the network at the start of the simulation.We iterate equations ( 10)- (12) for 2000 time steps to ensure that both the information spreading process and the epidemic spreading process could reach a steady state.The steady infection ratio of the entire network is represented by p I = p AI = 1 N i p AI i , where p AI i is the infection probability of node i.In the MC simulations, differently from MMCA where epidemic evolution is expressed as the change in the infected probability, we will show the state of each node, susceptible or infected actually at each time step.In the initial moment, we randomly choose a node to be in the AI state and the other nodes are in the US state.For the higher-order cliques, once there exists an aware higher-order neighbour, the node i accepts the information and to be alerted.If node i does not have higher-order neighbours but only N AL i aware pairwise neighbours, it will become aware with probability (1 The aware nodes change into the US state with probability δ.Similar to the information process, in the physical layer, a node i in the US state will be infected with probability (1 ) when in the US state, and with probability (1 ) when in the AS state, where N I i is the number of infected neighbours.The infected node could cure from epidemic with probability µ.The iteration will stop after 2000 time steps and average the results from multiple realizations to get the steady state fraction, N and p I = E[ N I ] N , where N A and N I represent the number of aware nodes and infected nodes when the iteration stops, E[•] is the note of expectation.
In figure 4, the higher-order threshold τ H c and the lower-order threshold τ L c are plotted against the third-order clustering coefficient C 3 on the G ER and G BA .To explore the relationship between τ c and C 3 , we set λ = 0.2, 0.5 and 0.8 as examples of the slow, middle and fast information spreading rates.We set γ = 0.2, 0.5 and 0.8 to describe the intensity of protection, corresponding to strong, medium and weak measures.As shown in figure 4, except for the numerical difference, the overall trends of the lines in (a) and (b), (c) and (d), (e) and (f) are alike, which indicates that the topology of the physical layer does not effect the inhibiting of the information layer on the epidemic threshold τ c .Thus, the following explanation does not distinguish the results in G ER and G BA .
Figure 4 shows that the epidemic threshold τ c monotonously increases with C 3 .We find that higher-order cliques could strongly inhibit epidemic spreading when γ = 0.2.However, when γ = 0.8, no matter what the information spreading rate λ is, the higher-order structures have no influence on the epidemic threshold τ c .Moreover, when the pairwise spreading rate λ is smaller, the effect of the higher-order structures on the epidemic threshold τ c is larger.
As we derived in section τ H c is consistently larger τ L c .When λ reduces, unaware nodes have a smaller probability of accepting information from their ordinary neighbors and transforming into aware state, and the difference ∆ between τ H c and τ L c is larger.It demonstrates that the higher-order interaction in cliques is rather important when the pairwise spreading rate λ is small.Conversely, as λ increases, the difference ∆ gradually decreases until τ H c and τ L c overlap, because information spreads so fast through pairwise edges that the impact of higher-order interaction is negligible.Efficient information spreading on the pairwise edges makes the influence of higher-order cliques less noticeable.Besides, a smaller γ could cause a larger ∆, because aware nodes will take more effective measures to protect themselves and prevent further epidemic spreading.
Based on the results shown in figure 4, according to given λ and γ, we make a subtraction and get the lines of ∆ = τ H c -τ L c in terms of C 3 in figure 5.The ∆ first increase and then decrease with the higher-order clustering coefficient C 3 .It means that the effect of the higher-order structure on increasing the epidemic threshold does not monotonically with C 3 .We obtain a turning point near C 3 = 0.35.The existence of turning point is due to the critical point in KCG related to network connectivity [52].
We study the relationship between the C 3 and the size of giant component in figure 6(a).We find that the information layer dose not form a giant component in the network and there are a large number of isolated nodes and small groups, when the C 3 is smaller than the turning point.When the C 3 is beyond the turning point, the size of giant component grows rapidly.Moreover, when the C 3 is large enough, the giant component contains the majority of nodes, which leads to the network with a high connectivity.To compare the proportion of edges in higher-order cliques or not, we give a column diagram with each column up to 1 in figure 6(b), and use dark blue and light blue to distinguish those two.The proportion of edges in higher-order cliques first increases and then decreases.
We study the effect of information spreading rate λ on the difference ∆ with various C 3 with fixed γ = 0.2 in ER networks and BA networks.Figure 7 shows that when C 3 is small and λ is less than 0.3, the line is almost horizontal, indicating that cliques are less effective.When λ is small but C 3 is large, cliques can effectively suppress the epidemic outbreak.Once C 3 exceeds 0.25, all lines begin to decline and give the result that the impact of higher-order cliques will diminish as the information spreading rate λ increases.In particular, when λ is above 0.8, the ∆ narrows to zero.It is noteworthy that in figure 7(a) the maximum is given by the line with C 3 = 0.4, while in figure 7(b), the line with C 3 = 0.35 yields the highest value.Consequently, we can get that the when C 3 is between 0.35 and 0.4, higher-order cliques can play the greatest advantage in inhibiting epidemic.
We than study the influence of the attenuation coefficient γ on the difference ∆. Figure 8 shows that the difference ∆ decreases monotonically with the attenuation coefficient γ when λ = 0.2 in ER and BA networks.The γ = 1 indicates that aware individuals do not take any protective measures, which means there is no inhibiton on the physical layer from the information layer, and higher-order cliques lose their effect.When γ = 0, in both ER networks and BA networks, the line with C 3 = 0.35 yields the maximum, this proves once again that higher-order cliques work best with C 3 around 0.35.
The influence of the steady-state fraction p I of infected individuals and the effective spreading rate τ is shown in figure 9, with κ = 1 or κ = λ.We provide the ratios from both MMCA and MC simulation when C 3 = 0.1, 0.35 and 0.6 as τ increases in an interval of 0.02 from 0 to 1. Nevertheless, it is surprised that there is no obvious difference between the two lines when C 3 = 0.1 and 0.6, however, steady-state fraction p I of infected individuals decreases when C 3 = 0.35, and this is consistent with the result from figure 9 that the maximum of ∆ appears when C 3 = 0.35.In the top two pictures, the black line and the red line almost overlap for few cliques could help to prevent epidemic.In the bottom two pictures, there are many pairwise edges in the network which strongly enhance pairwise interaction so that the advantage of the cliques is not important anymore.Therefore, the efficient of cliques in suppressing outbreaks from the disease-free equilibrium and in limiting the epidemic spreading is highest in the networks with a moderate higher-order clustering coefficient.

Conclusion
In this work, we have studied the relationship between higher-order clustering coefficient C 3 in the information layer and epidemic spreading in the physical layer with the UAU-SIS model in double-layer coupled networks.The results show that C 3 can increase the epidemic threshold and performs well in the environment with slow pairwise information spreading rate.We decuce an optimal third-order clustering coefficient, with which the inhibition effect of higher-order cliques on epidemic spreading is the most significant.We consider the higher-order structures, which reflect the reality of social behavior.It is suggested that individuals should strengthen contacts with their families and friends when facing epidemic.Closer relationships among population could enable individuals to aware diseases more quickly and take protective measures to defend themselves.However, each individual has judgment and will not follow others' words blindly.Considering the heterogeneity in individual judgment ability, the further detailed model needs to be explored in the future.We assume that there are two networks with different parameters, which make diagonal matrices We assume that the minimum of τ H c is attained at v H , and the minimums of lower-order thresholds locate at v L1 and v L2 for different λ 1 and λ 2 , respectively.We compare eigenvalue functions h P −1 , • , h Q 1 −1 , • and h Q 2 −1 , • at their respective minimums, when Moreover, by comparing different values of two vectors on the same function, we can observe that With all of the above inequalities and if M 1 i i > M 2 i i > 0, we obtain For the equilibrium equation with information decoupled, when λ changes from zero to one, mapping Φ will approach Ψ and q will also monotonically increase to p.The functions depend on λ continuously and we assume that λ 1 < λ 2 < 1 which can be used to deduce Ψ 1 (x) < Ψ 2 (x) < Φ(x), then we get q 1 ≺ q 2 ≺ p , m 1 > m 2 and that the diagonal elements in M 1 are all larger than the corresponding elements in M 2 .Consequently, the bigger λ is, the smaller M ii is.The difference ∆ increases as M i i increases, however, both γ and λ are negatively related to M i i .In other words, when attenuation coefficient γ and information spreading rate λ decrease, the gap between thresholds will widen, making higher-order cliques are more effective in inhibiting the disease.

Figure 1 .
Figure 1.Overviews of m-clique and m-wedge.From two-order to four-order, (a)-(c) show the cliques with all nodes connected, (d)-(f) are examples of m-wedge which is an m-clique and an adjacent edge

Figure 2 .
Figure 2.A diagram of two-layered multiplex networks with higher-order cliques.Information is exchanged following UAU model, and epidemic spreads is described with the SIS model.Blue, orange, green and red dots represent nodes in unware, aware, suscepitible and infected compartments, respectively.Cliques are shown as orange triangles in the information layer.

Figure 4 .
Figure 4. Epidemic threshold τc, including the higher-order threshold τ H c and the lower-order threshold τ L c , as a function of the third-order clustering coefficient C3.In (a), (c) and (e), virus spreads in the GER.In (b), (d) and (f), virus spreads in the GBA.The results of (a)-(b), (c)-(d) and (e)-(f) are obtained when λ = 0.2, 0.5 and 0.8, and δ = 1-λ.The solid triangles represent the higher-order thresholds τ H c , and the hollow triangles represent the lower-order thresholds τ L c .The red, green and blue lines represent the results obtained from the multiplex networks with γ = 0.2, 0.5, and 0.8, respectively.

Figure 5 .
Figure 5. Dependence of the epidemic threshold difference ∆ = τ H − τ L on the third-order clustering coefficient C3 in (a) and (b).Red, green and blue lines are given under γ = 0.2, 0.5 and 0.8.The upper half colored diamonds, fully colored diamonds and lower half colored diamonds point to λ = 0.2, 0.5 and 0.8, respectively, and δ = 1−λ.Panels (a) and (b) are obtained from ER network and BA network.

Figure 6 .
Figure 6.The number of nodes in the gaint component in KCGs with various C3 in (a).The proportion of edges in higher-order cliques and pairwise edges in KCGs with various C3 in (b), the dark blue parts represent the proportion of edges in higher-order cliques, while the light blue parts represent the proportion of pairwise edges.

Figure 7 .
Figure 7. Epidemic difference ∆ = τ H − τ L as a function of the information spreading rate λ in ER network (a) and BA network (b) with various C3 at the steady state, other parameters are set as γ = 0.2, δ = 1−λ.

8 .
Epidemic difference ∆ = τ H − τ L as a function of the attenuation coefficient γ in ER network (a) and BA network (b) with various C3 at the steady state, other parameters are set as λ = 0.2, δ = 0.8.

Figure 9 .
Figure 9.Comparison of the steady-state fraction P I of infected nodes with κ = 1 and κ = λ using the MC simulation(dots and stars) and the MMCA(solid line).The parameters are set to be µ = 0.5, λ = 0.2, δ = 0.8.The networks of information layer in each panel are the same KCG, and the lower layer is ER network in panels (a), (c) and (e) while BA network in panels (b), (d) and (f).The MC simulations are obtained by averaging 200 realizations.

11 − M 11 v 2 1 +
New J. Phys.25 (2023)  113043 Y Zhu et al where m i ∈ m = m 1 , m 2 , . . ., m N T .Obviously, m i and M ii are positively correlated, while γ and M ii are negatively correlated.Define an eigenvalue function h(C, x) = x T Cx x T Bx , where C is a variable matrix of the function h, and B is a constant adjacent matrix of the physical layer.Thresholds can be written as τ H c = min v∈ Rn /0n h P −1 , v = min v∈ Rn /0n . . .+ P −1 nn − M nn v 2 to show the n ) = 0 n and p A (t + 1) = Ψ p A (t) expands as