Introduction

Over the past twenty years, complex network theory provided important methods and tools for understanding pairwise interaction of the real world1,2,3. However, with the development of information technology, the interactions between nodes are no longer restricted to two nodes. A node can interact with many other nodes in the system4,5. Such as, collaborations networks6, ecological systems7 and brain networks8,9,10. The interaction between of many nodes is called higher-order interaction4,5. The simple complexes are the powerful tool to show the higher-order interaction relationships4,13,18. Therefore, many researchers pay more attention to the higher-order network based on simple complexes. Their works include the modelling11,12,13,14,15,16,17,18,19,20, topology properties4,5,21,22,23,24, synchronization performance25,26,27,28,29,30,31 spreading properties16,32,33,34,35,36,37,38,39,40 of higher-order networks. These research results show that the higher-order interaction will induce novel collective behavior. For instance, in the results of reference38 show that the higher-order interaction may be accelerate the spread of the disease.

The robustness is one of the critical dynamic properties of real complex systems3,41,42,43. Naturally, the robustness of higher-order network is also an important problem worth studying. There were some meaningful results on the robustness of the higher-order network. For example, Zhao et al.44 constructed a theoretical framework to study the percolation behavior of higher-order networks based on simplex structure. They revealed how high-order interactions between nodes lead to cascading failure processes through synergistic effects. In addition, the authors investigated the impact of the number of triangles on network robustness and found that a large number of 2-simplex can lead to the phenomenon of double phase transition. Furthermore, Zhao et al.45 extended their research to high-order simple complex networks with any dimension simplex. They revealed that high-density high-order structures had a weakening effect on the robustness of the system. Moreover, Lai et al.46 found that the simplex networks with interdependence have higher robustness than the common interdependent network based on graph. Furthermore, Peng et al.47 studied the robustness of higher-order interdependent networks by the percolation theory. They found that the number of 2-simplex is a critical factor lead to double transition phenomenon. These results had given us a new understanding on the robustness of network with higher-order structure. Specifically, in the research on the robustness of higher-order system, Dong et al.48 found an optimal level of interaction between subnetworks, which maximizes the system’s resilience to failures. Their findings highlight the need to consider real-world coupling patterns and possible optimizations for designing resilient systems.

In the research on complex network robustness, cascading failure analysis has been one of the effective ways to gain the network robustness49,50,51. Though analyzed the cascading failure process, we evaluated the ability that network withstand the attack, and obtained the influencing factors of network robustness, too. In previous results, there is a lack of the research results on robustness from cascading failure. As far, Yu and Ma et al.52 got the robustness of a higher-order network with 2-simplex and triple by studied the cascading failure on higher-order network. They proposed a capacity-load cascading failure model based on the high-order structures, and proposed four self-adaptive load redistribution strategies based on the multi-interaction of nodes. Their research results show that the 2-simplex higher-order structures help to increase the robustness of higher-order network. However, they only analyzed the influence of 2-simplex on the robustness of higher-order network. Furthermore, we want to find out the influence of other simplex structures on the robustness of higher-order network. And we also want to find out a method that could control the cascading failure process of the higher-order network by control the higher-order structure, thereby controlling the robustness of the higher-order network.

In this paper, we first constructed two higher-order networks with 0-simplex, 1-simplex and 2-simplex structures. Then, we proposed a new capacity load model by assign different weight coefficients to the simplex structure. At the same time, we also proposed a load redistribution method related to the simplex structure. In simulation, we analyzed the cascading failure processes of the synthetic higher-order networks and the higher-order network based on the real data by the proposed capacity load model. And we obtained the influence of 1-simplex and 2-simplex on the robustness. The analysis results show that we could control the robustness of network with higher-order structure by reset the weight coefficients of the simplex structure.

The structure of this paper is organized as follows. Section "Higher-order network model and evolution algorithm" provided specific generation process of scale-free higher-order network and random higher-order network. In Section "Higher-order network evolution algorithm", we proposed the capacity-load cascading failure model which is suitable to the cascading failure process in the higher-order networks. Section "Higher-order network capacity load cascading failure model" analyzed the results of the simulation experiments. In the section "Simulation experiments and results analysis", we summarized this paper and discussed the future work.

Higher-order network model and evolution algorithm

Simplex and simplicial complex

Higher-order networks are used to describe the systems with higher-order interactions4,5,11. The researchers constructed the higher-order network used the simplex and simplicial complex. A d-dimensional simplex (denoted as a d-simplex) is formed by a set of (d + 1) interacting nodes. For example, in Fig. 1, the node is 0-dimensional simplex in subgraph (a); the edge is 1-dimensional simplex in subgraph (b), which describes the pairwise interactions; the triangles are 2-dimensional simplex in subgraph (c), describing the interactions of three nodes; the tetrahedron is 3-dimensional simplex in subgraph (d). A graph containing various simplexes is called simplicial complex as shown in subgraph (e) of Fig. 1.

Fig. 1
figure 1

Four types of simplex and a simplicial complex. (a) is a 0-simplex, (b) is a 1-simplex, (c) is a 2-simplex, (d) is a 3-simplex and (e) is a simplicial complex ).

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The m-face α of a d-dimensional simplex is also simplex. It formed by a proper subset of the nodes of the d-dimensional simplex. In a higher-order network, the 0-face that is the node, the 1-face that id the edge, 2-face that is the 2-simplex, and the 3-face that is the 3-simplex. In additional, the 2-simplex include three 0-faces that are thee nodes, and include three 1-faces that are three edges. Let kd,m(α) denote the generalized degree of the m-face α in d-dimensional simplex. kd,m(α) is the number of d-dimensional simplexes adjacent from the m-face α. Where k1,0(α) represents the number of 1-simplexes adjacent from 0-face α, that is the number of edges adjacent from the node, where k2,0(α) represents the number of 2-simplexes adjacent from 0-face α, that is the number of triangles adjacent the node. For example, in subgraph (e) of Figure 1, k1,0(vi)=2, k2,0(vi)=2.

Higher-order network evolution algorithm

In order to explore the inferences of different simplex structures on the network robustness, inspired by the references18 and16, we constructed two higher-order networks with 0, 1 and 2-simplexes. The construction process is described as follows.

  1. (1)

    The evolution algorithms of scale-free higher-order network

First, we constructed a scale-free higher-order network only containing 2-simplexes based on the preference mechanism by the algorithm 1. Then, we used the algorithm 2 for added the 1-simplex into this higher-order network.

Algorithm 1
figure a

Construct a scale-free higher-order network only containing 2-simplexes

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Algorithm 2
figure b

The 1-simplex is added to the scale-free higher-order network which constructed by Algorithm 1.

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Let p1,0(k1,0), p2,0(k2,0) denotes the generalized degree distributions of nodes. The generalized degree distributions of scale-free higher-order network were characterized in Fig. 2. The results show, the generalized degree distributions of node all followed power-law distribution in scale-free higher-order network. The distribution of generalized degree k1,0(α) show that most nodes are adjacent from a few 1-simplexes, while a small number of nodes were adjacent from a large number 1-simplexes. The distribution of generalized degree k2,0 also show similar results. Furthermore, the results indicate that 1-simplexes and 2-simplexes were not uniformly distributed in scale-free higher-order networks.

  1. (2)

    The evolution algorithms of random higher-order network

Fig. 2
figure 2

Generalized degree distribution of a scale-free higher-order network (a) generalized degree distribution of k1,0; (b) generalized degree distribution of k2,0).

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Second, we constructed a random higher-order network based on ER evolution mechanism. The construction process was described by algorithm 3.

Algorithm 3
figure c

Construct a random higher-order network containing 1-simplexes and 2-simplexes.

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The generalized degree distributions of random higher-order network were characterized in Fig. 3. The results show, the generalized degree distributions of k1,0 and k2,0 all followed Poisson distribution in random higher-order network.

Fig. 3
figure 3

Generalized degree distribution of a random higher-order network (a) generalized degree distribution of k1,0; (b) generalized degree distribution of k2,0).

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Higher-order network capacity load cascading failure model

In order to explain intuitively the cascading failure process on the higher-order network, we described this process by the Fig. 4. In the initial stage (Subgraph (a)), the red node v1 was fails due to attack and its loads will redistributed to the neighboring nodes. In the first load redistribution stage (Subgraph (b)), according to the load redistribution principle, the loads of v1 redistributed to 1-simplexes and 2-simplexes which are adjacent from the node v1. In the second load redistribution stage (Subgraph (c)), these neighboring nodes of node v1 may be failed due to the increased loads, and their loads continue redistribution to 1-simplexes and 2-simplexes which are adjacent to them. This step will lead to the failures constantly spread in the higher-order network. When there are no more new failed nodes or the higher-order network has collapsed globally, this cascading failure process will be end, as show in the subgraph(c) of the Fig. 4.

Fig. 4
figure 4

Illustration of cascading failure in a higher-order network. (Green arrow : faulty node updates the load in 1-simplex; Blue arrow : faulty node updates the load in 2-simplex).

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According the above description, the loads of failed nodes were redistributed to 1-simplexes and 2-simplexes which are adjacent from these failed nodes. So in order to evaluate the influence of different simplex on the cascading failure process, we proposed a new cascading failure model based on capacity-load.

The new cascading failure model is described as following.

Initial load of a node

Inspired by the cascading failure model based on capacity-load idea in common graph, we considered the structure of 1-simplex and 2-simplex in higher-order networks and proposed the capacity-loading model of higher-order networks as show in Eq. (4):

$${L}_{i}\left(0\right)=\alpha {\left({R}_{1}\times {k}_{\text{1,0}}\left(i\right)+{R}_{2}\times {k}_{\text{2,0}}\left(i\right)\right)}^{\beta },i=\text{1,2},\dots ,N,\alpha \ge 1,\beta \ge 1.$$
(4)

where parameter α and parameter β is the adjustable parameter. The Eq. (4) indicates that the initial loads of a node i is related to its general degree k1,0(i) and k2,0(i) in higher-order network. The simplex has different effects on performance of higher-order network. So we set the weight factor R1 for the 1-simplex, and the weight factor R2 for the 2-simplex.

Capacity of a node

In a network, capacity is defined as the maximum load that one node can handle, and it is proportional to the initial load of the node. The capacity Cj of node j is denoted as Eq. (5).

$${C}_{j}=\left(1+T\right){L}_{j } \left(T\ge 0\right)$$
(5)

where, T is the capacity parameter. A lager T corresponds to larger node capacity. In a network, nodes with larger capacities mean they have stronger ability to handle loads, that is mean the network less prone to global collapse. So, we can enhance the robustness of higher-order network by lager T. However, in real network, increase in the capacity will lead to increase in cost. Hence, we need to find a critical threshold Tc , which is the minimum capacity that avoids a global collapse in a higher-order network. When T > Tc, the global collapse do not occur in the higher-order network, otherwise a global collapse will occurs. In many references3,41,42,43, the critical threshold Tc of T is widely used in evaluating the robustness of network, smaller Tc indicates more robustness of network.

Load redistributing strategy of a node

At time t, when node i fails, its loads are divided into same m parts. We consider the influences 1-simplex and 2-simplex, make m= R1 × k1,0(i) + R2 × k2,0(i). Such as, R1=1, R2=2 show that each 1-simplex undertake one pare load and each 2-simplex undertake two parts load in load redistributing.

The additional load received by non-failed node j will be Eq. (6). If non-failed node j is in 1-simplex associated with node i, then the additional loads of node j represented by above formula in Eq. (6). If non-failed node j is in 2-simplexex associated with node i, then the additional loads of node j showed by below formula in Eq. (6).

$$\Delta {L}_{i\to j}\left(t\right)=\left\{\begin{array}{c}{L}_{i}\left(t\right)\times \frac{{R}_{1}}{{R}_{1}\times {k}_{\text{1,0}}\left(i\right)+{R}_{2}\times {k}_{\text{2,0}}\left(i\right)}, j\in 1-simplex ; \\ {L}_{i}\left(t\right)\times \frac{{R}_{2}}{2({R}_{1}\times {k}_{\text{1,0}}\left(i\right)+{R}_{2}\times {k}_{\text{2,0}}\left(i\right))}, j\in 2-simplex.\end{array}\right.$$
(6)

According Eq. (6), the additional loads of non-failed node j ΔLi→j(t) from failed node i are dependent on the generalized degrees k1,0(i), k2,0(i) of the failed node i, the weight coefficient R1 or R2 of simplex and the initial load of failed node i. At the time t, the load of node j as shown in Eq. (7) .

$${L}_{j}\left(t\right)={L}_{j}\left(t-1\right)+\Delta {L}_{i\to j}\left(t\right)$$
(7)

Thus, if the Equation (8) be satisfied, then the node j after receiving the additional load is failure. And the load of failed node j also redistribute to its neighboring nodes, lead to new failed nodes.

$${L}_{j}\left(t-1\right)+\Delta {L}_{i\to j}\left(t\right)>{C}_{j},$$
(8)

In order to evaluate the robustness of the higher-order network, we consider the cascading failures that are caused by a single node failure. Initially, the node i is fails after being attacked and the load of node i will be redistribute the rest normal neighboring nodes of i in the higher-order network. If a normal neighboring node receives the additional load and Eq. (8) be satisfied. Then this neighboring node is fails. In this paper, we adopt two robustness evaluation metrics as following:

  1. 1)

    One evaluation metric is defined as I. Let I is the ratio of the number of failed nodes to the total number of nodes after the cascading failure stop, as shown in Eq. (9).

    $$I=\frac{{N}_{f}}{N} \left(0<I\le 1\right)$$
    (9)

where Nf (0 < Nf ≤ N) is the number of failed nodes when the higher-order network reaches a stable state, N is the total number of nodes, and I is the proportion of failed nodes. A smaller I indicate a stronger network’s resilience to cascading failures and, therefore, better robustness.

  1. 2)

    Other evaluation metric is defined as G. Let G is the ratio of the total number of nodes in the largest connected subgraph to the total number of nodes after the cascading failure stop, as shown in Eq. (10).Where Ne is the number of nodes in the largest connected subgraph. So G is called by the maximum connectivity ratio. A larger G indicate a stronger network’s resilience to cascading failures and, therefore, better robustness.

    $$G=\frac{{N}_{e}}{N} \left(0<G\le 1\right).$$
    (10)

Simulation experiments and results analysis

The researchers always adopted two attack strategies in the simulation of the cascading failure, which were random attacks and deliberate attacks, respectively. In this paper, we also adopted the two strategies to simulate the cascading failure process of two higher-order networks in simulation experiments. In random attack, we randomly selected a node and made it failure. In deliberate attacks, we selected node which has the highest sum of the generalized degree k1,0 and k2,0, and attacked it.

In order to simulate the cascading failure processes on higher-order network and reveal the influence factors of robustness, we assigned the simulation experiments of cascading failure. The capacity-load model was acted on the two type higher-order networks which were two synthetic high-order networks and constructed higher-order network based on real data. We counted the data which were related to the cascading failure. These data included the failed node number, the node number in the largest connected subgraph, the critical threshold Tc, different weight coefficient R1, R2 and so on. To achieve the validity and authenticity of the results, all experimental results were averaged over 50 measurements. We obtained the analysis results about the robustness of higher-order networks as following.

Sensitivity analysis of parameters α and β in capacity-load model

First, we analyzed the influences of parameters α and β on the threshold Tc, the results as shown in Fig. 5. Figure 5a and b show that scale-free higher-order network and random higher-order were less sensitive to α in the Eq. (4). With the increase in α, the critical robustness threshold Tc of two synthetic higher-order networks remained in a relatively value. Figure 5c and d demonstrate that two synthetic higher-order networks were more sensitive to β; the critical threshold of robustness Tc increased with β exponentially. The results in Fig. 5c and d imply that the robustness decreases as β increases. At β = 1, two synthetic higher-order networks robustness were optimal. Thus, we used the control variable method in the simulation analysis to select the parameter values that optimized the robustness of the higher-order networks. So the load parameter α was set to 10, and β was set to 1.

Fig. 5
figure 5

The effects of the parameters α, β on the robustness of two synthetic higher-order network.

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Impacts of attack strategies on the robustness

We analyzed the impacts of attack strategies on the robustness in two synthetic higher-order networks, the results as shown in Fig. 6. In order to avoid the influence of the weight coefficient on the results, let R1 = 1, R2 = 1 i.e. 1-simplex and 2-simplex are equally important. Where, subgraph (a), (b) are the results about scale-free higher-order network and subgraph (c), (d) are the results about random higher-order network.

Fig. 6
figure 6

The impact of attack strategies on the robustness of higher-order networks.

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In subgraph (a) of Fig. 6, the failure ratio I in the deliberate attack always was higher than in the random attack, implying when R1 = R2 = 1 the scale-free higher-order network constructed in this paper was more robust in random attacks. Moreover, the maximum connectivity ratio G of the deliberate attack was lower than the random attack in subgraph (b) of Fig. 6, which verified the similar results.

In subgraph(c) of Fig. 6, there were almost equal failure ratios I in two attack strategies of random higher-order network. The results indicate that the random higher-order network with 1-simplexes and 2-simplexes wasn’t sensitive to attack strategies. The similar results were validated by subgraph (d) of Fig. 6.

Impacts of the size of the higher-order networks on its robustness

In this paper, we constructed three scale-free higher-order networks with size of N = 1000, N = 2000, N = 3000, respectively. At the same time, we constructed three random higher-order networks with size of N = 800, N = 1000, N = 1500, respectively. We want to obtained the influence of network size on the robustness of higher-order networks.

In subgraph (a) of Fig. 7 shows the variation in the node failure ratio I with the capacity parameter T of scale-free higher-order networks with different size in two attack ways. It can be seen that scale-free higher-order networks with larger size have more robust in the same attack strategies. That is because the scale-free higher-order networks with larger size have more nodes to undertake load of the failed nodes, making a collapse of the higher-order network less likely. We obtained similar conclusion of robustness in scale-free higher-order network by the subgraph (b) of Fig. 7. After attack, the larger the size of higher-order networks, the more nodes in the largest connected component. The results implying that the more size of the higher-order networks, more robust in scale-free higher-order networks.

Fig. 7
figure 7

The impact of network size on the robustness of two synthetic higher-order networks.

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Subgraph (c) and (d) of Fig. 7 are the result about random higher-order networks. We observed form subgraph(c) of Fig. 7 that the failed nodes ratios I are decrease with the size of random higher-order networks increase. Meanwhile, the maximum connectivity ratios G in subgraph (d) are increase with the size of random higher-order network increase. The results show that the random higher-order networks with larger size have more robust.

Impact of the number of 2-simplex on the robustness of scale-free higher-order network

Let S2 is the number of 2-simplexes in the scale-free higher-order network. In order to find that the impact of S2 on the robustness of scale-free higher-order network, we fixed the number of 1-simplex, and constructed the scale-free higher-order networks with N = 1000 and different S2. In simulation, S2 were taken as 200, 500 and 800, respectively. Then we attacked these scale-free higher-order networks by two attack strategies. Subgraph (a) of Fig. 8 described the variation in the failed nodes ratios I with the capacity parameter T of scale-free higher-order networks with different S2 in two attack strategies. The results show that the more 2-simplexes there were in the scale-free higher-order network, the larger failed nodes ratios I. That is the more 2- simplexes there were in the scale-free higher-order networks, the more vulnerable they were. That is because a failed node redistribute its load to two non-failed nodes in the same 2-simplex, and lead to increase new failed nodes. Subgraph (b) of Fig. 7 show the more 2-simplexes there were in the scale-free higher-order networks, the smaller the maximum connectivity ratios G in the same capacity parameter T. That is the more 2-simplexes there were in the higher-order networks, the more vulnerable the higher-order networks were. Therefore, the robustness of scale-free higher-order networks can be improved by decreasing the number of the 2-simplex in the scale-free higher-order networks.

Fig. 8
figure 8

The impact of the number of 2-simplexes on the robustness of scale-free higher-order networks.

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Impact of the number of 2-simplex on the robustness of random higher-order network

According the constructive algorithm of random higher-order network in this paper, the number of 2-simplex in the random higher-order network was control by probability p2. In order to find that the impact of the number of 2-simplex on the robustness in random higher-order network, we fixed number of 1-simplex and constructed the random higher-order networks with N = 1000 and different p2. In simulation, p2 were taken as 0.0005, 0.0001, 0.001 and 0.005, respectively. Obviously, the larger p2 the larger number of 2-simplex in random higher-order network. We attacked these random higher-order networks by two attack strategies and obtained the curves of the failed nodes ratios I and the maximum connectivity ratios G in different capacity parameter T. The results of the failed nodes ratios I were drew in the Subgraph (a). And the results of the maximum connectivity ratios G were drew in the subgraph (b) of Fig. 9. We observed similar results as the scale-free higher-order networks in Fig. 9.

Fig. 9
figure 9

The impact of the number of 2-simplexes on the robustness of random higher-order networks.

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Thus, the number of 2-simplex will weaken the robustness of the scale-free higher-order network and random higher-order network.

Impact of parameters R1 and R2 on the robustness

Figures 10 and 11 show the impact of different weight coefficients on the robustness of higher-order networks.

Fig. 10
figure 10

The impact of weight coefficients on the robustness of higher-order networks. (R1 > R2).

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Fig. 11
figure 11

The impact of weight coefficients on the robustness of higher-order networks. (R1 < R2).

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Figure 10a, b show the simulation results of scale-free higher-order networks when the weight coefficient R1 > R2. We took the size N = 1000, S2 = 800, R1 = 1, R2 = 0.2, R2 = 0.5, R2 = 0.7. The simulation results in subgraph (a) of Fig. 10 show that for the same weight coefficient R2, the failed nodes ratios I in deliberate attack were higher than in random attack. The similar results were also shown in subgraph (b) of Fig. 10. The maximum connectivity ratios G in deliberate attack were lower than in random attack. Thus, when R1 > R2, the scale-free higher-order networks were more robust to random attack. it can be seen form subgraphs (a,b) of Fig. 10 that the smaller the value of R2, the more robust the scale-free higher-order networks were.

Figure 10c, d show the simulation results of random higher-order networks when the weight coefficient R1 > R2. Here, the size N = 1000, p1 = 0.01, p2 = 0.0001, R1 = 1, R2 = 0.2, R2 = 0.5, R2 = 0.7. The subgraph (c) drew the curves of failed nodes ratios I with capacity parameter T in different attack strategies of random higher-order networks. The subgraph (d) drew the curves of the maximum connectivity ratios G with capacity parameter T in different attack strategies. We observed similar results as the scale-free higher-order network from subgraphs (c) and (d). That were the failed nodes ratios I in random attack were lower in deliberate attack, and the maximum connectivity ratios G were higher than in deliberate attack. The results further indicated that when R1 > R2, the random higher-order networks were more robust to random attack, too. However, in subgraphs (c) and (d), we found different results from scale-free higher-order networks. In same attack strategy, the larger R2, the more robust the random higher-order networks were.

Figure 11a, b show the simulation results of scale-free higher-order networks when the weight coefficient R1 < R2, and R1 = 1, R2 = 2, R2 = 5, R2 = 7, R2 = 9 are taken. The simulation results in subgraph (a) of Fig. 11 show that for smaller T, the failed nodes ratios I in deliberate attack are higher than in random attack. However, as the capacity parameter T increases, the failed nodes ratios I in deliberate attack are lower than in random attack except for R2=2. In the subgraph (b) of Fig. 11, the same results are also shown, when the capacity parameter T is small, the maximum connectivity ratios G are lower in the face of deliberate attacks than that in the face of random attacks in the higher-order networks. When the capacity parameter T is large, except for R2 = 2, the maximum connectivity ratios G in the face of deliberate attacks are higher than that in the face of random attacks. In addition, it can be seen from the subgraphs (a) and (b) Fig. 11 that in the random attack, the scale-free higher-order networks are more robust in smaller R2, and more vulnerable in larger R2. However, in the deliberate attack, the higher-order network shown different results. When the capacity parameter T is smaller, the higher-order network shown more robust in smaller R2. However, as the capacity parameter T increased, the difference in I corresponding to different R2 was relatively smaller. That is, in deliberate attack, the weight coefficients R2 had a relatively small impact on the robustness of the higher-order network. In subgraph (b) of Fig. 9 also shows the similar results on the robustness of higher-order networks.

Subgraphs (c) and (d) drew the curves of the failed nodes ratios I and the maximum connectivity ratios G of random higher-order networks. The results in the subgraph (c) show that the value of I decreased with the increase of the value of T. Different from the scale-free higher-order network, there aren’t the transformation point in I. Moreover, the values of I decrease more quickly in random attack. The result indicate that when R1 < R2, the random higher-order network more robust in facing random attack. Meanwhile, in same attack, the smaller R2 is, the more robust the random higher-order network is. And we also observed that the impact of R2 on robustness of random higher-order was relatively small, especially for deliberate attack. The similar results of robustness in random higher-order network were verified by the maximum connectivity ratios G in subgraph (d) of Fig. 11.

Robustness analysis of higher-order networks base on real data

In order to confirm the practicability of the capacity load model which was proposed based on the higher-order structures. We applied the capacity load model to the public transportation data of HBN city in India53. We constructed a higher-order network based on the relationships between in bus stations and routes, where bus stations represent nodes in the higher-order network and routes represent edges. The robustness of the public transport higher-order network was analyzed by the proposed capacity load model in this paper and the proposed model has been validated to be available. Figure 12 shows the generalized degree distribution of the public transport higher-order network for Indian public transportation HBN city. The subgraph (a) of Fig. 12 showed the generalized degree distribution k1,0, and the generalized degree distribution k2,0 was showed in the subgraph (b) of Fig. 12. The results show that two generalized degree distributions both followed the power law distribution with fat tails. Therefore, the public transport higher-order network is a scale-free higher-order network.

Fig. 12
figure 12

Generalized degree distribution of the Indian public transportation HBN higher-order network.

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Figures 13 and 14 show the impact of different weight coefficients on the robustness of the Indian public transportation HBN higher-order network. Figure 13 shows the simulation results of R1 > R2, where R1 = 1, R2 = 0.2, R2 = 0.5, R2 = 0.7. The results in subgraph (a) of Fig. 13 show that for the same weight coefficient R2, as the capacity parameter T increases, the failed nodes ratio I in deliberate attack were higher than in random attack. The similar results were also shown in subgraph (b) of Fig. 13, as the capacity parameter T increases, the maximum connectivity ratio G in deliberate attack were lower than that in random attack. Therefore, when R1 > R2, the Indian public transportation HBN higher-order network were robust to random attack and vulnerable to deliberate attack. In addition, when the Indian public transportation HBN higher-order network facing the random attack, for smaller T, this network with smaller R2 was more robust, and for larger T, this network with smaller R2 was more vulnerable. The similar results were verified by maximum connectivity ratio G in subgraph (b) of Fig. 13. Thus, for higher-order networks based on the real data sets, we can be control its robustness though adjust weight coefficient of 2-simplex R2.

Fig. 13
figure 13

The influence of weight coefficients on the robustness of the higher-order HBN network for Indian public transportation. (R1 > R2).

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Fig. 14
figure 14

The influence of weight coefficients on the robustness of the higher-order HBN network for Indian public transportation. (R1 < R2).

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Figure 14 shows the simulation results in R1 < R2 and R1 = 1, R2 = 2, R2 = 5, R2 = 7, R2 = 9. The simulation results in subgraph (a) of Fig. 14 show that, in Indian public transportation HBN higher-order network, for the same weight coefficient R2, when the capacity parameter T is smaller, the failed nodes ratio I in deliberate attack was higher than random attack. However, as the capacity parameter T increases led to the opposite results. That is for larger T, there weren’t failed nodes in this higher-order network in deliberate attack. Subgraph (b) of Fig. 14 showed similar results about robustness of this Indian public transportation HBN higher-order network. Thus for R1 < R2, higher-order networks are robust to random attacks and vulnerable to deliberate attacks when the capacity parameter T is small. However the capacity parameter is large, higher-order networks are more robust to deliberate attacks and more vulnerable to random attacks. In addition, it can be seen from the Fig. 14 that in the random attack, when the capacity parameter T is small, the value of the weight coefficient R2 is smaller, and the higher-order network is more vulnerable. As the capacity parameter T increases, the value of the weight coefficient R2 is larger, and the higher-order network is more vulnerable. But in deliberate attack, the robustness of the Indian public transportation HBN higher-order network was insensitive to R2. In subgraph (b) of Fig. 14, the curve of the maximum connectivity ratio G also verified similar results on the robustness of this higher-order network.

Our analysis results for HBN higher-order network are consistent with the results of theoretical analysis in synthetic scale-free higher-order network. That indicated the proposed model in this paper can be used widely.

Conclusions and discussions

To study the impact of 1-simplexes and 2-simplexes on the robustness of higher-order networks, we constructed two synthetic higher-order networks based on simplex structure. The constructed higher-order networks contained 0-simplexes, 1-simplexes, and 2-simplexes. Moreover, we proposed a capacity load model which suitable to describe the cascading failure process of higher-order networks. We analyzed the cascading failure process by proposed model and obtained some results. We found the impact of different simplexes on the robustness of higher-order networks. The following conclusions are drawn:

  1. 1)

    The robustness of higher-order network had a negative correlation with the number of 2-simplex. So we can improve its robustness by decrease the proportion of 2-simplex.

  2. 2)

    The robustness of the higher-order network was sensitive for weight coefficients of the 1-simplex and 2-simplex. Thus, we can control the robustness of higher-order by adjusting the weight coefficients. For smaller capacity parameter T, we can improve the robustness of higher-order network by increase the difference between R1 and R2.

In the future, we will study the following works based on the foundation of this paper, as follows.

  1. 1)

    Considering the dynamic changes of nodes over time in real world, a cascading failure model will explored which is suitable to describe the cascading failure process of temporal higher-order network.

  2. 2)

    Study the impact of more attack strategies on robustness of higher-order network, such as coordinated attack strategy.

  3. 3)

    Validate the applicability of the proposed model on more higher-order networks which are constructed by real datasets.