Robustness analysis of random hyper-networks based on the internal structure of hyper-edges

1.   Introduction

In recent years, hypergraph-based hyper-networks are superior in representing multivariate complex relationships, thus it is widely used to model many real complex systems ^{[1,2,3,4,5,6,7,8,9]}. Meanwhile, research on the robustness of hyper-networks has also made great progress. Ma studied the cascading failures of a hyper-network. They found that hyper-networks under attack exhibit similar properties to ordinary networks. And when faced with the same external perturbations, the hyper-network is more robust ^[10]. Ma proposed a hyper-network cascade failure model based on the structure of the hyper-network. They obtained similar conclusions through the simulation analysis of this model ^[11]. Chen proposed a capacity-load cascading failure model based on hyper-edge diffusion. The model is applied to random hyper-networks and small-world hyper-networks. The comparison reveals that random hyper-networks have stronger robustness than small-world hyper-networks ^[12]. Luo found hyper-networks are more suitable than ordinary networks to represent City Bus Network. The node represents bus stops and the hyper-edge represents routes. The authors also found the bus hyper-network is robust to random attacks and vulnerable to deliberate attacks ^[13]. Wang constructed a hyper-network model for High-speed Rail Logistics in China. They identified the important nodes and routes by analyzing the robustness of the High-speed Rail Logistics hyper-network ^[14]. Zhang analyzed the differences robustness under different methods of brain function hyper-network construction ^[15]. Pearcy proposed a bacterial metabolic hyper-network capable of describing general metabolic processes. The concept of percolation process is extended to bacterial metabolic hyper-networks. And the robustness of the bacterial metabolic hyper-network was synthesized ^[16]. However, in the existing studies on the robustness of hyper-networks, most researchers have ignored the influence of the internal structure of the hyper-edges on the robustness of the hyper-networks. Yet, when the internal substructure of a complex system changes, it can also cause changes in the system performance. For instance, in the QQ group hyper-network ^[6]. The nodes represent users, and the hyper-edges represent different QQ groups. If the users in the group do not know each other, it will slow down the spread of information. Conversely, it will enhance the spread of information. Similarly, changes in the internal structure of the hyper-edge also have an impact on the robustness of the hyper-network. Researchers can gain more comprehensive understanding of factors, influencing the robustness of hyper-networks. It can also provide a reference for proposing a strategy to improve the robustness of the hyper-networks. But, there are fewer studies on how the internal structure of the hyper-edge affects the robustness of the hyper-network.

Although most complex systems in reality are not always random. But, random networks still hold an important network structure of interest to researchers ^{[17,18,19,20,21,22,23,24]}. Studying the properties of random networks can provide a reference for the research of other network properties. This enables a clear understanding of impact different network structures on system performance. In previous studies, scholars compared structural properties of some networks with random networks ^[23,24], an objective and accurate evaluation of the performance of the network structure is made. We carried out a study on the influence of the internal structure of the hyper-edges on the robustness of k-uniform random hyper-networks.

In this paper, we construct three random hyper-networks with nodes connected in different ways inside the hyper-edges. We propose a new load assignment method based on the capacity-load model ^[25]. This model can better reflect the relationship between the hyper-edges and the nodes of the hyper-network. And we obtained the influence of different structures inside the hyper-edge on the robustness of the random hyper-network by simulation analysis.

2.   Hypergraph-based hyper-network model

Berge ^[26,27] first gave the concept of hypergraph, definition is as follows: If the binary relation $H = (V, E)$ satisfies the condition: $Ø = e_{i} \in P(V), i \in \{1, 2, \cdots, m\}; \cup_{i = 1}^{m} = V.$

Where $V = \{v_{1}, v_{2}, \cdots, v_{n}\}, E = \{e_{1}, e_{2}, \cdots, e_{m}\}$, and $P(V)$ denotes the power set of the set $V$; then $H$ is a hypergraph. The elements in the set $V$ are the nodes of the hypergraph. The hyper-degree of node $v_{i}$ is denoted by $d_{H}(v_{i})$ ^[28] is defined as the number of hyper-edges containing node $v_{i}$. The node degree of node $v_{i}$ is similar to that of the ordinary network. It is still defined as the number of ordinary edges associated with node $v_{i}$, denoted as the $d(v_{i})$ ^[28]. The ordinary degree of a node $v_{i}$ within a separate hyper-edge $e_{i}$ is denoted by $k_{e_{i}}(v_{i})$. Figure 1 is a hypergraph containing 10 nodes and 5 hyper-edges. Take the example of node $v_{3}$. The hyper-degree $d_{H}(v_{3})$ of $v_{3}$ is 3. The node degree $d(v_{3})$ is 5. The ordinary degree $k_{e_{2}}(v_{3})$ inside the hyper-edge $e_{2}$ is 2.

Figure 1.  A hypergraph containing 10 nodes and 5 hyper-edges.

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2.1.   k-uniform random hyper-network model

A hypergraph is said to be k-uniform hypergraph if the number of nodes in each hyper-edge is k. A hyper-network with k-uniform hypergraph as topology is a k-uniform hyper-network. In this paper, we study the influence of the internal structure of the hyper-edges on the robustness of k-uniform random hyper-networks. The procedure for the construction of the k-uniform random hyper-network is given below, as described in Algorithm 1.

Algorithm 1 Modeling the k-uniform random hyper-network.
Input: Uniform variable, $k$; Initial number of isolated hyper-edges, $M_{0}$; Number of nodes, $N = M_{0} \times k$; Hyper-edge connection probability, $p \in [0, 1]$;
Output: hyper-network-matrix
1:   do Time step $T = 0$, hyper-network-matrix;
2:   for $T \leq { M_{0} \choose 2 }$;
3:     do Choose two $e_{1}$, $e_{2}$ randomly among $M_{0}$ isolated hyper-edges;
4:      do Generate a random numbers, $s \in (0, 1)$;
5:        if $s\leq p$;
6:          do Randomly select k nodes from $e_{1}$, $e_{2}$ (select at least 1 node in each hyper-edge);
7:         do Connecting them into a new hyper-edge;
8:        end if
9:      do index $\gets$ the number of $k$ nodes;
10:     for $i$ in index;
11:       do hyper-network-matrix $(M_{0} + T, i) = 1$;
12:       do $T = T + 1$;
13:     end for
14:   end for

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2.2.   Random hyper-network models based on different internal structures of hyper-edges

To better discover the influence of the internal structure of the hyper-edges in random hyper-networks on the robustness, we did further research. Based on the $k$-uniform random hyper-network, we construct three random hyper-network models with different connection methods of nodes inside the hyper-edges. Shown in Table 1.

Table 1.  Three random hyper-network models.

	Nodes connection method inside the hyper-edge
Hyper-network type	Randomly connected	Preferentially connected	Completely connected
	ER-R hyper-network	ER-P hyper-network	ER-C hyper-network

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The construction process of the three hyper-network models is described in Algorithm 2.

Algorithm 2 Modeling three random hyper-networks
Input: Uniform variable, $k$; Total hyper-edges, $M$; Total nodes, $N$; Node reconnection edges probability, $p \in [0, 1]$;
Output: inner-hyper-network-matrix
1:   do Disconnect the original connections between nodes inside the hyper-edges;
2:   switch (selecting the hyper-edge internal connection method);
3:     case1 Hyper-edge internal nodes are randomly connected;
4:     do Make $k$ nodes inside the hyper-edge $e_{i}$ become isolated nodes; ordinary edge $m_{k} = 0$; inner-hyper-network-matrix
5:     for $m_{k} \leq p \times k (k - 1)/2$;
6:       do Generate a random number $r$;
7:       do Randomly select a pair of nodes $v_{i}, v_{j}$ inside the same hyper-edge;
8:       if $r < p$
9:         if inner-hyper-network-matrix $(v_{i}, v_{j})! = 1$
10:             do inner-hyper-network-matrix $(v_{i}, v_{j}) = 1$
11:             do inner-hyper-network-matrix $(v_{j}, v_{i}) = 1$
12:         end if
13:       end if
14:     end for
15:     end
16:     case2 Hyper-edge internal nodes are preferentially connected;
17:     do Make $k$ nodes inside the hyper-edge $e_{i}$ become isolated nodes; time step $t = 0$; inner-hyper-network-matrix
18:     do Select 3 nodes to be completely connected. Make them the initial connected branches inside the hyper-edges $C_{0}$;
19:     do $\Gamma_{t}$ (The set of nodes contained in the connected branch $C_{t}$ at time $t$)
20:     for $\Gamma_{t} \leq k$
21:       do $t = t + 1$
22:       do Randomly select a node $v_{j}$ outside $C_{t}$ at time $t + 1$, $v_{j} (v_{j} \in V_{e_{i}}, v_{j} \notin C_{t})$
23:       do $C_{t+1} = C_{t} + v_{j}$
24:       do $\Pi_{i} = \dfrac{k_{e_{i}}(v_{i})}{\sum_{j = 1}^{\left| \Gamma_{t} \right|} k_{e_{i}}(v_{j})}, v_{i}, v_{j} \in C_{t}$
25:       do $v_{j}$ selects a node $v_{i}$ inside $C_{t}$ with a preferential connection probability of $\Pi(v_{i} \in V_{e_{i}}, v_{i} \in C_{t})$
26:         if inner-hyper-network-matrix $(v_{i}, v_{j})! = 1$
27:             do inner-hyper-network-matrix $(v_{i}, v_{j}) = 1$
28:             do inner-hyper-network-matrix $(v_{j}, v_{i}) = 1$
29:         end if
30:     end for
31:     end
32:     case3 Hyper-edge internal nodes are completely connected;
33:     do Make $k$ nodes inside the hyper-edge $e_{i}$ become isolated nodes; ordinary edge $m_{k} = 0$; inner-hyper-network-matrix
34:     do Each node is connected to $k - 1$ nodes with edges
35:     if inner-hyper-network-matrix $(v_{i}, v_{j})! = 1$
36:             do inner-hyper-network-matrix $(v_{i}, v_{j}) = 1$
37:             do inner-hyper-network-matrix $(v_{j}, v_{i}) = 1$
38:             do $\textbf{diag-inner-hyper-network-matrix} = 0$;
39:     end if
40:     end
41:   end

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3.   Hyper-network cascade failure model

Previous researchers have proposed some hyper-network cascade failure models. But, there is the problem of considering too single metrics. As an example, Chen proposed a hyper-network cascading failure model based on hyper-edge diffusion ^[12]. The authors converted the hyper-networks into line graphs for research. They default hyper-edge internal nodes are completely connected. When a hyper-edge is attacked and fails. Failure hyper-edge's load is spread to its neighboring unfailed hyper-edges. But it does not consider the case where the load is spread through the nodes inside the hyper-edges. To better characterize the influence of the internal structure of the hyper-edge on the robustness of the hyper-network. We propose a new load distribution method based on the capacity-load model.

In a hyper-network with $N$ nodes. The initial load of node $v_{i}$ is related to the hyper-degree $d_{H}(v_{i})$ and the node degree $d(v_{i})$ of this node. The initial load $L_{v_{i}}(0)$ is defined as:

To control the initial load of node $v_{i}$, let $\alpha$ is a load parameter and $\beta$ is an adjustable parameter. From (1), it shows that the initial load of the node is proportional to its hyper-degree and node degree.

Let node $v_{i}$ and node vj connected by ordinary edges inside a certain hyper-edge. At time $t$, node $v_{i}$ failed. Firstly, the load $L_{v_{i}}$ of the failed node $v_{i}$ will be equally divided by the hyper-degree $d_{H}(v_{i})$ of $v_{i}$. And assigns the equalized load to all the unfailed hyper-edges associated with the failed node. Then the load received by all hyper-edges associated with the failed node $v_{i}$ is $L_{v_{i}}/d_{H}(v_{i})$. Then, the load received by the hyper-edge $e_{i}$ is equalized again according to the $v_{i}$'s node degree $k_{e_{i}}(v_{i})$ in the hyper-edge $e_{i}$. The node $v_{j}$ receives the load as shown in (2).

The $\theta$ is the perturbation parameter, and $0 \leq \theta \leq 1$. From ($2$), a node $v_{j}$ inside the hyper-edge $e_{i}$ at time $t$ receives a load $\Delta L_{v_{i} \to v_{j}}(t)$ that is related to three factors: The hyper-degree of the failed node $v_{i}$ at time $t$; The ordinary node degree of node $v_{i}$ inside the hyper-edge $e_{i}$; and the initial load of node $v_{i}$. If node $v_{j}$ has not failed at time $t$, its load is the sum of the load of node $v_{j}$ at time $t - 1$ and the load redistributed from the failed node $v_{i}.$ That is:

The capacity is the largest amount of load can be handled by the node or hyper-edge. It is proportional to the initial load of the node. Let the capacity of node $v_{j}$ be $C_{v_{j}}$, which is expressed by (4):

$T$ is the capacity parameter. When the value of $T$ is larger, the capacity of the node is larger. With a larger node capacity, the node is more resilient to failures. But the cost of resilience increases. The critical threshold $T_{C}$ is the smallest value of the capacity parameter to avoid a global crash of the hyper-network. When $T > T_{C}$, there is no global crash in the whole hyper-network. When $T < T_{C}$, a global crash occurs in the whole hyper-network. So, the critical threshold $T_{C}$ of $T$ is an important measure of the robustness of the hyper-network. Obviously, the smaller $T_{C}$ the more robust hyper-network is.

If node $v_{j}$ fails after obtaining extra load, it will meet ($5$):

If (5) is true, then node $v_{j}$ will be overloaded and fail. When the load of $v_{j}$ is redistributed, it may cause other nodes to fail. Combining with (2), (5) can be expressed as:

To measure the robustness of the hyper-network. We initially attack the node $v_{i}$ and defeat it, and then redistribute its load. For other nodes after load redistribution, if ($6$) is fulfilled, the node fails. When all nodes inside a hyper-edge failed, the failure of this hyper-edge occurs. After the number of failed nodes in the hyper-network attains steady state or all nodes failed, we count the number of failed hyper-edges $F_{M} (0 \leq F_{M} \leq M)$ in the hyper-network. The hyper-edge failure ratio $f_{M}$ is shown in ($7$):

$M$ is the total number of hyper-edges in the hyper-network. From (7), it can be seen that the larger the $f_{M}$, the higher the number of failed hyper-edges in the hyper-network. It means the less robustness of the hyper-network.

4.   Simulation analysis

To analyze the influence of the node connection method inside the hyper-edge on the robustness of the $k$-uniform random hyper-network. We designed a simulation experiment as in Algorithm 3. The cascading failure process of $k$-uniform random hyper-network is simulated under both deliberate and random attack strategies. And record the relevant experimental data. Table 2 shows the experimental parameters in the simulation experiments. Each datum is averaged over 50 independent realizations.

Algorithm 3 Simulation steps
1:    do Construct a k-uniform random hyper-network with k of 20, 40, 60
2:    do Construct ER-R, ER-P and ER-C hyper-networks
3:    for $i = 1 : N$
4:      do Set node $v_{i}$ initial capacity and load
5:    end for
6:    switch (Choose an attack strategy)
7:      case1: Random attack strategy: Randomly select a node
8:      case2: Deliberate attack strategy: Select a node with the largest sum of hyper-degree and node degree in the hyper-network
9:    end
10:   while Hyper-network not in stable state
11:      do Attacking a node disables it
12:      do The load of the failed node is distributed to the non-failed nodes adjacent to it
13:      if All nodes in a hyper-edge are invalid
14:        do This hyper-edge fails and counts the number of failed hyper-edges
15:      end if
16:    end
17:   do Calculate and output the hyper-edge failure ratio

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Table 2.  Experimental simulation parameters and values.

Parameter Name	Parameter Meaning	Value range
$N$	Total number of hyper-network nodes	[80, 1600]
$M$	Total number of hyper-edges	50
$\alpha$	Load parameters	10
$\beta$	Adjustable parameters	1
$\theta$	Perturbation parameters	1
$L$	Node Load	–
$d$	Node-degree	–
$d_{H}$	Hyper-degree	–
$k$	Uniform number	20, 40, 60
$k_{e_{i}}$	Node-degree of the node inside a hyper-edge $e_{i}$	–
$C$	Node Capacity	–
$T$	Capacity Parameters	[0, 1]
$F_{M}$	Failure hyper-edge number	[0, $M$]
$f_{M}$	Hyper-edge failure ratio	[0, 1]
$T_{C}$	Critical Threshold	–

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4.1.   Robustness analysis of three k-uniform random hyper-networks

In this section, we make three $k$-uniform random hypernetworks under attack at different $k$. Subsequently, we analyze the variation of hyper-edge failure ratio $f_{M}$ with capacity parameter $T$ for three $k$-uniform random hypernetworks. The results show that: Under deliberate and random attacks, hyper-edge failure ratio tends to decrease with increasing value of capacity parameter. The hyper-network attains a critical point of global collapse under certain capacity parameters. When $T \leq T_{C}$, all three types of $k$-uniform random hyper-networks are in a state of global collapse. When $T > T_{C}$, the failure size starts to decrease and eventually attains the state of global non-failure.

4.1.1.   ER-R hyper-network robustness analysis

Figure 2 shows the variation of the hyper-edge failure ratio $f_{M}$ with the capacity parameter $T$, when the ER-R hyper-network is attacked with different sizes $k$. From Figures 2a and 2b it can be seen that the critical threshold $T_{C}$ of the ER-R hyper-network gradually reduces with the increase of the size $k$. Since the smaller the $T_{C}$, the more robust the hyper-network is, i. e., the robustness of the ER-R hyper-network increases with the size $k$ of the hyper-network.

Figure 2.  ER-R hyper-network hyper-edge failure ratio under different capacity parameters; (a) Random attack; (b) Deliberate attack.

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The diagram cannot clearly distinguish the difference between the critical threshold $T_{C}$ of the ER-R hyper-network when it is subjected to random attacks and deliberate attacks. So, we give the ER-R hyper-network specific critical thresholds when it is under attack, shown in Table 3. From the data in Table 3, it can be seen that ER-R hyper-network's $T_{C}$ under deliberate attack is always smaller than $T_{C}$ under random attack. Again, the smaller $T_{C}$ is, the more robust the hyper-network is, which gives: The ER-R hyper-network is robust to deliberate attacks and vulnerable to random attacks.

Table 3.  Critical threshold $T_{C}$ of ER-R hyper-network with different size $k$ ($k = 20, 40, 60$).

Hyper-network type	Attack strategy	Critical threshold $T_{C}$
		$k = 20$	$k = 40$	$k = 60$
ER-R hyper-network	Random attack	0.0668	0.03	0.02
	Deliberate attack	0.0652	0.0292	0.0192

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It can be seen that the ER-R hyper-network exhibits the same robustness as the random network when subjected to external attacks. That is, ER-R hyper-network is robust to deliberate attacks and vulnerable to random attacks. Next, we analyze the reasons for this robustness from the perspective of the ER-R hyper-network structure.

The connection between each hyper-edge of the $k$-uniform random hyper-network is a random connection. The distribution of the hyper-degree of the $k$-uniform random hyper-network at $k = 60$ is shown in Figure 3. When the connection between the nodes inside the hyper-edge is randomly connected, the node degree distribution and the hyper-degree distribution inside each hyper-edge are Poisson distributions. As shown in Figures 3 and 4. This connection enhances the random property of k-uniform random hyper-networks. Thus, $k$-uniform random hyper-networks exhibit properties consistent with random networks when attacked. That is, it is robust to deliberate attacks and vulnerable to random attacks.

Figure 3.  $k$-uniform random hyper-network hyper-degree distribution ($k = 60$).

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Figure 4.  ER- hyper-network node degree distribution ($k = 60$).

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4.1.2.   ER-P hyper-network robustness analysis

Figure 5 shows the variation of the hyper-edge failure ratio $f_{M}$ with the capacity parameter $T$, when the ER-P hyper-network is attacked with different sizes $k$. From Figures 5(a) and 5(b), it can be seen that the robustness of the ER-P hyper-network increases with the increase of the hyper-network size $k$.

Figure 5.  ER-P hyper-network hyper-edge failure ratio under different capacity parameters; (a) Random attack; (b) Deliberate attack.

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Next, we give the ER-P hyper-network specific critical thresholds when it is under attack, shown in Table 4. From the data in Table 4, it can be seen that ER-P hyper-network's $T_{C}$ under deliberate attack is always bigger than $T_{C}$ under random attack. That is, the ER-P hyper-network is robust to random attacks and vulnerable to deliberate attacks.

Table 4.  Critical threshold $T_{C}$ of ER-P hyper-network with different size $k$ ($k = 20, 40, 60$).

Hyper-network type	Attack strategy	Critical threshold $T_{C}$
		$k = 20$	$k = 40$	$k = 60$
ER-P hyper-network	Random attack	0.114	0.068	0.0516
	Deliberate attack	0.1524	0.1464	0.13

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It can be seen that the ER-P hyper-network exhibits the contrary robustness as the random network when subjected to external attacks. That is, ER-P hyper-network is robust to random attacks and vulnerable to deliberate attacks. Why does this kinetic phenomenon occur? Next, we analyze the reasons for this robustness from the perspective of the ER-P hyper-network structure. When the connection between nodes inside the hyper-edges is preferentially connected, the node degree distribution inside each hyper-edge is a power-law distribution. Since the hyper-degree distribution is still Poisson, it makes the entire node degree distribution of the ER-P hyper-network like a power-law distribution. As shown in Figure 6. This hyper-edge internal structure reduces the randomness of the k-uniform random hyper-network, so that ER-P hyper-network exhibits the consistent properties of a scale-free network when it is under attack. That is, it is robust to random attacks and vulnerable to deliberate attacks.

Figure 6.  ER-P hyper-network node degree distribution ($k = 60$).

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4.1.3.   ER-C hyper-network robustness analysis

Figure 7 shows the variation of the hyper-edge failure ratio $f_{M}$ with the capacity parameter $T$, when the ER-C hyper-network is attacked with different sizes $k$. From Figures 7(a) and 7(b), it can be seen that the robustness of the ER-C hyper-network increases with the increase of the hyper-network size $k$.

Figure 7.  ER-C hyper-network hyper-edge failure ratio under different capacity parameters; (a) Random attack; (b) Deliberate attack.

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Again, we give the ER-C hyper-network specific critical thresholds when it is under attack, shown in Table 5. From the data in Table 5, it can be seen that ER-C hyper-network's $T_{C}$ under deliberate attack is always bigger than $T_{C}$ under random attack. That is, the ER-C hyper-network is robust to random attacks and vulnerable to deliberate attacks.

Table 5.  Critical threshold $T_{C}$ of ER-C hyper-network with different size $k$ ($k = 20, 40, 60$).

Hyper-network type	Attack strategy	Critical threshold $T_{C}$
		$k = 20$	$k = 40$	$k = 60$
ER-C hyper-network	Random attack	0.0392	0.0204	0.0138
	Deliberate attack	0.0472	0.0206	0.015

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The ER-C hyper-network, like the ER-P hyper-network, both exhibit the opposite robustness to the random network when subjected to attacks. That is, the ER-C hyper-network is robust to random attacks and vulnerable to deliberate attacks. Here we analyze the reasons in detail.

When the nodes inside the hyper-edge are completely connected to each other, the inside of each hyper-edge is a regular network. Due to the load assignment method defined in this paper, there will be some nodes with large hyper-degree in the load assignment with overlapping load assignment. Because the random hyper-network is a uniform hyper-network, the difference in hyper-degree from node to node is small. Attacking the node whose hyper-degree is large, the neighboring nodes of the failed node will receive more load. When a neighboring node receives the load assigned by the failed node, it is more likely to crash. So, the ER-C hyper-network is robust to random attacks and vulnerable to deliberate attacks. The difference between the critical threshold $T_{C}$ of random and deliberate attacks is particularly small influenced by the hyper-network structure.

In addition, Chen studied random hyper-network that also used full connections within the hyper-edge ^[12]. However, it uses the method of converting the hyper-network into a line graph network to study. In other words, the effect of the internal structure of the hyperedges on the robustness of the hypernetwork is ignored. Moreover, Chen proposed a capacity-load model with equal load distribution that does not consider the effect brought by the structure. i.e., the effect of the hyper-edge inside and outside together on the hyper-network robustness. We reproduced the simulation of Chen and compared it with the method in this paper, as shown in Figure 8. To ensure the validity of the comparison, the two methods are compared at the same hyper-network size $k = 20$. When the $k$-uniform random hyper-network under random attack, in Chen's method the critical threshold $T_{C} = 0.075$ and in this papers' method the critical threshold $T_{C} = 0.039$. When the $k$-uniform random hyper-network under deliberate attack, in Chen's method the critical threshold $T_{C} = 0.07$ and in this papers' method the critical threshold $T_{C} = 0.047$. Thus, it can be seen that the robustness of this paper's method is better than Chen's method under attack.

Figure 8.  Robustness comparison of $k$-uniform random hyper-networks under two capacity-load models; (a) Random attack; (b) Deliberate attack.

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This shows that, it can be seen that the robustness of the three $k$-uniform random hyper-networks is closely related to the connection method of nodes inside the hyper-edges. When the nodes inside the hyper-edges are randomly connected, the $k$-uniform random hyper-network exhibits robustness consistent with that of a random network. When the nodes inside the hyper-edges are preferentially connected and completely connected, the $k$-uniform random hyper-network exhibits robustness in contrast to the random network.

Analyzing again the critical threshold $T_{C}$ for the three $k$-uniform random hyper-networks under attack reveals that: For the same hyper-network size $k$, the ER-C hyper-network has the best robustness, followed by the ER-R hyper-network, and the ER-P hyper-network has the worst robustness. Analysis of its causes: The robustness of $k$-uniform random hyper-networks under attack is related to the number of ordinary edges inside the hyper-edges. The size of the ordinary edges with different connection methods of nodes inside the hyper-edge is shown in Figure 9.

Figure 9.  The variation of ordinary edge size $m_{k}$ inside the hyper-edge with hyper-network size $k$ for the three connection methods.

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5.   Conclusions

The innovations of this paper are mainly 2 points. First point is: Studying the robustness of $k$-uniform random hyper-networks from perspective of the internal structure of hyper-edges. Second point is: Based on the capacity-load model, we propose a capacity-load model that is more suitable for describing the load distribution inside and outside the hyper-edge. Applying this model to $k$-uniform random hyper-networks, its robustness under attack is obtained.

Based on the innovations of this paper, we have obtained conclusions as follows. When the connection between nodes inside a hyper-edge is same as the connection between hyper-edges, the $k$-uniform random hyper-network is robust to deliberate attacks and vulnerable to random attacks. When the connection between nodes inside a hyper-edge is not consistent with the connection between hyper-edges, the $k$-uniform random hyper-network is robust to random attacks and vulnerable to deliberate attacks. The ER-C hyper-network has the best robustness, followed by the ER-R hyper-network, and the ER-P hyper-network has the worst robustness. In other words, the larger number $m_{k}$ of ordinary edges inside the hyper-edge, the better robustness of the $k$-uniform random hyper-network. The larger size $k$ of the $k$-uniform random hyper-network, the more robust it is. To summarize, the internal structure of hyper-edges has an important impact on the robustness of hyper-networks.

Acknowledgments

This work is supported by Natural Science Foundation of Qinghai Province in China (NO. 2019-ZJ-7012) and the National Natural Science Foundation of China (NOS. 11801296, 61603206).

Conflict of interest

All authors have no conflicts of interest to declare.