The relationship between the internal structure of the hyper-edge and robustness of the hypernetwork has not yet been investigated. Aiming at this problem, this paper proposes a hypernetwork capacity-load model with non-uniform load distribution. And obtained the robustness of the non-uniform scale-free hyper-network under different internal structures of the hyperedge. The simulation reveals that the robustness of the non-uniform scale-free hyper-network is closely related to the internal structure of the hyper-edge. The non-uniform scale-free hypernetwork is most robust when the nodes inside the hyper-edges are fully connected. The results show that the internal structure of hyper-edge has a large impact on the overall robustness of the non-uniform scale-free hyper-network.
Nowadays, complex networks have become an effective tool for modeling all kinds of complex
systems [
In this paper, based on non-uniform scale-free hyper-networks [
The concept of hyper-graph, defined as follows. If the binary relation H=(V,E) satisfies the condition
[
where the elements in the set V are calledthe nodes or vertices of the hyper-graph, and the elements
in E are called the hyper-edges of the hyper-graph. P(V) denotes the power set of the set V; then H is a
hyper-graph. The hyper-graph of a node i in a hyper-graph is defined as the number of hyper-edges
containing the node i, denoted as dH(i) [
Inspired by the capacity-load model proposed by Motter [
(1) In a hyper-network with N nodes, the initial load of node i is related to the hyper-degree dH(i) and node degree d(i) of that node, and its initial load Li(0) is defined as
Li (0) = α (dH (i) + d (i))β ,α ≥ 1,β ≥ 1
In order to control the initial load of node i, let α be the load parameter and β be the adjustable parameter.
(2) The load on the failed node i will be distributed to node j in 2 steps.
Step 1: First assign to its associated unfailed hyper-edges according to the priority probability. where Εi denotes the set of all associated hyper-edges of the faulty node i.
Step 2: After the faulty node i assigns the load to its adjacent unfailed hyper-edges according to the above equation, it continues to assign according to the priority probability Πnode.
where Γi denotes the set of all unfailed neighbor nodes of the faulty node i within the hyper-edge ei.
Then the load received by node j is shown in equation (4).
From equation (4), it can be seen that the additional load received by node j within the hyper-edge ∏ hyper−edge =
o(ei ) o(em ) m∈Ei ∏node =
kei (i) r∈Γi kei (r) ΔLji = Li × Πhyper−edge × Πnode ei is related to the order of the hyper-edge it is on, the commonness of node j within the hyper-edge ei, and the initial load of node i.
In a realistic hyper-network, the capacity is the maximum value of the load that a node or hyperedge can handle, proportional to the initial load of the node. Let the capacity of node j be, expressed by equation (5).
C j = (1+ T )Lj T ≥ 0 (5)
Here T is the capacity parameter, the larger the value of T, the higher the capacity of the node and the more resilient it is to failures, but the cost of resilience increases. The critical threshold TC is the minimum capacity value to avoid global collapse of the hyper-network. When T>TC, the entire hypernetwork does not experience a global collapse. When T<TC, the whole hyper-network will experience a global collapse. Therefore, the critical threshold TC of T is an important indicator of the robustness of the hyper-network. Obviously, a smaller TC indicates a more robust hyper-network.
If node j fails after obtaining additional load, it should satisfy the following inequality.
Lj + ΔLji > C j
If equation (6) holds, then node j will overload and fail, which may cause other nodes to fail when node j 's load is redistributed.
To measure the robustness of the hyper-network, node i is initially attacked and made to fail, and then its load is redistributed. For other nodes after load redistribution, the node fails if equation (6) is satisfied. When all nodes within a hyper-edge fail, then this hyper-edge fails. After the number of failed nodes in the hyper-network reaches a steady state or all nodes fail (global collapse), the number of failed hyper-edges FM (0 ≤ FM≤ M) in the hyper-network is counted and the percentage of hyper-edge failure fM is calculated as shown in equation (7).
F fM = M , 0 ≤ fM ≤ 1
M
Where M is the total number of hyper-edges in the hyper-network. From equation (7), it can be seen that a larger fM indicates a larger number of failed hyper-edges in the hyper-network, i.e., the less robustness of the hyper-network.
In analyzing the relationship between the internal structure of non-uniform scale-free hyper-network hyper-edge and the robustness of the hyper-network, this paper constructs three hyper-networks: the non-uniform scale-free hyper-network with preferentially connected nodes inside the hyperedge is denoted as NON-BA-P hypernetwork; the non-uniform scale-free hyper-network with stochastically connected nodes inside the hyperedge is denoted as NON-BA-S hypernet-work; the non-uniform scale-free hyper-network with fully connected nodes inside the hyperedge is denoted as NON-BA-F hypernetwork. And simulates the cascading failure process of the non-uniform scale-free hyper-network with different structures inside the three hyper-edges under two strategies of deliberate attack and random attack simulations, and the related experimental data are recorded. The random attack in the simulation experiment is to randomly select a node in the hyper-network to attack; while the deliberate attack is to select the node in the hyper-network that satisfies the maximum sum of node degree value and node hyperdegree to attack.
The size of each type of hyper-network is related to the parameter max-node, so in order to perform simulation analysis at different sizes, three sizes of networks with max-node of 20, 40 and 60 are used in this paper. The focus of this paper is to discover the influence of the internal structure of the hyperedge on the robustness of the hyper-network, and to eliminate the influence of other uncertainties on the robustness, we use the control variable method, i.e., the parameter α is taken as 10 and the parameter β is taken as 1. To ensure the validity and authenticity of the results, the experimental results are taken as the average of 100 times results. 0.8 0.7 0.6 fM0.5 0.4 0.3 0.2 0.1 0.9 0.8 0.7 0.6 fM0.5 0.4 0.3 0.2 0.1
0.8 0.7
Figs.3(a) and (b) further verify that the critical threshold TC of the NON-BA-F hyper-network is 0.8 0.7
We found through simulation experiments that the three non-uniform scale-free hyper-networks show a decreasing trend of hyper-edge failure ratio with the increase of capacity parameter value under two strategies of deliberate attack and random attack, and reach the critical threshold of global collapse under a certain capacity parameter. When T≤TC, all three non-uniform scale-free hyper-networks are in the state of global collapse, and when T>TC, the failure scale starts to decrease and finally reaches the state of global non-failure. In order to observe the change of the critical threshold more conveniently and intuitively, we give the data tables of the three non-uniform scale-free hyper-networks when the maximum number of nodes within the hyper-edge is 20,40,60, respectively, as shown in Table 1.
We find that different structures inside the hyper-edge have different effects on the robustness of the hyper-network. When the internal structure of the hyper-edge is fully connected, the non-uniform scalefree hyper-network is the most robust; followed by when the internal structure of the hyper-edge is randomly connected, the non-uniform scale-free hyper-network robustness is at a medium level; when the internal structure of the hyper-edge is preferentially connected, the non-uniform scale-free hypernetwork robustness is the worst.
In order to break through the limitations of the existing research on the structural robustness of hyper-networks and explore the relationship between the internal structure of hyper-edges and the robustness of hyper-networks, we propose three hyper-edges with different internal structure of non-uniform scale-free hyper-networks models, and propose a non-uniformly distributed capacity-load model of hyper-networks, and analyze the influence of the internal structure of hyper-edges on the overall robustness of non-uniform scale-free hyper-networks. The following conclusions are obtained: the internal structure of the hyper-edge has an important influence on the robustness of the non-uniform scale-free hyper-network, and the robustness of the non-uniform scale-free hyper-network is strongest when the internal structure of the hyper-edge is fully connected; the robustness of the non-uniform scale-free hyper-network is worst when the internal structure of the hyper-edge is preferentially connected. And when the maximum size number max-node of nodes inside the hyper-edge is larger, the robustness of the non-uniform scale-free hyper-network is stronger.
This work is supported by Natural Science Foundation of Qinghai Province in China (NO. 2019-ZJ7012) and the National Natural Science Foundation of China (NOS. 11801296, 61603206). 6 References