#### 5.1. Storage Efficiency Analysis

In this section we discuss the parameters of the hypergraph-based blockchain network. We study the effect of network parameters on storage efficiency. As discussed in

Section 3.3.1, the main parameters are the graph’s degree N and the count of the upper limit of nodes in hyperedge C. These parameters affect the average storage in each node (SEN), or the used memory in each node. The methodology of the designed experiments is that: For a given value of

C and a number of total nodes in the network we calculate the average storage capacity of nodes and draw the figures. In the following experiments, we assume that the number of transactions in the network is 10

^{4}. The results are shown in

Figure 7. From

Figure 7, we can see that with a given N, if the value of C increases, the average record storage capacity of each node continuously increases. The reason is that with the increase of C, the number of nodes in each hyperedge increases, while the number of hyperedges decreases. Each transaction record is stored by more nodes, so that the average storage in each node increases. On the other hand, it also can be seen that the smaller the number of nodes in the graph, the greater the storage capacity of each node. With the same number of transaction records and C values, the smaller the number of nodes, the smaller the corresponding number of hyperedges. Under the premise that the distribution of nodes in each hyperedge is the same, there are more transaction data records in each hyperedge in a graph with fewer hyperedges, resulting in a larger average storage capacity for each node. In particular, when C is equal to the number of nodes in the hypergraph, the proposed model will degenerate into the original blockchain model.

Furthermore, we found that for different N, the curves in the images are very similar. Therefore, we superimposed different curves of N from 1 to 10, and the results are shown in

Figure 8. It can be seen from

Figure 8 that the graph’s degree N of the hypergraph has almost no effect on the average storage of each node. That means the storage of each node is only related to the number of transactions and C. In order to prove the advantage of the hypergraph-based blockchain model, we compared the total used memory capacity in the model with the original blockchain model. The results are shown in

Figure 9. The original blockchain stores all records in all nodes, so the used memory capacity for the whole network is in proportion to the number of transactions. It can be clearly seen from

Figure 9 that the model proposed in this paper is much smaller than the original blockchain model in terms of overall network storage consumption, and it also shows that the total storage quantity is basically independent with the graph’s degree N.

In the original blockchain model, if 51% of nodes are hacked at the same time, the deception can be established. However, the risk is reduced in the proposed model. Therefore, a bigger C must be set and the arbitration threshold should be set lower.

#### 5.2. Network Evolution

The security analysis in

Section 3.5 illustrates the importance of the co-rank, which is denoted by C/2 in

Section 3.3.1, and the experiments in

Section 5.1 show that not only the storage efficiency but also security factors rely on the network parameter C. In this section, we discuss the network evolution with C.

In

Section 3.6, we designed a hyperedge splitting algorithm for the evolution of network structure, which affects the average cardinality of the graph. The algorithm guarantees the cardinality of each hyperedge between co-rank(C/2) and rank(C). Therefore, as the number of network nodes increases, the number of network hyperedges will change, but a lower bound and an upper bound will be guaranteed. Let

h_{min}(n) denote the min count of hyperedges in a

n vertexes hypergraph and

h_{max}(

n) to the max. Depending on the algorithm, we get the following formulas:

Equation (1) means that all vertexes always insert into one hyperedge and split the hyperedge. Then for all hyperedges in the graph, there are only C/2 vertexes in each edge. For the Formula (2), vertexes are inserted into the edge with the fewest number of vertexes each time, only when all hyperedges have C nodes does the splitting happen. As a general situation, a new vertex is inserted into a hyperedge randomly. Then the nodes are uniformly distributed, meaning that when there are

k hyperedges in the graph, after inserting

k*C/2 nodes, the k nodes inserted next will cause the

k edges to split. Thus, if we let

h(

n) denote the count of hyperedges, it should meet the following formula:

We designed experiments to verify this characteristic of the network, and the results are shown in

Figure 10.

Figure 10 shows that with the increase of C, the capacity of each hyperedge increases, and the change of hyperedge number is more and more stepwise, which is in line with the description of Equation (3). Particularly, in

Figure 10f, the height and length of each step is almost twice that of the previous level, which is consistent with the analysis results.

The experiments in

Section 5.1 show that for minimizing the entire network storage capacity, we need a small C, but as discussed in

Section 3.5, for security, we need a big C; this is a contradiction. Therefore, we need to find the minimum value of C to maintain security. The problem is that for a given verification threshold

t, how many forged nodes must be joined in order to construct a hyperedge with a forged node ratio exceeding

t. Let

n(

i) denote the nodes counted in a hypergraph with

i hyperedges. Based on experiments in

Section 5.2, we can approximately consider that:

Obviously,

n(1) ∈ [1,

C/2], and C is much bigger than 1. Therefore, from iteration Formula (4) we get the general term of

n(i):

From

n(

i) to

n(

i + 1) the new node ratio goes to 50% for each hyperedge, and the count of nodes we added is calculated in Equation (6):

Under equal probability conditions, if we want a hyperedge with a forged node ratio exceeding

t, we at least add

${\mathrm{log}}_{2}\frac{1}{1-t}$ edges. The total nodes added,

T(

n)

, is calculated in Equation (7):

The edge count

i is related with the node count in the network in Equation (3) and can be approximately considered to be

$\mathrm{i}={\mathrm{log}}_{2}\left(N+1\right)$ (approximate calculation according to

Figure 10f) where

N denotes the count of nodes in the network. Equation (7) becomes to Equation (8):

In Equation (8), T(n) denotes how difficult it is to forge a hyperedge. It is related with the rank C, network scale N, and the given verification threshold t. For the given parameters of T(n), N and t, we get the smallest C by solving Equation (8).