Large size networks may have small diameter, such as WWW. It is well known as the small-world effect, an ubiquitous feature of real-world networks. Mathematically, the small-world feature^[23] is described by the slow growth of average distance 3 We compare the average path length L_actual of the real network with that of its random counterpart L_random, which has small-world property with the same numbers of nodes and edges. If L_actual is close to L_random, we could say that the real-world network also has the small-world feature.

Different types of empirical data show that most of the real network’s degree distributions are subject to the power-law distribution or approximate power-law distribution. The feature of scale-free networks^[16] can be expressed as p(k) ∼ k^−γ, where k is the nodes degree, i.e., the number of links that a node has, and p(k) represents the probability of finding a node with degree k in the network.^[24]

The community structure represents the basic functional activity unit in the network, which shows the characteristics of “clustering and grouping”. The edges within the community are thicker, while the edges between different communities are sparse. In the real network evolution process, nodes with similar functions will gradually self-organize into a community. As the nodes in the community have similar function, different nodes can be replaced with each other in the production management. It is conducive to improve network reliability. At the same time, the detection of the community is also conducive to analyze communication differences of different communities, and also facilitate the production management.^[25]

The community testing continues to receive attention.^[26] Traditional approaches to community detection include spectral analysis,^[27–29] modularity based methods,^[30] edge clustering,^[31,32] clique percolation,^[33] and so on.^[43–45] In this work, we extend the spectral analysis methods to the analysis of networks of industrial Internet.

Spectral analysis methods mainly include Laplace matrix decomposition^[34,45] and normal matrix decomposition methods.^[35,36] Laplace matrix L = K − A, in which K is the degree of network diagonal matrix. The diagonal elements are the degrees of the nodes. The non-diagonal elements are zero. The matrix A is the connection matrix. If nodes i and j have direct edges of connection, a_ij = 1. Otherwise, a_ij = 0. The normal matrix is defined as N = K⁻¹ A.

The matrix L has a trivial eigenvalue 0, with the corresponding eigenvector 1_{N × 1}. The nodes of the same community are approximately equal in the eigenvector of the nonzero eigenvalue. If there are g communities, then there are g − 1 eigenvalues closed to zero, and the corresponding eigenvectors can be used as a basis for community division. Then, similar to matrix L, the normal matrix N has a trivial eigenvalue 1, with the corresponding eigenvector being 1_{N × 1}. If there are g communities, there will be g − 1 eigenvalues closed to 1, and the nodes of the same community are approximately equal in the corresponding eigenvector. The principles of spectral analysis for Laplace matrix and normal matrix are similar.

In order to improve the detection accuracy and be more likely to form a joint relationship in the real network between the nodes, we often introduce weights to the unweighted network, and make the weight of the edge inversely proportional to the degree of the vertex.^[33,34] In this way, we introduce parameter α, 4 5 When α ≈ 0, L_α degenerates into Laplace matrix. N_α cannot degrade into N, we introduce a symmetric matrix to avoid the appearance of complex elements in the eigenvectors.

The smartphone supply chain network and iPhone supply chain network are described in Table 1. After investigating the basic feature of the major brands of smartphone supply chain network and iPhone supply network, we can conclude that the characteristics of these two networks are similar. From Table 1, we can see that the edge density is very low in the network, both of them are a sparse network, and the average distances between the nodes are relatively short. In the real world, there is a famous concept of “six degrees of separation”, which means that the average distance between any two nodes is less than 6, to describe the small-world feature of real networks. It indicates that the average distances between nodes of the smartphone supply chain network and the iPhone supply chain network also meet the small-world feature. On one hand, it is noted that Apple as the core of the entire network has an absolute influence on the average distance, which indirectly shows Apple’s controllability to the supply chain system. On the other hand, the clustering coefficients of these two networks are very high, which indicates that the aggregation of nodes is high. Through the network topology, we can find that the high aggregation is mainly due to the same kind of components from the multiple suppliers, there is an intensive cooperation among them, and they can be replaced by each other, so high aggregation also leads to the robustness of the entire supply chain system. Thus, if one supplier has an unexpected accident, it will not affect the entire product chain.

Table 1.

Table 1.

Table 1. Basic features of the two supply chain networks: network size N, average degree ⟨ k ⟩, edge density ρ, average distance L, and cluster coefficient C. . Network N ⟨ k ⟩ ρ L C Smartphone 154 12.506 0.082 3.1792 0.7772 iPhone 78 4.41 0.057 1.9427 0.8248

Table 1. Basic features of the two supply chain networks: network size N, average degree ⟨ k ⟩, edge density ρ, average distance L, and cluster coefficient C. .

In order to further analyze the characteristics of small-world, we analyze the degree distribution, community, and other features to verify the efficiency of the network.^[42]

The network statistics are shown in Table 2. The results show that the smartphone component supply chain network and the iPhone component supply chain network have large clustering coefficients of regular networks, and also have small average path lengths of random networks. By comparing with the corresponding random network, satisfying Eq. (3), it shows that the supply relationship in the supply chain network of these two parts has significant small-world feature.

Table 2.

Table 2.

Table 2. Comparison of smartphone and iPhone supply chain networks and random networks. . Network Cactual Crandom Lactual Lrandom Smartphone 0.7772 0.042 3.1792 2.489 iPhone 0.8248 0.065 1.9427 3.031

Table 2. Comparison of smartphone and iPhone supply chain networks and random networks. .

The existence of small-world feature suggests that the suppliers in the supply chain network can easily communicate and cooperate with each other. A higher clustering coefficient indicates a tighter connection among the nodes in the supply chain network. With the rapid development of information technology, more and more enterprises use information technology and Internet media to establish a connection with each other. Enterprises can establish a closer relationship among various suppliers, and more frequently exchange information. In this way, a high coefficient clustering of suppliers can be selected as the core suppliers to focus on cooperation and development in component supply chain management. The average path length can represent the delivery time of the product in the supply chain network. The smaller the average path length in the component supply chain network, the higher the efficiency of the exchange and cooperation between the suppliers, thus, it can be used to determine the close relationship between two suppliers (nodes). In order to maintain the advantages in the competitive environment, suppliers should take the following measures: portfolio rebalancing, speed up the information exchange, improve ability to respond and adapt to change.

Figure 3(a) shows the degree distribution of smartphone components supply chain network. The degree distribution of iPhone components supply chain network is shown in Fig. 3(b). From Fig. 3, it is found that most of the nodes’ degrees are very small, in other words, most of the nodes have few neighbors, and it is noted that the distribution has a long tail. That is heterogeneity of the scale-free network. So it can be seen from the figure that the two networks approximately follow the power-law distribution.

Fig. 3.

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	Fig. 3. Degree distributions of (a) the smartphone’s and (b) iPhone’s supply chain networks.

The component supply chain network describes the connection between the main suppliers, and we analyze the basic characteristics of the supply chain network from the network point of view. The supply chain networks have a few nodes with a lot of links. That is to say, there is a lot of cooperation and information exchange channel. In the supply chain network, due to the priority of the connection, the core suppliers have a greater degree of connectivity. Therefore, to a certain extent, the core suppliers directly affect the overall size of the supply chain system. It is particularly important to enhance the core suppliers’ connection channels to value creation for the whole supply chain system.

At the same time, in the complex network research, it is found that the scale-free network has high communication efficiency and high survivability or robustness. The high communication efficiency is attributed to the large degree nodes as the information transit center, it is conducive to production management and information communication between the various suppliers, and to reduce storage costs and product backlog. The high survivability refers to that the network can still keep connectivity under random failure of the nodes, maintaining the good functional activity. In the real production, if a supplier who supplies problematic components, there will soon be an alternative plan immediately which can ensure production is orderly. However, if the large degree nodes fail, it will have a huge impact on the network function. Thanks to the proportion of large degree nodes being very low, we can focus on protection and special management of supplies with large degree nodes. It is conducive to improve the network survivability.

This paper conducts community detection on the iPhone supply chain network. Figures 4 and 5 describe the distribution of the network nodes location through the eigenvectors.

Fig. 4.

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	Fig. 4. Eigenvector distribution of Lα. Eigenvector 2 is the eigenvector corresponding to the second smallest eigenvalue, and eigenvector 3 is the eigenvector corresponding to the third smallest eigenvalue. (a) α = 0, (b) α = 0.3, (c) α = 0.6, (d) α = 0.9.

Fig. 5.

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	Fig. 5. Eigenvector distribution of Nα. Eigenvector 2 is the eigenvector corresponding to the second largest eigenvalue, and eigenvector 3 is the eigenvector corresponding to the third largest eigenvalue. (a) α = 0.1, (b) α = 0.4, (c) α = 0.7, (d) α = 1.

The nodes position coincidence is high in different α, and it has a few similarities. Furthermore, figure 6 depicts the clustering results obtained by L_α and N_α, and there is a similar result when α is another value. It is noted that the clustering results of the same method are different when α varies. Meanwhile, the results of various methods are different when α is the same. Because some nodes are isolated nodes, or locate at the boundary of the community, this makes these nodes difficult to detect. However, on the whole, we observe that the network is divided into three large communities that represent the major component suppliers of iPhone. It is noted that nodes within the same community represent suppliers providing the same components. Any failure of a supplier would not influence the production of iPhone, and thus the robustness of the network is improved. Analogously, the process of the industrial Internet evolution will also form the similar pattern: nodes (suppliers) in the same community produce similar components and compete with each other. The growth of one supplier would lead to the decrease of other suppliers, since the total demands are limited. That is to say, suppliers are sensitive to other similar suppliers and the market. Therefore, the industrial network has high efficiency of information communication (the community is relatively dense), which benefits reducing the product backlog and improving the efficiency of the whole social resource. For the smartphone network, the result is similar to Figs. 4 and 5, and we do not show them in the paper.

Fig. 6.

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	Fig. 6. (color online) Clustering results of spectral analysis, the same color represents a category. (a), (b) Clustering results using Lα when α = 0, 0.6; (c), (d) clustering results using Nα when α = 0.4, 1.