Towards Evaluating the Robustness of the Open-Source Product Community under Multiple Attack Strategies

Abstract

As an emerging product innovation model, open-source innovation has undergone rapid development in recent years. The sustainability and stability of the open-source product community (OSPC) is crucial for product innovation, and the effect that users have on the OSPC’s robustness is an important and closely scrutinized topic. This paper explores the robustness of the OSPC from the aspect of user knowledge contribution. We first construct an OSPC network and analyze its characteristics. An improved node evaluation method is then proposed to identify different types of knowledge contribution nodes. Further, seven node- and edge-based attack strategies are designed to simulate network robustness changes, with evaluation indicators being proposed for structural robustness and knowledge robustness. The results reveal that our proposed node evaluation method can effectively identify nodes of different knowledge contribution types. Additionally, the network is found to have different robustness performance when facing multiple deliberate attacks on three important knowledge contribution node types. Moreover, the network shows different robustness characteristics when facing deliberate attacks on betweenness and weight edges. Our findings can benefit product innovation and OSPC managers by enhancing the robustness of the OSPC network.

1. Introduction

With the arrival of the digital economy era, the open-source community (OSC) has become the leading mode of innovation in the global software industry. In this unique collaborative mode, users from all over the world voluntarily communicate, collaborate, and share knowledge on the OSC platform via the Internet, generating high-quality products, services, and ideas. Open-source systems, such as Linux, Apache, and Firefox, have been proven to have good robustness and reliability. The sustainable development of the OSC is driven by the participation, collaboration, and innovation of dedicated users, whose online participation is closely related to many high-quality products [1,2]. When participating in open-source projects, users can keep up with the latest developments in relevant technical knowledge and expand their technical capabilities, which further excites their enthusiasm for participation. Therefore, it is of great value to OSC managers to ensure the continuous participation of existing users in community projects and attract new users to the community.

The OSC’s adherence to the concepts of openness and sharing allows users to freely create, yet this can also bring uncertainty to project progress and product stability. For example, users may reduce their contribution or even leave the OSC when they encounter conflicts or disagreements during the development process, when the community’s management mechanism does not match their needs, or when the project lacks attractiveness or is subject to malicious competition from competitors. A lack of user participation and contribution is the main reason for OSC failure [3]. Therefore, for the future development of an OSC, it is crucial for managers to be able to detect such risks in a timely manner and develop effective response strategies when users exhibit the behavior of leaving the community. Concurrently, differences in users’ skills and motivations also create significant variety in the level of user participation and contribution to community projects. Some scholars found that the collaborative behavior of users has significant heterogeneity, that is, the minority of users participate in many projects and engage in a large amount of knowledge collaboration with other users, while the majority of users participate in few projects and engage only in a small amount of knowledge collaboration [4]. As a result, in the community evolution process, users of varying importance are found, such as core leaders, core contributors, ordinary users, and “free riding” users [2,5]. The high level of activity and contribution of key users means that their departure can have a significant impact on the OSC. Therefore, identifying key users—and developing targeted incentive strategies to attract and retain them—is of great significance to the sustainable development of an OSC.

As a complex system, an OSC is extremely vulnerable to user churn. Robustness is the ability of a system to maintain a certain performance level under parameter perturbations [6]. Many scholars evaluate this robustness using complex network design attack strategies [7,8,9,10,11,12], which involve the mapping of attacks on network systems based on real-world scenarios. Yang et al. [13] proposed five attack strategies based on the degree and capacity of nodes, targeting Barabási–Albert (BA), Watts–Strogatz small world (WS), and Erdös–Rényi (ER) networks, and Cui et al. [14] considered intra-degree priority and inter-degree priority attack modes as network attack strategies. Zhao et al. [15] defined six forms of malicious attacks for the analysis of knowledge network robustness. Existing research has achieved good results in designing attack strategies by removing key network nodes or edges to observe changes in network performance. However, there is limited research on the further refinement of these key nodes and edges. In the open-source product community (OSPC), users join the community for its rich knowledge resources: knowledge flows and spreads within the community based on the collaborative relationships between users, becoming one of the most valuable community resources [16]. A user’s individual characteristics can give them advantages in disseminating information and knowledge, collaborating on tasks, or integrating projects. When such users leave the community or reduce their contribution to the community, it can affect the evolution of community products and the attractiveness of the community [17]. Therefore, it is necessary to categorize nodes and edges based on their different advantages to determine their importance in the network. Attack strategies can then be designed from multiple perspectives to analyze how network robustness is impacted by attacks on nodes and edges.

Key network nodes determine the evolution direction of community products and networks [18]. The accurate and effective identification of key nodes is crucial for the robustness and stability of the network, such as in terms of virus transmission and information diffusion [19,20,21]. In recent years, an increasing number of scholars have focused on the identification of key nodes in complex networks. Traditional centrality methods such as degree centrality, betweenness centrality, presence centrality, eigenvector centrality, and k-shell are widely used to identify key network nodes. Due to the inability of any single centrality method to accurately identify key nodes, some scholars have proposed improved methods based on these centrality indicators. For example, Kitsak et al. [22] stated that a node’s position affects its importance in the network; they proposed k-shell decomposition centrality to identify important nodes in complex networks, yet the nodes obtained using this method are too coarse-grained. Xie et al. [23] proposed a complex network key node identification method that measures the impact of (a) the global structure using the k-shell method and (b) the local structure by assessing the degree of influence (i.e., comprehensiveness) of neighboring and secondary neighbor nodes on specified nodes. Yang et al. [24] comprehensively considered both the global position and local structure using the number of iterations in the k-shell decomposition process; they proposed an improved k-shell decomposition method to identify influential nodes. Some scholars combine several classical centrality measures according to the method of the technique for order preference by similarity to an ideal solution (TOPSIS) to identify influential network nodes. For instance, Yan et al. [25] identified influential nodes using several different basic centrality methods, such as the entropy weight method and gravity law, and Yang et al. [26] proposed a comprehensive measurement method that considered degree centrality, compactness centrality, and intermediate degree centrality. Wang et al. [27] evaluated node importance by considering the topological structure, industry characteristics, and directionality of the air cargo network using the TOPSIS method. These methods provide valuable references for evaluating important network nodes. However, in OSPC networks, different types of knowledge contribution nodes have different advantages and measurement indicators. As such, based on the TOPSIS method, we propose corresponding methods for evaluating node importance while also considering the network topology characteristics, overall network performance, and node characteristics.

The existing literature on network robustness has focused primarily on the structural robustness of networks, including the integrity of the largest connected subgraph or changes in network efficiency when the network is under attack [7,28]. However, for knowledge-based communities such as OSPCs, the depth and breadth of knowledge are key factors in attracting new users to the community [15]. The reciprocity of knowledge in the community plays a vital role in users’ explicit and tacit knowledge contribution behaviors. Park and Gabbard [29] stated that when scientists observe mutually beneficial and constructive projects in virtual communities, they are more likely to participate in knowledge sharing. Duanmu and Fai [30] found that various internal factors (e.g., the recipient’s evaluation of knowledge, existing knowledge stock, and ability to acquire knowledge directly) affect the effectiveness of knowledge collaboration. Therefore, evaluating community quality from the knowledge perspective is very important. Lei et al. [31] considered the semantic features of a network when evaluating its robustness, but this was still only at the structural level. Some scholars believe that evaluating network robustness solely from the perspective of network topology structure is not comprehensive enough: it is also necessary to consider the functional robustness of complex networks [32]. Wang et al. [33] proposed a new robustness evaluation method for power networks and used the Institute of Electrical and Electronics Engineers (IEEE) 118 bus test case to verify its effectiveness. Zhang et al. [34] constructed a knowledge supernetwork for business incubators (BIs) and explored the robustness of the knowledge supernetwork among incubating firms. These studies provide a reference for us to evaluate network robustness from a knowledge perspective.

The structure of this paper is organized as follows. In Section 2, we construct a knowledge collaboration network (KCN) model for OSPCs based on complex network modeling methods. Here, some indicators are extracted to evaluate the importance of different knowledge contribution nodes in the KCN. In Section 3, we describe the evaluation method for determining the importance of knowledge contribution nodes. Further, we introduce the robustness analysis method from two aspects: attack strategies and robustness evaluation indexes. In Section 4, we analyze and discuss changes in KCN robustness under various attack strategies. Finally, concluding remarks are given in Section 5.

2. Model Construction and Characteristics Analysis of the Community Network

2.1. Data Collection and Pre-Processing

The OSPC aims to use the skills and knowledge resources of the public to collaboratively create innovative products in a quick, efficient manner. In this knowledge collaboration innovation model, an open-source product is created through the joint completion of multiple innovation tasks. Users’ viewpoints, opinions, and task comments can be considered as knowledge output by the users, with the implementation of open-source products being driven by a multitude of knowledge. For example, the OSPC Local Motors (Supplementary Materials, https://localmotors.com/ (accessed on 8 January 2018)) was founded in 2008 and was once the world’s largest automotive design communication community. It consists of a dedicated online communication community and an offline micro-factory. In its online community, a large number of automotive design enthusiasts express their ideas, participate in regular creative exchange activities organized by the community, and discuss the design and production of products. In its offline micro-factory, community producers are responsible for testing and validating online creative ideas and feeding back the test results to the online community for further discussion. They also participate in the manufacturing of products with customers who purchase community products after the product is finalized. Using this innovative model, the community released multiple outstanding innovative products. Although the community was closed in 2022 due to transformation reasons, its product innovation model provides an important reference for the sustainability of other open-source communities’ products. Accordingly, this paper uses the data on 11 projects from the Local Motors community (dated between May 2008 and January 2018) as research source data. The project numbers, user numbers, user comment relationships, comment content, comment times, and other content were retained when processing the data. According to statistics, a total of 1663 participating project users were obtained, with 25,431 comment relationships between users. In the OSPC, users exhibit multiple behaviors. In addition to contributing knowledge to open-source products, users also engage in some social activities. For example, some user comments are just gratitude or encouragement to contributors, while others are chat-ups and advertising messages. That is, not all the information posted by users is related to knowledge. Therefore, we use machine learning technology to filter out information unrelated to knowledge contribution, which retained 1428 users, 8955 effective knowledge collaboration relationships, and a total of 15,331 effective knowledge collaboration between users.

2.2. Network Construction and Network Characteristics

The direct and indirect micro-interaction behaviors between users have also led to the emergence of macro-complex networks in the OSC. Based on complex network theory, users who contributed knowledge are abstracted as network nodes, which are represented by V. The collaboration, communication, attention, and other behaviors between users are abstracted as network edges, which are represented by E, and the number of collaborations between users is represented by edge weight W. In this paper, the OSPC network model G = (V, E_(u − u), W) is constructed, which includes 1428 nodes, 8461 edges, and a total weight of 15331. The topology parameters and network characteristics are obtained using the social network analysis method, as shown in

Table 1. Topology parameters and network characteristics.

Topology Parameter	Node	Edge	Average Degree	Average Path Length	Clustering Coefficient	Density	Overall Network Efficiency
	1428	8461	5.93	3.11	0.43	0.0075	0.1322
Network characteristic	Small-world characteristic		Scale-free property		Assortativity
	Yes		Yes		heterogeneous network

Note: According to Davis et al. [35], small-world parameter SW = [Cactual/Lactual] * [Lrandom/Crandom]. Therefore, Lrandom = ln(n)/ln(k) and Crandom = k/n, where n is the number of nodes and k is the average degree.

.

Table 1 shows that the network density is only 0.0075 (i.e., the network edges account for only 0.75% of all possible edges). The network is very sparse, and the connections between users are not close, which means that the flow of information in the network is not strong. The average path length of the network is 3.11, indicating that any two users can establish a collaborative relationship through an average of three edges. The overall network efficiency is only 0.1322, implying that the collaboration efficiency of users is low and the transmission rate of information in the network is not high. The network clustering coefficient is 0.43, which is significantly higher than the clustering coefficient of a random network of the same size (0.004). This indicates that users do not exist randomly, but they tend to engage in group collaboration. In addition, the small average path length and large clustering coefficient of this user collaboration network demonstrate a clear small-world characteristic.

2.3. Evaluation Indexes for Open-Source Product Community Networks

In complex networks, degree centrality is the most direct metric for characterizing node centrality in network analysis. The higher a node’s degree, the more it exchanges information or cooperates with other nodes, and the more important it is in the network. In a directed network, due to the directionality of edges, the node degree includes the out-degree and the in-degree. The specific calculation formulae are:

$$D_{out}=\frac{k_{out}\left(i\right)}{N-1}$$

(1)

$$D_{in}=\frac{k_{in}\left(i\right)}{N-1}$$

(2)

where N is the number of nodes in the network; $D_{out}$ and $D_{in}$ represent the out-degree centrality and in-degree centrality of node i in the network, respectively; and $k_{out}$ and $k_{in}$ represent the out-degree and in-degree of node i in the network, respectively.

Betweenness centrality (BC) reflects the role and influence of corresponding nodes or edges in the network. It is expressed as:

$${BC}_i=\sum _{s\ne i\ne t}\frac{n_{st}^i}{g_{st}}$$

(3)

where $g_{st}$ is the number of shortest paths from node s to node t and $n_{st}^i$ is the number of shortest paths passing through node i in $g_{st}$ shortest paths from node s to node t.

Closeness centrality (CC) reflects the degree of closeness between one node and the other nodes in a network. The larger the CC value, the closer the node is to other nodes, which gives it a better view of the information flow and makes it easier to transmit information to other nodes. The specific calculation method is:

$$CC{ }_i=\frac{N}{\sum _{j=1}^N{d}_{ij}}$$

(4)

where $d_{ij}$ is the distance between node i and node j.

The structural hole refers to the phenomenon where a certain individual (or individuals) in a social network has direct contact with some individuals but not others. From the overall perspective of the network, it seems that there is a “hole” in the network structure. The structural hole in the network acts as a buffer, preventing people on both sides of it from communicating information, and making the information flow on both sides of it heterogeneous. Individuals who cross the structural hole can obtain the unique information flow on both sides of it [36].

Burt’s structural hole calculation considers four factors: effective size, efficiency, constraint, and hierarchy [37]. The most used of these is constraint, which describes the ability of nodes to utilize structural holes in individual networks. The smaller the coefficient value, the greater the degree of structural holes, and the more capable nodes are of occupying and crossing through structural holes. The calculation formula for constraint can be found in Zhou et al. [18].

In a weighted network, the strength of a node (i.e., node weight) considers both the number of nodes nearest to it and the weight between them, which is a comprehensive reflection of the node’s local information. For a directed weighted network, the respective out-strength and in-strength of node i are:

$$s_i^{out}=\sum _{j=1}^N{\omega }_{ij}$$

(5)

$$s_i^{in}=\sum _{j=1}^N{\omega }_{ji}$$

(6)

In addition, when nodes communicate or collaborate with each other, it is considered that there is an edge between them. To depict the difference in collaboration intensity between nodes, each edge is assigned a corresponding weight value (i.e., the edge weight). The value of an edge weight represents the number of collaborations between nodes. The more neighboring nodes a node has, and the greater the edge weight with these neighboring nodes, the stronger the information dissemination ability of both the node and its neighboring nodes.

Based on these indicators, this paper evaluates the importance of nodes in the OSPC network from three aspects: their knowledge influence, knowledge collaboration ability, and knowledge dissemination ability. It also evaluates the importance of edges, based on the edge betweenness and edge weight, as shown in

Table 2. Indexes for evaluating different types of knowledge nodes and edges.

Node and Edge Type	Evaluating Indexes
knowledge influence node	node betweenness, node strength (node weight), eigenvector centrality, structural hole
knowledge collaboration ability node	out-degree, in-degree, number of project topics involved
knowledge dissemination ability node	closeness centrality, clustering coefficient, edge weight of node
knowledge influence edge	edge betweenness
knowledge weight edge	edge weight

.

3. Methodology

3.1. Node Importance Evaluation Method

This paper uses a comprehensive evaluation method for identifying knowledge influence nodes, knowledge collaboration ability nodes, and knowledge dissemination ability nodes using the evaluation indexes in Table 2, according to multi-attribute decision indicators [38].

The specific method is as follows. Assuming there are N nodes in the complex network, the corresponding set of decision plans can be represented as $S=\left\{S_1, \dots ,S_N\right\}$. If m indicators evaluate the importance of each node, then the corresponding indicator attributes set is recorded as $I=\left\{I_1, \dots ,I_m\right\}$. The value of the jth indicator of the ith node is $S_i\left(I_j\right),i=1,\dots ,N;j=1,\dots m$, forming the decision matrix $\mathrm{X}$:

$$\mathrm{X}=\left(\begin{array}{ccc}{S}_1\left(I_1\right)& \cdots & S_1\left(I_m\right)\\ ⋮& \ddots & ⋮\\ S_N\left(I_1\right)& \cdots & S_N\left(I_m\right)\end{array}\right)$$

(7)

Due to a large number of indicators, complex relationships exist between indicators, and the dimensions of each indicator are different. To uniformly measure the size of each indicator, they are divided into benefit-type indicators (where the higher the indicator value, the stronger the capability) and cost-type indicators (where the higher the indicator value, the worse the capability), and the indicator matrix is standardized [39]:

For a benefit-type indicator:

$$r_{ij}=\frac{S_i\left(I_j\right)}{{S_i\left(I_j\right)}^{max}}$$

(8)

For a cost-type indicator:

$$r_{ij}=\frac{{S_i\left(I_j\right)}^{min}}{S_i\left(I_j\right)}$$

(9)

The normalized decision matrix is denoted as $R={\left(r_{ij}\right)}_{N\times m}$. Let the weight of the jth indicator be $w_j=(j=1,\dots ,m,\sum w_j=1)$, which forms a weighted normalization matrix with normalization decision matrix R:

$$Y={(y}_{ij})=\left(w_j{r}_{ij}\right)=\left(\begin{array}{ccc}{w}_1{r}_{11}& \cdots & w_m{r}_{1m}\\ ⋮& \ddots & ⋮\\ w_1{r}_{N1}& \cdots & w_m{r}_{Nm}\end{array}\right)$$

(10)

where the weight of each indicator is obtained using the analytic hierarchy process.

The positive ideal decision nodes $A^{+}$ and negative ideal decision nodes $A^{-}$ are determined from matrix $Y$:

$$A^{+}=\left\{\underset{i\in L}{max}(y_{i1},\dots ,y_{im})\right\}=\left\{y_1^{max},\dots ,y_m^{max}\right\}$$

(11)

and

$$A^{-}=\left\{\underset{i\in L}{min}(y_{i1},\dots ,y_{im})\right\}=\left\{y_1^{min},\dots ,y_m^{min}\right\}$$

(12)

where $L=\left\{1,\dots ,N\right\}$.

Further, the distance from each node to the positive ideal decision node $A^{+}$ and negative ideal decision node $A^{-}$, denoted as $D_i^{+}$ and $D_i^{-}$, respectively, are:

$$D_i^{+}={\left[\sum _{j=1}^m{(y_{ij}-y_j^{max})}^2\right]}^{1/2}$$

(13)

and

$$D_i^{-}={\left[\sum _{j=1}^m{(y_{ij}-y_j^{min})}^2\right]}^{1/2}$$

(14)

The closeness $Z_i$ of the ideal node, according to equations $D_i^{+}$ and $D_i^{-},$ is then obtained and sorted, where the larger the closeness, the higher the importance of the node in the network. The calculation formula for $Z_i$ is as follows:

$$Z_i=\frac{D_i^{-}}{(D_i^{-}+D_i^{+})},0{\le Z}_i\le 1$$

(15)

Taking the calculation of knowledge influence nodes as an example, four evaluation indicators are used to identify knowledge influence nodes according to Table 2: node betweenness (BC), node strength (NS), eigenvector centrality (EC), and structural hole (C). The calculation formulae for these four indicators are shown in Section 2.3 and are calculated using the software Ucinet 6.0. The results calculated in the order of BC, NS, EC, and C are shown as:

$$\mathrm{X}=\left[\begin{array}{c}\begin{array}{c}17421.414\\ 0\\ 0\\ 11991.555\\ 6634.145\\ 0\\ \dots \end{array} \begin{array}{c}258\\ 31\\ 17\\ 231\\ 89\\ 110\\ \dots \end{array} \begin{array}{c}0.029\\ 0.005\\ 0.002\\ 0.024\\ 0.012\\ 0.011\\ \dots \end{array} \begin{array}{c}0.126\\ 0.217\\ 0.687\\ 0.092\\ 0.161\\ 0.077\\ \dots \end{array}\\ \end{array}\right]$$

Among them, BC, NS, EC, and C are benefit-type indicators, while C is a cost-type indicator. Therefore, the normalized decision matrix R is calculated as:

$$\mathrm{R}=\left[\begin{array}{c}\begin{array}{c}0.0760\\ 0\\ 0\\ 0.0523\\ 0.0289\\ 0\\ \dots \end{array} \begin{array}{c}0.1943\\ 0.0233\\ 0.0128\\ 0.1739\\ 0.0670\\ 0.0828\\ \dots \end{array} \begin{array}{c}0.0443\\ 0.0076\\ 0.0031\\ 0.0366\\ 0.0183\\ 0.0168\\ \dots \end{array} \begin{array}{c}0.2460\\ 0.1429\\ 0.0451\\ 0.3370\\ 0.1925\\ 0.4026\\ \dots \end{array}\\ \end{array}\right]$$

The entropy weight method is used to calculate the weight of each indicator. The calculated weights are: $w_{\mathrm{BC}}=0.6107$, $w_{\mathrm{NS}}=0.0678$, $w_{\mathrm{EC}}=0.0678$, and $w_{\mathrm{C}}=0.2535$. Based on the weight values of each indicator and the normalized decision matrix R, the weighted normalization matrix can be obtained as:

$$\mathrm{Y}=\left[\begin{array}{c}\begin{array}{c}0.0464\\ 0\\ 0\\ 0.0320\\ 0.0177\\ 0\\ \dots \end{array} \begin{array}{c}0.0132\\ 0.0016\\ 0.0009\\ 0.0118\\ 0.0046\\ 0.0056\\ \dots \end{array} \begin{array}{c}0.0030\\ 0.0005\\ 0.0002\\ 0.0025\\ 0.0012\\ 0.0011\\ \dots \end{array} \begin{array}{c}0.0622\\ 0.0361\\ 0.0114\\ 0.0853\\ 0.0487\\ 0.1019\\ \dots \end{array}\\ \end{array}\right]$$

Then, the positive ideal decision nodes $A^{+}$ and negative ideal decision nodes $A^{-}$ are, respectively:

$$A^{+}=(0.6107, 0.068, 0.069, 0.2536)$$

$$A^{-}=(0, 0.00005, 0, 0.0056)$$

Furthermore, the $D_i^{+},$ $D_i^{-},$ and $Z_i$ for each node are calculated according to Equations (13)–(15) and the $Z_i$are ranked. The top ten knowledge influence nodes are shown in

Table 3. Top ten knowledge contribution nodes and edges.

Knowledge Influence Nodes		Knowledge Collaboration Ability Nodes		Knowledge Dissemination Ability Nodes		Knowledge Influence Edges		Knowledge Weight Edges
Node Number	${\mathit{Z}}_{\mathit{i}}$	Node Number	${\mathit{Z}}_{\mathit{i}}$	Node Number	${\mathit{Z}}_{\mathit{i}}$	Edge	Influence	Edge	Weight
711	0.861	711	0.970	130	0.865	1089–711	19,253.1	1231–711	65
19	0.627	149	0.964	142	0.856	19–711	16,779.9	149–872	43
149	0.601	26	0.852	149	0.854	1–7	14,667.8	649–837	43
130	0.594	130	0.812	19	0.854	8–26	9389.5	711–1231	43
26	0.537	142	0.798	26	0.843	19–149	8640.0	149–19	41
142	0.480	19	0.629	649	0.829	19–142	8076.2	644–149	38
7	0.329	77	0.543	77	0.797	130–711	7696.2	130–649	37
77	0.318	7	0.419	711	0.796	7–1373	7538.3	421–711	30
899	0.280	90	0.343	7	0.783	711–26	6966.1	461–711	29
649	0.250	43	0.339	42	0.781	711–142	6874.4	483–445	28

. Similarly, Table 3 shows the top ten knowledge collaboration ability nodes, knowledge dissemination ability nodes, knowledge influence edges, and knowledge weight edges.

Table 3 shows that eight of the top ten knowledge contribution nodes are the same for all three node types (i.e., 7, 19, 26, 77, 130, 142, 149, and 711). This demonstrates that these nodes have important positions and influence in the network, but their importance to different knowledge contributions varies. Among the knowledge influence edges, the connection between nodes 1089 and 711 has the greatest influence, indicating that a large part of the knowledge transmission path passes through this edge during the transmission process (i.e., this edge has significant control over the knowledge collaboration process). Therefore, the collaborative relationship between nodes 1089 and 711 has a greater control advantage in the entire network. In addition, among the top ten influential edges, six nodes can be found in the top ten knowledge contribution nodes, and the connections between these important nodes are very close. For example, the edge influence between nodes 19 and 711 ranks second, showing that this edge has a great impact and control advantage on the network’s knowledge-sharing process. Further, the edge weight ranking listed in Table 3 demonstrates that nodes 711, 149, and 130 cooperate most closely with other nodes.

3.2. Robustness Evaluation of Knowledge Collaboration Networks

3.2.1. Network Robustness Attack Strategies

To comprehensively analyze the robustness of OSPC networks under attack, we considered five deliberate attack strategies and two random attack strategies. Regarding the deliberate attack strategies, these relate either to network nodes being deliberately attacked (i.e., users leaving the community after being affected by various factors), or network edges being deliberately attacked (i.e., users being less willing to contribute knowledge when they are negatively disturbed, and existing knowledge contributions are deleted). Regarding random attack strategies, these relate either to network nodes being randomly attacked (i.e., users leaving the community at will) or network edges being randomly attacked (i.e., users arbitrarily deleting their existing contributions). The detailed attack strategies are shown in

Table 4. Attack strategies for network nodes and edges.

Attack Type	Attack Strategy	Strategy Description
Random attack strategies	Random node is attacked (RA)	Randomly select n nodes for removal to simulate the irregular loss of users.
	Random edge is attacked (RD)	Randomly select n edges for removal to simulate the irregular loss of user knowledge contribution.
Deliberate attack strategies	Knowledge influence node is attacked (KI)	Sort the knowledge influence nodes in the initial network in descending order based on their importance, remove the node with the highest importance value and its connected edges, and calculate the current network robustness. Reorder the knowledge influence nodes in the current network in descending order based on their importance, and repeat the above steps n times to simulate the continuous loss of knowledge influence users in the community.
	Knowledge collaboration ability node is attacked (KC)	Sort the knowledge collaboration nodes in the initial network in descending order based on their importance, remove the node with the highest importance value and its connected edges, and calculate the current network robustness. Reorder the knowledge collaboration nodes in the current network in descending order based on their importance, and repeat the above steps n times to simulate the continuous loss of knowledge collaboration users in the community.
	Knowledge dissemination ability nodes is attacked (KD)	Sort the knowledge propagation nodes in the initial network in descending order based on their importance, remove the node with the highest importance value and its connected edges, and calculate the current network robustness. Reorder the knowledge dissemination nodes in the current network in descending order based on their importance, and repeat the above steps n times to simulate the continuous loss of knowledge dissemination users in the community.
	Knowledge influence edge is attacked (KE)	Sort the edges in the network in descending order based on their betweenness, remove the edge with the highest betweenness, and calculate the current network robustness. Repeat the above steps n times to simulate the continuous loss of existing important knowledge contributions.
	Knowledge weight edge is attacked (KW)	Sort the important knowledge edges in descending order based on their weights in the network, remove the edge with the highest weight value, and calculate the current network robustness. Repeat the above steps n times to simulate the continuous reduction of user knowledge contribution.

.

3.2.2. Network Robustness Evaluation Indexes

The robustness of complex networks can be measured using various metrics, such as the largest connected subgraph, average path length, and natural connectivity, among others [6,40]. This paper measures two types of robustness for OSPS networks in the face of different attack strategies: structural robustness, which is measured using the relative size of the largest connected subgraph and network efficiency, and knowledge robustness, which is measured using knowledge redundancy and knowledge capacity.

In the KCN, the relative size of the largest connected subgraph reflects the degree of knowledge collaboration among users in the community. When the network is under attack, the relative size of the largest connected subgraph is defined as the ratio of the network’s largest connected subgraph size $N^{\prime }$ to the overall network size $N$ [6]. This reflects the topology changes in the network after being attacked, which can intuitively reflect the degree of damage to the network. The solution process is as follows:

$$S=\frac{N\prime }{N}$$

(16)

where the value of S is within [0, 1], and the larger the value, the higher the network robustness. A value of S = 0 indicates that users complete tasks independently (i.e., knowledge collaboration is the worst). A value of S = 1 indicates that users engage in various forms of knowledge collaboration (i.e., knowledge collaboration is the best).

Cowan et al. [41] stated that a shorter network path distance is conducive to knowledge collaboration between nodes, and the information dissemination efficiency between nodes is also high. Network efficiency is used to represent the speed of knowledge transfer in a network. When the distance between two nodes is shorter, the speed of knowledge transfer from one node to the other is faster. When network nodes are lost, the collaborative relationship between them also changes, which affects the efficiency of knowledge dissemination in the network. Therefore, network efficiency can further evaluate the robustness characteristics of the network in terms of knowledge transfer efficiency when encountering node loss.

Network efficiency is defined as:

$$E=\frac{1}{N(N-1)}\sum _{i\ne j}\frac{1}{d_{ij}}$$

(17)

where $N$ represents the total number of nodes and $d_{ij}$ represents the shortest path between node i and node j. The value range of E is [0, 1], where E = 0 implies that the network efficiency is the worst (i.e., the network is composed of isolated nodes that do not exchange or propagate information) and E = 1 implies that the network efficiency is the best (i.e., information exchange in the network is smooth).

In addition to the two structural indexes shown above, we use knowledge redundancy to evaluate the characteristics of knowledge changes in the OSPC after being attacked.

The OSPC network focuses on the acquisition, sharing, and innovation of knowledge resources. In addition to knowledge exchange and sharing within the community, there is a continuous stream of new knowledge joining the knowledge network. Recognizing the importance and value of knowledge in the community, users are willing to join the community and contribute knowledge. Due to the openness of knowledge resources, there is a high degree of resource duplication in the network, resulting in a phenomenon of “redundancy” in the network structure [42]. According to structural hole theory [37], the higher the redundancy in the network and the fewer structural holes it has, the stronger the network robustness and the higher the efficiency of knowledge dissemination. Therefore, when the network is attacked, it affects the redundancy of the network, which then affects network robustness. The redundancy of a node can be calculated as:

$$r_i=\raisebox{1ex}{$(m_i-1)$}\!\left/ \!\raisebox{-1ex}{$m_i$}\right.$$

(18)

where there are different $r_i$ for different nodes.

The redundancy of all nodes in the network can be calculated as:

$$\overline{R_i}=\frac{1}{N}\sum _{i=0}^N{r}_i$$

(19)

where i represents the number of different nodes and N is the number of network nodes.

Further, in a KCN, the larger the knowledge capacity, the more knowledge resources are involved in the network collaboration process, which is more conducive to the development of the community. Since this paper studies the knowledge capacity involved in collaborative processes, node weight is used to describe the total amount of collaborative knowledge contributed by a knowledge subject. The sum of all node weights is the knowledge capacity of the entire KCN.

4. Robustness Simulation and Results

4.1. Robustness Simulation Process

Based on the description of various attack strategies and robustness evaluation indexes, we design a simulation flowchart to simulate the seven attack strategies faced by the OSPC network. Then, we use Matlab 2016b programming to simulate the changes in both the structural robustness and knowledge robustness under each strategy. The simulation flowchart is shown in Figure 1.

4.2. Results Analysis

In this section, we draw comparison charts based on the experimental simulation results using Origin 9.0 software and provide corresponding analyses and discussions.

4.2.1. Structural Robustness of the Network under Node Attack Strategies

Figure 2a shows that as nodes in the OSPC network are attacked, the relative size of the largest connected component (S) follows a downward trend for all four attack strategies, i.e., random attack (RA), knowledge influence (KI), knowledge collaboration ability (KC), and knowledge dissemination ability (KD) node loss. However, the value of S stays significantly higher when the network is faced with random attack (RA) than deliberate attacks (KI, KC, and KD). For example, S decreases by 22.8% when 200 nodes are attacked under the RA strategy, whereas only 9 nodes need to be attacked to have the same impact under the KI and KC strategies. When the top 10 network nodes are attacked under deliberate attack strategies KI, KC, and KD, S decreases by 23.4%, 24.5%, and 23.9%, respectively (i.e., the value of S follows KI > KD > KC). This suggests that network robustness decreases the quickest when knowledge collaboration nodes are attacked. This is because the order of knowledge collaboration is disrupted, and the degree of collaboration significantly drops. As further nodes are attacked, the decline rate in S is similar for all three deliberate attack strategies, though it decreases the quickest under the KD attack strategy. When the top 100 nodes are attacked under the KI, KC, and KD strategies, S decreases by 73.4%, 62.9%, and 74%, respectively (i.e., the value of S follows KC > KI > KD). This shows that the network robustness decreases the quickest when knowledge dissemination nodes are attacked. This is because of the widespread destruction of the network, which hinders the dissemination of information.

Figure 2b shows that the network efficiency follows a downward trend for all four node attack strategies, with this trend being significantly slower under random attack strategy RA than deliberate attack strategies KI, KC, and KD. This again proves that the network has high robustness in the face of random node loss and low robustness in the face of deliberate node loss. When the top six nodes are attacked under the KI, KC, and KD strategies, the network efficiency declines by 36.6%, 30.6%, and 25.3%, respectively (i.e., the network efficiency value follows KD > KC > KI). When the top 40 nodes are attacked under the KI, KC, and KD strategies, the network efficiency declines by 70.3%, 67.1%, and 68.5%, respectively (i.e., the network efficiency value follows KC > KD > KI). As further knowledge nodes are attacked, the network efficiency decreases rapidly for all three deliberate attack strategies, particularly under the KI and KC strategies. When the top 150 nodes are attacked, the network efficiency approaches zero under all three deliberate attack strategies.

4.2.2. Knowledge Robustness of the Network under Node Attack Strategies

Figure 3a shows that as nodes in the OSPC network are attacked, network redundancy follows a downward trend for all four node attack strategies, particularly the deliberate attack strategies (KI, KC, and KD). This shows that the network has high knowledge robustness when facing random attacks and low knowledge robustness when facing deliberate attacks. When the top 10 network nodes are attacked under deliberate attack strategies KI, KC, and KD, the network redundancy decreases by 18.9%, 18.6%, and 18.8%, respectively. As further nodes are attacked, the network redundancy rapidly decreases. For example, for the network redundancy to drop close to 50%, 73, 83, and 72 nodes need to be attacked under the KI, KC, and KD strategies, respectively. Further, when the top 200 nodes are attacked under the KI, KC, and KD strategies, the network redundancy decreases by 84.2%, 77.6%, and 65.0%, respectively.

Figure 3b shows that the knowledge capacity of the network follows a downward trend for all four node attack strategies, with this trend being significantly slower under random attack strategy RA than deliberate attack strategies KI, KC, and KD. When the top 10 knowledge nodes are attacked under the KI, KC, and KD strategies, the knowledge capacity of the network decreases by 36.2% 36.7%, and 36.8%, respectively. These values are close because the top 10 nodes for all three attack strategies include 8 identical important nodes, as previously shown in Table 3. When the top 20 knowledge nodes are attacked under the KI, KC, and KD strategies, the knowledge capacity of the network decreases by 51.5%, 50.0%, and 49.1%, respectively. This shows that these top 20 nodes have made a significant contribution to network knowledge.

4.2.3. Structural Robustness of the Network under Edge Attack Strategies

Figure 4 shows that the value of the network’s structural robustness indexes is higher under the random edge attack strategy RD than deliberate edge attack strategies KE and KW (i.e., the structural robustness index values follow RD > KW > KE). Figure 4a shows that when the top 80 edges are attacked under the knowledge influence edge (KE) strategy, the relative size of the largest connected component (S) starts to rapidly decline. In contrast, when the same number of edges is attacked under the knowledge weight edge (KW) strategy, S does not significantly change. When the top 380 edges are attacked under the KW strategy, S begins to show an obvious downward trend. However, in general, the value of S decreases slowly under this attack strategy. Figure 4b shows that when the top 180 edges are attacked under the KE strategy, the network efficiency decreases by 10.2%. In contrast, 700 edges need to be attacked under the KW strategy to achieve the same effect. Further, when 860 edges are attacked under the KE and KW strategies, the network efficiency decreases by nearly 50% and 12%, respectively. The analysis results demonstrate that the KCN has high structural robustness in the face of random edge loss and low structural robustness in the face of deliberate edge loss. In addition, the structural robustness of the network is greatly reduced when an edge with a larger betweenness is attacked. When further edges are attacked, attacks on edges with larger betweenness have a greater impact on structural robustness than attacks on edges with larger weights. This illustrates that edges with larger betweenness have a greater influence on the whole knowledge network, playing a more important role in network robustness.

4.2.4. Knowledge Robustness of the Network under Edge Attack Strategies

Figure 5 shows that the knowledge robustness indexes of the network decrease slightly under the random edge attack strategy RD and significantly more under deliberate edge attack strategies KE and KW. Figure 5a shows that the knowledge redundancy value under the attack strategies follows RD > KW > KE. When 200 network edges are attacked, the network redundancy under the KE and KW strategies decreases by 2.7% and 0.91%, respectively. As the number of attacked network edges increases, the network redundancy declines quickly under the KE strategy. When 1000 network edges are attacked, the network redundancy under the KE and KW strategies decreases by 9.7% and 5.7%, respectively, which illustrates that the top betweenness edges and weight edges occupy fewer structural hole positions, and the network has high robustness. Figure 5b shows that the knowledge capacity under the RD attack strategy is significantly higher than under the KW and KE attack strategies. The knowledge capacity shows a downward trend for both deliberate edge attack strategies (KW and KE), and the difference between the two is not significant. When 1000 edges are attacked, the knowledge capacity under the KW and KE strategies drops by 16.5% and 15.2%, respectively. When network edges are attacked, the impact is greater on the knowledge capacity of the network than the knowledge redundancy, which indicates that although these edges occupy fewer structural holes, they carry more knowledge resources.

4.3. Discussion

The results of this study demonstrate that the OSPC exhibits different robustness characteristics when facing different attack strategies, resulting in varying degrees of impact on the sustainable development of the community. Specifically, from the perspective of structural robustness, the OSPC network has high structural robustness when faced with a random node attack strategy and low structural robustness when faced with a deliberate attack strategy. This has also been proven in open-source software projects [43]. Under the three proposed deliberate node attack strategies, when a lower number of key nodes are attacked, network connectivity declines the quickest when the knowledge collaboration ability nodes are attacked, and network efficiency declines the quickest when the knowledge influence nodes are attacked. As a greater number of key nodes are attacked, network connectivity declines the quickest when the knowledge dissemination ability nodes are attacked, and network efficiency still declines the quickest when knowledge influence nodes are attacked. This shows that managers should formulate management strategies for different knowledge contribution users according to the needs of community development.

From the perspective of knowledge robustness, the OSPC network has a higher knowledge robustness when faced with a random node attack strategy and a lower knowledge robustness when faced with a deliberate node attack strategy. Under the three proposed deliberate node attack strategies, when a lower number of key nodes are attacked, changes in the knowledge robustness of the network are relatively close for all three strategies. As a higher number of key nodes are attacked, the knowledge robustness is lowest when the knowledge influence nodes are attacked, which indicates that knowledge influence nodes have stronger knowledge ability.

The structural robustness and knowledge robustness of the network are higher when faced with a random edge attack strategy than when faced with a deliberate attack strategy. Under the two proposed deliberate edge attack strategies, the structural robustness and redundancy are significantly lower when faced with the knowledge-influence edge attack strategy than when faced with the knowledge-weight edge attack strategy. However, the knowledge capacity is similar for both deliberate edge attack strategies. This shows that the knowledge influence edge has more control over the network structure and carries more redundant knowledge.

The perfect management and governance mechanism for the community is the guarantee to drive the innovation of community members [44,45]. According to the above analysis results, the following management implications are made:

As a type of knowledge-based community, the OSPC needs a strong knowledge background to attract users to join. Therefore, the community should provide users with a friendly knowledge innovation environment that includes knowledge communication channels, detailed knowledge documentation and development tools, user interaction interfaces that make it easy for users to contribute knowledge, and more personalized knowledge competitions. Our research results show that different types of knowledge contribution users have different effects on network robustness. As such, managers should formulate personalized incentive strategies to encourage users to actively contribute to the community and help users increase knowledge, such as by providing corresponding honorary certificates, invitations to community activities, or other rewards for different types of users who contribute more knowledge.

Project failure caused by insufficient user cooperation is the most common phenomenon in the product innovation process. Developers often fail to find a suitable partner quickly, resulting in project stagnation. In addition, some users lose interest in the community because they cannot find suitable projects. Our research found that the relationships among users are not close and the information flow in the network is not strong. Therefore, community managers should make personalized collaborator recommendations and project recommendations for users according to their contribution characteristics to help them quickly find partners and integrate into projects.

5. Conclusions

This paper studies the structural characteristics of the OSPC network, proposing (a) evaluation methods to identify critical knowledge contribution nodes and edges in the network, (b) network robustness evaluation indexes from varying perspectives, and (c) multiple attack strategies to analyze both the structural robustness and knowledge robustness of the network when faced with attacks on nodes and edges. The results indicate that the proposed node and edge important evaluation indexes can better identify critical knowledge contribution nodes and edges in the OSPC network. The impact of node and edge loss on network robustness varies under different attack strategies, which can provide corresponding references for community defense. Furthermore, we found that when key users reduce their contributions to the community or even leave it, the structure and knowledge of the community are threatened. Actually, it is inevitable that users will have conflicts or differences in opinion during the development process. To prevent malicious disputes between users, malicious evaluations of key users, and other bad behaviors (all of which can lead to key users leaving the community), community managers should formulate more transparent community rules and guidelines to maintain community order. Further, they should establish an early warning mechanism, capture feedback and the needs of key users in a timely manner, and formulate corresponding countermeasures to improve the innovation experience of key users.