2.2.1. Construction of the Global Wood Forest Products Trade Network
The global wood forest products trade network is constructed with countries (regions)involved in this trade as network nodes, their trade relationships as edges, and trade volumes and trade closeness as weights. A complex network model is established based on these parameters. The following definitions are applied to the global wood forest products trade network:
where
m represents the four different types of wood forest products networks categorized as upstream, midstream, downstream, and recycling.
V is the set of nodes representing all countries (regions);
E is the set of edges representing trade relationships among countries (regions);
W is the function set of trade volumes between countries (regions);
is the function set of trade closeness between countries (regions); and
T represents the year.
In this study, trade volume (in kilograms) rather than trade value (in US dollars) is chosen as one of the edge weights to avoid uncertainties arising from inflation and price fluctuations, thereby more authentically revealing trade relationships and dependencies with in the supply chain, which is crucial for analyzing network resilience evolution.
Additionally, trade closeness is incorporated as another edge weight to explore its impact on the shortest path length. In weighted networks, the path length is not only related to the number of connecting edges but also to the weight of trade closeness. Generally, the larger the trade volume between countries (regions), the lower the transaction costs, the closer the trade ties, leading to a smaller trade closeness weight and a shorter shortest pathlength. The specific calculation formula is:
where
represents the trade closeness between country (region)
I and
j;
is the maximum trade volume among all edges in the network; and
is the trade volume of wood forest products between country (region)
i and
j.
From the above formula, in the global wood forest products trade network, the edge with the maximum trade volume has a trade closeness weight of 1, while other edges have trade closeness weights greater than “1”. In reality, trade closeness is undoubtedly influenced by more complex factors, such as political relations, geographical locations, cultural differences, and numerous other difficult-to-quantify factors. However, within the specific analytical framework of this study, we focus solely on trade volume as the core factor.
2.2.2. Related Indicators of Network Structural Resilience
Network structural resilience refers to the ability of a network to maintain its overall functionality and continuous operation in the face of failures, attacks, or disruptions. Drawing on existing research [
43,
44] and innovatively incorporating trade volume and trade closeness weights, this study measures the structural resilience of the weighted network from four aspects: transitivity, clustering, hierarchy, and assortativity.
(1) Transitivity – Weighted Global Efficiency
Weighted global efficiency refers to the average of the reciprocals of the weighted shortest path lengths between all node pairs in the network, reflecting the speed and capacity of information transmission throughout the network. Based on existing research [
43], the calculation formula for weighted global efficiency is:
where
represents the weighted (trade closeness) global efficiency;
N is the total number of nodes in the network; and
is the weighted shortest path between nodes
i and
j.
In the global wood forest products trade network, a high weighted global efficiency indicates the presence of more redundant paths and backup mechanisms to with stand potential failures or shocks. Conversely, a low weighted global efficiency suggests vulnerability due to uneven trade volume distribution or poor network connectivity, making it susceptible to difficulties when faced with disruptions.
(2) Clustering - Weighted Average Clustering Coefficient
The weighted average clustering coefficient is the arithmetic mean of the weighted clustering coefficients of all nodes in the network. It reflects the intensity of tight clustering among nodes based on trade volumes. According to existing research findings [
43], this coefficient is calculated by taking the ratio of the total actual trade volumes among a country’s (or region’s) direct trading neighbors to the theoretical maximum possible trade volumes among those neighbors, and then averaging these ratios across all nodes. The formula is as follows:
where
represents the weighted (trade volume) clustering coefficient of node
i;
represents the trade volume between node s and node t;
represents the degree of node
i ;
represents the maximum edge trade volume;
represents the weighted average clustering coefficient.
In the global wood product trade network, a high weighted average clustering coefficient indicates that nodes tend to form closely connected clusters, with frequent trade and tight cooperation within these clusters. This results in good local connectivity and stability, enhancing the resilience and robustness of the local network.
(3) Hierarchy - Weighted Degree Distribution
Hierarchy is measured through the network’s degree distribution index [
44]. Weighted degree distribution refers to the probability distribution of node degrees, considering trade volumes as weights. The larger the absolute value of the slope of the weighted degree distribution, the more significant the hierarchy in terms of weighted degrees among nodes. For the wood product trade network, a power-law curve is plotted using the weighted (trade volume) degree of nodes against their rank (where nodes are ordered by degree from largest to smallest). The formula is as follows:
where
represents the weighted (trade volume) degree of node
i in the network;
represents the ranking of node
i’s weighted degree among all weighted node degrees;
C is a proportional constant;
is the slope of the weighted degree distribution curve, measuring the hierarchical nature of the network.
In the wood product trade network, a larger span of node weighted degree rankings implies a higher level of hierarchy, indicating the presence of one or more core nodes forming a centralized network structure. This structure enhances the network’s “robustness,” fostering cohesion and competitiveness. However, it may also lead to path dependencies among non-core nodes, thereby increasing the network’s “vulnerability.” In contrast, networks with lower hierarchy have lower risk sensitivity but lack cohesion, organization, and competitiveness.
(4) Assortativity – Weighted Assortativity Coefficient
The assortativity coefficient is a Pearson correlation coefficient based on “degree,” used to measure the relationship between connected node pairs [
45]. The weighted assortativity coefficient refers to the tendency of countries (regions) in the network to connect with countries (regions) of similar total trade volumes, taking into account the trade volume weighting. The assortativity coefficient ranges from -1 to 1. If the assortativity coefficient is positive, it indicates an assortative network, where hub countries (regions) are more likely to connect with each other, exhibiting a polarization effect. Conversely, if the assortativity coefficient is negative, it represents a disassortative network, where hubs tend to establish trade relations with countries (regions) of lower total trade volumes, exhibiting a trickledown effect [
42]. If the assortativity coefficient is zero, it signifies that there is no significant correlation in total trade volumes between interacting countries (regions). The formula for calculating the weighted assortativity coefficient is as follows [
46]:
where
represents the weighted (trade volume) assortativity, where
and
are the degrees of nodes
j and
k connected by the i-th edge,
represents the trade volume weightof the i-th edge, and
is the sum of the trade volume weights of all edges.
In the global wood forest products trade network, an assortative network strengthens connections among hub countries (regions), providing stability and rapid recovery capabilities, but may hinder cross-community information flow, limiting responsiveness. In contrast, a disassortative network facilitates information exchange and resource sharing between hub and peripheral countries (regions), dispersing risks, enhancing diversity and adaptability, but may lead to over-reliance on a few hub countries (regions) or connections, affecting network stability and resilience.
2.2.3. Indicators Related to Node Resilience in Networks
Node resilience is a multidimensional and comprehensive concept that encapsulates not only a node’s self-recovery and anti-interference capabilities in the face of failures, attacks, or disruptions but also its role and effectiveness in risk propagation within the network. Based on existing research [
33], this study comprehensively assesses node resilience from three major aspects: node strength, transit efficiency, and diffusion influence, innovatively incorporating weighted factors such as trade volume and trade intimacy to portray nodes’ resilience more comprehensively in the network.
(1)Node Strength – Weighted Degree
Node strength, also known as weighted degree, represents the total trade volume between a country (or region) and all its direct trading partners within a given time period. In the directed global trade network of wood forest products, it is further classified into weighted out-degree and weighted in-degree, with the formulas as follows [
47]:
where
represents the weighted out-degree of node
i ;
represents the weighted in-degree of node
i;
represents the trade volume from node
i to node
j;
represents the trade volume from node
j to node
i; both
and
indicate the trade relationship betweenthe two nodes.
In the global trade network of wood forest products, countries (or regions) with high node strength establish strong connections with multiple other countries, enhancing their redundancy and flexibility. This facilitates swift response and recovery from external disturbances, strengthening their resilience. During random attacks, these high-strength nodes rely on their stronger anti-interference and recovery capabilities to mitigate impacts while maintaining stable trade activities to support other nodes. However, under targeted attacks, their central positions in the network may render them more vulnerable, accelerating risk propagation and exacerbating the overall network’s exposure.
(2) Transit Capability – Weighted Betweenness Centrality
Transit capability is characterized by weighted betweenness centrality [
33], which measures the extent to which a country (or region) lies on the weighted shortest paths between other trading pairs, evaluating its importance and influence in the network from a “bridge” perspective. The formula is as follows:
where
represents the weighted (trade closeness) betweenness centrality of node
i;
is the number of weighted shortest paths from node
s to node
t, and
is the number of weighted shortest paths among the
weighted shortest paths from node
s to node
t that pass through node
i.
In the global trade network, countries with high weighted betweenness centrality, due to their direct trade links with multiple countries, form multiple redundant paths. This allows them to maintain trade via alternative routes during random disruptions, demonstrating strong anti-interference ability. However, targeted attacks on these transit hubs can disrupt critical network pathways, impeding information and resource flows, potentially triggering cascading failures and network collapse.
(3) Diffusion Capability – Weighted Closeness Centrality
Diffusion capability is represented by weighted closeness centrality, which considers the reciprocal of the sum of weighted shortest path lengths from a node to all other nodes in the network, incorporating trade intimacy [
47]. A higher value indicates closer trade relations with other nodes, enabling faster information or resource dissemination. The formula is as follows:
where
represents the weighted (trade tightness) closeness centrality of node
i;
N represents the total number of nodes in the network;
represents the weighted shortest path length from node
i to node
j;
represents the sum of the weighted shortest paths from node
i to all other nodes in the network.
In the global trade network of wood forest products, countries (or regions) with high weighted closeness centrality exhibit strong market influence, bargaining power, and risk diversification strategies due to their extensive and intimate trade connections. These enable them to rapidly acquire and disseminate information, effectively integrate and allocate resources. Under random disturbances, nodes with high weighted closeness centrality, owing to their positional advantages, can swiftly receive and transmit information, respond promptly, and maintain stable operation, ensuring network connectivity and information transmission efficiency. Random interference is unlikely to simultaneously target multiple such nodes. However, under deliberate attacks, the failure of these highly diffusive nodes can lead to information silos, impeding information transmission speed and efficiency, potentially causing a significant decline in network connectivity, thereby affecting overall network functionality and stability.