KG Estimation Levels
In order to test the research hypotheses, knowledge graphs have been estimated with an increasing degree of specificity. In first, the Knowledge Graph (KG) resulting from the whole research sample od SMEs (global-level) has been estimated. Subsequently, a KG has been estimated for each level of the GIC-related variables (i.e., Environmental Consciousness, Innovation Capability and Economic Performance), by splitting firms into 6 groups (low Vs high level of each variable). At last, a KG for each firm has been estimated (firm-level).
Global-Level Knowledge Graph Estimation
According to the first research aim, a Knowledge Graph representing the overall knowledge structure of the energy SMEs, operating in the 8 countries considered, has been estimated. To this end, the estimation procedure described above has been applied to the whole research sample of 2,902 firms.The resulting Knowledge Graph has then been visualized through the Qgraph Qgraph R package using a circular layout. Edges showing weights less than |0.2| have been removed from the displayed graph for visualization purposes.
Group-Level Knowledge Graph Estimation
As a precondition for testing the hypotheses H1, H2, and H3, multiple KGs have been estimated following the same procedure as described above. Two KGs have been estimated for representing the GIC of SMEs with low Vs high EC. Therefore, firms have been split into two groups according to their EC score, and a knowledge graph has been estimated for each group: \(KG_{low}^{ec}\) for the low EC group, and \(KG_{high}^{ec}\) for the high EC group. The same procedure has been replicated by grouping SMEs by INC, enabling the estimation of \(KG_{low}^{inc}\) and \(KG_{high}^{inc}\), the network representations of the GIC for firms with low Vs high INC. Last, \(KG_{low}^{ep}\) and \(KG_{high}^{ep}\), the network representations of the GIC for firms with low Vs high EP have been estimated. All the resulting graphs have been visualized by means of the Qgraph R package, using a circular layout.
Firm-Level Knowledge Graph Estimation
In order to calculate firms' GIC score, a KG has been estimated for each of the SMEs included in the research sample. The global network metrics have then been computed considering modules of edge weights. The indices of Global Efficiency, Density, and Global Clustering Coefficient have been calculated by means of the Qgraph R package. The Small-Worldness Index has been calculated with the igraph R package.
Testing on Graphs Isomorphism
In order to test whether the structure of KG significantly differs (i.e., are not isomorphic) when considering firms with low Vs high EC (H1), low Vs high INC (H2), and low Vs high EP (H3), multiple pairwise comparisons have been performed by using the Network Comparison Test R package (van Borkulo et al., 2015). The NCT is a permutation-based technique developed to test the invariance hypothesis between pairs of networks (Van Borkulo et al., 2022). The NCT tests for two specific hypotheses: the first pertaining to the network structure, tests whether there are significant differences in terms of number and disposition of edges between two networks; the second, tests whether the two networks significantly differ in terms of the overall strength of edges.
Network Structure Invariance
The Network Structure Invariance (M ) tests, for a pair of weighted networks G^1 and G^2 with adjacency matrices Ω^1 and Ω^2 respectively, the hypothesis that G^1 significantly differs from G^2, assuming as null hypothesis:
\begin{equation} H^0: Ω^1=Ω^2 \end{equation}
Network Structure Invariance
The Global Strength Invariance (S ), evaluates the difference between the sum of edge weights of G^1 and G^2, two networks having nodes i and j with \(i ≠ j). It should be noted that, according to Opsahl et al. (2010), the absolute sum of the weights (\(|w_ij|\)) of each network are considered for the Global Strength (W ) measure. The S test statistic is then calculated as:
\begin{equation} |\sum_{i=1}^p \sum_{j>i} (|w_{ij}^1|-|w_{ij}^2|)|. \end{equation}
Assuming that, in a given population (p):
\begin{equation} H^0: \sum_{i=1}^p \sum_{j>i}|w_{ij}^1| = \sum_{i=1}^p \sum_{j>i}|w_{ij}^2|. \end{equation}
And, consequently, \(S = 0\).
The NCT is based on a three-step algorithm aimed at (i) deriving the relevant test statistics M and S of the observed networks' structure; (ii) recalculating M and S in N permutations by varying edges positioning and weights in order to derive a reference distribution for each of the test statistics; and (iii) estimating the significance of the observed networks’ test statistics by comparing them with the reference distributions. For the analyses of this research, 1,000 permutations have been considered to test on both the Network Structure Invariance and the Global Strength Invariance of KG.