2. Data and French multinationals’ network reconstruction

Multinational firms can be conceptualized as networks of production units—plants or subsidiaries—spanning multiple countries and sectors. In Coase’s foundational image, firms create “islands of conscious power” within a market “ocean” of decentralized transactions (Coase, Reference Coase1937). Extending this vision, Ghoshal and Bartlett (Reference Ghoshal and Bartlett1990) explicitly characterize multinational corporations as “inter-organizational networks embedded in broader external networks” of suppliers, customers, and institutional environments. The OECD similarly describes multinationals as “networks within (Global Value Chains) networks” (Cadestin et al., Reference Cadestin, De Backer, Desnoyers-James, Miroudot, Ye and Rigo2018).

To examine these corporate networks empirically, we use the LiFi database compiled by INSEE, the French National Statistical Institute. This survey exhaustively reports equity linkages between French corporate groups and their domestic or foreign affiliates. We focus exclusively on foreign affiliates to capture the international component of firm networks. Due to a methodological change in 2012—when LiFi began integrating information from the Outward Foreign Affiliates Statistics survey—we restrict our analysis to the 2012–2022 period.

For each French group (parent company), the LiFi dataset provides the NAF 2-digit industry code, along with country and industry identifiers for each affiliate. In 2012, the dataset recorded 9,817 multinational groups with 46,146 foreign affiliates operating across 2,064 unique country-industry pairs. By 2022, coverage had expanded to 18,905 groups and 62,450 affiliates in 2,420 pairs.Footnote ¹

To harmonize industry classifications and reduce dimensionality, we map NAF 2-digit codes to the 26-sector aggregation used in the Eora Multi-Region Input–Output framework.Footnote ² Combining this with 189 countries, we define a fixed set of 4,914 possible country-sector nodes.

Each firm’s annual network is constructed as a symmetric, binary adjacency matrix $\mathbf{F}_{i,t}$ of dimension $4,914 \times 4,914$ , where:

\begin{align*} F_{o,d} = \begin{cases} 1 & \text{if firm $i$ owns affiliates in both country-sector nodes $o$ and $d$ at time $t$}, \\ 0 & \text{otherwise}. \end{cases} \end{align*}

Edges are undirected and reflect co-location of affiliates across nodes within a single firm. We then aggregate these firm-level matrices to construct a weighted, undirected network of French multinationals for each year $t$ , summarized by an adjacency matrix $\mathbf{M}_t$ , where:

\begin{align*} M_{o,d} = \sum _{i=1}^{N_t} F_{o,d;i} \end{align*}

Here, $F_{o,d;i}$ is the $\{o,d\}$ element of firm $i$ ’s matrix and $N_t$ is the number of multinationals observed in year $t$ . The resulting matrix $\mathbf{M}_t$ captures how frequently French MNEs co-locate in each pair of country-industries—providing a measure of network intensity at the system level.

Despite over 12 million possible node-pair combinations,Footnote ³ these networks are sparse: less than 2% of potential edges are activated in any given year—yet this still represents nearly 180,000 distinct edges annually. Most node-pairs are linked by only a single firm, and fewer than 10% of edges are shared by more than 10 firms.

For example, in 2012 the most common edge connected the wholesale trade sectors of Germany and Spain, used by 88 firms. In 2022, the most frequent connection was similar in magnitude, linking the same country-sectors for 97 firms. These patterns reflect both strategic complementarity in affiliate placement and structural concentration in multinational production networks.

To clarify how we construct the multinational co-location network, replace with Figure 1 displays a toy example that proceeds in three steps: the initial bipartite graph linking French parent firms to foreign country–industry cells (left panel); the projection of each firm’s footprint into the country–industry space (middle panel); and the aggregated, weighted co-location network across all firms (right panel).

Figure 1. Illustrative example of network reconstruction.Note: Left: bipartite network between French firms and foreign country–industry nodes. Middle: firm-level projection into the country–industry layer. Right: aggregated co-location network where edge weights count the number of firms co-locating in a pair of nodes.

In the example, Firms 1 and 2 both operate in China–Electronics and Germany–Automotive. Their shared presence generates a weight of 2 on the China–Electronics–Germany–Automotive edge in the aggregated network.

We repeat this procedure for each year to obtain annual co-location networks. In the next section, we analyze them to characterize structural properties and their evolution over time.

3. A robust network during COVID-19

3.1 Network topology

We begin with aggregate topology over 2012–2022, tracking density, degree centralization, degree assortativity, and clustering. Taken together, these indicators summarize the architecture of the ownership co-location network before, during, and after the COVID-19 shock.

3.1.1 Network density

Network density measures the share of realized edges—i.e., country-industry pairs co-occupied by at least one French MNE—relative to the total number of possible node pairs. As in many large real-world networks, our network is sparse: only about 1.5% of the 12 million possible node pairs are realized in any given year. This reflects the economic irrelevance or impracticality of most potential country-industry combinations.

Nevertheless, in Figure 2 we observe a substantial increase in density between 2012 and 2017—from 1.04% to 1.78% confirming the broadening of internationalization strategies by French MNEs, that is now well documented (Joyez, Reference Joyez2017; de Warren, Reference de Warren2020, e.g.). After a slight decline, the network stabilizes at a post-2017 density slightly above 1.6%. Crucially, the pandemic period (2020–2022) is not associated with any abrupt change in density, suggesting a high level of structural continuity.

Figure 2. Density of the French multinational network (2012–2022).

3.1.2 Degree centralization

Centralization quantifies the extent to which a network is organized around a small number of highly connected nodes. Following Freeman (Reference Freeman1979), we compute the degree centralization index, which ranges from 0 (uniform distribution of degrees) to 1 (star-like concentration). As shown in Figure 3, centralization increased modestly from 0.21 in 2012 to 0.279 in 2018—an approximate 33% rise—before stabilizing and slightly declining post-2018. Once again, we find no significant structural break associated with the COVID-19 crisis.

Figure 3. Degree centralization of the French multinational network (2012–2022).

3.1.3 Degree assortativity

Assortativity measures the tendency of nodes to connect with others of similar degree. Defined as the Pearson correlation coefficient between the degrees of linked node pairs (Newman, Reference Newman2002), its values range from −1 (disassortative mixing) to + 1 (perfect assortativity). Figure 4 shows that the French MNE network is generally assortative: central nodes tend to connect with other central nodes. Between 2012 and 2017,Footnote ⁴ assortativity declined—consistent with firms integrating more peripheral nodes into their networks. This trend reverses slightly after 2017, suggesting a consolidation or return to more clustered network cores. Notably, no sharp discontinuity appears in 2020, 2021 or 2022.

Figure 4. Degree assortativity of the French multinational network (2012–2022).

3.1.4 Clustering

While each firm-level network is fully transitive by construction; if a firm operates in three nodes, all three node pairs are linked; this property does not carry over to the aggregate. When multiple firms overlap on different node pairs, the resulting triads can be unevenly weighted. To assess this, we compute the weighted global clustering coefficient following Fagiolo (Reference Fagiolo2007), which incorporates the distribution of edge weights within triads.

Figure 5 shows a low but stable level of weighted clustering over the entire period, with no abnormal shifts in 2020–2022. This pattern is consistent with overall structural stability during the pandemic.

Figure 5. Weighted clustering coefficient of multinationals’ network over time.

Overall, the evidence points to a remarkably robust structure of French multinationals’ ownership networks during COVID-19. While these networks do evolve gradually, the pandemic did not trigger a significant reconfiguration of connectivity, hierarchy, or clustering. Moreover, the movements in these indices in 2020–2021 are comparable to the typical year-to-year variation observed throughout the 2010s, rather than indicating a break.

3.2 A stable core

Although the aggregate structure of multinational networks remains stable through the COVID-19 period, assessing the persistence of the most central nodes and edges is essential. Structural robustness entails not only steady global topology but also continuity in the composition of the network’s key components.

Following Opsahl et al. (Reference Opsahl, Agneessens and Skvoretz2010), we measure node centrality as the geometric mean of a node’s degree $k_i$ (number of connections) and strength $s_i$ (sum of incident edge weights). This metric captures both the breadth and intensity of a node’s ties, giving higher scores to nodes that are simultaneously well connected and influential:

(1) \begin{equation} \text{cent}_i = k_i^{0.5} \cdot s_i^{0.5} \end{equation}

We compute this centrality index annually and select the ten most central nodes in 2019, the last pre-pandemic year. Figure 6 traces their ranks over 2012–2022. Two patterns emerge. First, composition is highly persistent: eight of the 2019 top ten were already in the top ten in 2012 and remain there throughout the decade. Second, central nodes are geographically concentrated in neighboring European countries and sectorally concentrated in two industries, Wholesale Trade and Financial and Business Services.

Figure 6. Central nodes stability.

Table 1. Centrality autoregression

t-stats in parenthesis, ^{* * *},^{* *} and ^* respectively indicate significance at 1,5 and 10%.

This persistence suggests that the structural core of the French multinational network remained unaffected by the COVID-19 shock. To quantify this stability across the full distribution of nodes, we estimate an autoregressive model of centrality scores:

\begin{equation*} \text{cent}_{i,t} = \beta _0 + \beta _1 \, \text{cent}_{i,t-1} \end{equation*}

Table 1 reports OLS estimates of this model over the 2013–2022 period. The results confirm a very high degree of persistence in node centrality, with $\beta _1$ coefficients consistently close to or above 0.9 and R $^2$ values near 0.95. Although a mild downward trend in the autoregressive coefficient is observed after 2017, no discontinuity is visible in 2020 or 2022, reinforcing the impression of structural robustness.

Figure 7. Main edges stability.

Building on the node-centrality results, we ask whether the most intensively used edges—that is, the most common country-industry pairings—also remained stable during the COVID-19 period. Even if central nodes are unchanged, the network’s internal organization could be reshaped through the rewiring of connections.

We identify the ten most frequent edges in 2019 and track their ranks over 2012–2022 (Figure 7). These edges are somewhat more volatile than the top nodes: about half do not remain continuously in the top ten. Nevertheless, the five most central edges are remarkably persistent. They typically connect the central nodes identified earlier, which is consistent with the positive assortativity documented in Section 4; when not linking leading European nodes among themselves, they commonly tie those nodes to major industries in the United States. The composition of top edges in 2019 is also very close to that in 2021,Footnote ⁵ underscoring the durability of key linkages. This pattern aligns with prior evidence on geographic homophily and the intensification of trade networks (Zhou, Reference Zhou2011, Reference Zhou2013), as well as with gravity-model findings for international investment (Kleinert and Toubal, Reference Kleinert and Toubal2010).

Despite the stability of major nodes and edges during COVID-19, robustness is not uniform across the network. We define the core as the top 5% of nodes by centrality in 2019. This set comprises 105 nodes out of 2,304 and accounts for more than 72% of total centrality, a share that remains stable over the sample period and is consistent with the Freeman centralization results. Rank dynamics also differ sharply across tiers. For the 105 core nodes, the standard deviation of rank between 2012 and 2022 is only 22.6, driven largely by the rise of China. By contrast, peripheral nodes exhibit much greater volatility, with a standard deviation exceeding 460.2. This contrast indicates a concentrated, persistent backbone alongside a highly fluid periphery.

In the next section, we further examine the similarity and changes over the complete network, and not only the core nodes.

4. An adaptive network

4.1 Weighted Jaccard similarity index

To gauge the stability of network edges over time, we use the Jaccard Similarity Index (Jaccard, Reference Jaccard1901), defined as the ratio of the intersection to the union of two sets. While the original index applies to binary data, extensions to weighted networks are essential for our setting, where edge weights reflect the frequency of co-location across firms.

Following Ružička (Reference Ružička1958), we adopt the weighted Jaccard similarity. Let $\mathbf{M}{t}=(m{1,t},\ldots ,m_{n,t})$ and $\mathbf{M}{t'}=(m{1,t'},\ldots ,m_{n,t'})$ denote the edge-weight vectors of the multinational network in years $t$ and $t'$ , with each vector rescaled to lie in $[0,1]$ . The weighted Jaccard index is

\begin{equation*} WJS(t, t') = \frac {\sum _{i=1}^{n} \min (m_{i,t}, m_{i,t'})}{\sum _{i=1}^{n} \max (m_{i,t}, m_{i,t'})} \end{equation*}

A value of $WJS = 1$ indicates identical edge weights across the two networks, while $WJS = 0$ implies complete dissimilarity, with non-overlapping edge activations. This generalization naturally reduces to the standard Jaccard index when all values are binary.

Figure 8. Weighted Jaccard similarity index: Year-to-year and with respect to 2019.

Figure 8 presents the evolution of similarity in French multinational network structures across time, using the weighted Jaccard index. We report both year-to-year changes and the distance of each yearly network from 2019—the last full pre-COVID year.

Overall, year-to-year similarity is high and stable over time, with no discontinuity in 2020. However, we find a roughly 30% rebalancing of edges each year, which is substantial, particularly for a network whose topological indicators remain stable. The architecture does not stand still: core metrics mask significant micro-level turnover, so the network cannot be described as identical from one year to the next.

These results are not inconsistent: substantial edge turnover can coexist with structural robustness. Following the logic of Granovetter (Reference Granovetter1973), small adjustments to critical bridges can reshape topology, whereas extensive churn among redundant links may leave the global structure largely unchanged. Given the stability of the core, most rewiring is likely concentrated at the periphery. Having established a stable backbone alongside nontrivial churn, we now test whether the determinants of edge weights have changed, which would signal shifts in firms’ FDI and co-location strategies. To this end, we estimate the drivers of edge weights to assess whether the post-2020 co-location pattern reflects a reweighting of the decision rules that generate ties.

5. Evolution of co-location strategies

Since conventional regression methods such as OLS are not well-suited for dyadic network data due to the violation of the independence assumption inherent in such models (Krackhardt, Reference Krackhardt1988). The interconnected nature of network structures introduces dependencies that invalidate standard inferential procedures.

Table 2. Quadratic assignment procedure

MRQAP with classic node-label permutations ( $R=5{,}000$ ); valued, undirected symmetric network; upper triangle only (diagonal excluded). $p$ -values in parentheses; ${}^{***}$ , ${}^{**}$ , and ${}^{*}$ denote 1%, 5%, and 10% significance.

To address this limitation, several statistical approaches have been developed for relational data. Among these, the quadratic assignment procedure (QAP) is particularly appropriate for our case, as it accommodates continuous-valued networks where edge weights represent the frequency of co-location. While exponential random graph models are better suited to binary or categorical outcomes, QAP provides a robust framework for linear models involving dyadic data (Dekker et al., Reference Dekker, Krackhardt and Snijders2007).

The QAP involves a permutation-based significance test that retains the inherent interdependence of network observations by permuting rows and columns of the adjacency matrix simultaneously (Krackardt, Reference Krackardt1987). Extending this, the MRQAP estimates linear models for network data and tests coefficients via random (or node-label) permutations. We begin by estimating a simple autoregressive model to evaluate the persistence of edge weights across time, using unconstrained permutations:

(2) \begin{equation} \ln (\mathbf{M}_{ij,t}) = \beta _0 + \beta _1 \ln (\mathbf{M}_{ij,t-1}) + \epsilon _{ij} \end{equation}

All variables are log-transformed to reduce the skewness and concentration in the distribution of edge weights. As shown by Dekker et al. (Reference Dekker, Krackhardt and Snijders2007), log transformations enhance MRQAP’s performance, particularly in networks with count-based or highly concentrated edge weights.

Beyond temporal dependence, we also explore whether other structural determinants—especially gravity-related variables—affect the network’s evolution. Inspired by gravity models of trade and investment (Kleinert and Toubal, Reference Kleinert and Toubal2010), we extend the model to include bilateral distance and GDP-based variables:

- $\mathbf{Dist}_{ij}$ : Bilateral geographic distance between host countries $i$ and $j$ , sourced from the CEPII Gravity dataset (Conte et al., Reference Conte, Cotterlaz and Mayer2022). - $\mathbf{SumGDP}_{ij,t-1}$ : Sum of the GDP of countries $i$ and $j$ in year $t-1$ . - $\mathbf{absdiffGDP}_{ij,t-1}$ : Absolute GDP difference between the two countries. - $\mathbf{SameIndus}_{ij,t}$ : Dummy variable equal to 1 if the edge links two subsidiaries in the same industry.

This extended model is expressed as:

(3) \begin{align} \ln (\mathbf{M}_{ij,t}) = \beta _0 + \beta _1 \ln (\mathbf{M}_{ij,t-1}) &+ \beta _2 \ln (\mathbf{Dist}_{ij}) + \beta _3 \ln (\mathbf{SumGDP}_{ij, t-1}) \nonumber \\[3pt] &+ \beta _4 \ln (\mathbf{absdiffGDP}_{ij, t-1}) + \beta _5 \ln (\mathbf{SameIndus}_{ij, t}) + \epsilon _{ij} \end{align}

Specifically, to estimate this relationship, we implement MRQAP with 5,000 node-label permutations suited to valued, undirected, symmetric matrices, restricting estimation to the upper triangle (diagonal excluded). The edge weights enter as $log(w_{ij}+1)$ .Footnote ⁶

The estimation results for both models are reported in Tables 2 and 3.

Table 3. Quadratic assignment procedure

MRQAP with classic node-label permutations ( $R=5{,}000$ ); valued, undirected symmetric network; upper triangle only (diagonal excluded). $p$ -values in parentheses; ${}^{***}$ , ${}^{**}$ , and ${}^{*}$ denote 1%, 5%, and 10% significance.

Both specifications consistently confirm the persistence of edge weights over time. The autoregressive coefficient $\beta _1$ is positive and highly significant throughout the period, indicating that the frequency with which firms co-locate in particular country-industry pairs is strongly path-dependent. However, this persistence weakens in 2021 and 2022, where the coefficient value drops by roughly half, suggesting a partial break in historical co-location patterns following the COVID-19 crisis.

The extended model offers further insight. While the influence of bilateral distance remains negative—consistent with gravity theory—its statistical significance sharply declines after 2020. The GDP-related variables, which were highly significant in earlier years (with a positive effect from total GDP and a negative effect from GDP disparity), lose significance and diminish in magnitude after 2020. This suggests a departure from previous co-location strategies, in which firms preferred pairing large and economically similar countries.

After the COVID shock, these economic fundamentals appear to have lost their explanatory power. Notably, the variable $\mathbf{SameIndus}_{ij,t}$ , typically associated with a tendency to cluster activity within the same industry—also shows a marked change: while it was positive and significant in earlier years, it turns negative in 2021 and 2022. This may reflect a shift in strategic orientation, with MNEs reallocating activity across industries rather than reinforcing existing sectoral specialization.

These results point to a strategic reconfiguration of the French multinational network after COVID-19. Earlier sections documented stable core structures—density, centralization, and persistence of top nodes—yet the MRQAP evidence shows that the determinants of tie formation at the margin have shifted. Post-2020 patterns depart from the gravity-driven architecture that characterized co-location choices in the preceding decade.

In short, the pandemic did not unsettle the core topology of multinational production networks, but it changed the logic of marginal expansion: where and how firms extend or reweight their international footprint. This underscores the value of combining structural diagnostics with dyadic modeling to capture both the stability of the backbone and the evolution of decision rules that shape new or reweighted ties.