## wp17148 - Annex I

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---

### Definitions and Network Measures
- Degree: number of total trade links (partners); in-degree and out-degree reflect imports and exports respectively.  
- Strength: intensity of a trade connection in a weighted network, i.e. the sum of all (import and export) bilateral trade flows of a country in the network, frequently scaled by the sum of all trade flows in the network; in-strength and out-strength refer to import and export link intensity.  
- Eigenvector centrality: measures connectivity to many or important links; in- and out-eigencentrality measure connectivity to central importers and exporters respectively.  
- Closeness centrality: the average geodesic distance to other nodes in the network.  
- Betweenness centrality: the number of shortest paths connecting all nodes that pass through the node of interest; high-betweenness is akin to being a network broker.  
- Data limitation: available bilateral trade data includes only trade in goods; network and traditional indicators may notably underestimate integration of countries heavily reliant on services.

### A. Traditional Measures of LAC Integration (key statistics and patterns)
- LAC accounted for about 8 percent of global economic activity in 2015, but only about 5.1 percent of global exports of goods and services.  
- In 2015, LAC’s trade (exports and imports) represented only 44 percent of regional GDP.  
- Trade openness across LAC countries ranges from 25 to 125 percent of GDP.  
- Brazil: trade accounts for only 27½ percent of GDP.  
- Argentina: trade accounts for only 24 percent of GDP.  
- Mexico is relatively more open due to proximity and low barriers with the United States after NAFTA (1994).

### B. Overview of the Trade Network (regional position and clustering)
- Central nodes in the WTN: United States, China and major European economies (most central under weighted centrality).  
- LAC clusters closer to the United States; China has gained prominence as an export market for LAC commodity exporters.  
- The Caribbean region remains peripheral in the WTN given small economies and goods-only data.

### C. Integration on the Extensive Margin (Degree) — findings
- LAC trades with about 70 percent of all countries as of 2015, somewhat more diversified than the world average and lagging only North America, Europe and MENA.  
- Five large LAC countries (Brazil, Colombia, Argentina, Peru, and Chile) account for about 60 percent of the region’s GDP and have trade links that on average cover about 91 percent of all potential global trading partners (at or above the 75th percentile of global degree).  
- Top 13 most integrated countries in Asia have trade relationships with 96 percent of potential partners.  
- Mexico has trade links with only half of its potential partners. Panama, Belize, Guyana and Suriname also linked with only half of the countries.  
- Correlation between number of trade links and (log) size of the economy: 0.8.  
- Caribbean region: less than half of all possible trade links realized (weakest market diversification in goods).  
- Degree growth: LAC increased its number of reported partners by about 70 percent on average over the last 20 years; largest growth in late 1980s to late 1990s.

### D. Integration on the Intensive Margin (Strength) — findings
- LAC lags behind North America, Asia, and Europe in strength (sum of nominal trade flows).  
- Mexican trade flows stand out: Mexico ranks 13th in the world in terms of its strength score in the WTN (2015).  
- Bilateral export flows from Mexico to the United States are the third largest in the world in 2015; exports from China and Canada to the United States rank first and second respectively.  
- Exports from Brazil to China ranked 75th in the world (2015).

### E. Other Centrality Measures and Correlations
- Eigencentrality is positively correlated with measures related to country participation in global value chains, such as foreign value added in gross exports (though the upstream component shows only a weak correlation).  
- Average density of the WTN was 0.56 in 2015 (more than half of potential connections realized).  
- Most centrality measures (closeness, eigencentrality) strongly correlate with degree; betweenness has an exponential relationship with other measures (threshold effect for becoming a broker).  
- Regional rankings by unweighted centrality: North America most connected, followed by Europe; Middle East and Latin America close third; Asia, Africa and the Caribbean follow.

### F. Integration Within Latin America (regional subnetworks and hubs)
- LAC regional subnetwork density: 97 percent of export and import links realized (near-perfect connectivity among member countries).  
- Large countries (Mexico, Brazil) do not act as dominant regional hubs; smaller countries (Costa Rica, Dominican Republic, Trinidad and Tobago) often have higher relative regional degree.  
- Only about 15 percent of LAC total exports are destined to regional markets (LAC), lower than developed economies in Asia and Europe where regional destinations account for well over 50 percent of exports.  
- Bolivia, Paraguay and Uruguay: more regionally integrated given concentration of exports to Brazil (Mercosur).  
- Mexico serves as a hub for some Central American countries but not a broad regional hub.  
- Fragmentation in regional trade agreements: six regional blocs with limited coverage and diverse rules of origin; limited trade/absence of trade agreement between Mexico and Brazil.

### G. Network Integration and Growth Effects (empirical relationships)
- More central positions in the trade network are associated with stronger growth outcomes, even after controlling for trade openness.  
- Degree centrality (more total trade partners, especially exports) and eigencentrality (connectivity to connected importers/exporters) matter for higher economic growth, likely reflecting technology and knowledge diffusion benefits.  
- Connectivity to the most connected importers in the WTN (top 3 by in-eigencentrality) could boost growth beyond volume-based measures.  
- Trade connectivity with regional partners (shares of intraregional exports to total exports) associated with larger growth effects than connectivity outside own region.

### IV. Actual vs. Benchmark Trade Networks — Binary (Unweighted) Methodology and Results
- Binary benchmark: LOGIT model where link(i→j) is 1 if export from i to j; explanatory variables include ln(GDP) of exporter and partner, ln(distance), contiguity, common language, colonial relationship, common colonizer post 1945, landlocked dummy, and regional trade agreement dummy.  
- Cut-off probability chosen to match simulated network density to actual density each year: cut-off averages about 46 percent in the 1980s, increasing gradually to 52 percent by 2010-15.  
- Gravity model findings (LOGIT): increase in own or partner GDP by 1 percent increases odds ratio of having a trade link by 0.6-0.7 percent; distance and landlocked reduce probability; common language and former colonial relation increase probability.  
- Predicted degree closely matches actual degree evolution for LAC and Asia; Asia marginally exceeded predictions consistently.  
- Mexico, Venezuela and Panama are significantly less diversified than predicted by the logit-gravity model; smaller countries (Belize, Suriname, Nicaragua, Paraguay) are better connected relative to predictions.

### IV. Actual vs. Benchmark Trade Networks — Weighted Networks (PPML) Methodology and Results
- Weighted benchmark: PPML gravity model predicting nominal exports:
  - Specification: exports = exp(α1 ln GDP exporter + α2 ln GDP partner + α3 ln weighted distance + ... + ε) (see text for full covariates).
- PPML regression results:
  - Ln GDP Exporter: 0.818***  
  - Ln GDP Partner: 0.839***  
  - Ln Weighted Distance: -0.630***  
  - Contiguity: 0.599***  
  - Common language: 0.463***  
  - Colonial relationship: -0.115***  
  - Common colonizer: 0.888***  
  - Landlocked: -0.411***  
  - Regional trade agreement: 0.209***  
  - Pseudo R2: 0.8891  
  - Observations: 902,150  
  - Significance notation: * p< 0.05, ** p< 0.01, *** p < 0.001
- Weighted-network findings:
  - Most LAC countries are more weakly integrated in strength than predicted by the gravity model.  
  - Strength gap scaled by predicted value shows underperformance of large countries is relatively small (e.g., Brazil about 6 percent under-trading), while smaller economies (Panama, Belize) show larger percentage underperformance.  
  - Countries over-trading relative to predictions (in world network): Paraguay, Bolivia, Mexico, Chile, Nicaragua. Paraguay and Bolivia over-trade the most relative to flow size.  
  - Intra-regional weighted comparisons: most LAC countries under-trade regionally relative to gravity predictions; Chile and Mexico over-trade globally but under-trade with regional partners; El Salvador and Costa Rica under-trade globally but have strong regional ties; Bolivia, Paraguay and Nicaragua over-trade both globally and regionally.  
  - Temporal pattern: actual weighted strength matched model predictions until about 2010; since then LAC underperformed relative to predicted flows. Asia has consistently over-performed relative to predictions.

### V. Conclusions (summary of substantive findings and implications)
- Network indicators add insight beyond trade-to-GDP openness: countries with low trade-GDP ratios can be highly integrated by number of trade links.  
- On the extensive margin LAC is relatively well connected (market diversification), with many countries trading with a large share of partners; yet concentrated export patterns (geographic or commodity composition) limit centrality.  
- On the intensive margin LAC is weaker: trade flow intensity (strength) lags other regions and has trended below gravity-model predictions since about 2010.  
- Large LAC economies under-trade in absolute terms but relatively modestly compared to model predictions (e.g., Brazil ~6 percent shortfall); smaller countries show larger percent deviations.  
- LAC’s regional network is dense (97 percent) but connection strength is limited and the region is outward oriented (only about 15 percent of exports destined regionally).  
- There is scope for larger countries to assume more central regional roles, but factors enabling such shifts are left for future research.

### Annex I. An Overview of Network Measures — technical definitions and formulas
- Network definitions and conventions:
  - A network is a structure made up of a set of objects (called nodes or vertices) that are connected together.
  - A node or vertex is the fundamental unit of which networks are formed; connections between nodes are called edges or links.
  - A weighted or valued network is represented by giving edge values equal to the weights of the corresponding connections.
  - A directed graph is a network in which each edge has a direction, pointing from one vertex to another.
- Degree:
  - Definition: The degree of a node is the number of connections it has to other nodes, often expressed as a fraction of total possible connections it can have.
  - Formula: ݇ ௜ = ═╥ܣ ௜௝ ௧ୀଵ
    - where ܣ ௜௝ =1 if there is a connection between j and i.
- Strength:
  - Definition: The sum of weights attached to ties belonging to a node (e.g. value of total trade).
  - Formula: ݏ ௜ = ═╥ܣ ௜௝ ݓ ௜௝ ௧ୀଵ
    - where ܣ ௜௝ =1 if there is a connection between j and i and ݓ ௜௝ denotes the weight of such connection (in the source cases it is the value of the trade flows).
- Eigenvector Centrality:
  - Definition: Eigenvector centrality gives each vertex a score proportional to the sum of the scores of its neighbors. Connections to high-scoring nodes contribute more to a node's score than equal connections to low-scoring nodes.
  - Directional notes: Inward eigencentrality considers only vertices that point out to vertex v; outward eigencentrality considers those that vertex v points to.
  - Formula (relative centrality score of vertex v): ݔ ௩ = 1 ߣ ═╥ܣ ௜௝ ݔ ௝ = ௧∈ெሺ௩ሻ 1 ߣ ܽ╥ ௩௧ ݔ ௧ ௧∈ீ
    - For a given graph G=(V,E) with |V| number of vertices, A = (ܽ ௩,௧) is the adjacency matrix, i.e. ܽ ௩,௧ =1 if vertex v is linked to vertex t, and ܽ ௩,௧ =0 otherwise. M(v) is a set of neighbors of v and λ is a constant.
  - Weighted eigencentrality assigns weights to each connection (links) considered in calculation.
  - Sample numeric centrality values shown in the source:
    - A(0.33) C(0.50) D(0.33) E(0.50) G(0.33) F(0.33) B(0.33)
    - A(0.74) C(1.00) D(0.85) E(1.00) G(0.74) F(0.74) B(0.74)
- Closeness Centrality:
  - Definition: Closeness centrality is calculated as the sum of the length of the shortest paths between the vertex and all other vertices in the graph. The more central a node is, the closer it is to all other nodes.
  - Formula: C(x) = ௡ ═╥ d(y,x) ೤
    - where d(y,x) is the distance (length of the shortest path) between vertex x and y. n is the total number of shortest paths between the vertices and all other vertices.
- Betweenness Centrality:
  - Definition: Betweenness centrality for each vertex is the number of the shortest paths that pass through the vertex. For every pair of vertices in a graph, there exists a shortest path between them such that either the number of edges that the path passes through (for undirected graphs) or the sum of the weights of the edges (for directed graphs) is minimized.
  - Formula: ܥ ஻ (ݒ) = ═╥ ߪ ௦௧ (ݒ)ݒ (ߪ ௦௧)
    - where ߪ ௦௧ is the total number of shortest paths joining any two nodes/vertices s and t and ߪ ௦௧ (ݒ) is the number of those paths that pass through v.
  - Computation steps:
    1. For each pair of vertices (s,t), compute the shortest paths between them.
    2. For each pair of vertices (s,t), determine the fraction of shortest paths that pass through the vertex in question (here, vertex v).
    3. Sum this fraction over all pairs of vertices (s,t).
- Authority Centrality:
  - Definition: The authority centrality of a vertex is defined to be proportional to the sum of the hub centralities of the vertices that point to it.
  - Formula: ݔ ௜ ═╥ߙ ܣ ௜௝ ݕ ௝ ௝
    - where ݔ ௜ is the authority centrality and ݕ ௝ is the hub centrality, and ܣ ௜௝ =1 if vertices j points to vertices i.
- Hub Centrality:
  - Definition: The hub centrality of a vertex is proportional to the sum of the authority centralities of the vertices it points to.
  - Formula: ݕ ௜ ═╥ߚ ܣ ௝௜ ݔ ௝ ௝
    - where ݔ ௜ is the authority centrality and ݕ ௝ is the hub centrality, and ܣ ௝௜ =1 if vertices i points to vertices j.
  - Sample numeric authority/hub values shown in the source:
    - A(0.00) C(0.73) D(1.00) E(0.73) G(0.00) F(0.00) B(0.00)
    - A(0.00) C(0.89) D(1.00) E(0.89) G(0.00) F(0.00) B(0.00)

*Source: wp17148 - Annex I (IMF working paper content provided in source PDF).*

### Annex I:

### wp17148 - Annex I

### Definitions and Network Measures
- Degree: number of total trade links (partners); in-degree and out-degree reflect imports and exports respectively.  
- Strength: intensity of a trade connection in a weighted network, i.e. the sum of all (import and export) bilateral trade flows of a country in the network, frequently scaled by the sum of all trade flows in the network; in-strength and out-strength refer to import and export link intensity.  
- Eigenvector centrality: measures connectivity to many or important links; in- and out-eigencentrality measure connectivity to central importers and exporters respectively.  
- Closeness centrality: the average geodesic distance to other nodes in the network.  
- Betweenness centrality: the number of shortest paths connecting all nodes that pass through the node of interest; high-betweenness is akin to being a network broker.  
- Data limitation: available bilateral trade data includes only trade in goods; network and traditional indicators may notably underestimate integration of countries heavily reliant on services.

### A. Traditional Measures of LAC Integration (key statistics and patterns)
- LAC accounted for about 8 percent of global economic activity in 2015, but only about 5.1 percent of global exports of goods and services.  
- In 2015, LAC’s trade (exports and imports) represented only 44 percent of regional GDP.  
- Trade openness across LAC countries ranges from 25 to 125 percent of GDP.  
- Brazil: trade accounts for only 27½ percent of GDP.  
- Argentina: trade accounts for only 24 percent of GDP.  
- Mexico is relatively more open due to proximity and low barriers with the United States after NAFTA (1994).

### B. Overview of the Trade Network (regional position and clustering)
- Central nodes in the WTN: United States, China and major European economies (most central under weighted centrality).  
- LAC clusters closer to the United States; China has gained prominence as an export market for LAC commodity exporters.  
- The Caribbean region remains peripheral in the WTN given small economies and goods-only data.

### C. Integration on the Extensive Margin (Degree) — findings
- LAC trades with about 70 percent of all countries as of 2015, somewhat more diversified than the world average and lagging only North America, Europe and MENA.  
- Five large LAC countries (Brazil, Colombia, Argentina, Peru, and Chile) account for about 60 percent of the region’s GDP and have trade links that on average cover about 91 percent of all potential global trading partners (at or above the 75th percentile of global degree).  
- Top 13 most integrated countries in Asia have trade relationships with 96 percent of potential partners.  
- Mexico has trade links with only half of its potential partners. Panama, Belize, Guyana and Suriname also linked with only half of the countries.  
- Correlation between number of trade links and (log) size of the economy: 0.8.  
- Caribbean region: less than half of all possible trade links realized (weakest market diversification in goods).  
- Degree growth: LAC increased its number of reported partners by about 70 percent on average over the last 20 years; largest growth in late 1980s to late 1990s.

### D. Integration on the Intensive Margin (Strength) — findings
- LAC lags behind North America, Asia, and Europe in strength (sum of nominal trade flows).  
- Mexican trade flows stand out: Mexico ranks 13th in the world in terms of its strength score in the WTN (2015).  
- Bilateral export flows from Mexico to the United States are the third largest in the world in 2015; exports from China and Canada to the United States rank first and second respectively.  
- Exports from Brazil to China ranked 75th in the world (2015).

### E. Other Centrality Measures and Correlations
- Eigencentrality is positively correlated with measures related to country participation in global value chains, such as foreign value added in gross exports (though the upstream component shows only a weak correlation).  
- Average density of the WTN was 0.56 in 2015 (more than half of potential connections realized).  
- Most centrality measures (closeness, eigencentrality) strongly correlate with degree; betweenness has an exponential relationship with other measures (threshold effect for becoming a broker).  
- Regional rankings by unweighted centrality: North America most connected, followed by Europe; Middle East and Latin America close third; Asia, Africa and the Caribbean follow.

### F. Integration Within Latin America (regional subnetworks and hubs)
- LAC regional subnetwork density: 97 percent of export and import links realized (near-perfect connectivity among member countries).  
- Large countries (Mexico, Brazil) do not act as dominant regional hubs; smaller countries (Costa Rica, Dominican Republic, Trinidad and Tobago) often have higher relative regional degree.  
- Only about 15 percent of LAC total exports are destined to regional markets (LAC), lower than developed economies in Asia and Europe where regional destinations account for well over 50 percent of exports.  
- Bolivia, Paraguay and Uruguay: more regionally integrated given concentration of exports to Brazil (Mercosur).  
- Mexico serves as a hub for some Central American countries but not a broad regional hub.  
- Fragmentation in regional trade agreements: six regional blocs with limited coverage and diverse rules of origin; limited trade/absence of trade agreement between Mexico and Brazil.

### G. Network Integration and Growth Effects (empirical relationships)
- More central positions in the trade network are associated with stronger growth outcomes, even after controlling for trade openness.  
- Degree centrality (more total trade partners, especially exports) and eigencentrality (connectivity to connected importers/exporters) matter for higher economic growth, likely reflecting technology and knowledge diffusion benefits.  
- Connectivity to the most connected importers in the WTN (top 3 by in-eigencentrality) could boost growth beyond volume-based measures.  
- Trade connectivity with regional partners (shares of intraregional exports to total exports) associated with larger growth effects than connectivity outside own region.

### IV. Actual vs. Benchmark Trade Networks — Binary (Unweighted) Methodology and Results
- Binary benchmark: LOGIT model where link(i→j) is 1 if export from i to j; explanatory variables include ln(GDP) of exporter and partner, ln(distance), contiguity, common language, colonial relationship, common colonizer post 1945, landlocked dummy, and regional trade agreement dummy.  
- Cut-off probability chosen to match simulated network density to actual density each year: cut-off averages about 46 percent in the 1980s, increasing gradually to 52 percent by 2010-15.  
- Gravity model findings (LOGIT): increase in own or partner GDP by 1 percent increases odds ratio of having a trade link by 0.6-0.7 percent; distance and landlocked reduce probability; common language and former colonial relation increase probability.  
- Predicted degree closely matches actual degree evolution for LAC and Asia; Asia marginally exceeded predictions consistently.  
- Mexico, Venezuela and Panama are significantly less diversified than predicted by the logit-gravity model; smaller countries (Belize, Suriname, Nicaragua, Paraguay) are better connected relative to predictions.

### IV. Actual vs. Benchmark Trade Networks — Weighted Networks (PPML) Methodology and Results
- Weighted benchmark: PPML gravity model predicting nominal exports:
  - Specification: exports = exp(α1 ln GDP exporter + α2 ln GDP partner + α3 ln weighted distance + ... + ε) (see text for full covariates).
- PPML regression results (Table 3, coefficients and statistics):
  - Ln GDP Exporter: 0.818***  
  - Ln GDP Partner: 0.839***  
  - Ln Weighted Distance: -0.630***  
  - Contiguity: 0.599***  
  - Common language: 0.463***  
  - Colonial relationship: -0.115***  
  - Common colonizer: 0.888***  
  - Landlocked: -0.411***  
  - Regional trade agreement: 0.209***  
  - Pseudo R2: 0.8891  
  - Observations: 902,150  
  - Significance notation: * p< 0.05, ** p< 0.01, *** p < 0.001
- Weighted-network findings:
  - Most LAC countries are more weakly integrated in strength than predicted by the gravity model.  
  - Strength gap scaled by predicted value shows underperformance of large countries is relatively small (e.g., Brazil about 6 percent under-trading), while smaller economies (Panama, Belize) show larger percentage underperformance.  
  - Countries over-trading relative to predictions (in world network): Paraguay, Bolivia, Mexico, Chile, Nicaragua. Paraguay and Bolivia over-trade the most relative to flow size.  
  - Intra-regional weighted comparisons: most LAC countries under-trade regionally relative to gravity predictions; Chile and Mexico over-trade globally but under-trade with regional partners; El Salvador and Costa Rica under-trade globally but have strong regional ties; Bolivia, Paraguay and Nicaragua over-trade both globally and regionally.  
  - Temporal pattern: actual weighted strength matched model predictions until about 2010; since then LAC underperformed relative to predicted flows. Asia has consistently over-performed relative to predictions.

### V. Conclusions (summary of substantive findings and implications)
- Network indicators add insight beyond trade-to-GDP openness: countries with low trade-GDP ratios can be highly integrated by number of trade links.  
- On the extensive margin LAC is relatively well connected (market diversification), with many countries trading with a large share of partners; yet concentrated export patterns (geographic or commodity composition) limit centrality.  
- On the intensive margin LAC is weaker: trade flow intensity (strength) lags other regions and has trended below gravity-model predictions since about 2010.  
- Large LAC economies under-trade in absolute terms but relatively modestly compared to model predictions (e.g., Brazil ~6 percent shortfall); smaller countries show larger percent deviations.  
- LAC’s regional network is dense (97 percent) but connection strength is limited and the region is outward oriented (only about 15 percent of exports destined regionally).  
- There is scope for larger countries to assume more central regional roles, but factors enabling such shifts are left for future research.

*Source: wp17148 - Annex I (IMF working paper content provided in source PDF).*

### Annex I. An Overview of Network Measures

### Annex I. An Overview of Network Measures

### Network definitions and conventions
- A network is a structure made up of a set of objects (called nodes or vertices) that are connected together.
- A node or vertex is the fundamental unit of which networks are formed; connections between nodes are called edges or links.
- A weighted or valued network is represented by giving edge values equal to the weights of the corresponding connections.
- A directed graph is a network in which each edge has a direction, pointing from one vertex to another.

### Degree
- Definition: The degree of a node is the number of connections it has to other nodes, often expressed as a fraction of total possible connections it can have.
- Formula: ݇ ௜ = ═╥ܣ ௜௝ ௧ୀଵ
  - where ܣ ௜௝ =1 if there is a connection between j and i.

### Strength
- Definition: The sum of weights attached to ties belonging to a node (e.g. value of total trade).
- Formula: ݏ ௜ = ═╥ܣ ௜௝ ݓ ௜௝ ௧ୀଵ
  - where ܣ ௜௝ =1 if there is a connection between j and i and ݓ ௜௝ denotes the weight of such connection (in the source cases it is the value of the trade flows).

### Eigenvector Centrality
- Definition: Eigenvector centrality gives each vertex a score proportional to the sum of the scores of its neighbors. Connections to high-scoring nodes contribute more to a node's score than equal connections to low-scoring nodes.
- Directional notes: Inward eigencentrality considers only vertices that point out to vertex v; outward eigencentrality considers those that vertex v points to.
- Formula (relative centrality score of vertex v): ݔ ௩ = 1 ߣ ═╥ܣ ௜௝ ݔ ௝ = ௧∈ெሺ௩ሻ 1 ߣ ܽ╥ ௩௧ ݔ ௧ ௧∈ீ
  - For a given graph G=(V,E) with |V| number of vertices, A = (ܽ ௩,௧) is the adjacency matrix, i.e. ܽ ௩,௧ =1 if vertex v is linked to vertex t, and ܽ ௩,௧ =0 otherwise. M(v) is a set of neighbors of v and λ is a constant.
- Weighted eigencentrality assigns weights to each connection (links) considered in calculation.
- Sample numeric centrality values shown in the source:
  - A(0.33) C(0.50) D(0.33) E(0.50) G(0.33) F(0.33) B(0.33)
  - A(0.74) C(1.00) D(0.85) E(1.00) G(0.74) F(0.74) B(0.74)

### Closeness Centrality
- Definition: Closeness centrality is calculated as the sum of the length of the shortest paths between the vertex and all other vertices in the graph. The more central a node is, the closer it is to all other nodes.
- Formula: C(x) = ௡ ═╥ d(y,x) ೤
  - where d(y,x) is the distance (length of the shortest path) between vertex x and y. n is the total number of shortest paths between the vertices and all other vertices.

### Betweenness Centrality
- Definition: Betweenness centrality for each vertex is the number of the shortest paths that pass through the vertex. For every pair of vertices in a graph, there exists a shortest path between them such that either the number of edges that the path passes through (for undirected graphs) or the sum of the weights of the edges (for directed graphs) is minimized.
- Formula: ܥ ஻ (ݒ) = ═╥ ߪ ௦௧ (ݒ)ݒ (ߪ ௦௧)
  - where ߪ ௦௧ is the total number of shortest paths joining any two nodes/vertices s and t and ߪ ௦௧ (ݒ) is the number of those paths that pass through v.
- Computation steps:
  1. For each pair of vertices (s,t), compute the shortest paths between them.
  2. For each pair of vertices (s,t), determine the fraction of shortest paths that pass through the vertex in question (here, vertex v).
  3. Sum this fraction over all pairs of vertices (s,t).

### Authority Centrality
- Definition: The authority centrality of a vertex is defined to be proportional to the sum of the hub centralities of the vertices that point to it.
- Formula: ݔ ௜ ═╥ߙ ܣ ௜௝ ݕ ௝ ௝
  - where ݔ ௜ is the authority centrality and ݕ ௝ is the hub centrality, and ܣ ௜௝ =1 if vertices j points to vertices i.

### Hub Centrality
- Definition: The hub centrality of a vertex is proportional to the sum of the authority centralities of the vertices it points to.
- Formula: ݕ ௜ ═╥ߚ ܣ ௝௜ ݔ ௝ ௝
  - where ݔ ௜ is the authority centrality and ݕ ௝ is the hub centrality, and ܣ ௝௜ =1 if vertices i points to vertices j.
- Sample numeric authority/hub values shown in the source:
  - A(0.00) C(0.73) D(1.00) E(0.73) G(0.00) F(0.00) B(0.00)
  - A(0.00) C(0.89) D(1.00) E(0.89) G(0.00) F(0.00) B(0.00)

*Source: Annex I. An Overview of Network Measures (wp17148)*

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_Source: https://www.imf.org/-/media/files/publications/wp/2017/wp17148.pdf_