Applied Informatics and Computational Economics Lab
2 July 2026
Required Packages
library(tidyverse) # data wrangling + ggplot2library(igraph) # graphs, centrality, communities, random graphslibrary(ggraph) # network visualization, grammar-of-graphics stylelibrary(tidygraph) # tidy interface to igraphlibrary(kableExtra) # results tables# optional, for downloading the raw data yourself:# library(eurostat) # get_eurostat("ext_tec03")
import pandas as pdimport numpy as npimport networkx as nx # graphs, centrality, communitiesimport matplotlib.pyplot as plt
Why no Stata in this lecture?
Network analysis is dominated by R (igraph) and Python (networkx). Stata has no built-in network toolbox, and the community suite nwcommands (Grund, 2015) is unmaintained and not on SSC. Simple measures (degree, strength, reciprocity) reduce to collapse and merge on an edge list, but anything path-based — betweenness, closeness, communities — is impractical. For this topic, use the right tool: R or Python.
Literature Review — Networks in Economics
Jackson (2008), Social and Economic Networks, Princeton University Press — the canonical graduate text. doi:10.2307/j.ctvcm4gh1
Jackson, Rogers & Zenou (2017), “The Economic Consequences of Social-Network Structure”, Journal of Economic Literature 55(1), 49–95. doi:10.1257/jel.20150694
Acemoglu, Carvalho, Ozdaglar & Tahbaz-Salehi (2012), “The Network Origins of Aggregate Fluctuations”, Econometrica 80(5), 1977–2016. doi:10.3982/ECTA9623
Carvalho (2014), “From Micro to Macro via Production Networks”, Journal of Economic Perspectives 28(4), 23–48. doi:10.1257/jep.28.4.23
Ballester, Calvó-Armengol & Zenou (2006), “Who’s Who in Networks. Wanted: The Key Player”, Econometrica 74(5), 1403–1417. doi:10.1111/j.1468-0262.2006.00709.x
Newman (2003), “The Structure and Function of Complex Networks”, SIAM Review 45(2), 167–256. doi:10.1137/S003614450342480
De Benedictis, Nenci, Santoni, Tajoli & Vicarelli (2014), “Network Analysis of World Trade using the BACI-CEPII Dataset”, Global Economy Journal 14(3-4), 287–343. doi:10.1515/gej-2014-0032
Chaney (2014), “The Network Structure of International Trade”, American Economic Review 104(11), 3600–3634. doi:10.1257/aer.104.11.3600
Fagiolo, Reyes & Schiavo (2009), “World-Trade Web: Topology, Country Centrality, and Dynamics”, Physical Review E 79, 036115. doi:10.1103/PhysRevE.79.036115
Serrano & Boguñá (2003), “Topology of the World Trade Web”, Physical Review E 68, 015101. doi:10.1103/PhysRevE.68.015101
This lecture expands the lab’s earlier notebook International Trade of EU (course material, 09-trade.Rmd), and borrows its Eurostat application. Data access via the eurostat R package: Lahti, Huovari, Kainu & Biecek (2017), The R Journal 9(1), 385–392. doi:10.32614/RJ-2017-019
The Problem & Motivation
Aggregate trade statistics answer “how much?” — they cannot answer “with whom, and how is the system wired?”
Bilateral trade flows are a network: countries are nodes, flows are directed, weighted edges
Shocks propagate along edges — Acemoglu et al. (2012): granular shocks survive aggregation because of network asymmetry
Two countries with identical export/GDP ratios can occupy completely different positions: a hub vs a spoke
Global value chains make the indirect exposure (friend-of-a-friend trade) a first-order policy question
Network position predicts growth, shock resilience, and gains from trade beyond gravity covariates (De Benedictis & Tajoli, 2011)
Why aggregates mislead
Suppose countries \(i\) and \(j\) both export 40% of GDP. If \(i\) sells to 25 partners and \(j\) sells 90% of everything to a single neighbour, a demand shock in that neighbour is a catastrophe for \(j\) and noise for \(i\). The aggregate export ratio is identical; the network position is not. Centrality, reciprocity and clustering are the statistics that see this difference.
Part I — Network Fundamentals
Graphs — Basic Definitions
A graph (network) is a pair of sets:
\[
G = (V, E), \qquad V = \{1, 2, \dots, n\}, \qquad E \subseteq V \times V
\]
\(V\): the nodes (vertices) — here, countries
\(E\): the edges (links) — here, trade flows
Directed graph: \((i,j) \in E\) does not imply \((j,i) \in E\) — exports have a direction
Weighted graph: each edge carries a value \(w_{ij} > 0\) — the size of the flow
An edge list is the tidy representation: one row per edge (from, to, weight)
The number of possible directed edges among \(n\) nodes is:
\[
E_{\max} = n(n-1)
\]
Directed or undirected?
Trade is naturally directed: Germany exporting to Malta is not the same edge as Malta exporting to Germany. But some tools (eigenvector centrality, most community detectors) are defined for undirected graphs, so we sometimes collapse the network, summing the two directed weights. Always state which convention you are using — results change.
The Adjacency Matrix
The same object appears in the companion deck Spatial Econometrics under a different name: row-normalise \(A\) and it becomes the spatial weight matrix\(W\). Here it encodes who trades with whom; there it encodes who is near whom, and the econometrics of \(\rho W y\) follows.
The algebraic representation of \(G\) is the \(n \times n\)adjacency matrix:
A B C D E
A 0.0 5.0 3.0 0.0 0.0
B 5.0 0.0 4.0 0.0 0.0
C 3.0 4.0 0.0 2.0 0.0
D 0.0 0.0 2.0 0.0 6.0
E 0.0 0.0 0.0 6.0 0.0
The matrix is a mirror image across the diagonal — \(A_{ij}=A_{ji}\) for every pair.
Visualizing the Undirected Network
Code
library(ggraph)set.seed(14159)ggraph(g_toy_u, layout ="circle") +geom_edge_link(aes(width = weight), alpha =0.5, color ="#185FA5") +geom_node_point(size =14, color ="#1D9E75") +geom_node_text(aes(label = name), color ="white", fontface ="bold") +scale_edge_width(range =c(0.5, 3), name ="weight") +theme_void()
Code
import matplotlib.pyplot as pltpos = nx.spring_layout(G_toy_u, seed=14159)w = [G_toy_u[u][v]["weight"] /2for u, v in G_toy_u.edges()]_ = plt.figure(figsize=(7, 5))_ = nx.draw_networkx_nodes(G_toy_u, pos, node_size=1400, node_color="#1D9E75")_ = nx.draw_networkx_labels(G_toy_u, pos, font_color="white", font_weight="bold")_ = nx.draw_networkx_edges(G_toy_u, pos, width=w, edge_color="#185FA5", alpha=0.6)_ = plt.axis("off")plt.tight_layout()plt.show()
No arrowheads: an undirected link means the relation runs both ways by construction.
A Toy Directed Network — Edge List & Adjacency
The same five generic nodes, now joined by directed links. Order matters, so \((A,B)\) and \((B,A)\) are different edges and the adjacency matrix is asymmetric.
strength(g_toy, mode ="out") # total outgoing weightstrength(g_toy, mode ="in") # total incoming weightstrength(g_toy_u) # undirected: a single strength
# invert weights: strong links = short distancesfor u, v in G_toy.edges(): G_toy[u][v]["dist"] =1/ G_toy[u][v]["weight"]# networkx uses incoming distances, so reverse to match igraph mode="out"clo = nx.closeness_centrality(G_toy.reverse(), distance="dist")print("closeness:", {k: round(v, 2) for k, v in clo.items()})
for u, v in G_toy.edges(): G_toy[u][v]["dist"] =1/ G_toy[u][v]["weight"]bet = nx.betweenness_centrality(G_toy, weight="dist")print("betweenness:", {k: round(v, 2) for k, v in bet.items()})
It is not only how many neighbours you have — it is how important they are.
Eigenvector: your importance is proportional to the importance of your neighbours
Katz–Bonacich: sum walks of every length, discounting longer ones by \(\phi\) — the economics workhorse
PageRank: a random-surfer normalization built for directed graphs
Eigenvector centrality needs an undirected graph, so we collapse directions first
Katz–Bonacich and the key player
Ballester, Calvó-Armengol & Zenou (2006) show that in a linear-quadratic game on a network, equilibrium effort is exactly the Katz–Bonacich centrality vector: \(x^* = (I-\phi A)^{-1}\mathbf{1}\). The key player — the node whose removal reduces aggregate activity most — is found from the same matrix. In trade terms: \(\sum_k \phi^k A^k\) adds up direct exposure (\(k=1\)), exposure through one intermediary (\(k=2\)), and so on, discounting each extra step by \(\phi\). This is precisely how demand shocks travel through input-output and trade linkages in Acemoglu et al. (2012) — the “influence vector” there is a Katz–Bonacich-type object. One formula, three literatures: peer effects, production networks, international trade.
Eigenvector — importance proportional to neighbours’ importance, with \(\lambda\) the largest eigenvalue of \(A\) (Perron–Frobenius):
\[
x_i = \frac{1}{\lambda} \sum_{j} A_{ij} \, x_j
\qquad \Longleftrightarrow \qquad
A x = \lambda x
\]
Katz–Bonacich — discounts walks by length:
\[
b = (I - \phi A)^{-1} \mathbf{1} = \sum_{k=0}^{\infty} \phi^k A^k \mathbf{1}, \qquad 0 < \phi < 1/\lambda
\]
PageRank — random-surfer normalization for directed graphs:
# average over reachable ordered pairs (matches igraph mean_distance)dists = [d for src, tgt in nx.all_pairs_shortest_path_length(G_toy)for node, d in tgt.items() if d >0]print("mean distance:", round(sum(dists) /len(dists), 3))
A community is a group of nodes more densely connected internally than externally.
Modularity\(Q\) compares observed within-group weight to what a random rewiring with the same strengths would give
The Louvain algorithm greedily maximizes \(Q\) — fast, standard, works on weighted graphs
Defined for undirected graphs, so we collapse directions first
Rule of thumb: \(Q > 0.3\) indicates meaningful community structure
Limitations of modularity
Modularity has a resolution limit: it cannot detect communities smaller than roughly \(\sqrt{2m}\) edges, and Louvain is a greedy heuristic whose output depends on the random start — always fix the seed and check stability. Modularity is also defined for undirected graphs, so we collapse directions first. For small networks (\(n < 30\), like ours) treat the partition as descriptive, not as inference.
Modularity of a partition \(\{c_i\}\) (Newman), with \(s_i\) the node strengths:
We fix the size (\(n=250\) nodes) and the mean degree (\(\bar{k}\approx 4\)), then change only the wiring rule. Any difference in clustering, path length or degree spread is therefore due to structure alone.
Erdős–Rényi\(G(n,p)\): connect every pair with the same probability — pure chance
Watts–Strogatz: a ring lattice with a few links rewired — a small world
Barabási–Albert: new nodes attach to already-popular ones — grows hubs
import pandas as pddef mean_dist(G): # average over reachable pairs ds = [d for _, tgt in nx.all_pairs_shortest_path_length(G)for _, d in tgt.items() if d >0]returnround(sum(ds) /len(ds), 2)def giant_diameter(G): # diameter of the largest component C =max(nx.connected_components(G), key=len)return nx.diameter(G.subgraph(C))def measure(G): deg = [d for _, d in G.degree()]returndict(edges=G.number_of_edges(), mean_deg=round(np.mean(deg), 2), cluster=round(nx.transitivity(G), 3), mean_dist=mean_dist(G), diameter=giant_diameter(G), assort=round(nx.degree_assortativity_coefficient(G), 3), max_deg=max(deg))tab = pd.DataFrame([measure(cg_er), measure(cg_ws), measure(cg_ba)], index=["ER", "WS", "BA"])print(tab.to_string())
The degree distribution is the clearest fingerprint. On a log-log plot of the tail (complementary CDF, \(P(K \ge k)\)):
ER: a Poisson bump — degrees cluster tightly around the mean, the curve falls off fast
WS: even narrower — almost every node has the same degree
BA: a straight line = a power-law tail — a handful of nodes have enormous degree
Code
# complementary CDF of the degree sequence: P(K >= k)ccdf_df <-function(g, label) { d <-degree(g) ks <-sort(unique(d[d >0])) cc <-numeric(length(ks))for (i inseq_along(ks)) cc[i] <-mean(d >= ks[i])data.frame(k = ks, ccdf = cc, model = label)}df <-rbind(ccdf_df(c_er, "ER"), ccdf_df(c_ws, "WS"), ccdf_df(c_ba, "BA"))ggplot(df, aes(k, ccdf, color = model)) +geom_point(size =2) +geom_line() +scale_x_log10() +scale_y_log10() +scale_color_manual(values =c(ER ="#185FA5", WS ="#1D9E75", BA ="#C0132C")) +labs(x ="degree k (log)", y ="P(K >= k) (log)", color =NULL,title ="Degree distributions: only BA has a heavy tail")
Code
def ccdf(G): d = np.array([x for _, x in G.degree() if x >0]) ks = np.sort(np.unique(d)) cc = np.array([np.mean(d >= k) for k in ks])return ks, cc_ = plt.figure(figsize=(8, 4.6))for G, lab, col in [(cg_er, "ER", "#185FA5"), (cg_ws, "WS", "#1D9E75"), (cg_ba, "BA", "#C0132C")]: ks, cc = ccdf(G) _ = plt.loglog(ks, cc, "o-", color=col, label=lab)_ = plt.xlabel("degree k (log)"); _ = plt.ylabel("P(K >= k) (log)")_ = plt.title("Degree distributions: only BA has a heavy tail")_ = plt.legend(); plt.tight_layout(); plt.show()
What the Numbers Say
Three wiring rules, three very different worlds:
Erdős–Rényi — low clustering, short paths, Poisson degrees, assortativity \(\approx 0\). The null model: what “no structure” looks like.
Watts–Strogatz — high clustering and short paths at the same time: the small-world property of social and collaboration networks.
Barabási–Albert — a fat-tailed degree distribution: a few dominant hubs, the shortest paths of all, and negative assortativity (hubs attach to the periphery).
Why economists care: real trade and production networks are hub-dominated, closer to BA than to ER. That is exactly why idiosyncratic shocks to central sectors/countries do not wash out but drive aggregate volatility — Acemoglu, Carvalho, Ozdaglar & Tahbaz-Salehi (2012, doi:10.3982/ECTA9623). Small-world structure, in turn, governs how fast shocks and information diffuse.
Further reading: Newman, Networks: An Introduction (2010, doi:10.1093/oso/9780198805090.001.0001); Jackson, Social and Economic Networks (2008, Princeton UP, JSTOR). Numbers differ slightly between R (igraph) and Python (networkx) because the two libraries use different random generators — the qualitative fingerprints are identical.
Part IV — Application: The EU Trade Network
Data — Eurostat Table ext_tec03
Annual international trade of EU countries by partner country (Trade by Enterprise Characteristics). We use total exports in thousands of euros, 2012–2024.
You could download it live (as the original lab notebook did):
Dimensions: freq, unit (THS_EUR = thousand €, NR_ENT = number of enterprises), stk_flow (EXP/IMP), nace_r2 (sector), partner, geo (reporter), time, values.
Building the Edge List
We keep: exports, total economy, thousand euros, EU-27 reporters and partners — one directed weighted edge per country pair. Weights are converted to shares of the exporter’s world exports so that small and large countries are comparable.
eu27 <-c("AT","BE","BG","CY","CZ","DE","DK","EE","EL","ES","FI","FR","HR","HU","IE","IT","LT","LU","LV","MT","NL","PL","PT","RO","SE","SI","SK")exp_tot <- trade %>%filter(unit =="THS_EUR", stk_flow =="EXP", nace_r2 =="TOTAL")world <- exp_tot %>%filter(geo %in% eu27, partner =="WORLD") %>%select(geo, time, world = values)edges_all <- exp_tot %>%filter(geo %in% eu27, partner %in% eu27, geo != partner, !is.na(values)) %>%inner_join(world, by =c("geo", "time")) %>%mutate(share =round(100* values / world, 2)) %>%select(time, from = geo, to = partner, value = values, share)write.csv(edges_all, "../data/trade_edges.csv", row.names =FALSE)head(edges_all)
rows: 8862 years: 2012-2024
time from to value share
1 2012 BE AT 3209881 0.94
2 2013 BE AT 3363683 0.97
3 2014 BE AT 3607532 1.04
4 2015 BE AT 3549116 1.01
5 2016 BE AT 3556102 1.01
6 2017 BE AT 3814261 1.03
With positive trade between almost every pair, the raw network is nearly complete — every binary statistic is trivially 1. Thresholding (De Benedictis et al., 2014 call this “binarization”) keeps only flows that constitute real dependence. The cutoff is a researcher choice: too low and the network is complete, too high and it fragments. The Variations section shows how results move with the threshold — always report this sensitivity. Weighted statistics (strength, weighted PageRank) can also be computed with no threshold at all; use both.
Visualizing the EU Trade Network
Code
V(g_eu)$instr <-strength(g_eu, mode ="in", weights =E(g_eu)$share)set.seed(14159)ggraph(g_eu, layout ="stress") +geom_edge_fan(aes(width = share), alpha =0.35, color ="#185FA5",arrow =arrow(length =unit(2.5, "mm"), type ="closed"),end_cap =circle(4, "mm")) +geom_node_point(aes(size = instr), color ="#D85A30") +geom_node_text(aes(label = name), repel =TRUE, fontface ="bold",size =4.5) +scale_edge_width(range =c(0.2, 2.2), name ="share (%)") +scale_size(range =c(3, 16), name ="in-strength") +labs(title ="EU trade network 2023 (edges: >5% of exporter's world exports)") +theme_void()
Code
instr =dict(G_eu.in_degree(weight="share"))pos = nx.spring_layout(G_eu, seed=14159, k=1.2)_ = plt.figure(figsize=(11, 7))_ = nx.draw_networkx_nodes(G_eu, pos, node_size=[60+14* instr[v] for v in G_eu], node_color="#D85A30")_ = nx.draw_networkx_edges(G_eu, pos, alpha=0.3, edge_color="#185FA5", width=[0.2+ G_eu[u][v]["share"] /8for u, v in G_eu.edges()], arrowsize=10, connectionstyle="arc3,rad=0.1")_ = nx.draw_networkx_labels(G_eu, pos, font_size=9, font_weight="bold")_ = plt.title("EU trade network 2023 (edges: >5% of exporter's world exports)")_ = plt.axis("off")plt.tight_layout()plt.show()
country in_deg in_str betw pagerank
DE 25 412.4 157 0.281
FR 9 96.6 54 0.156
NL 8 61.0 25 0.095
PL 4 30.8 79 0.065
IT 10 83.9 15 0.065
AT 3 16.2 0 0.057
BE 4 41.0 5 0.055
ES 3 38.9 26 0.049
CZ 2 19.7 56 0.016
PT 1 8.5 0 0.014
Code
for u, v in G_eu.edges(): G_eu[u][v]["dist"] =1/ G_eu[u][v]["share"]cen = pd.DataFrame({"in_deg": dict(G_eu.in_degree()),"in_str": dict(G_eu.in_degree(weight="share")),"betw": nx.betweenness_centrality(G_eu, weight="dist"),"pagerank": nx.pagerank(G_eu, weight="share")})print(cen.sort_values("pagerank", ascending=False).round(3).head(10))
Germany is the hub on every measure — 25 of 26 partners send it more than 5% of their exports. France, the Netherlands, Italy and Poland form a second tier; Poland’s high betweenness reflects its gateway role for the Baltics and Central Europe.
Geography emerges from trade flows alone: a Central-European bloc anchored on Germany, a Benelux group, a Mediterranean group around France–Italy–Spain, and a Nordic–Baltic group — no coordinates were used.
The Network over Time, 2012–2024
Did the EU trade network become denser and more reciprocal?
Code
years <-sort(unique(edges_all$time))dens <-numeric(length(years))rec <-numeric(length(years))for (i inseq_along(years)) { ey <- edges_all %>%filter(time == years[i], share >5) gy <-graph_from_data_frame(ey %>%select(from, to), directed =TRUE) dens[i] <-edge_density(gy) rec[i] <-reciprocity(gy)}ts <-data.frame(year = years, density = dens, reciprocity = rec)ggplot(ts, aes(x = year)) +geom_line(aes(y = density, color ="density"), linewidth =1.2) +geom_line(aes(y = reciprocity, color ="reciprocity"), linewidth =1.2) +geom_point(aes(y = density, color ="density"), size =3) +geom_point(aes(y = reciprocity, color ="reciprocity"), size =3) +scale_color_manual(values =c(density ="#185FA5",reciprocity ="#D85A30"), name =NULL) +scale_x_continuous(breaks =seq(2012, 2024, 2)) +labs(y =NULL, title ="EU trade network structure over time",subtitle ="Edges: >5% of exporter's world exports")
Gravity explains flow sizes; networks explain the architecture of who-trades-with-whom. Head & Mayer (2014), Handbook of International Economics 4. doi:10.1016/B978-0-444-54314-1.00003-3
Chaney (2014): firms find new markets through existing contacts — trade networks grow by “friend-of-a-friend” dynamics, generating the observed geographic diffusion of exports. doi:10.1257/aer.104.11.3600
Production networks: sectoral shocks travel along input-output edges; asymmetric networks (hubs!) turn micro shocks into aggregate volatility — Acemoglu et al. (2012). doi:10.3982/ECTA9623
Global value chains: upstreamness/downstreamness are path-based network measures on the world input-output matrix — Antràs & Chor (2022), Handbook of International Economics 5. doi:10.1016/bs.hesint.2022.02.005
Supply-chain resilience: Covid and the Suez blockage renewed interest in centrality-as-vulnerability — Baldwin & Freeman (2022), Annual Review of Economics 14, 153–180. doi:10.1146/annurev-economics-051420-113737
Reading strategy: use network statistics as measurement (dependent or explanatory variables), and gravity/GE trade models as theory — the strongest papers do both.
Under \(G(n,p)\) with \(\hat{p} = L/[n(n-1)]\), each edge’s reverse exists independently with probability \(\hat{p}\), so the null expectation of reciprocity is \(\hat{p}\) itself
A raw reciprocity of \(0.46\) means nothing by itself — in a dense network even random edges reciprocate often
The test asks whether mutual trade dependence is structural
Test statistic, with \(\hat{r} = L^{\leftrightarrow}/L\):
\[
z = \frac{\hat{r} - \hat{p}}{\sqrt{\hat{p}(1-\hat{p}) / L}}
\]
Do trade hubs follow a scale-free law\(P(k) \sim k^{-\alpha}\)? The CSN procedure (Clauset, Shalizi & Newman, 2009, doi:10.1137/070710111):
“Scale-free” claims are routinely made from eyeballing log-log plots; CSN showed most fail a formal test
Fit the tail by maximum likelihood, then judge the fit with a Kolmogorov–Smirnov distance and a bootstrap p-value
Economics alternative: the log rank-size regression with the Gabaix–Ibragimov \((\text{rank}-\tfrac{1}{2})\) correction (doi:10.1198/jbes.2009.06157) — Zipf’s law econometrics
Maximum likelihood exponent (for \(k \geq k_{\min}\)):
verdict: power law NOT rejected (but see the warning below!)
Code
deg_in = np.array([d for _, d in G_eu.in_degree() if d >0])kmin =2tail = deg_in[deg_in >= kmin]n_t =len(tail)alpha =1+ n_t / np.sum(np.log(tail / (kmin -0.5)))# KS distance against the fitted continuous power lawxs = np.sort(tail)F_emp = np.arange(1, n_t +1) / n_tF_fit =1- (xs / kmin) ** (1- alpha)D = np.max(np.abs(F_emp - F_fit))D_crit =1.36/ np.sqrt(n_t) # approximate 5% KS critical valueprint("alpha (MLE) :", round(alpha, 2), " (k_min =", kmin, ")")print("KS statistic :", round(D, 3))print("5% critical :", round(D_crit, 3))print("criterion : reject power law if D > critical value")print("verdict:","REJECT the power law"if D > D_critelse"power law NOT rejected (small n -- weak evidence!)")
Power laws need large samples
With 27 nodes the KS test has almost no power: failing to reject means very little. Clauset, Shalizi & Newman recommend \(n_{\text{tail}} \gtrsim 50\), and even then a lognormal often fits equally well. We include the test here to teach the method — for serious scale-free claims use the world trade network (150+ countries, e.g. the BACI-CEPII data) or firm-level transaction networks.
Part VI — Variations
Variation — Threshold Sensitivity
Every binarized result should survive a threshold change. Recompute the key statistics at 2%, 5% and 10%:
Imports instead of exports (stk_flow == "IMP"): does the import network have the same hubs? Correlate in-strength across the two
Maximum spanning tree backbone: keep the \(n-1\) strongest edges that connect everyone — the skeleton of EU trade
Add the neighbours: geo also covers CH, NO, UK, TR — how does Brexit-era UK sit in the network over 2012–2024?
Enterprise counts (unit == "NR_ENT"): a network of how many firms trade, not how much — extensive vs intensive margin
Multilayer view: treat each NACE sector as a layer; compare centrality rankings across layers
Rolling communities: run Louvain year by year — are blocs stable over 2012–2024?
Weighted throughout: redo the whole analysis with no threshold using strength and weighted PageRank only
Exercises — Estimation
Rebuild the 2023 network with a 2% threshold and recompute in-degree, in-strength and PageRank. Which countries change rank the most, and why?
Construct the import network (stk_flow == "IMP") for 2023 and compare its top-5 in-strength countries with the export network’s.
Compute out-degree over time for Greece (EL) 2012–2024. Does its export diversification inside the EU rise or fall?
Compute weighted betweenness using \(1/w_{ij}\) distances for all 27 countries in 2019 and 2023. Which country’s intermediation role changed most?
Build the network on enterprise counts (unit == "NR_ENT", shares of the country’s world total) and correlate its in-strength ranking with the euro-value ranking (Spearman).
Estimate the Katz–Bonacich vector\(b = (I - \phi A)^{-1}\mathbf{1}\) on the binarized 2023 network for \(\phi = 0.5/\lambda_{\max}\) and \(\phi = 0.9/\lambda_{\max}\), using solve() in R or np.linalg.solve in Python. How sensitive is the ranking to \(\phi\)?
Extract the maximum spanning tree of the collapsed weighted network (mst() on negative weights in igraph) and plot it. Which edges form the backbone?
Run Louvain for every year 2012–2024 with a fixed seed. Report the modularity time series and the years in which bloc membership changes.
Exercises — Testing
Repeat the reciprocity z-test at thresholds 2%, 5% and 10%. Is the rejection robust?
Repeat the clustering Monte Carlo test with \(B = 5000\) replications. Does the p-value change materially from \(B = 500\)?
Design a Monte Carlo test for assortativity: is the EU network’s degree assortativity significantly negative compared to \(G(n, \hat{p})\)?
Apply the CSN power-law test to the out-strength distribution (a continuous variable) instead of in-degree. Compare \(\hat{\alpha}\) and the KS p-value.
Run the Gabaix–Ibragimov rank-size regression in all three languages for 2013 and 2023, and test whether \(\hat{\alpha}\) changed between the two years.
Simulate 500 directed ER graphs with the same \(n\) and \(L\) and compare the observed maximum in-degree (Germany’s 25) with the simulated distribution. What is the Monte Carlo p-value of such a hub arising by chance?
Use a degree-preserving rewiring null (rewire(g, keeping_degseq(niter = 1000)) in igraph) instead of ER for the clustering test. Does the conclusion survive when hubs are kept fixed?
Test whether the Louvain partition beats chance: permute country labels 500 times, recompute modularity of the permuted partition, and compare with \(Q_{\text{obs}}\).
Further Reading
Jackson (2008), Social and Economic Networks — chapters 2–3 map directly onto Part I–II of this lecture. doi:10.2307/j.ctvcm4gh1
Kolaczyk & Csárdi (2014), Statistical Analysis of Network Data with R — the igraph companion. doi:10.1007/978-1-4939-0983-4
De Benedictis, Nenci, Santoni, Tajoli & Vicarelli (2014) — a full worked tour of world trade network statistics; our application follows its spirit on EU data. doi:10.1515/gej-2014-0032
Clauset, Shalizi & Newman (2009) — before you ever claim “scale-free”. doi:10.1137/070710111
