JM gratefully acknowledge financial support from NSF Grant SBE0738187. We thank the International Institute for Applied Systems Analysis (IIASA) and the Young Sci-entist Summer Program (YSSP) where this research be-gan, with financial support from The National Academies.
We thank the Santa Fe Institute for support during the continuation of this research. We thank Aaron Clauset, Ben Good, and Doyne Farmer for several helpful conver-sations and suggestions.
Appendix A. Hierarchical clustering with the over-lap distance
Define the overlap distance between nodes as
dij = 1−pij, (A.1) wherepij is the probability that nodesiandjare grouped in the same community. Since pij is the probability that iandj are cogrouped,dij is simply the probability thati andj are not cogrouped. To determine distances between clusters of nodes, let
dAB= 1
|A||B| X
i∈A,j∈B
dij (A.2)
= 1− 1
|A||B| X
i∈A,j∈B
pij (A.3)
= 1−pAB, (A.4)
where pAB ≡ |A||B|1
P
i∈A,j∈Bpij is the probability that a randomly picked pair from clustersAandBare cogrouped.
This choice of dAB is known as the “average linkage cri-terion”. In the present context it enables a simple inter-pretation of both node and cluster distances in terms of probabilities.
The elementspij of the coclassification matrix cannot take on arbitrary values; the laws of probability impose interdependent constraints on matrix elements. Given the cogrouping probabilities pik andpjk ofiandj with some third node k, one can show that pij is bound above and below as
max(0, pik+pjk−1)≤pij ≤1− |pik−pjk|. (A.5)
We can use these bounds to show two useful properties of the overlap distance. First, using the lower bound, one can show that
dij≤dik+djk; (A.6) i.e. the overlap distance obeys triangle inequality.
Second, using the upper bound one may show that the overlap distance is equal to the Chebychev orL∞distance applied to columns of the coclassification matrix:
dij = max
k |pik−pjk|. (A.7) TheL∞distance is the largest absolute difference between elements of columnsi andj. Rewriting the upper bound as |pik−pij| ≤1−pij, we see that maxk|pik−pjk| is at most 1−pij. To see that they are in fact equal, letk=i and note that|pik−pjk|=|pii−pji|= 1−pij. Since the argument of maxk achieves the largest possible value for at least one value ofk, dij = maxk|pik−pjk|= 1−pij. Appendix B. Industry flow statistics
11
TableB.2:Industrystatistics.Allvaluesareaveragesacrosscountries.Thethruflowcolumngivestheaveragesizeofeachindustryiastheaverage normalizedthroughflow¯ti=P ctc i/P c1.(SeeEq.(11).)Thesumoverallindustriessumsto100%,withinasmallerrorduetorounding.Theexport,final, andintermediaterevenuescolumnsgivethepercentageofanindustry’smoneyin-flowsreceivedfromeachcategory.Thethreecolumnssumto100%for eachindustry.Theimports,valueadded,andintermediateexpenditurecolumnsgivethepercentageofanindustry’smoneyout-flowsgoingtoeachcategory. Thesethreecolumnsalsosumto100%foreachindustry.Thefinalcolumngivestheself-flowaiiasapercentageofthetotalthroughflowtigoingthroughi. Moneyin-flowsMoneyout-flows iIndustry%Thruflow%Exports%Final%Int.rev.%Imports%Val.add.%Int.exp.%Self 1Foodproducts,beverages,andtobacco5.8416.850.632.69.525.065.514.0 2Agriculture,hunting,forestry,andfishing4.3010.922.466.76.150.643.313.1 3Hotelsandrestaurants2.887.369.922.84.747.947.41.1 4Chemicalsexcludingpharmaceuticals2.3534.812.552.820.031.348.712.5 5Rubberandplasticsproducts1.1225.38.965.820.435.044.66.9 6Textiles,textileproducts,leather,andfootwear2.2532.734.532.818.733.447.918.6 7ManufacturingNEC,recycling1.0723.848.327.912.839.747.53.6 8Woodandproductsofwoodandcork0.7918.49.072.613.533.752.814.6 9Construction7.681.376.522.27.539.952.67.8 10Othernon-metallicmineralproducts1.3216.57.176.49.639.351.19.7 11Fabricatedmetalproducts,exceptmachin.andequip.1.8718.215.066.812.638.349.29.1 12Iron&steel1.8233.00.566.519.325.854.919.2 13Machineryandequipment,NEC2.5733.332.634.215.537.846.77.9 14Electricalmachineryandapparatus,NEC1.2936.816.147.118.835.146.07.5 15Non-ferrousmetals0.4836.62.361.124.724.750.617.0 16Buildingandrepairingofshipsandboats0.4238.634.027.416.035.348.75.6 17Motorvehicles,trailers,andsemi-trailers2.3738.136.525.424.925.149.914.1 18Railroadequipmentandtransportequipment,NEC0.2528.938.532.519.733.047.47.1 19Radio,television,andcommunicationequipment0.9844.829.825.525.731.243.19.6 20Medical,precision,andopticalinstruments0.4536.934.129.016.940.342.84.2 21Office,accountingandcomputingmachinery0.5654.327.418.335.826.138.13.8 22Aircraftandspacecraft0.2347.416.935.627.735.137.26.6 23Wholesaleandretailtrade,repairs10.267.756.236.14.057.039.14.0 24Finance,insurance4.254.824.171.13.360.136.612.9 25Postandtelecommunications1.725.130.564.54.266.729.14.6 26Otherbusinessactivities4.348.014.477.74.355.640.19.1 27Computerandrelatedactivities0.806.131.562.46.054.439.55.4 28Othercommunity,social,andpersonalservices2.893.859.336.95.153.041.97.7 29Education2.540.693.16.21.975.722.40.8 30Pulp,paper,paperproducts,printing,andpublishing2.5415.316.768.013.536.949.620.4 31Realestateactivities6.030.273.825.91.174.924.03.4 32Publicadmin.anddefense;compulsorysocialsec.5.190.593.46.04.364.231.51.3 33Rentingofmachineryandequipment0.443.319.876.93.056.640.46.2 34Transportandstorage4.8421.226.951.99.348.941.812.5 35Healthandsocialwork3.970.487.911.64.662.632.84.0 36Pharmaceuticals0.4825.438.036.615.337.846.96.6 37Researchanddevelopment0.378.035.256.86.156.637.33.7 38Miningandquarrying1.8421.45.173.57.055.437.64.7 39Coke,refinedpetroleumproducts,andnuclearfuel1.6420.922.956.134.320.045.76.0 40Electricity,gas,andwatersupply2.681.632.765.78.050.441.612.2 12
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Table B.3: Comparison of Weibull and lognormal fits to flow weight distribution.
Country Weibull Lognormal ∆log-likelihood Best fit
λ k m s
Australia 8.81×10−5 0.408 -10.7 2.96 67.2 Weibull
Brazil 1.88×10−4 0.485 -9.71 2.44 36.7 Weibull
Canada 1.21×10−4 0.483 -10.1 2.34 2.99 Weibull
China 1.58×10−4 0.471 -9.9 2.44 21.9 Weibull
CzechRepublic 1.14×10−4 0.471 -10.2 2.49 48.4 Weibull
Denmark 6.83×10−5 0.433 -10.8 2.57 -15.3 lognormal
Finland 1.31×10−4 0.489 -9.99 2.12 -71.3 lognormal
France 1.49×10−4 0.518 -9.86 2.20 12.3 Weibull
Germany 1.44×10−4 0.514 -9.87 2.12 -30.1 lognormal
Greece 4.14×10−5 0.343 -11.7 3.45 35 Weibull
Hungary 1.34×10−4 0.541 -9.87 1.94 -63.4 lognormal
Italy 1.29×10−4 0.461 -10.2 2.56 37 Weibull
Japan 1.14×10−4 0.423 -10.5 3.07 124 Weibull
Korea 1.07×10−4 0.442 -10.4 2.58 11.8 Weibull
Netherlands 1.05×10−4 0.494 -10.2 2.00 -120 lognormal
Norway 7.70×10−5 0.463 -10.6 2.32 -53.2 lognormal
Poland 1.50×10−4 0.490 -9.88 2.23 -24.2 lognormal
Spain 9.93×10−5 0.469 -10.3 2.33 -25.2 lognormal
UnitedKingdom 8.31×10−5 0.439 -10.6 2.63 29.4 Weibull UnitedStates 9.23×10−5 0.440 -10.5 2.52 -26.2 lognormal
pooled 1.08×10−4 0.456 -10.3 2.54 545.6 Weibull
14