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Autocorrelation in Social Networks: A Preliminary Investigation of Sampling Issues Antonio Páez Darren M. Scott Erik Volz Sunbelt XXVI – International Network for Social Network Analysis Network Autocorrelation Analysis
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Spatial analysis ◘Central tenet: First Law of Geography “Everything is related to everything else, but near things are more related than distant things” (Tobler, 1970) ◘Spatial analysis (Miller, 2004)
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Spatial statistical models ◘Statistical representation of this principle ○Spatially autoregressive model Y= X + WY +WY + Spatial spillovers Economic externalities … (e.g. Fingleton, 2003;2004)
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Connectivity matrix W ◘Key element of the model ○Defines the spatial structure of the study area ○Position relative to other units
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First Law: General principle ◘Distance in social space “Everyone is related to everyone else, but near people are more related than distant people” ◘Akerlof’s social distance (1997) “Agents who are initially close interact more strongly while those who are socially distant have little interaction”
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Social network analysis ◘Network models Y= X + WY +WY + Social influence (e.g. Leenders, 2002; Marsden and Friedkin, 1994)
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Geo-referencing ◘Nature of connectivity is relatively unambiguous even if definition of weights is not
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Social referencing ◘Identification of network connections
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Specification of W ◘Research questions Within a linear autoregressive framework: ○What is the effect of under-specifying matrix W… ? (how much effort should go into trying to observe/identify network connections?) ○What is the effect of different network topologies…? On quality of estimators, model identification (Previous work by Stetzer, 1982; Griffith, 1996)
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Experimental setup ◘Assumptions ○Closed system (interactions with the rest of the world are negligible) ○All individuals are observed, their attributes can be obtained ○Not all network connections are identified »Deliberate effort to minimize observation cost: select individuals and identify all their connections
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Experimental setup ◘Simulate networks with different topologies (Matrix W ) ○Poisson distribution / exponential distribution ○Degree distribution: 1.5, 3.5, 5.5, 7.5 ○Clustering: 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 Random networks with tunable degree distribution and clustering – Volz, 2004
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Experimental setup ◘Simulate data ○ : 0.1, 0.3, 0.5, 0.7, 0.9 ○ 1 =2.0; 2 =1.0 ○ X 1 : const; X 2 : uniform (1,10) (see Anselin and Florax, 1995) ○ : standard normal ○ n =100 (number of observations) Y = WY +X 1 1 +X 2 2 +
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Experimental setup ◘Randomly sample from connectivity matrix W (e.g. 95% of individual connections) ○ s : 0.95, 0.90, 0.85, 0.80, 0.75, 0.70, 0.65, 0.60, 0.55, 0.50 ◘Estimate coefficients ○1,000 replications @ each level of sampling ◘Calculate mse: bias – variance ◘Model identification: likelihood ratio test
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Results Degree Distribution ( d ) Clustering ( c ) 1.53.57.55.5 0.2 0.3 0.4 0.5 0.6 0.7 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.2 0.3 0.4 0.5 0.6 0.7 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.2 0.3 0.4 0.5 0.6 0.7 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.2 0.3 0.4 0.5 0.6 0.7 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 = 0.10.1 0.30.3 0.50.5 0.7 0.90.70.9 =
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Summary and conclusions ◘Specification of connectivity matrix in social network settings ◘Resources available for observing network connections – sampling strategies ◘Simulation experiment using networks with controlled topologies: quality of estimators, power of identification tests
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Summary and conclusions ◘Main control is degree of network autocorrelation ◘Clustering: relatively small effect
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Summary and conclusions ◘Weak network autocorrelation ( = 0.1 ~ 0.3) ○Effect of under-specification on coefficients is relatively small ○Tests may fail to identify the effect ◘Moderate network autocorrelation ( = 0.5) ○Effect on coefficients becomes noticeable @ s~0.75, and this effect is sharper with increasing degree distribution ○Tests correctly reject null hypothesis of no autocorrelation 90% of time @ p=0.05
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Summary and conclusions ◘Strong network autocorrelation ( = 0.7 ~ 0.9) ○Quality of estimators deteriorates very rapidly, even @ s~0.90 ○Tests lose power at higher degree distributions ◘Further research ○Alternative sampling schemes (e.g. snowball, referral) ○Over-specification of connectivity matrix W ○“Seeding” matrix W
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d = 1.5; c = 0.2; = 0.1 d c
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d = 1.5; c = 0.2; = 0.3 d c
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d = 1.5; c = 0.2; = 0.5 d c
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d = 1.5; c = 0.2; = 0.7 d c
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d = 1.5; c = 0.2; = 0.9 d c
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d = 1.5; c = 0.3; = 0.1 d c
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d = 1.5; c = 0.3; = 0.3 d c
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d = 1.5; c = 0.3; = 0.5 d c
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d = 1.5; c = 0.3; = 0.7 d c
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d = 1.5; c = 0.3; = 0.9 d c
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d = 1.5; c = 0.4; = 0.1 d c
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d = 1.5; c = 0.4; = 0.3 d c
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d = 1.5; c = 0.4; = 0.5 d c
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d = 1.5; c = 0.4; = 0.7 d c
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d = 1.5; c = 0.4; = 0.9 d c
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d = 1.5; c = 0.5; = 0.1 d c
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d = 1.5; c = 0.5; = 0.3 d c
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d = 1.5; c = 0.5; = 0.5 d c
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d = 1.5; c = 0.5; = 0.7 d c
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d = 1.5; c = 0.5; = 0.9 d c
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d = 1.5; c = 0.6; = 0.1 d c
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d = 1.5; c = 0.6; = 0.3 d c
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d = 1.5; c = 0.6; = 0.5 d c
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d = 1.5; c = 0.6; = 0.7 d c
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d = 1.5; c = 0.6; = 0.9 d c
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d = 1.5; c = 0.7; = 0.1 d c
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d = 1.5; c = 0.7; = 0.3 d c
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d = 1.5; c = 0.7; = 0.5 d c
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d = 1.5; c = 0.7; = 0.7 d c
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d = 1.5; c = 0.7; = 0.9 d c
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d = 3.5; c = 0.2; = 0.1 d c
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d = 3.5; c = 0.2; = 0.3 d c
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d = 3.5; c = 0.2; = 0.5 d c
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d = 3.5; c = 0.2; = 0.7 d c
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d = 3.5; c = 0.2; = 0.9 d c
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d = 3.5; c = 0.3; = 0.1 d c
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d = 3.5; c = 0.3; = 0.3 d c
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d = 3.5; c = 0.3; = 0.5 d c
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d = 3.5; c = 0.3; = 0.7 d c
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d = 3.5; c = 0.3; = 0.9 d c
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d = 3.5; c = 0.4; = 0.1 d c
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d = 3.5; c = 0.4; = 0.3 d c
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d = 3.5; c = 0.4; = 0.5 d c
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d = 3.5; c = 0.4; = 0.7 d c
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d = 3.5; c = 0.4; = 0.9 d c
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d = 3.5; c = 0.5; = 0.1 d c
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d = 3.5; c = 0.5; = 0.3 d c
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d = 3.5; c = 0.5; = 0.5 d c
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d = 3.5; c = 0.5; = 0.7 d c
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d = 3.5; c = 0.5; = 0.9 d c
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d = 3.5; c = 0.6; = 0.1 d c
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d = 3.5; c = 0.6; = 0.3 d c
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d = 3.5; c = 0.6; = 0.5 d c
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d = 3.5; c = 0.6; = 0.7 d c
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d = 3.5; c = 0.6; = 0.9 d c
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d = 3.5; c = 0.7; = 0.1 d c
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d = 3.5; c = 0.7; = 0.3 d c
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d = 3.5; c = 0.7; = 0.5 d c
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d = 3.5; c = 0.7; = 0.7 d c
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d = 3.5; c = 0.7; = 0.9 d c
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d = 5.5; c = 0.2; = 0.1 d c
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d = 5.5; c = 0.2; = 0.3 d c
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d = 5.5; c = 0.2; = 0.5 d c
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d = 5.5; c = 0.2; = 0.7 d c
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d = 5.5; c = 0.2; = 0.9 d c
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d = 5.5; c = 0.3; = 0.1 d c
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d = 5.5; c = 0.3; = 0.3 d c
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d = 5.5; c = 0.3; = 0.5 d c
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d = 5.5; c = 0.3; = 0.7 d c
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d = 5.5; c = 0.3; = 0.9 d c
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d = 5.5; c = 0.4; = 0.1 d c
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d = 5.5; c = 0.4; = 0.3 d c
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d = 5.5; c = 0.4; = 0.5 d c
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d = 5.5; c = 0.4; = 0.7 d c
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d = 5.5; c = 0.4; = 0.9 d c
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d = 5.5; c = 0.5; = 0.1 d c
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d = 5.5; c = 0.5; = 0.3 d c
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d = 5.5; c = 0.5; = 0.5 d c
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d = 5.5; c = 0.5; = 0.7 d c
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d = 5.5; c = 0.5; = 0.9 d c
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d = 5.5; c = 0.6; = 0.1 d c
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d = 5.5; c = 0.6; = 0.3 d c
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d = 5.5; c = 0.6; = 0.5 d c
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d = 5.5; c = 0.6; = 0.7 d c
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d = 5.5; c = 0.6; = 0.9 d c
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d = 5.5; c = 0.7; = 0.1 d c
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d = 5.5; c = 0.7; = 0.3 d c
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d = 5.5; c = 0.7; = 0.5 d c
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d = 5.5; c = 0.7; = 0.7 d c
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d = 5.5; c = 0.7; = 0.9 d c
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d = 7.5; c = 0.2; = 0.1 d c
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d = 7.5; c = 0.2; = 0.3 d c
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d = 7.5; c = 0.2; = 0.5 d c
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d = 7.5; c = 0.2; = 0.7 d c
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d = 7.5; c = 0.2; = 0.9 d c
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d = 7.5; c = 0.3; = 0.1 d c
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d = 7.5; c = 0.3; = 0.3 d c
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d = 7.5; c = 0.3; = 0.5 d c
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d = 7.5; c = 0.3; = 0.7 d c
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d = 7.5; c = 0.3; = 0.9 d c
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d = 7.5; c = 0.4; = 0.1 d c
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d = 7.5; c = 0.4; = 0.3 d c
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d = 7.5; c = 0.4; = 0.5 d c
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d = 7.5; c = 0.4; = 0.7 d c
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d = 7.5; c = 0.4; = 0.9 d c
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d = 7.5; c = 0.5; = 0.1 d c
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d = 7.5; c = 0.5; = 0.3 d c
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d = 7.5; c = 0.5; = 0.5 d c
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d = 7.5; c = 0.5; = 0.7 d c
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d = 7.5; c = 0.5; = 0.9 d c
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d = 7.5; c = 0.6; = 0.1 d c
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d = 7.5; c = 0.6; = 0.3 d c
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d = 7.5; c = 0.6; = 0.5 d c
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d = 7.5; c = 0.6; = 0.7 d c
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d = 7.5; c = 0.6; = 0.9 d c
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d = 7.5; c = 0.7; = 0.1 d c
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d = 7.5; c = 0.7; = 0.3 d c
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d = 7.5; c = 0.7; = 0.5 d c
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d = 7.5; c = 0.7; = 0.7 d c
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d = 7.5; c = 0.7; = 0.9 d c
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