Motivation for the Measure of Entropy (adapted from Shannon, 1948 and Cover and Thomas, 1991) In this technical appendix we adapt measures of entropy to characterize the Shannon’s uncertainty of potential know-how flow and subgroups. We then standardize the to from measures for comparison across schools. Formally, let represent the number of possible links along which information could n flow. The larger the number of links, the greater the uncertainty. If the links have equal probability of occurring, then uncertainty is proportional to 1/(probability of any given link). To create an additive measure of uncertainty across multiple independent links, for each link take the natural log of 1/probability, and then sum. Thus the combined uncertainty of links from A to B and from B to C is + + Ln[1/p(A,B)] Ln[1/p(B,C)]= Ln[p(A,B)-1] Ln[p(B,C)-1]= −Ln[p(A,B)] Then, because each event is not equally likely to occur, weight the events by their −Ln[p(B,C)]. probability. This generates an expected value, or p(A,B)Ln[p(A,B)] average: − − Summing across all pairs, entropy=, as in the main text. p(B,C)Ln[p(B,C)]. Deconstructing Resource Flow to Potential Receivers and from Potential Providers The flow of resources across the network can be differentiated into a component based on the aggregated flows each potential receiver and each potential provider. Formally, let to from , and (A1) . (A2)
Motivation for the Measure of Entropy (adapted from Shannon, 1948 and Cover and Thomas, 1991) In this technical appendix we adapt Shannon’s measures of entropy to characterize the uncertainty of potential know-how flow to and from subgroups. We then standardize the measures for comparison across schools. Formally, let n represent the number of possible links along which information could flow. The larger the number of links, the greater the uncertainty. If the links have equal probability of occurring, then uncertainty is proportional to 1/(probability of any given link). To create an additive measure of uncertainty across multiple independent links, for each link take the natural log of 1/probability, and then sum. Thus the combined uncertainty of links from A to B and from B to C is Ln[1/p(A,B)] + Ln[1/p(B,C)]= Ln[p(A,B)-1] + Ln[p(B,C)-1]= −Ln[p(A,B)] −Ln[p(B,C)]. Then, because each event is not equally likely to occur, weight the events by their probability. This generates an expected value, or average: − p(A,B)Ln[p(A,B)] − p(B,C)Ln[p(B,C)]. Summing across all pairs, entropy=, as in the main text. Deconstructing Resource Flow to Potential Receivers and from Potential Providers The flow of resources across the network can be differentiated into a component based on the aggregated flows to each potential receiver and from each potential provider. Formally, let , and (A1) . (A2)
Frank%20-%20What%20is%20a%20Good%20Social%20Network%20for%20Policy%20Implementation%20(MS%206437)%20acceptance%20version.docx