Appendix Figure 1: This figure illustrates a hypothetical friendship network. In this figure, each circle represents a student and each directional arrow indicates a friendship nomination from one student to another. Dark grey circles indicate students who are in Friendship Group #1 and light grey circles indicate students in Friendship Group #2. Patterned lines indicate intervention participants. The SNAmeasures for this network are listedin the first column of Appendix Tables 1 and 2.
Appendix Table 1
Description and Formulas of the Social Integration Measures of Diffusion Potential
Measure / Formula / Hypothetical NetworkConnectedness
Structural Cohesion / The average number of “node-independent paths” connecting each pair of students in the network (see Moody & White, 2003). Ties were treated as undirected (i.e., a tie existed between two students as long as one student named the other student as a friend) and students with no friendship tieswere excluded from the calculation / Excluding students with no friendship ties (L and M), all students except F are connected to every other student through two node-independent paths. F is connected to every other student through one node-independent path. The structural cohesion of this network is 1.86.
Social Distance / Meannumber of nominations needed to get from one student to every other student (i.e., mean geodesic distance; See Wasserman & Faust, 1994). Ties were treated as directed.Pairs of students who were not connected were excluded from the calculation / e.g., Geodesic distance from D to J = 6 (D → E → A → C → K → H→ J)
Average social distance for the network = 2.08
Clustering
Segregation Index /
Where Expectedoutgrpis calculated from the marginals of a matrix that compares the # of ties within and between each pair of groups in the network (See Freeman, 1978) / Group / 0 / 1 / 2 / Totals
0 / 0 / 0 / 0 / 0
1 / 0 / 15 / 1 / 16
2 / 0 / 2 / 7 / 9
Totals / 0 / 17 / 8 / 25
Expected out-of-group nominations = (9*17)/25 + (16*8)/25 = 11.2
Observed out-of-group nominations = 3
Segregation Index = 0.73
Transitivity Ratio /
where transitive triplesare those triples in which when there is a tie from person i to person j and also a tie from person j to person h, then there is also a tie from person i to person h(see Wasserman & Faust, 1994) / e.g., Transitive Triple: A → B→ E; A → E; Intransitive Triple: A → C → K (because there is no tie between A and K)
Overall, there are 15 transitive triples and 26 intransitive triples. Therefore, the transitivity ratio for this network is 15/(15+26) = 0.37
Hierarchy
Indegree Centralization /
Where max(Cx) = maximum observed value in the network for indegreecentrality, Ci= indegreecentrality for student i, and the denominator is the largest sum of differences possible (See Freeman, 1979). Indegree centrality is defined as the number of nominations that student i received from other students in the network / E has an indegree of 6 (This student was named by A, B, C, D, G, and H)
Indegree centralization for the network = 0.37
Betweenness Centralization /
Where max(Cx) = maximum observed value in the network for betweennesscentrality, Ci= betweennesscentrality for student i, and the denominator is the largest sum of differences possible (See Freeman, 1979). Betweenness centrality is defined as:
Where gj(i)k is the shortest path between students j and k that go through student iandgjkis the total number of shortest paths between j and k / C has the highest betweenness centrality (CB = 32.5)
G’s betweenness centrality = 0, because this student doesn’t lie on the shortest path between any 2 students
Betweenness centralization for the network = 0.19
Appendix Table 2
Description and Formulas of the Location of Intervention Participant Measures of Diffusion Potential
Measure / Formula / Hypothetical NetworkDistribution of Participants in the Network
Proportion of Groups with 1+ SFP10-14Participant / / Group 1 (dark grey circles) has two SFP10-14 participants (students A and E), but Group 1 (light grey circles) does not have any SFP10-14 participants: 1/2 = 0.50
Proportion of Non-participants within 2 Steps of an SFP10-14 Participant / / Eight non-participants can reach an SFP10-14 participant within 2 steps and two non-participants (IL) cannot
(8 non-participants within 2 steps) / (10 non-participants) = 0.80
Participants’ Relative Status
Indegreea /
Where MCent.Participantsis the mean indegree centrality across all participants and MCent.Non-Participants is the mean indegree centrality across all non-participants. Indegree centrality is defined as the number of nominations that student i received from other students in the network / If students A (indegree = 2), E (indegree = 6), and M (indegree = 0) participated in SFP10-14, then the mean indegree among SFP 10-14 participants = 2.67 (SD = 3.06).
The mean indegree among non-participants = 1.7 (SD = 1.34).
The pooled SD = 1.78.
Therefore, Cohen’s Dfor indegree centrality = (2.67 – 1.7) / 1.78 = 0.54. Thus, the mean indegree centrality of the participants is approximately ½ SD greater than the mean centrality of non-participants.
Betweennessb /
Where MCent.Participantsis the mean betweenness centrality across all participants and MCent.Non-Participants is the mean betweenness centrality across all non-participants. Betweenness centrality is defined as:
Where gj(i)k is the shortest path between students j and k that go through student iandgjkis the total number of shortest paths between j and k / MBetweennessParticipants=13.83, (SD = 12.00)
MBetwennessNonParticipants= 7.45 (SD = 11.69)
The pooled SD = 11.74
Cohen’s D= (13.83 – 7.45) / 11.74 = 0.54
aTo account for skew in indegree centrality, we transformed this measure by taking the square root of indegree centralitybefore calculating Cohen’s D. We do not show that transformation as part of the example to simplify presentation of the calculations. bTo account for skew in betweenness centrality, we transformed this measure by taking the cube root of betweenness centrality before calculating Cohen’s D. Because the range of betweenness centrality also was much smaller in larger networks, we multiplied this measure by network size divided by 150 (mean network size).We do not show that transformation as part of the example to simplify presentation of the calculations.