Figure 1:
This graph has seven cycles (one is Hamiltonian). Four are chordless and three of those are also geodesic (and, in addition, convex). The cycle in red (around the “outside” of the graph) is chordless, but not geodesic.
Figure 2:
Patricia’s 1990 network. Visualization created using the network R package (Butts, 2008, 2015).
Figure 3:
Patricia’s 1992 network. Nodes marked with a star are marked as “Christian Alters” in Patricia’s original diagram, and nodes colored yellow are included inside the “Sphere of the Blue Flame” in Patricia’s original diagram. Visualization created with Pajek (Batagelj and Mrvar, 2004; Mrvar and Batagelj. 2016).
Figure 4:
Patricia’s 1993 network. Nodes marked with a star are from the box labelled “(Behind)” Patricia’s original diagram, and nodes colored yellow are included inside the “Sphere of the Blue Flame in Patricia’s original diagram. Visualization created with Paiek (Batagelj and Mrvar, 2004; Mrvar and Batageli, 2016)
Table 1:
Summary statistics of the networks.
| Network | N | Components | Mean degree | Density | Clustering coefficient | Assortativity coefficient |
|---|---|---|---|---|---|---|
| Patricia 1990 | 14 | 1 | 2.57 | 0.19780 | 0.40000 | -0.25000 |
| Patricia 1992 | 85 | 2 | 2.21 | 0.02633 | 0.10345 | -0.46399 |
| Patricia 1993 | 107 | 5 | 2.13 | 0.02010 | 0.10266 | -0.37400 |
| Grey’s Anatomy | 44 | 4 | 2.09 | 0.04863 | 0.00000 | -0.22567 |
| Dolphins | 62 | 1 | 5.13 | 0.08408 | 0.30878 | -0.04359 |
| Zachary karate club | 34 | 1 | 4.59 | 0.13904 | 0.25568 | -0.47561 |
| Kapferer tailor shop | 39 | 1 | 8.10 | 0.21323 | 0.38506 | -0.18269 |
| Law firm friendship | 71 | 3 | 11.24 | 0.16056 | 0.44862 | 0.07948 |
| High school friendship | 134 | 3 | 6.06 | 0.04556 | 0.47540 | 0.28718 |
1 Note: All networks are undirected. The statistics were computed using the igraph (Csárdi and Nepusz, 2006) package in R (R Core Team, 2016).
Table 2:
ERGM models of Patricia’s 1990 network.
| Effect | Model 1 | Model 2 | Model 3 |
|---|---|---|---|
| Edges | -2.023 (0.507)*** | -2.030 (0.503)*** | -2.391 (0.669)*** |
| GWESP | 0.697 (0.341)* | 0.710 (0.355)* | 0.861 (0.505) |
| Degree 2 | 1.029 (0.573) | ||
| Degree 2-3 | 1.006 (0.568) | ||
| Degree 3 | 1.501 (0.584)* | ||
| AIC | 90.62 | 90.61 | 84.12 |
| BIG | 98.16 | 98.14 | 91.65 |
1 Note: Estimated with the statnet software (Handcock et al., 2008; Hunter et al., 2008; Handcock et al., 2016a,b). Estimated standard errors in parentheses. The GWESP α parameter is fixed at 0. *** p < 0.0001, ** p < 0.001, * p < 0.05.
Table 3:
ERGM models of Patricia’s 1992 network.
| Effect | Model 1 | Model 2 | Model 3 |
|---|---|---|---|
| Edges | -4.588 (0.311)*** | -6.047 (0.561)*** | -8.392 (0.775)*** |
| GWDEGREE | 1.397 (0.503)** | 1.908 (0.560)*** | 1.838 (0.572)** |
| GWESP | 0.793 (0.173)*** | 0.764 (0.178)*** | 0.608 (0.177)*** |
| Activity Christian | 0.734 (0.196)*** | 0.772 (0.217)*** | |
| Homophil y Christian | 0.377 (0.205) | 0.305 (0.215) | |
| Activity Integrated | 0.232 (0.310) | 0.150 (0.331) | |
| Homophily Integrated | 0.427 (0.350) | 0.290 (0.380) | |
| Activity Sphere | 0.646 (0.200)** | ||
| Homophily Sphere | 2.744 (0.513)*** | ||
| AIC | 854.40 | 843.20 | 782.60 |
| BIC | 872.90 | 886.40 | 838.20 |
1 Note: Estimated with the statnet software (Handcock et al., 2008; Hunter et al., 2008; Handcock et al., 2016a,b). Estimated standard errors in parentheses. The GWDEGREE decay parameter is fixed at 0.5 and the GWESP α parameter is fixed at 0. *** p < 0.0001, ** p < 0.001, *p < 0.05.
Table 4:
ERGM models of Patricia’s 1993 network.
| Effect | Model 1 | Model 2 | Model 3 | Model 4 |
|---|---|---|---|---|
| Edges | -4.941 (0.282)*** | -6.609 (0.470)*** | -9.914 (0.869)*** | -10.169 (0.901)*** |
| GWDEGREE | 1.501 (0.443)*** | 2.384 (0.547)*** | 2.314 (0.550)*** | 2.249 (0.538)*** |
| GWESP | 0.870 (0.163)*** | 0.835 (0.168)*** | 0.656 (0.169)*** | 0.643 (0.172)*** |
| Activity Christian | 0.708 (0.190)*** | 0.747 (0.203)*** | 0.789 (0.203)*** | |
| Activity Integrated | 0.540 (0.243)* | 0.507 (0.258)* | 0.580 (0.247)* | |
| Homophily Christian | 0.473 (0.203)* | 0.453 (0.205)* | 0.512 (0.214)* | |
| Homophily Integrated | 0.725 (0.252)** | 0.670 (0.260)** | 0.828 (0.271)** | |
| Activity Sphere | 0.660 (0.192)*** | 0.729 (0.194)*** | ||
| Homophily Sphere | 3.611 (0.730)*** | 3.600 (0.744)*** | ||
| Activity Behind | 0.631 (0.377) | |||
| AIC | 1094.00 | 1062.00 | 975,60 | 975.00 |
| BIC | 1114.00 | 1108.00 | 1035.00 | 1041.00 |
1 Note: Estimated with the statnet software (Handcock et al., 2008: Hunter et al., 2008; Handcock et al., 2016a,b). Estimated standard errors in parentheses. The GWDEGREE decay parameter is fixed at 0.5 and the GWESP α parameter is fixed at 0. *** p < 0.0001, ** p < 0.001, *p < 0.05.
Figure 5:
Largest geodesic cycle size (top), and distribution of geodesic cycle sizes (bottom) for Patricia’s 1990 network. In the top plot, the dashed red line is the value in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series or ERGM as labelled on the x-axis. In the bottom plots, the points shown as red diamonds joined by the red line are the values in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series 2.5k distribution (left) and from the ERGM (right). In both ERGM plots the ERGM is Model 3 in Table 2.
Figure 6:
Largest geodesic cycle size (top), and distribution of geodesic cycle sizes (bottom) for Patricia’s 1992 network n the top plot, the dashed red line is the value in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series or ERGM as labelled on the x-axis. In the bottom plots, the points shown as red diamonds joined by the red line are the values in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series 2.5k distribution (left) and from the ERGM Model 3 (right). ERGM model numbers refer to those in lame 3.
Figure 7:
Largest geodesic cycle size (top), and distribution of geodesic cycle sizes (bottom) for Patricia’s 1993 network. In the top plot, the dashed red line is the value in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series or ERGM as labelled on the x-axis. In the bottom plots, the points shown as red diamonds joined by the red line are the values in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series 2.5k distribution (left) and from the ERGM Model 3 (right). ERGM model numbers refer to those in Table 4.
Figure 8:
Distribution of geodesic cycle sizes (bottom) for Patricia’s 1993 network. The points shown as red diamonds joined by the red line are the values in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series 1k distribution (top) and ERGM Model 1 (bottom). ERGM model numbers refer to those in Table 4.
Figure 9:
Largest geodesic cycle size (top), and distribution of geodesic cycle sizes (bottom) for the Grey’s Anatomy sexual contact network. In the top plot, the dashed red line is the value in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series or ERGM as labelled on the x-axis. In the bottom plots, the points shown as red diamonds joined by the red line are the values in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series 2.5k distribution (left) and from the ERGM (right). In both ERGM plots the ERGM is Model 3 in Table B2.
Figure 10:
Largest geodesic cycle size (top), and distribution of geodesic cycle sizes (bottom) for the dolphin social network. In the top plot, the dashed red line is the value in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series or ERGM as labelled on the x-axis. In the bottom plots, the points shown as red diamonds joined by the red line are the values in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series 2.5k distribution (left) and from the ERGM (right). In both ERGM plots the ERGM is Model 1 in Table B3.
Figure 11:
Largest geodesic cycle size (top), and distribution of geodesic cycle sizes (bottom) for the Lazega law firm friendship network. In the top plot, the dashed red line is the value in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series or ERGM as labelled on the x-axis. In the bottom plots, the points shown as red diamonds joined by the red line are the values in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series 2.5k distribution (left) and from the ERGM (right). In both ERGM plots the ERGM is Model 1 in Table B4.
Figure 12:
Largest geodesic cycle size (top), and distribution of geodesic cycle sizes (bottom) for the Zachary karate club network. In the top plot, the dashed red line is the value in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series or ERGM as labelled on the x-axis. In the bottom plots, the points shown as red diamonds joined by the red line are the values in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series 2.0k distribution (left) and from the ERGM (right). In both ERGM plots the ERGM is Model 2 in Table B5.
Figure 13:
Largest geodesic cycle size (top), and distribution of geodesic cycle sizes (bottom) for the Kapferer tailor shop network. In the top plot, the dashed red line is the value in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series or ERGM as labelled on the x-axis. In the bottom plots, the points shown as red diamonds joined by the red line are the values in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series 2.5k distribution (left) and from the ERGM (right). In both ERGM plots the ERGM is Model 2 in Table B6.
Figure 14:
Largest geodesic cycle size (top), and distribution of geodesic cycle sizes (bottom) for the high school friendship network. In the top plot, the dashed red line is the value in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series or ERGM as labelled on the x-axis. In the bottom plots, the points shown as red diamonds joined by the red line are the values in the observed network, with the box plots showing the values in 100 networks simulated from the dk-series 2.5k distribution (left) and from the ERGM (right). In both ERGM plots tile ERGM is Model 2 in Table B7.
Figure A1:
Grey’s Anatomy sexual contact network. Male actors are colored blue, and female pink. Visualization created using the network R package (Butts, 2008, 2015).
Figure A2:
Dolphin social network. Visualization created using the network R package (Butts, 2008, 2015).
Figure A3:
Lazega law firm friendship network. Nodes are colored according to the office the person works at. Visualization created using the network R package (Butts, 2008, 2015).
Figure A4:
Zachary karate club network. Nodes are colored according to role (Instructor [green, Mr. Hi], Member [orange], or President [purple, John A.]). Visualization created using the network R package (Butts, 2008, 2015).
Figure A5:
Kapferer tailor shop network. Visualization created using the network R package (Butts, 2008, 2015).
Figure A6:
High school friendship network. Male students are colored blue, and female pink (there is one unknown colored gray). Visualization created using the network R package (Butts, 2008, 2015).
Figure A7:
Patricia’s 1992 network. Nodes marked with a dot are included inside the “Sphere of the Blue Flame” in Patricia’s original diagram. Coloring is according to network communities found with the Louvain algorithm (Blondel et al., 2008). It is apparent that the Sphere of the Blue Flame is largely included in. but does not correspond exactly with, a network community using this method, with Miranda, Stephanie, Bryony and Millie being part of another community according to the algorithm. In addition Naomi and Dordy are in the Sphere, but not in the corresponding community according to the Louvain algorithm. Dordy in particular would never be in the same community as the rest of the Sphere according to any network community algorithm, unless it also includes JC and the surrounding nodes, as Dordy is at a geodesic distance of six (via JC) from the nearest other member of the Sphere. The inclusion of Dordy in the Sphere appears to be due to spatial rather than network logic (according to the layout of Patricia’s original drawing). See Martin (2017) for more discussion of this point. Community detection and visualization were done with Pajek (Batagelj and Mrvar, 2004: Mrvar and Batagelj, 2016).
Table B1:
Parameters for undirected networks.
| Effect | Description |
|---|---|
| Edges | Baseline density |
| Degree k | Nodes of degree k. A positive parameter indicates over-representation of nodes of degree k. “Degree k – l indicates nodes of degree between the values of k and l inclusive. |
| GWDEGREE | Geometrically weighted degree distribution. A positive parameter indicates anti-preferential attachment (Hunter, 2007). See also Levy (2016). |
| GWDSP | Geometrically weighted dyadwise shared partner. |
| GWESP | Geometrically weighted edgewise shared partner. A positive parameter indicates transitivity (closure). |
| Homophily c | Homophily on categorical attribute c. A positive parameter value indicates an edge preferentially forming between nodes with the same value of the categorical attribute. This may be shown instead as “a – b” for differential homophily: homophily specifically between values a and b of a categorical attribute, rather than uniform homophily on an attribute. |
| Heterophily c | Heterophily on continuous attribute c. This is based on the absolute value of the difference of the attribute values of two nodes. |
| a Activity | Activity on binary attribute a. A positive parameter value indicates that nodes with the binary attribute are more likely to have an incident edge. |
| a Interaction | interaction on binary attribute a. A positive parameter value indicates that two nodes with the binary attribute are more likely to have an edge between them. |
Table B2:
ERGM models of the Grey’s Anatomy sexual network.
| Effect | Model 1 | Model 2 | Model 3 |
|---|---|---|---|
| Edges | –1.442 (0.241)*** | –0.287 (0.564) | –0.844 (0.636) |
| Homophily Sex | –3.133 (0.718)*** | –3.428 (0.741)*** | –3.542 (0.732)*** |
| Degree 1 | 2.026 (0.500)*** | 3.533 (1.058)*** | 3.393 (1.000)*** |
| Degree 2 | 1.828 (0.911)* | 1.743 (0.858)* | |
| Degree 3 | 0.988 (0.805) | 0.983 (0.769) | |
| Heterophily Birth year | –0.132 (0.030)*** | –0.142 (0.032)*** | |
| Attending – Attending | 1.172 (0.508)* | 1.085 (0.533)* | |
| Attending – Chief | 1.137 (0.682) | 1.004 (0.699) | |
| Attending – Non-Staff | –0.714 (0.642) | –0.834 (0.648) | |
| Attending – Nurse | 0.109 (0.988) | –0.058 (1.215) | |
| Attending – Other | 0.490 (0.789) | 0.345 (0.874) | |
| Attending – Resident | 1.041 (0.502)* | 1.004 (0.502)* | |
| Chief-Non-Staff | –0.156 (1.183) | –0.517 (1.316) | |
| Chief – Resident | 0.438 (1.080) | 0.427 (1.096) | |
| Intern – Intern | 5.003 (1.904)** | 4.611 (1.753)** | |
| Non-Staff-Non-Staff | –1.289 (1.312) | –1.431 (1.266) | |
| Non-Staff – Resident | 0.395 (0.593) | 0.398 (0.604) | |
| Nurse – Resident | 1.147 (0.847) | 1.554 (0.861) | |
| Homophily Black | 2.326 (0.754)** | ||
| Homophily White | 0.856 (0.392)* | ||
| AIC | 302.10 | 278.40 | 271.30 |
| BIC | 316.70 | 365.80 | 368.40 |
1 Note: Estimated with the statnet software (Handcock et al., 2008; Hunter et al., 2008; Handcock et al., 2016a,b). Estimated standard errors in parentheses. Models based on those described in Lind (2012); Leavitt and Clark (2014); Weissman (2019). *** p < 0.0001, ** p < 0.001, * p < 0.05.
Table B3:
ERGM models of the dolphin social network.
| Effect | Model 1 | Model 2 |
|---|---|---|
| Edges | –0.821 (0.642) | –1.553 (3.122) |
| GWDEGREE | –2.148 (0.647)*** | –0.522 (3.260) |
| GWDSP (α = 0.7) | –0.305 (0.067)*** | |
| GWESP (α = 0.1) | 0.984 (0.151)*** | |
| GWDSP | –0.250 (0.368) | |
| GWDSP α | 0.834 (0.669) | |
| GWESP | 0.630 (0.421) | |
| GWESP α | 1.082 (0.324)*** | |
| AIC | 1014.00 | 1014.00 |
| BIC | 1036.00 | 1047.00 |
1 Note: Estimated with the statnet software (Handcock et al., 2008; Hunter et al., 2008; Handcock et al., 2016a,b) using the “Stepping” algorithm (Hummel et al., 2012). Estimated standard errors in parentheses. The GWDEGREE decay parameter is fixed at 0.5. *** p < 0.0001, ** p < 0.001, * p < 0.05.
Table B4:
ERGM models of the Lazega law firm friendship network.
| Effect | Model 1 |
|---|---|
| Edges | –5.256 (0.317)*** |
| GWDEGREE | 1.290 (0.874) |
| GWESP | 0.597 (0.072)*** |
| GWESP α | 1.398 (0.030)*** |
| Homophily GENDER | 0.535 (0.103)*** |
| Homophily LAW SCHOOL | 0.137 (0.130) |
| Homophily OFFICE | 0.767 (0.111)*** |
| Homophily PRACTICE | 0.485 (0.105)*** |
| Homophily STATUS | 0.759 (0.104)*** |
| Heterophily AGE | –0.019 (0.009)* |
| Heterophily SENIORITY | –0.019 (0.009)* |
| AIC | 1697.00 |
| BIC | 1761.00 |
1 Note: Estimated with the statnet software (Handcock et al., 2008; Hunter et al., 2008; Handcock et al., 2016a,b). Estimated standard errors in parentheses. The GWDEGREE decay parameter is fixed at 0.1. *** p < 0.0001, ** p < 0.001, * p < 0.05.
Table B5:
ERGM models of the Zachary karate club network.
| Effect | Model 1 | Model 2 |
|---|---|---|
| Edges | –3.830 (0.405)*** | –2.095 (0.491)*** |
| GWDEGREE | 5.566 (3.376) | 0.988 (1.228) |
| GWESP (α = 0.5) | 1.102 (0.211)*** | 0.358 (0.230) |
| Instructor | 2.345 (0.527)*** | |
| President | 2.369 (0.543)*** | |
| Faction abs. diff. 1 | –0.246 (0.316) | |
| Faction abs. diff. 2 | –1.542 (0.492)** | |
| Faction abs. diff. 3 | –2.179 (0.605)*** | |
| Faction abs. diff. 4 | –2.672 (0.626)*** | |
| AIC | 419.40 | 346.80 |
| BIC | 432.40 | 385.80 |
1 Note: Estimated with the statnet software (Handcock et al., 2008; Hunter et al., 2008; Handcock et al., 2016a,b). Estimated standard errors in parentheses. The GWDEGREE decay parameter is fixed at 0.2. The faction identifier is coded in the Zachary karate club data from the ergm.count package (Krivitsky, 2016) as -2 (strongly Mr. Hi’s), -1 (weakly Mr, Hi’s), 0 (neutral). +1 (weakly John’s), and +2 (strongly John’s), and so the absolute difference in faction id ranges from 0 (same faction) to 4 (strongly Mr Hi’s and strongly John’s). *** p < 0.0001, ** p < 0.001, * p < 0.05.
Table B6:
ERGM models, reproducing those in Hummel et al. (2012), of the Kapferer tailor shop network.
| Effect | Model 1 | Model 2 |
|---|---|---|
| Edges | –3.082 (0.567)*** | –2.997 (0.523)*** |
| GWDEGREE | 0.360 (0.935) | |
| GWDSP (o = 0.25) | –0,129 (0.051)* | –0.130 (0.052)* |
| GWESP (α = 0.25) | 1.491 (0.343)*** | 1.436 (0.286)*** |
| AIC | 732.70 | 732.20 |
| BIC | 751.10 | 746.00 |
1 Note: Estimated with the statnet software (Handcock et al., 2008; Hunter et al., 2008; Handcock et al., 2016a,b) using the “Stepping” algorithm (Hummel et al., 2012). Estimated standard errors in parentheses. The GWDEGREE decay parameter is fixed at 0.25. *** p < 0.0001, ** p < 0.001, * p < 0.05.
Table B7:
ERGM models of the high school friendship network.
| Effect | Model 1 | Model 2 | Model 3 |
|---|---|---|---|
| Edges | –7.000 (0.570)*** | –8.542 (0.471)*** | –8.574 (0.457)*** |
| GWDEGREE | 2.447 (0.385)*** | 3.062 (0.266)*** | 3.037 (0.259)*** |
| GWDEGREE decay | 1.563 (0.113)*** | ||
| GWDSP (α = 0.5) | –0.014 (0.031) | 0.049 (0.023)* | 0.047 (0.022)* |
| GWESP | 1.210 (0.079)*** | ||
| GWESP α | 1.157 (0.029)*** | ||
| GWESP (α = 1.2) | 1.344 (0.069)*** | 1.334 (0.067)*** | |
| Homophily Class | 1.057 (0.086)*** | 1.090 (0.084)*** | 1.081 (0.080)*** |
| Homophily Sex | 0.171 (0.086)* | ||
| AIC | 2410.00 | 1937.00 | 1975.00 |
| BIC | 2459.00 | 1972.00 | 2018.00 |
1 Note: Estimated with the statnet software (Handcock et al., 2008: Hunter et al., 2008; Handcock et al., 2016a,b). Estimated standard errors in parentheses. In models 2 and 3, the GWDEGREE decay parameter is fixed at 1.7, the GWESP α parameter is fixed at 1.2, and the GWDSP α parameter is fixed at 0.5. Models without homophily on class included do not converge (there is a very noticeable community structure in the network, caused by the classes; a parameter needs to be included in the ERGM to account for this). *** p < 0.0001, ** p < 0.001, * p < 0.05.