# FrequentlyAskedQuestions: gwtermsNotes.txt

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1 | From morrism@U.WASHINGTON.EDU Mon Nov 10 11:40:54 2014 |

2 | Date: Mon, 10 Nov 2014 11:38:18 -0800 |

3 | From: martina morris <morrism@U.WASHINGTON.EDU> |

4 | To: SOCNET@LISTS.UFL.EDU |

5 | Subject: Re: Interpretation of GWDEGREE |

6 | |

7 | ***** To join INSNA, visit http://www.insna.org ***** |

8 | |

9 | Hi Mark, |

10 | |

11 | Here's how I like to think about this. |

12 | |

13 | There are two parameters in the gw-terms, and the overall effect on the |

14 | odds of a tie is a product of the two, given the way the statistic is |

15 | constructed. That's why these are called "curved terms". |

16 | |

17 | You can think of the gwdegree term as having the form |

18 | |

19 | beta * f(y, alpha) |

20 | |

21 | This has the usual form parameter*statistic, except that there is a second |

22 | parameter, alpha, in the statistic itself. The statistic essentially sums |

23 | the number of nodes of each degree, except that alpha modifies the value |

24 | of that number, as a function of degree. |

25 | |

26 | Alpha essentially imposes a rate of decay by degree, so the higher degree |

27 | nodes contribute less to the statistic than the lower. It can be |

28 | interpreted as the declining marginal return for each additional tie (or |

29 | additional shared partner for gw(e/n/d)sp). So yes, this does relate to |

30 | the preferential attachment concept (more below). |

31 | |

32 | Beta controls the overall propensity for degree (or shared partners). |

33 | |

34 | A good way to start to interpret the parameters is to set alpha=0, and |

35 | look at the change statistics (you can do this by calculating the f(y, |

36 | alpha) statistic with and without a proposed tie). Setting alpha=0 has |

37 | the effect of making only the first tie for a node count as a change; so |

38 | the possible values of the change statistic are |

39 | |

40 | 0 (if both nodes already have other ties), |

41 | 1 (if one node was an isolate), and |

42 | 2 (if both nodes were isolates). |

43 | |

44 | Beta then multiplies this, so it can be interpreted as how the odds of a |

45 | tie change, as a function of the change in the number of nodes that are no |

46 | longer isolates when it is toggled on. |

47 | |

48 | Of course, interpretation depends on the other terms in the model, and in |

49 | general you would have an edges term in to control overall density. In |

50 | that case, beta would reflect a propensity against/for isolates (for |

51 | positive/negative estimates respectively), relative to a random graph |

52 | with this density. |

53 | |

54 | When alpha > 0, there is no discontinuity at 1 vs more, but instead a |

55 | continuous decline in the value of additional partners, where the rate of |

56 | decline falls as alpha increases. For alpha=inf, there is no declining |

57 | marginal return, the odds of a tie don't depend on the degrees of the |

58 | nodes (and for shared partners, you're back to the triangle term). |

59 | |

60 | So, in answer to your question, it's the alpha parameter that is the |

61 | "anti-preferential attachment" component. As it varies from 0 to inf., it |

62 | never represents preferential attachment -- at inf., ties are just |

63 | independent of degree. But the smaller the value of alpha, the more |

64 | anti-preferential the degree distribution will be. |

65 | |

66 | I found it helped me to understand these terms by making up an excel |

67 | spreadsheet to calculate the term itself, and the change statistics. If |

68 | you think something like this might help, I can clean mine up and make it |

69 | available. |

70 | |

71 | best, |

72 | Martina |

73 | |

74 | On Mon, 10 Nov 2014, Lubell, Mark wrote: |

75 | |

76 | > ***** To join INSNA, visit http://www.insna.org ***** |

77 | > |

78 | > Dear SOCNET: |

79 | > |

80 | > My research group is having an internal debate about how to interpret the geometrically weighted degree parameter for ERGM models, as implemented in Statnet. If anybody has a good paper or presentation discussing interpretation (beyond the various papers introducing the calculation and estimation), I would love to know about them. |

81 | > |

82 | > In particular, is GWDEGREE an anti-preferential attachment term such that a positive coefficient produces a low variance degree distribution, or does a positive coefficient produce a high variance degree distribution with a centralized network? And if you have a low variance degree distribution....what is the best way to think about the social processes generating a decentralized network? |

83 | > |

84 | > Thanks, Mark Lubell |

85 | > UC Davis |

86 | > |

87 | > _____________________________________________________________________ |

88 | > SOCNET is a service of INSNA, the professional association for social |

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92 | > |

93 | |

94 | **************************************************************** |

95 | Professor of Sociology and Statistics |

96 | Director, UWCFAR Sociobehavioral and Prevention Research Core |

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108 | _____________________________________________________________________ |

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