1. Social Networks in Elephants Eric Vance ISDS Duke University Eric Vance ISDS Duke University SAMSI Social Networks Workshop CMU, March 2, 2006

2. Scientific Questions  How does the social structure of elephants change in the Wet Season v. the Dry Season?

3. Scientific Questions  What is the role of kinship amongst elephants?

4. Scientific Questions: Kinships  Mother/Daughter Relationships  Sister Relationships  Measured DNA Relatedness  Mother/Daughter Relationships  Sister Relationships  Measured DNA Relatedness

5. Bilinear Mixed Effects Model  Peter Hoff (2005) modeled the pairwise relationships between actors using a latent social space   ij =  0 +  s x i +  r x j +  d X ij +  ij  ij = a i +b j +  ij +z i 'z j  Application to Political Science: Model the interactions between countries  Peter Hoff (2005) modeled the pairwise relationships between actors using a latent social space   ij =  0 +  s x i +  r x j +  d X ij +  ij  ij = a i +b j +  ij +z i 'z j  Application to Political Science: Model the interactions between countries

6. Elephant Social Structure  Males leave their families around ages 14-17  Families fission into subgroups, and fuse back together daily  Adult females and juveniles are led by a Matriarch  Males leave their families around ages 14-17  Families fission into subgroups, and fuse back together daily  Adult females and juveniles are led by a Matriarch

7. Amy, Matriarch of Family AA

8. Data Collection  Researchers in Kenya ride into the national park to observe families of elephants  Elephants are identified by sight  Clusters of females occupying the same physical area are recorded

9. Observations on Pairs of Elephants  If two elephants Amy and Angelina are seen together then the observation y AmyAng = 1  If Audrey is not nearby, then y AmyAud = 0, and y AngAud = 0

10. Modeling Elephant Interactions  We model the probability p AmyAng of two elephants being seen together  Logistic regression: logit(p ij ) =  ij   ij is the linear predictor  We model the probability p AmyAng of two elephants being seen together  Logistic regression: logit(p ij ) =  ij   ij is the linear predictor

11. The Model for  ij  Intercept  0 is the common baseline for the probability of any two elephants from Family AA being seen together   is a random effect for an elephant’s intrinsic sociability  Sociable elephants will be seen more often with other elephants   ij is unexplained error or white noise  Intercept  0 is the common baseline for the probability of any two elephants from Family AA being seen together   is a random effect for an elephant’s intrinsic sociability  Sociable elephants will be seen more often with other elephants   ij is unexplained error or white noise How often are elephants together?

12. The Model for  ij  Three kinship terms  k k ij :  1. Mother/Daughter pair indicator k 1ij  2. Sister pair indicator k 2ij  3. Noisy measure of the similarity in DNA k 3ij  Three kinship terms  k k ij :  1. Mother/Daughter pair indicator k 1ij  2. Sister pair indicator k 2ij  3. Noisy measure of the similarity in DNA k 3ij What is the role of kinships?

13. The Model for  ij  z i 'z j is a pairwise interaction effect  Each elephant has an (unobservable) position in a latent Social Space  z i 'z j is the inner product of the position vectors in Social Space of elephant i and elephant j  The inner product of two vectors is the similarity of their directions, scaled by their magnitudes: z i 'z j = |z i ||z j |cos(   z i 'z j is a pairwise interaction effect  Each elephant has an (unobservable) position in a latent Social Space  z i 'z j is the inner product of the position vectors in Social Space of elephant i and elephant j  The inner product of two vectors is the similarity of their directions, scaled by their magnitudes: z i 'z j = |z i ||z j |cos( 

14. The Model for  ij  ij =  0 +  i +  j +  k k ij +  ij +z i 'z j

15. Inner product z i 'z j  If z i 'z j = 0, elephants i and j interact as often as the baseline  0, their sociabilities  i,  j, and their kinships k ij predict  If z i 'z j > 0, i and j like each other and are together more often than otherwise predicted  If z i 'z j < 0, i and j dislike each other  If z i 'z j = 0, elephants i and j interact as often as the baseline  0, their sociabilities  i,  j, and their kinships k ij predict  If z i 'z j > 0, i and j like each other and are together more often than otherwise predicted  If z i 'z j < 0, i and j dislike each other

16. Elephant Social Space Sun  I choose the dimension of Social Space d = 2 Shade Hills Flat

17. Priors  ij =  0 +  i +  j +  k k ij +  ij + z i 'z j  Intercept:  0  N(0, 100)  Sociabilities:  i  N(0,  2 soc ),  2 soc  IG(.5,.5)  Kinship Coefficients:  k  N(0, 100  I 3 )  Pairwise error:  ij  N(0,  2  ),  2   IG(.5,.5)  Social space: z i  N(0,  2 z  I 2 ),  2 z  IG(.5,.5)  ij =  0 +  i +  j +  k k ij +  ij + z i 'z j  Intercept:  0  N(0, 100)  Sociabilities:  i  N(0,  2 soc ),  2 soc  IG(.5,.5)  Kinship Coefficients:  k  N(0, 100  I 3 )  Pairwise error:  ij  N(0,  2  ),  2   IG(.5,.5)  Social space: z i  N(0,  2 z  I 2 ),  2 z  IG(.5,.5)

18. Data  Researchers in Kenya observe the AA family 637 times over three years  418 observations during Dry Season  219 observations during Wet Season  Run a separate model for each season  Researchers in Kenya observe the AA family 637 times over three years  418 observations during Dry Season  219 observations during Wet Season  Run a separate model for each season

19. Results for Family AA

20. Intercept  0 Posterior Densities  ij =  0 +  i +  j +  k k ij +  ij + z i 'z j

21. AA Sociability  Posterior Densities  ij =  0 +  i +  j +  k k ij +  ij + z i 'z j

22. Kinship Coefs  k Posterior Densities  ij =  0 +  i +  j +  k k ij +  ij + z i 'z j

23. Kinship Coefs  k Posterior Densities  ij =  0 +  i +  j +  k k ij +  ij + z i 'z j

24. Kinship Coefs  k Posterior Densities  ij =  0 +  i +  j +  k k ij +  ij + z i 'z j

25. Pairwise Effects z i `z j  ij =  0 +  i +  j +  k k ij +  ij + z i 'z j

26. Elephants in Social Space: Dry Season  ij =  0 +  i +  j +  k k ij +  ij + z i 'z j

27. Wet Season Social Space  ij =  0 +  i +  j +  k k ij +  ij + z i 'z j

28. Dry Season, No Kinships  ij =  0 +  i +  j +  k k ij +  ij + z i 'z j

29. Wet Season, No Kinships  ij =  0 +  i +  j +  k k ij +  ij + z i 'z j

30. Conclusions  Elephants interact more often in the Wet season  Sociabilities and kinship effects are similar in both seasons  Kinships are very important in explaining the variability in how elephants interact  Kinship effects are similar in both seasons  Elephants interact more often in the Wet season  Sociabilities and kinship effects are similar in both seasons  Kinships are very important in explaining the variability in how elephants interact  Kinship effects are similar in both seasons