Social Networks in Elephants Eric Vance ISDS Duke University Eric Vance ISDS Duke University SAMSI Social Networks Workshop CMU, March 2, 2006 Scientific Questions How does the social structure of elephants change in the Wet Season v. the Dry Season? Scientific Questions What is the role of kinship amongst elephants? Scientific Questions: Kinships Mother/Daughter Relationships Sister Relationships Measured DNA Relatedness Mother/Daughter Relationships Sister Relationships Measured DNA Relatedness 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 Elephant Social Structure Males leave their families around ages Families fission into subgroups, and fuse back together daily Adult females and juveniles are led by a Matriarch Males leave their families around ages Families fission into subgroups, and fuse back together daily Adult females and juveniles are led by a Matriarch Amy, Matriarch of Family AA 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 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 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 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 elephants 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 elephants intrinsic sociability Sociable elephants will be seen more often with other elephants ij is unexplained error or white noise How often are elephants together? 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? 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( The Model for ij ij = 0 + i + j + k k ij + ij +z i 'z j 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 Elephant Social Space Sun I choose the dimension of Social Space d = 2 Shade Hills Flat 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) 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 Results for Family AA Intercept 0 Posterior Densities ij = 0 + i + j + k k ij + ij + z i 'z j AA Sociability Posterior Densities ij = 0 + i + j + k k ij + ij + z i 'z j Kinship Coefs k Posterior Densities ij = 0 + i + j + k k ij + ij + z i 'z j Kinship Coefs k Posterior Densities ij = 0 + i + j + k k ij + ij + z i 'z j Kinship Coefs k Posterior Densities ij = 0 + i + j + k k ij + ij + z i 'z j Pairwise Effects z i `z j ij = 0 + i + j + k k ij + ij + z i 'z j Elephants in Social Space: Dry Season ij = 0 + i + j + k k ij + ij + z i 'z j Wet Season Social Space ij = 0 + i + j + k k ij + ij + z i 'z j Dry Season, No Kinships ij = 0 + i + j + k k ij + ij + z i 'z j Wet Season, No Kinships ij = 0 + i + j + k k ij + ij + z i 'z j 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