• Dynamics of interaction in small groups
  • Differential propensity to act
• Investigate association with known covariates
  • regarding the individual
  • previous history of events
  • context

The problem of interest for today's talk involves a set of entities and events among them occurring over time. Each of these events can be called edges or relational events. One example we will return to throughout today's talk will be one where the nodes represent students in a particular classroom and the edges represent direction acts of communication. The individuals might be distinguished by some covariate, such as age, as represented by the different colors here. One of the most fundamental goals is to characterize the differential propensity to interact, and better understandhow these differences are associated with known covariates. For example, we might want to ask whether interactions between same-gender students are more likely and, furthermore, how this varies by context. To answer these types of questions, today we will discuss a statistical model for interaction in small groups of people. ----- The problem of interest for today's talk involves a set of entities and events among them occurring over time. Each event represents some interaction directed from one entity to another. I might refer to these entities as nodes, vertices, or individuals. Each of these events can be called edges or relational events. One example we will return to throughout today's talk will be one where the nodes represent students in a particular classroom and the edges represent direction acts of communication. The individuals might be distinguished by some covariate, such as age, as represented by the different colors here. This type of interaction data is of particular interest to Sociologists who study social networks and the dynamics of small groups. One of the most fundamental goals is to characterize the differential propensity to interact, and better understandhow these differences are associated with known covariates. For example, we might want to ask whether interactions between same-gender students are more likely and, furthermore, how this varies by context. To answer these types of questions, today we will discuss a statistical model for interaction in small groups of people. I will be talking about some of the challenges that arise and how we propose to solve them.

• Previous work: Models for network event data
  • Background
  • Relational event framework
  • Simulated example
• Contribution: Hierarchical models for multiple sequences of events
• Application: High school classroom dynamics
• Future directions

• How do we choose the resolution for aggregating?
• A network for each day? A network for each hour?
• What happens if the behavior of interest is at a smaller time scale?

• $N$ people: $i,j \in \{1,\ldots,N\}$
• Each interaction is an edge with a timestamp: $(i,j,t)$
• Observe a sequence of edges: $\{(i_k,j_k,t_k), k\in [1,M]\}$
• Often edges can occur more than once
• Can benefit from thinking of each edge as a process with some rate

Assume event rates change only when events occur.

Rate of k^th observed event

Represents the fact that no event occurred between event k-1 and event k

For more info see (Butts, 2008).

$\lambda_{ij}(t) = \exp\{\beta'X_{ij}(t)\}$

$\log \lambda_{ij}(t) = \beta_0 + \beta_1 X_{ij1} + \beta_2 X_{ij2} + \beta_3 X_{ij3} + \beta_4 X_{ij4} + \beta_5 X_{ij5} + \beta_6 X_{ij6}$

$\log \lambda_{ij}(t) = \beta_0 + $ $\beta_1 X_{ij1}$$ + \beta_2 X_{ij2} + \beta_3 X_{ij3} + \beta_4 X_{ij4} + \beta_5 X_{ij5} + \beta_6 X_{ij6}$

$\log \lambda_{ij}(t) = \beta_0 + \beta_1 X_{ij1} + $$\ \beta_2 X_{ij2}$$ + \beta_3 X_{ij3} + \beta_4 X_{ij4} + \beta_5 X_{ij5} + \beta_6 X_{ij6}$

$\log \lambda_{ij}(t) = \beta_0 + \beta_1 X_{ij1} + \beta_2 X_{ij2} +$ $\ \beta_3 X_{ij3}$$ + \beta_4 X_{ij4} + \beta_5 X_{ij5} + \beta_6 X_{ij6}$

$\log \lambda_{ij}(t) = \beta_0 + \beta_1 X_{ij1} + \beta_2 X_{ij2} + \beta_3 X_{ij3} + \beta_4 X_{ij4} + \beta_5 X_{ij5} + \beta_6 X_{ij6}$

Additive effects that model turn-taking in conversation (Gibson, 2003)

$\log \lambda_{ij}(t) = \beta_0 + \beta_1 X_{ij1} + \beta_2 X_{ij2} + \beta_3 X_{ij3} + $ $\ \ \beta_4 X_{ij4}$$ + \beta_5 X_{ij5} + \beta_6 X_{ij6}$

E.g. $X_{ij4}$ immediately after event (12,15)

$\log \lambda_{ij}(t) = \beta_0 + \beta_1 X_{ij1} + \beta_2 X_{ij2} + \beta_3 X_{ij3} + \beta_4 X_{ij4} + $ $\ \ \beta_5 X_{ij5}$$ + \beta_6 X_{ij6}$

E.g. $X_{ij5}$ immediately after event (12,15)

$\log \lambda_{ij}(t) = \beta_0 + \beta_1 X_{ij1} + \beta_2 X_{ij2} + \beta_3 X_{ij3} + \beta_4 X_{ij4} + \beta_5 X_{ij5} + $ $\ \ \beta_6 X_{ij6}$

E.g. $X_{ij6}$ immediately after event (12,15)

$\log \lambda_{ij}(t) = \beta_0 + \beta_1 X_{ij1} + \beta_2 X_{ij2} + \beta_3 X_{ij3} + \beta_4 X_{ij4} + \beta_5 X_{ij5} + \beta_6 X_{ij6}$

E.g. Entire matrix $\lambda$ immediately after event (12,15)

• Previous work: Models for event data in networks
• Contribution: Hierarchical models for multiple sequences of events
  • Motivation and specification of our hierarchical model
  • Prediction experiments
  • MCMC with parallel tempering
• Application: High school classroom dynamics
• Future directions

• Some event sequences may have few events
• Effects may have few relevant events

$\begin{align} p(\theta,\mu,\sigma | \mathbf{Y}, \mathbf{X}) \propto & \prod_{j=1}^J \ p(Y_j|\theta_j,\mathbf{X}_j) \prod_{p=1}^P \ p(\theta_{jp}|\mu_p,\sigma_p) p(\mu_p,\sigma_p)\end{align}$

At each time $t$, rank $\lambda_{ij}(t)$

Precision @ k: Proportion of the next events that were ranked higher than k

• Metropolis-Hastings: construct a Markov chain whose stationary distribution is the posterior distribution of our parameters
• MH can get stuck easily: Peaked modes, some parameters might be at the boundary, and some $\theta_{jp}$ are more variable than others.
• Finding a good proposal distribution is tricky

Unnormalized distribution of interest: $g(\Theta)$

$$ \begin{align} \pi(\Theta_1,\ldots,\Theta_J) \propto& \prod_{j=1}^J h_j(\Theta_j) \\ h_j(\Theta_j) \propto & \exp \{-g(\Theta_j)/t_j \} \end{align} $$ where $t_0 < \cdots < t_J$ and $h_j(\Theta_j)$ is the target distribution for chain $j$.

Swap between chains $j$ and $k$ at iteration $t$ with acceptance probability: $$\begin{align} \min \left\{ 1, \frac{h_j(\Theta_{k}^{(t)})h_{k}(\Theta_{j}^{(t)})}{h_j(\Theta_j^{(t)})h_{k}(\Theta_{k}^{(t)})} \right\} \end{align} $$

• Previous work: Models for event data in networks
• Contribution: Hierarchical models for multiple sequences of events
• Application: High school classroom dynamics
  • Background
  • Parameter estimates
  • Model selection via DIC
  • Model checking via the posterior predictive
• Future directions

• Collected via participant observation (McFarland, 2001)
• 650 classroom sessions
• Covariates about the class: subject, teacher, etc.
• Covariates about the individuals: race, extracurriculars, etc.

Deviance:

$D(y,\theta) = -2\log p(y|\theta)$

Effective # parameters:

$p_D = \frac{1}{L} \sum_{l=1}^L D(y,\theta^l) - D(y,\hat{\theta})$

Criterion:

$DIC = \frac{1}{L} \sum_{l=1}^L D(y,\theta^l) + p_D$

(Spiegelhalter,2002)

• Participation shift effects are important
• Mixing terms important (e.g. $(i,j)$ are friends)

• Investigate whether the observed data is reasonable under our model.
• Interested in a particular statistic of a sequence $T(Y_j)$.
• Simulate $Y_j^{(i)} \sim REM(\theta_j^{(i)},\mathbf{X}_j)$
• Compare $T(Y_j)$ to distribution of $T(Y_j^{(i)})$

• Previous work: Models for event data in networks
• Contribution: Hierarchical models for multiple sequences of events
• Application: High school classroom dynamics
• Future directions

• Model the heterogeneity among the classroom sessions
• Extend to allow for each event to include more than two individuals

Special thanks to Carter Butts and Padhraic Smyth.

Using a $t$ family for $\theta$:

$\begin{align}\frac{\theta_{jp} - \mu_{p}}{\sigma_p} \sim& t_{\nu}\\ \nu \sim & \mbox{Exponential}(r)\end{align}$

Multilevel modeling

$\mu_{p} = X_j \beta_p$

Informative hyperpriors:

$\begin{align} \mu_p \sim & \mbox{Normal}(\rho,\tau)\\ \sigma_p \sim & \mbox{Gamma}(\alpha,\beta) \end{align}$

$\begin{align} \hat{\theta}_j =& \frac{y_{.j}/\sigma_j^2 + \mu/\tau^2}{1/(\sigma_j^2+\tau^2)} \\ \hat{v}_j =& \frac{1}{1/(\sigma_j^2+\tau^2)} \\ \hat{\mu} =& \frac{\sum_j \frac{1}{1/(\sigma_j^2+\tau^2)} y_{.j}}{\sum_j \frac{1}{1/(\sigma_j^2+\tau^2)}}\\ y_{.j} | \mu, \tau \sim & N(\mu, \sigma_j^2 + \tau^2) \\ p(\theta|\mu,\tau,y) \sim& N(\hat{\theta}_j, \hat{v}_j) \\ p(\mu,\tau|y)=&\frac{p(\mu,\tau,\theta|y)}{p(\theta|\mu,\tau,y)} \forall \theta \\ \end{align} $

$p(\tau|y) \propto p(t) v_{\mu}^{1/2} \prod_j (\sigma_j^2 + \tau^2)^{-1/2} \exp(-(y_{.j}-\mu)^2/(2(\sigma_j^2+\tau^2)))$

Only a big effect on AB-BA. Reference prior putting weight on smaller sigmas. Gamma(2,20) a strong prior, but AB-BA resisting.

Under strong prior, some of the lower level effects $\theta_{jp}$ become more pronounced.