Dynamic Social Behavior Model (DSBM)

• Dynamic Social Behavior Model (DSBM) is a Bayesian hierarchical model that infers latent, time-varying social networks from movement data.
• It separates the latent social state, behavioral expression, and observation process to accurately recover social ties from noisy telemetry.
• Utilizing attraction and alignment mechanisms, the model captures directional movement patterns and outperforms proximity-based methods in inferring social connectivity.

Searching arXiv for the cited DSBM-related papers and neighboring terminology. Searching arXiv for “(Scharf et al., 2015) dynamic social networks based on movement”, “(Xu et al., 2014) dynamic stochastic blockmodels”, and related DSBM terminology. Dynamic Social Behavior Model (DSBM) denotes a class of dynamic models in which latent social structure and observed behavior are separated and linked over time. In the usage most directly aligned with the term, it is a Bayesian hierarchical, discrete-time, continuous-space hidden Markov model in which an unobserved, time-varying social network controls movement through attraction and alignment, allowing social relationships to be inferred from telemetry rather than from direct behavioral observation (Scharf et al., 2015). In adjacent network science, the same acronym also appears as “Dynamic Stochastic Blockmodel,” a distinct state-space model for evolving block interaction structure in social networks (Xu et al., 2014).

1. Scope and defining idea

In the movement-based formulation, the DSBM is designed for settings in which direct behavioral observations are expensive, opportunistic, invasive, or unavailable, but repeated movement measurements are available. The central hypothesis is that sociality leaves a movement signature: socially connected individuals tend to move toward one another and to travel in similar directions. The model therefore separates three layers: the latent social behavior/state, the behavioral expression through movement, and the observation process through noisy telemetry (Scharf et al., 2015).

This separation is methodologically important. Trajectories are not treated as the social network itself; they are treated as noisy consequences of an unobserved dynamic network. A plausible implication is that the DSBM is best understood as a latent-state model of social affiliation rather than as a proximity heuristic or a pure movement model. The distinction matters because the model is intended to recover who is socially connected to whom, when ties form or dissolve, and how those ties persist, rather than merely to threshold distances between individuals.

2. Latent dynamic social network

The core latent object is a dynamic network $\mathbf W(t)$ indexed at discrete times $t=1,\dots,T$, with binary, undirected, unweighted edges

$$w_{ij}(t)= \begin{cases} 1, & \text{if individuals } i \text{ and } j \text{ are socially connected at time } t,\ 0, & \text{otherwise.} \end{cases}$$

Each edge evolves as a pairwise independent Bernoulli Markov chain. At the initial time,

$w_{ij}(1)\sim \mathrm{Bern}(p_1),$

and for $t=2,\dots,T$,

$$w_{ij}(t)\mid w_{ij}(t-1)\sim \begin{cases} \mathrm{Bern}(p_{1|0}), & w_{ij}(t-1)=0,\ \mathrm{Bern}(p_{1|1}), & w_{ij}(t-1)=1. \end{cases}$$

Here $p_1$ is the marginal edge probability at $t=1$, $p_{1|0}$ is the formation probability, and $p_{1|1}$ is the persistence probability (Scharf et al., 2015).

The model assumes that expected network density is constant in time and reparameterizes the transition law through a temporal-stability parameter $t=1,\dots,T$0: $t=1,\dots,T$1 This yields a particularly transparent interpretation: $t=1,\dots,T$2 corresponds to temporal independence, because then $t=1,\dots,T$3, while $t=1,\dots,T$4 corresponds to a near-static network in which edge states almost never change. Within this construction, the DSBM encodes social behavior as a time-varying binary relation with explicit formation and persistence dynamics rather than as a static latent partition or a distance threshold.

3. Movement process: attraction and alignment

Let $t=1,\dots,T$5 denote the latent position of individual $t=1,\dots,T$6 at synchronized time $t=1,\dots,T$7. Conditional on the social network, the positions evolve according to a Gaussian Markov random field whose mean and precision depend on current ties. The model uses two mechanisms.

The first is attraction. For each individual, the centroid of connected neighbors is

$t=1,\dots,T$8

where $t=1,\dots,T$9 is the size of the ego-network if the individual has at least one connection, and equals a positive constant $$w_{ij}(t)= \begin{cases} 1, & \text{if individuals } i \text{ and } j \text{ are socially connected at time } t,\ 0, & \text{otherwise.} \end{cases}$$0 otherwise. The attraction direction is the unit vector

$$w_{ij}(t)= \begin{cases} 1, & \text{if individuals } i \text{ and } j \text{ are socially connected at time } t,\ 0, & \text{otherwise.} \end{cases}$$1

The use of a unit vector is deliberate: attraction specifies direction while keeping step magnitude approximately bounded. On average, attraction moves an individual a distance $$w_{ij}(t)= \begin{cases} 1, & \text{if individuals } i \text{ and } j \text{ are socially connected at time } t,\ 0, & \text{otherwise.} \end{cases}$$2 toward its ego-network centroid per time step (Scharf et al., 2015).

The second mechanism is alignment. Its contribution to the conditional mean is

$$w_{ij}(t)= \begin{cases} 1, & \text{if individuals } i \text{ and } j \text{ are socially connected at time } t,\ 0, & \text{otherwise.} \end{cases}$$3

Only currently connected individuals contribute; the term averages over connected neighbors, subtracts off the attraction component of each neighbor’s displacement, and then aligns residual motion directions. The authors interpret $$w_{ij}(t)= \begin{cases} 1, & \text{if individuals } i \text{ and } j \text{ are socially connected at time } t,\ 0, & \text{otherwise.} \end{cases}$$4 as no alignment and $$w_{ij}(t)= \begin{cases} 1, & \text{if individuals } i \text{ and } j \text{ are socially connected at time } t,\ 0, & \text{otherwise.} \end{cases}$$5 as near-perfect alignment, while excluding $$w_{ij}(t)= \begin{cases} 1, & \text{if individuals } i \text{ and } j \text{ are socially connected at time } t,\ 0, & \text{otherwise.} \end{cases}$$6 because it yields an intrinsic conditional autoregressive model with improper covariance. In the joint GMRF form, the precision blocks are

$$w_{ij}(t)= \begin{cases} 1, & \text{if individuals } i \text{ and } j \text{ are socially connected at time } t,\ 0, & \text{otherwise.} \end{cases}$$7

These two mechanisms provide the model’s explicit social-behavior semantics: connected individuals tend both to move toward one another and to travel in parallel.

4. Observation model and inference

The latent positions are not observed directly. Instead, telemetry yields noisy observations $$w_{ij}(t)= \begin{cases} 1, & \text{if individuals } i \text{ and } j \text{ are socially connected at time } t,\ 0, & \text{otherwise.} \end{cases}$$8 at irregular, asynchronous times, with individual-specific schedules and measurement error. Direct joint inference over the full posterior is described as computationally infeasible because of irregular observation times, measurement error, asynchronous sampling, and high-dimensional latent trajectory vectors. The paper therefore uses multiple imputation based on a separate continuous-time movement model (Scharf et al., 2015).

The approximation is

$$w_{ij}(t)= \begin{cases} 1, & \text{if individuals } i \text{ and } j \text{ are socially connected at time } t,\ 0, & \text{otherwise.} \end{cases}$$9

with imputed paths $w_{ij}(1)\sim \mathrm{Bern}(p_1),$0 generated by the continuous-time correlated random walk model of Johnson et al. (2008), implemented in the R package crawl. That imputation model uses an Ornstein–Uhlenbeck process for velocity, treats individuals as conditionally independent at the imputation stage, and accounts for measurement error and irregular observation times. The resulting two-step procedure draws $w_{ij}(1)\sim \mathrm{Bern}(p_1),$1 imputed paths and, during MCMC for the social model, conditions each iteration on one imputed path; in the application discussed in the paper, $w_{ij}(1)\sim \mathrm{Bern}(p_1),$2 was sufficient.

Inference for the social model itself is performed by MCMC. Because $w_{ij}(1)\sim \mathrm{Bern}(p_1),$3 is latent and binary for each pair and time, the number of latent network variables is

$w_{ij}(1)\sim \mathrm{Bern}(p_1),$4

The authors identify the number of individuals $w_{ij}(1)\sim \mathrm{Bern}(p_1),$5 as the primary computational bottleneck and report that, once $w_{ij}(1)\sim \mathrm{Bern}(p_1),$6 exceeds a few dozen, MCMC on a laptop becomes infeasible. They suggest two extensions: imposing a maximum interaction radius $w_{ij}(1)\sim \mathrm{Bern}(p_1),$7, so distant edges are deterministically zero, and imposing stronger structure on $w_{ij}(1)\sim \mathrm{Bern}(p_1),$8, such as clique or subgroup structure, to reduce dimensionality. A further inferential difficulty is the temporal-stability parameter $w_{ij}(1)\sim \mathrm{Bern}(p_1),$9, which is described as hard to estimate and prior-sensitive; in simulation it shows moderate systematic bias toward zero, although posterior inference for $t=2,\dots,T$0 is reported as robust.

5. Empirical behavior and validation

In simulation, the model is fit to latent paths $t=2,\dots,T$1 directly, so the study isolates recovery of dynamic social structure rather than errors induced by telemetry preprocessing. Performance is compared to a naive proximity network

$t=2,\dots,T$2

with $t=2,\dots,T$3 chosen favorably by matching the true mean network density as closely as possible. Under simulation truth $t=2,\dots,T$4, $t=2,\dots,T$5, $t=2,\dots,T$6, $t=2,\dots,T$7, $t=2,\dots,T$8, and $t=2,\dots,T$9, the posterior medians are $$w_{ij}(t)\mid w_{ij}(t-1)\sim \begin{cases} \mathrm{Bern}(p_{1|0}), & w_{ij}(t-1)=0,\ \mathrm{Bern}(p_{1|1}), & w_{ij}(t-1)=1. \end{cases}$$0, $$w_{ij}(t)\mid w_{ij}(t-1)\sim \begin{cases} \mathrm{Bern}(p_{1|0}), & w_{ij}(t-1)=0,\ \mathrm{Bern}(p_{1|1}), & w_{ij}(t-1)=1. \end{cases}$$1, $$w_{ij}(t)\mid w_{ij}(t-1)\sim \begin{cases} \mathrm{Bern}(p_{1|0}), & w_{ij}(t-1)=0,\ \mathrm{Bern}(p_{1|1}), & w_{ij}(t-1)=1. \end{cases}$$2, $$w_{ij}(t)\mid w_{ij}(t-1)\sim \begin{cases} \mathrm{Bern}(p_{1|0}), & w_{ij}(t-1)=0,\ \mathrm{Bern}(p_{1|1}), & w_{ij}(t-1)=1. \end{cases}$$3, $$w_{ij}(t)\mid w_{ij}(t-1)\sim \begin{cases} \mathrm{Bern}(p_{1|0}), & w_{ij}(t-1)=0,\ \mathrm{Bern}(p_{1|1}), & w_{ij}(t-1)=1. \end{cases}$$4, and $$w_{ij}(t)\mid w_{ij}(t-1)\sim \begin{cases} \mathrm{Bern}(p_{1|0}), & w_{ij}(t-1)=0,\ \mathrm{Bern}(p_{1|1}), & w_{ij}(t-1)=1. \end{cases}$$5, respectively, with all credible intervals covering the true values except $$w_{ij}(t)\mid w_{ij}(t-1)\sim \begin{cases} \mathrm{Bern}(p_{1|0}), & w_{ij}(t-1)=0,\ \mathrm{Bern}(p_{1|1}), & w_{ij}(t-1)=1. \end{cases}$$6. The inferred network substantially outperforms the proximity rule by avoiding spurious ties when individuals pass close by by chance and by detecting socially connected movement even when connected individuals remain farther apart than the proximity threshold (Scharf et al., 2015).

The principal empirical application uses telemetry from seven killer whales tagged near the Antarctic Peninsula during one week in February 2013. The sample includes three type A whales, one type B1 whale, and three type B2 whales. After continuous-time correlated-random-walk preprocessing and synchronization to a regular time grid, posterior summaries are $$w_{ij}(t)\mid w_{ij}(t-1)\sim \begin{cases} \mathrm{Bern}(p_{1|0}), & w_{ij}(t-1)=0,\ \mathrm{Bern}(p_{1|1}), & w_{ij}(t-1)=1. \end{cases}$$7 with 95% CI $$w_{ij}(t)\mid w_{ij}(t-1)\sim \begin{cases} \mathrm{Bern}(p_{1|0}), & w_{ij}(t-1)=0,\ \mathrm{Bern}(p_{1|1}), & w_{ij}(t-1)=1. \end{cases}$$8, $$w_{ij}(t)\mid w_{ij}(t-1)\sim \begin{cases} \mathrm{Bern}(p_{1|0}), & w_{ij}(t-1)=0,\ \mathrm{Bern}(p_{1|1}), & w_{ij}(t-1)=1. \end{cases}$$9 with CI $p_1$0, $p_1$1 with CI $p_1$2, $p_1$3 with CI $p_1$4, $p_1$5 with CI $p_1$6, and $p_1$7 with CI $p_1$8.

The inferred network is reported as strongly consistent with known killer whale ecology. The three type B2 whales are strongly connected throughout the study and move as a group; among the three type A whales, only whales 5 and 6 show connection, mainly during the first few days; and there is essentially no support for inter-type social ties. Among the 15 inter-type pairs, there are no posterior means above 0.5 at any time. The biological interpretation is that sociality in this dataset is expressed mainly through coordinated directional travel rather than strong direct convergence in space: the posterior $p_1$9 is near 1, while $t=1$0 is small.

6. Related meanings and neighboring frameworks

The acronym and the broader idea of dynamic social behavior are used differently across adjacent arXiv literature.

Model	Core object	Relation to DSBM
“Dynamic social networks based on movement” (Scharf et al., 2015)	Latent dynamic network $t=1$1 controlling movement	Direct movement-based DSBM
“Dynamic stochastic blockmodels for time-evolving social networks” (Xu et al., 2014)	Time-varying block interaction probabilities and class memberships	Same acronym in adjacent literature, but “Dynamic Stochastic Blockmodel”
“Dynamic Behavioral Mixed-Membership Model for Large Evolving Networks” (Rossi et al., 2012)	Time-varying mixed-membership role vectors $t=1$2 and transition matrix $t=1$3	Behavioral role model that can serve as a DSBM when behavior is graph-structural
“LISBET: a machine learning model for the automatic segmentation of social behavior motifs” (Chindemi et al., 2023)	Transformer embeddings plus HMM states for motifs	Deep-learning instantiation of dynamic behavior representation and segmentation

The contrast with the Dynamic Stochastic Blockmodel is especially sharp. That model is a state-space extension of the stochastic blockmodel for discrete-time dynamic networks, with latent block interaction logits evolving linearly and noisy block-density observations handled by an extended Kalman filter (Xu et al., 2014). It models dynamic social behavior at the group interaction level, but its state variable is blockwise connection structure rather than latent social ties inferred from movement. By contrast, the movement-based DSBM treats the network itself as latent and uses attraction and alignment in continuous space as the behavioral channel (Scharf et al., 2015).

Broader dynamic-behavior models extend the same general architecture in other directions. DBMM models evolving graph-structural roles through nonnegative matrix factorization and transition operators $t=1$4, emphasizing scalability and interpretability for very large networks (Rossi et al., 2012). LISBET separates representation learning from latent-state segmentation by first learning socially meaningful pose embeddings with a transformer and then fitting a Gaussian HMM whose states define behavioral motifs (Chindemi et al., 2023). Taken together, these works place the Dynamic Social Behavior Model within a wider family of dynamic latent-state models in which social behavior is represented as an evolving hidden structure linked to observable trajectories, interactions, or pose sequences.