Individuality in the hive - Learning to embed lifetime social behaviour of honey bees

1 INTRODUCTION . Animals living in large groups often coordinate their behaviors , resulting in emergent properties at the group level , from flocking birds to democratic elections . In most animal groups , the role an individual plays in this process is thought to be reflected in the way it interacts with group members . Technological advances have made it possible to track all individuals and their interactions in animal societies , ranging from social insects to primate groups ( Mersch et al. , 2013 ; Gernat et al. , 2018 ; Mathis et al. , 2018 ; Graving et al. , 2019 ; Pereira et al. , 2019 ) . These datasets have unprecedented scale and complexity , but understanding these data has emerged as a new and challenging problem in itself ( Pinter-Wollman et al. , 2014 ; Krause et al. , 2015 ; Brask et al. , 2020 ) . A popular approach to understand high-dimensional data is to learn semantic embeddings ( Frome et al. , 2013 ; Asgari & Mofrad , 2015 ; Camacho-Collados & Pilehvar , 2018 ; Nelson et al. , 2019 ) . Such embeddings can be learned without supervision , are interpretable , and are useful for accomplishing downstream tasks . Individuals in animal societies can be described with semantic embeddings extracted from social interaction networks using matrix factorization methods . For example , in symmetric non-negative matrix factorization ( SymNMF ) , the dot products of any two animals ’ factor vectors reconstruct the interaction matrix ( Wang et al. , 2011 ; Shi et al. , 2015 ) , see Figure 1 a and b ) . If the embeddings allow us to predict relevant behavioral properties , they serve our understanding as semantic representations . However , in temporal settings where the interaction matrices change over time , there is no straightforward extension of this algorithm . The interaction matrices at different time points can be factorized individually , but there is no guarantee that the embeddings stay semantically consistent over time , i.e . the prediction of relevant behavioral properties will deteriorate . In living systems , interaction dynamics are highly variable ; individuals differ in when they appear in the data and how long they live . Different non-overlapping groups of individuals , e.g . from different years , may not interact with each other at all . How can we find a common semantic embedding even in these extreme cases ? How do we learn embeddings that generalize to different groups and still provide insights into each individual ’ s functional role ? If animals take on roles partially determined by a common factor , such as age , how can we learn this dependency ? Several approaches to extend NMF to temporal settings have been proposed in a variety of problem settings . Yu et al . ( 2016 ) and Mackevicius et al . ( 2019 ) propose a factorization method for time series analysis . Gauvin et al . ( 2014 ) focus on the analysis of communities that are determined by their temporal activity patterns . Jiao et al . ( 2017 ) consider the case of communities from graphs over time and enforce temporal consistency with an additional loss term . Yu et al . ( 2017 ) and Wang et al . ( 2017 ) represent networks as a function of time . Temporal matrix factorization can be seen as a tensor decomposition problem , for which methods with many applications have been proposed in the literature , see Kolda & Bader ( 2009 ) for a review . In particular , time-shifted tensor decomposition methods have been used in multi-neuronal spike train analysis when recordings of multiple trials from a population of neurons are available ( Mørup et al. , 2008 ; Williams , 2020 ) . However , to our knowledge , no method yet considers the case of entities following a common trajectory depending on an observable property ( e.g. , the age of an individual ) , which we show to be a powerful inductive bias . We approach this question using honey bees , a popular model system for studying individual and collective behavior ( Elekonich & Roberts , 2005 ) . Bees allocate tasks across thousands of individuals without central control , using an age-based system : young bees care for brood , middle-aged bees perform within-nest labor , and old bees forage outside ( Seeley , 1982 ; Johnson , 2010 ) . Colonies are also organized spatially : brood is reared in the center , honey and pollen are stored at the periphery , and foragers offload nectar near the exit . Therefore , an individual ’ s role is partially reflected in its location , which allows us to evaluate whether the embeddings our method learns are semantically meaningful . We propose jointly learning two meaningful representations of honey bee social behavior : 1 ) an individuality embedding that characterizes the lifetime behavior of each individual and 2 ) a daily representation of the individual ’ s functional position in the social network that can be derived from the individuality embeddings . We show that these representations can be learned in an unsupervised fashion , using only interaction matrices of the individuals over time . We analyze a dataset obtained by tracking thousands of individually marked honey bees in two colonies , at high temporal and spatial resolution over a total of 155 days , covering entire lifespans and multiple generations . We evaluate how well the embeddings capture the semantic differences of individual honey bee development by evaluating their predictiveness for different tasks and behaviorally relevant metrics . Our method can be used to study dynamic groups in which individual units change their latent properties over time , and group composition is not fixed due to birth , death , or migration of individuals . We provide a new perspective on how to extract fundamental factors that underlie behavioral and developmental plasticity in animal groups . 2 METHODS . 2.1 TEMPORAL NMF ALGORITHM . SymNMF factorizes a matrix A ∈ RN×N+ such that it can be approximated by the product FF T , where F ∈ RN×M+ and M N : F̂ = argmin F≥0 ∥∥∥A− FF T∥∥∥2 Ai , j ≈ f ( i ) · f ( j ) T f ( i ) = Fi , : f ( i ) ∈ RM+ ( 1 ) When applied to social networks , f ( i ) can represent the role of an entity within the social network A ( Wang et al. , 2011 ; Shi et al. , 2015 ) - however , in temporal settings , factorizing the matrices for different times separately will result in semantically inconsistent factors F . Here we present a novel temporal NMF algorithm ( TNMF ) which extends SymNMF to temporal settings in which A ∈ RT×N×N+ changes over time t. We assume that the entities i ∈ { 0 , 1 , . . . , N } follow to some extent a common trajectory depending on an observable property ( for example the age of an individual ) . We represent an entity at a specific point in time t using a factor vector f+ ( t , i ) such that Ât , i , j = f+ ( t , i ) · f+ ( t , j ) T Â ∈ RT×N×N+ f+ ( t , i ) ∈ RM+ ( 2 ) In contrast to SymNMF , we do not directly factorize At to find the optimal factors that reconstruct the matrices . Instead , we decompose the problem into learning an average trajectory of factors m ( c ( t , i ) ) and structured variations from this trajectory o ( t , i ) that depend on the observable property c ( t , i ) : f ( t , i ) = m ( c ( t , i ) ) + o ( t , i ) f+ ( t , i ) = max ( 0 , f ( t , i ) ) ( 3 ) c : NT×N → N m : N→ RM+ o : NT×N → RM This decomposition is an inductive bias that allows the model to learn semantically consistent factors for entities , even if they do not share any data points ( e.g. , there is no overlap in their interaction partners ) , as long as the relationship between functional role and c ( t , i ) is stable . Note that in the simplest case c ( t , i ) = t , TNMF can be seen as a tensor decomposition model , i.e . the trajectory of all entities is aligned with the temporal dimension t of A . In our case , c ( t , i ) maps to the age of individual i at time t. While many parameterizations for the function o ( t , i ) are possible , we only consider one particular case in this work : We learn a set of individuality basis functions b ( c ( t , i ) ) ( shared among all entities ) that define a coordinate system of possible individual variations and the individuality embeddings φ , which capture to what extent each basis function applies to an entity : o ( t , i ) = K∑ k=0 φi , k · bk ( c ( t , i ) ) φ : RN×K bk : NT → R ( 4 ) where K is the number of learned basis functions . This parameterization allows us to disentangle the forms of individual variability ( individuality basis functions ) and the distribution of this variability ( individuality embeddings ) in the data . We implement the functions m ( c ( t , i ) ) and b ( c ( t , i ) ) with small fully connected neural networks with non-linearities and several hidden layers . The parameters θ of these functions and the entities ’ embeddings φ are learned jointly using minibatch stochastic gradient descent : θ̂ , φ̂ = argmin θ , φ ∥∥∥A− Â∥∥∥2 ( 5 ) Note that non-negativity is not strictly necessary , but we only consider the non-negative case in this work for consistency with prior work ( Wang et al. , 2011 ; Shi et al. , 2015 ) . Furthermore , instead of one common property with discrete time steps , the factors could depend on multiple continuous properties , i.e . c : RT×N → RP , e.g . the day and time in a intraday analysis of social networks . 2.2 REGULARIZATION . Due to the inductive bias , the model performs well without additional explicit regularization in the datasets considered in this work ( see Table 2 ) . However , we find that the model ’ s interpretability can be improved using additional regularization terms without significantly affecting its performance . We encourage sparsity in both the number of used factors and individuality basis functions by adding L1 penalties of the mean absolute magnitude of the factors f ( t , i ) and basis functions b ( c ( t , i ) ) to the objective . We encourage individuals ’ lifetimes to be represented with a sparse embedding using an L1 penalty of the learned individuality embeddings φ . We also introduce an optional adversarial loss term to encourage the model to learn embeddings that are semantically consistent over time , i.e . to only represent two entities that were present in the dataset at different times with different embeddings if this is strictly necessary to factorize the matrices A . We jointly train a discriminative network d ( φi ) that tries to classify the time of the first occurrence of all entities based on their individuality embeddings φ . The negative cross-entropy loss of this model is added as a regularization term to equation 5 in a training regime similar to generative adversarial networks ( Goodfellow et al. , 2014 ) . See appendix A.2.1 for more details and A.4 for an ablation study of the effect the individual regularization terms have on the results of the model .

Summary: This paper proposes a NMF formulation ||A-FF^T||^2 where A and F are different types of information extracted from social datasets. In the honeybee example the authors highlight, A represents the spatial relationship between bees, and F encodes the age of the bees. The authors setup F to be decomposable into two types of embeddings, one which characterizes the group activity and the other which characterizes the individual activity.