diff --git "a/_dFIT4oBgHgl3EQf9yvA/content/tmp_files/load_file.txt" "b/_dFIT4oBgHgl3EQf9yvA/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/_dFIT4oBgHgl3EQf9yvA/content/tmp_files/load_file.txt" @@ -0,0 +1,996 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf,len=995 +page_content='Proceedings of Machine Learning Research – Under Review:1–20, 2023 Full Paper – MIDL 2023 submission DBGDGM: Dynamic Brain Graph Deep Generative Model Alexander Campbell∗1,2 ajrc4@cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='uk Simeon Spasov∗1 ses88@cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='uk Nicola Toschi1 Pietro Li`o3,4 1 Department of Computer Science and Technology, University of Cambridge, United Kingdom 2 The Alan Turing Institute, United Kingdom 3 University of Rome Tor Vergata, Italy 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Martinos Center for Biomedical Imaging, Harvard Medical School, United States Editors: Under Review for MIDL 2023 Abstract Graphs are a natural representation of brain activity derived from functional magnetic imaging (fMRI) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' It is well known that clusters of anatomical brain regions, known as functional connectivity networks (FCNs), encode temporal relationships which can serve as useful biomarkers for understanding brain function and dysfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Previous works, however, ignore the temporal dynamics of the brain and focus on static graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In this paper, we propose a dynamic brain graph deep generative model (DBGDGM) which simul- taneously clusters brain regions into temporally evolving communities and learns dynamic unsupervised node embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Specifically, DBGDGM represents brain graph nodes as embeddings sampled from a distribution over communities that evolve over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We parameterise this community distribution using neural networks that learn from subject and node embeddings as well as past community assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Experiments demonstrate DBGDGM outperforms baselines in graph generation, dynamic link prediction, and is com- parable for graph classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Finally, an analysis of the learnt community distributions reveals overlap with known FCNs reported in neuroscience literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Keywords: Dynamic graph, generative model, functional magnetic resonance imaging 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Introduction Functional magnetic resonance imaging (fMRI) is a non-invasive imaging technique pri- marily used to measure blood-oxygen level dependent (BOLD) signal in the brain (Huettel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' A natural representation of fMRI data is as a discrete-time graph, henceforth referred to as a dynamic brain graph (DBG), consisting of a set of fixed nodes correspond- ing to anatomically separated brain regions and a set of time-varying edges determined by a measure of dynamic functional connectivity (dFC) (Calhoun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' DBGs have been widely used in graph-based network analysis for understanding brain function (Hirsch and Wohlschlaeger, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Raz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2016) and dysfunction (Alonso Mart´ınez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Dautricourt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' ∗ Contributed equally © 2023 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Campbell, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Spasov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Toschi & P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Li`o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='11408v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='LG] 26 Jan 2023 Campbell Spasov Toschi Li`o Recently, there is growing interest in using deep learning-based methods for learning representations of graph-structured data (Goyal and Ferrara, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Hamilton, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' A graph representation typically consists of a low-dimensional vector embedding of either the entire graph (Narayanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2017) or a part of it’s structure such as nodes (Grover and Leskovec, 2016), edges (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019), or sub-graphs (Adhikari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Although originally formulated for static graphs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' not time-varying), several existing methods have been extended (Mahdavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020), and new ones proposed (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Sankar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020), for dynamic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The embeddings are usually learnt in either a supervised or unsupervised fashion and typically used in tasks such as node classification (Pareja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020) and dynamic link prediction (Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' To date, very few deep learning-based methods have been designed for, or existing methods applied to, representation learning of DBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Those that do, tend to use graph neural networks (GNNs) that are designed for learning node- and graph-level embeddings for use in graph classification (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Dahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Although node/graph- level embeddings are effective at representing local/global graph structure, they are less adept at representing topological structures in-between these two extremes such a clusters of nodes or communities (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Recent methods that explicitly incorporate community embeddings alongside node embeddings have shown improved performance for static graph representation learning tasks (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Cavallari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' How to leverage the relatedness of graph, node, and community embeddings in a unified framework for DBG representation learning remains under-explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We refer to Appendix A for a summary of related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Contributions To address these shortcomings, we propose DBGDGM, a hierarchical deep generative model (DGM) designed for unsupervised representing learning on DBGs derived from multi-subject fMRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Specifically, DBGDGM represents nodes as embed- dings sampled from a distribution over communities that evolve over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The community distribution is parameterized using neural networks (NNs) that learn from graph and node embeddings as well as past community assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We evaluate DBGDGM on multi- ple real-world fMRI datasets and show that it outperforms state-of-the-art baselines for graph reconstruction, dynamic link prediction, and achieves comparable results for graph classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Problem formulation We consider a dataset of multi-subject DBGs derived from fMRI data D ≡ G(1:S, 1:T) = {G(s, t)}S, T s, t=1 that share a common set of nodes V = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' , vN} over T ∈ N timepoints for S ∈ N subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Each G(s, t) ∈ G(1:S, 1:T) denotes a non-attributed, unweighted, and undirected brain graph snapshot for the s-th subject at the t-th timepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We define a brain graph snapshot as a tuple G(s, t) = (V, E(s, t)) where E(s, t) ⊆ V × V denotes an edge set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The i-th edge for the s-th subject at the t-th timepoint e(s, t) i ∈ E(s, t) is defined e(s, t) i = (w(s,t) i , c(s,t) i ) where w(s,t) i is a source node and c(s,t) i is a target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We assume each node corresponds to a brain region making the number of nodes |V| = V ∈ N fixed over subjects and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We also assume edges correspond to a measure of dFC allowing the number of edges |E(s, t)| = E(s, t) ∈ N vary over subjects and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We further assume there exists 2 DBGDGM: Dynamic Brain Graph Deep Generative Model K ∈ N clusters of nodes, or communities, the membership of which dynamically changes over time for each subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Let z(s, t) i ∈ [1 : K] denote the latent community assignment of the i-th edge for the s-th subject at the t-th timepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For each subject’s DBG our aim is to learn, in an unsupervised fashion, graph α(s) ∈ RHα, node φ(s, t) 1:N = [φ(s, t) n ] ∈ RN×Hφ, and community ψ(s, t) 1:K = [ψ(s, t) k ] ∈ RK×Hψ representations of dimensions Hα, Hφ, Hψ ∈ N, respectively, for use in a variety of downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Method Figure 1: Plate diagram for DBGDGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' La- tent and observed variables are denoted by white-and gray-shaded circles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Solid black squares denote non-linear map- pings parameterized by NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' DBGDGM defines a hierarchical deep gen- erative model and inference network for the end-to-end learning of graph, node, and community embeddings from multi- subject DBG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Specifically, DBGDGM treats the embeddings and edge commu- nity assignments as latent random vari- ables collectively denoted Ω(s, t) = {α(s), φ(s, t) 1:N , ψ(s, t) 1:K , {z(s, t) i }E(s, t) i=1 }, which along with the observed DBGs, defines a proba- bilistic latent variable model with joint den- sity pθ(G1:S, 1:T , Ω1:S, 1:T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Generative model Graph embeddings We begin the gen- erative process by sampling graph embed- dings from a prior α(s) ∼ pθα(α(s)) imple- mented as a normal distribution following pθα(α(s)) = Normal(0Hα, IHα) (1) where 0Hα is a matrix of zeros and IHα is a identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Each embedding is a vector α(s) ∈ RHα representing subject-specific information that remains fixed over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Node and community embeddings Next, let φ(s, t) n ∈ RHφ and ψ(s, t) k ∈ RHψ denote the n-th node and the k-th community embedding, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' To incorporate tempo- ral dynamics, we assume node and community embeddings are related through Markov chains with prior transition distributions φ(s, t) n ∼ pθφ(φ(s, t) n |φ(s, t−1) n , α(s)) and ψ(s, t) k ∼ pθψ(ψ(s, t) k |ψ(s, t−1) k , α(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We specify each prior to be a normal distribution following pθφ(φ(s, t) n |φ(s, t−1) n , α(s)) = Normal(φ(s, t−1) n , σφIHφ) (2) pθψ(ψ(s, t) k |ψ(s, t−1) k , α(s)) = Normal(ψ(s, t−1) k , σψIHψ) (3) where the graph embeddings are used for initializing the means, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', φ(s, 0) n = α(s), ψ(s, 0) k = α(s) and the standard deviations σφ, σψ ∈ R>0 are hyperparameters controlling how smoothly each embedding changes between consecutive timepoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 3 Campbell Spasov Toschi Li`o Edge generation We next describe the edge generative process of a graph snapshot G(s, t) ∈ G(1:S, 1:T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Similar to Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (2019),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' for each edge e(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i = (w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' c(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ) ∈ E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) we first sample a latent community assignment z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ∈ [1 : K] from a conditional prior z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ∼ pθz(z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i |w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ) implemented as a categorical distribution pθz(z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i |w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ) = Categorical(π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) θz ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) θz = MLPθz(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) wi ) (4) where MLPθz : RHφ → RK is a Lz-layered multilayered perception (MLP) that parame- terizes community probabilities using node embeddings indexed by w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In other words, each source node w(s, t) i is represented as a mixture of communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' A linked target node c(s, t) i ∈ [1 : N] is then sampled from the conditional likelihood c(s, t) i ∼ pθc(c(s, t) i |z(s, t) i ) which is also implemented as a categorical distribution pθc(c(s, t) i |z(s, t) i ) = Categorical(π(s, t) θc ), π(s, t) θc = MLPθc(ψ(s, t) zi ) (5) where MLPθc : RHψ → RN is a Lc-layered MLP that parameterizes node probabilities using community embeddings indexed by z(s, t) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' That is, each community assignment z(s, t) i is represented as a mixture of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' By integrating out the latent community assignment variable p(c(s, t) i |w(s, t) i ) = � z(s, t) i ∈[1:K] pθc(c(s, t) i |z(s, t) i )pθz(z(s, t) i |w(s, t) i ) (6) we define the likelihood of node c(s, t) i being a linked neighbor of node w(s, t) i , in a given graph snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Factorized generative model Given this model specification, the joint probability of the observed data and the latent variables can be factorized following pθ(G1:S 1:T , Ω1:S,1:T ) = S � s=1 � pθα(α(s)) T � t=1 � V� n=1 pθφ(φ(s, t) n |φ(s, t−1) n ) K � k=1 pθψ(ψ(s,t) k |ψ(s,t−1) k ) E(s, t) � i=1 pθz(z(s, t) i |φ(s, t) wi )pθc(c(s, t) i |ψ(s, t) zi ) �� (7) where θ = {θc , θz} is the set of generative model parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', NN weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The gener- ative model of DBGDGM summarized in Appendix B 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Inference network To learn the embeddings, we must infer the posterior distribution over all latent variables conditioned on the observed data pθ(Ω(1:S, 1:T)|G(1:S, 1:T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' However, exact inference is in- tractable due the log marginal likelihood requiring integrals that are hard to evaluate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 4 DBGDGM: Dynamic Brain Graph Deep Generative Model log pθ(G(1:S, 1:T)) = � Ω log pθ(G(1:S, 1:T), Ω(1:S, 1:T))dΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' As a result, we use variational infer- ence (Jordan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 1999) to approximate the true posterior with a variational distribution qλ(Ω(1:S,1:T)) with parameters λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' To do this, we maximize a lower bound on the log marginal likelihood of the DBGs, referred to as the ELBO (evidence lower bound), defined as LELBO(θ, λ) = Eqλ � log pθ(G1:S, 1:T , Ω1:S, 1:T ) qλ(Ω(1:S, 1:T)) � ≤ log pθ(G(1:S, 1:T)) (8) where Eqλ[·] denotes the expectation taken with respect to the variational distribution qλ(Ω(1:S, 1:T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' By maximizing the ELBO with respect to the generative and variational parameters θ and λ we train our generative model and perform Bayesian inference, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Structured variational distribution To ensure a good approximation to true posterior, we retain the Markov properties of the node and community embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' This results in a structured variational distribution (Hoffman and Blei, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Saul and Jordan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 1995) which factorizes following qλ(Ω(1:S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 1:T)) = S � s=1 � qλα(α(s)) T � t=1 � V� n=1 qλφ(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) n | φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) n ) K � k=1 qλψ(ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) k | ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) k ) E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) � i=1 qλz(z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i | φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) wi ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) ci ) �� (9) where each distribution is specified to mimic the structure of the generative model so that qλα(α(s)) = Normal(µ(s) λα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' σ(s) λα ) (10) qλφ(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) n |φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) n ) = Normal(µ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λφ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' σ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λφ ) {µ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λφ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' σ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λφ } = GRUλφ(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) n ) (11) qλψ(ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) k |ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) n ) = Normal(µ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' σ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λψ ) {µ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' σ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λψ } = GRUλψ(ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) k ) (12) qλz(z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i |φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) wi ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) ci ) = Categorical(π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λz ) π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λz = MLPλz(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) wi ⊙ φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) ci ) (13) where GRUλj : RHj → RHj is a Lj-layered GRU for each j ∈ {φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' ψ} and MLPλz : RHφ → RK is Lz-layered MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Furthermore, we use MLPs to initialize the GRUs with the graph embeddings such that φ(s, 0) n = MLPλφ(α(s)) and ψ(s, 0) k = MLPλψ(α(s)) where MLPλj : RNα → RNj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' This allows for subject-specific variation to be incorporated in the temporal dynamics of the node and community embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Another difference with the generative model is now the variational distribution of the community assignment qλz(·) in- cludes information from neighboring nodes via c(s, t) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Finally, we use the same NN from the generative model to parameterize the variational distribution of the community assignment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', λz = θz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' This not only spares additional trainable parameters for the variational dis- tribution but also further links the variational parameters of qλ(·) to generative parameters of pθ(·) resulting in more robust learning (Farnoosh and Ostadabbas, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The set of pa- rameters for the inference network is therefore λ = {λα = {µ(s) λα, σ(s) λα }S s=1, λφ, λψ, λz = θz}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 5 Campbell Spasov Toschi Li`o Model HCP UKB NLL (↓) MSE (↓) NLL (↓) MSE (↓) CMN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='999 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='029 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='050 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='005 * 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='861 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='017 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='050 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='003 * VGAE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='857 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='017 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='051 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='002 * 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='851 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='027 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='061 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='002 * OSBM 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='808 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='026 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='051 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='003 * 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='726 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='039 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='052 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='003 * VGRAPH 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='569 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='046 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='022 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='004 * 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='716 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='037 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='020 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='003 * VGRNN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='674 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='034 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='011 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='003 * 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='649 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='035 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='014 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='002 * ELSM 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='924 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='040 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='081 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='002 * 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='809 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='024 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='115 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='003 * DBGDGM 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='587 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='001 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='002 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='586 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='004 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='003 AUROC (↑) AP (↑) AUROC (↑) AP (↑) CMN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='665 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='007 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='654 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='006 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='678 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='004 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='668 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='005 * VGAE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='661 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='010 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='674 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='008 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='688 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='010 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='607 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='009 * OSBM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='655 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='027 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='675 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='024 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='678 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='032 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='682 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='033 * VGRAPH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='689 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='004 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='682 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='002 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='664 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='002 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='621 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='001 * VGRNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='689 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='007 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='698 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='006 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='698 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='009 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='696 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='007 * ELSM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='669 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='004 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='662 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='002 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='661 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='001 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='662 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='002 * DBGDGM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='768 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='732 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='786 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='762 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='038 Table 1: Graph reconstruction (top) and dynamic link prediction (bottom) results (mean ± standard deviation over 5 runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' First and second-best results shown in bold and underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Statistically significant difference from DBGDGM marked *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Training objective Substituting the variational distribution from (9) and the joint dis- tribution from (7) into the ELBO (8) gives the full training objective which can be optimized using stochastic gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We estimate all gradients using the reparameterization trick (Kingma and Welling, 2013) and the Gumbel-softmax trick (Jang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Mad- dison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We refer to Appendix B further details on the ELBO and learning the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Experiments We evaluate DBGDGM against baseline models on the tasks of graph reconstruction, dy- namic link prediction, and graph classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Each task is designed to evaluate the use- fulness of the learnt embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Datasets We construct two multi-subject DBG datasets using publicly available fMRI scans from the Human Connectome Project (HCP) (Van Essen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2013) and UK Biobank (UKB) (Sudlow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We randomly sample S = 300 subjects ensuring an even male/female split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' To create DBGs, we parcellate each scan into V = 360 region-wise BOLD signals using the Glasser atlas (Glasser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2016), apply sliding-window Pearson correlation (Calhoun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2014) with a non-overlapping window of size and stride of 30, and threshold the top 5% values of the lower triangle of each correlation matrix as connected following Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The described procedure gives T = 16 graph snapshots for each subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Biological sex is taken as graph-level labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We refer to Appendix C for further details on each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 6 DBGDGM: Dynamic Brain Graph Deep Generative Model Baselines We compare DBGDGM against a range of different unsupervised probabilistic baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For static baselines, we include variational graph autoencoder (VGAE) (Kipf and Welling, 2016b), a deep generative version of the overlapping stochastic block model (OSBM) (Mehta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019), and vGraph (VGRAPH) (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For dynamic baselines we include variational graph recurrent neural network (VGRNN) (Hajiramezanali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019) and evolving latent space model (ELSM) (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For the graph re- construction and link prediction tasks, we also include a heuristic baseline based on common neighbors between nodes at previous snapshots (CMN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Finally, for graph classification we include a support vector machine which takes as import static FC matrices (FCM) (Abra- ham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Further details about baseline model can be found in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Implementation We split both datasets into 80/10/10% training/validation/test data along the time dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We train all models using the Adam optimizer (Kingma and Ba, 2014) with decoupled weight decay (Loshchilov and Hutter, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' All baseline hy- perparameters are set following their original implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For DBGDGM, choose the number of communities K based on validation NLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Finally, we train all models 5 times using different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Implementation details can be found in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Evaluation metrics For graph reconstruction, we evaluate the probability of the edges in the test dataset using negative log-likelihood (NLL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We also compare the mean-squared error (MSE) between actual and reconstructed node degree over all test snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For dy- namic link prediction, we sample an equal number of positive and negative edges in the test dataset and measure performance using area under the receiver operator curve (AUROC) and average precision (AP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Finally, for graph classification we predict the biological sex for each subjects’ DBG and evaluate on accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' To predict graph labels, we average node embeddings per subject for the baselines and the community embeddings for DBGDGM before training a SVM using 10-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For comparing models, we use the al- most stochastic order (ASO) test (Dror et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019) with significance level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='05 and correct for multiple comparisons (Bonferroni, 1936).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Results Dynamic graph reconstruction and link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We summarize the average test results of all models over 5 runs using optimally tuned hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' From Table 1, it is clear that DBGDGM outperforms baselines on both tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For graph reconstruction, DBGDGM shows an 18% and 30% relative improvement in NLL on HCP and UKB, re- spectively, compared to the second-best baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For dynamic link prediction, the relative improvement is > 11% in AUCROC and > 5% in AP compared to second-best baselines de- pending on dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We attribute these statistically significant gains to DBGDGM’s ability to learn dynamic brain connectivity more effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Graph classification For graph classification, DBGDGM achieves ∼ 75% accuracy for HCP and ∼ 73% for UKB (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We outperform 4 baselines and show indiscernible performance to VGAE and OSBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' To show the interpretative power of DBGDGM, we re- run the graph classification experiment for HCP with the embeddings of each community separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We find a community which comprises brain regions in the Cingulo-opercular 7 Campbell Spasov Toschi Li`o Figure 2: Graph classification results (5 runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Statistical significance from DBGDGM marked *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Figure 3: Overlap between communities learned by DBGDGM and FCNs from Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (CON) and the Somatomotor (SMN) networks, which achieves 68% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' This finding is in agreement with studies that show SMN is predictive of gender (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Interpretability analysis We use the learnt distributions over the nodes to calculate overlap between each community and known functional connectivity networks (FCNs) from Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (2019) (see Appendix F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Figure 3 shows that DBGDGM finds communities that sig- nificantly overlap with existing FCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In particular, nodes in community 1 almost fully corresponds to the visual network (VIS1 + VIS2), which is in keeping with the nature of the experiment (the resting state data was acquired with eyes open and cross-hair fixation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Remarkably, the second and third most homogeneous communities correspond to a large degree to the DMN, which is well known to dominate resting state activity as a whole (Yeshurun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The inspection of additional communities and respective predictive power, along with their evolution in time at the region-of-interest granularity, has the poten- tial to unveil the yet largely unexplored relationships between dynamic brain connectivity changes and, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' psychiatric or neurological disorders (Heitmann and Breakspear, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='HCP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='AUD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='CON ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='DAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='80% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='DMN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='FPN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='LAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='ORA ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='VGRNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='ELSM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='DBGDGM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='VGRAPH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='DBGDGM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='VGRNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='FCMDBGDGM: Dynamic Brain Graph Deep Generative Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Conclusion We propose DBGDGM, a hierarchical DGM designed for unsupervised representing learning of DBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Specifically, DBGDGM jointly learns graph-, community-, and node-level embed- dings that outperform baselines on classification, interpretability, and dynamic link predic- tion with statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Moreover, an analysis of the learnt dynamic community- node distributions shows significant overlap with existing FCNs from neuroscience literature further validating our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Acknowledgments This work is supported by The Alan Turing Institute under the EPSRC grant EP/N510129/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Data were provided [in part] by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Re- search;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' and by the McDonnell Center for Systems Neuroscience at 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+page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' URL https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='com/science/article/pii/S105381191401012X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Chao Zhang, Chase C Dougherty, Stefi A Baum, Tonya White, and Andrew M Michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Functional connectivity predicts gender: Evidence for gender differences in resting brain connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Human brain mapping, 39(4):1765–1776, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Wenbin Zhang, Liming Zhang, Dieter Pfoser, and Liang Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Disentangled dynamic graph deep generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In Proceedings of the 2021 SIAM International Conference on Data Mining (SDM), pages 738–746.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' SIAM, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Lekui Zhou, Yang Yang, Xiang Ren, Fei Wu, and Yueting Zhuang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Dynamic network embedding by modeling triadic closure process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In Proceedings of the AAAI conference on artificial intelligence, volume 32, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Related work Dynamic graph generative models Classic generative models for graph-structured data are designed for capturing a small set of specific properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', degree distribution, eigenvalues, modularity) of static graphs (Erdos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 1960;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Barab´asi and Albert, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Nowicki and Snijders, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' DGMs that exploit the learning capacity of NNs are able to learn more expressive graph distributions (Mehta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Kipf and Welling, 2016b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Sarkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Recent DGMs for dynamic graphs are majority VAE-based (Kingma and Welling, 2013) and cannot learn community representations (Hajiramezanali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Gracious et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The few that do, are designed for static graphs (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Cavallari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Learning representations of dynamic brain graphs Unsupervised representation learning methods for DBGs tend to focus on clustering DBGs into a finite number of con- nectivity patterns that recur over time (Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Spencer and Goodfellow, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Community detection is another commonly used method but mainly applied to static brain graphs (Pavlovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Esfahlani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Extensions to DBGs are typically not end-to-end trainable and do not scale to multi-subject datasets (Ting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Martinet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Recent deep learning-based methods are predominately GNN-based (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Dahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Unlike DBGDGM, these methods are supervised and focus on learning deterministic node- and graph-level representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Method B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Generative model Algorithm 1 summarizes the generative model for DBGDGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 15 Campbell Spasov Toschi Li`o Algorithm 1: DBGDGM generative model Input: {E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t)}S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' T s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t=1 Hyperparameters: K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Hα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Hψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Hφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Lψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Lφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Lz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' σ2 ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' σ2 φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Initialize: D ← ∅ for s ← 1 to S do α(s) ∼ p(α(s)) = Normal(0Hα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' IHα) for t ← 1 to T do for k ← 1 to K do ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='t) k ∼ p(ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) k |ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) k ) = Normal(ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' σψIHψ) end for n ← 1 to V do φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='t) n ∼ p(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) n |φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) n ) = Normal(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' σφIHφ) end ˜E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) ← ∅ for i ← 1 to |E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t)| do z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ∼ p(z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i |w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ) = Categorical(fθπ(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) wi )) c(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ∼ p(c(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i |z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ) = Categorical(fθπ(ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) zi )) ˜E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) ← ˜E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) ∪ {(w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' c(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i )} end G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) ← (V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' ˜E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t)) D ← D ∪ {G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t)} end end B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Training objective and learning the parameters Substituting the variational distribution from (9) and the joint distribution from (7) into the ELBO (8) gives the full training objective defined as LELBO(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' λ) = S � s=1 T � t=1 E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) � i=1 � Eqλz qλψ � log pθ(c(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i |w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) zi ) � − Eqλφ � DKL[qλz(z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i | φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) wi ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) ci )||pθz(z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i | φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) wi )] �� − S � s=1 � DKL[qλα(α(s))||pθα(α(s))] T � t=1 � (14) − V � n=1 Eqλφ � DKL[qλφ(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) n | φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) n )||pθφ(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) n | φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) n )] � − K � k=1 Eqλψ � DKL[qλψ(ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) k | ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) k )||pθψ(ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) k | ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) k )] ��� 16 DBGDGM: Dynamic Brain Graph Deep Generative Model where DKL[·||·] denotes the Kullback-Leibler (KL) divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' By maximizing (14), the parameters (θ, λ) of the generative model and inference network can be jointly learnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Learning the parameters In order to use efficient stochastic gradient-based optimiza- tion techniques (Robbins and Monro, 1951) for learning (θ, λ), the gradient of the ELBO has to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The main challenge of this is obtaining gradients of the variables under expectation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', Eq∗[·], since they are sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' To allow gradients to flow through these sampling steps, we use the reparameterization trick (Kingma and Welling, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Rezende et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2014) for the normal distributions and the Gumbel-softmax trick (Jang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Maddison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2016) for the categorical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' All gradients are now easily com- puted via back-propagation (Rumelhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 1986) making DBGDGM end-to-end train- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In addition, we analytically calculate the KL terms for both normal and categorical distributions, which leads to lower variance gradient estimates and faster training as com- pared to noisy Monte Carlo estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Datasets To create multi-subject DBG datasets, we use real fMRI scans from the UK Biobank (Sud- low et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2015) and Human Connectome Project (Van Essen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Both data sources represent well-characterized population cohorts that have undergone standardized neuroimaging and clinical assessments to ensure high quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' UK Biobank1 (UKB) The UKB dataset consists of S = 300 resting-rate fMRI scans (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 3D image of the brain taken over consecutive timepoints) randomly sampled from the v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='3 January 2017 release ensuring an equal male/female split (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' sex balanced) with an age range of 44 − 57 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The total number of images for each scan is 490 timepoints (6 minutes duration with a repetition time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='74s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The dataset is minimally preprocessed following the pipeline described in Alfaro-Almagro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Human Connectome Project2 (HCP) The HCP dataset similarly consists of S = 300 sex balanced resting-state fMRI scans randomly sampled from the S1200 release with an age range of 22 − 35 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Only images from the first scanning-session using left-right phase encoding are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The total number of images for each scan is 1, 200 timepoints (15 minutes duration with a repetition time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='72s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The dataset is minimally preprocessed following the pipeline described in Glasser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (2013) Further preprocessing The fMRI scans from each dataset are further preprocessed to create DBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Firstly, each scan is transformed into a multivariate timeseries of BOLD signals using the Glasser atlas (Glasser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2016) to average voxels within V = 360 brain regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Next, to ensure comparability with UKB, we truncate the length of HCP timeseries to 490 timepoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Following the commonly used sliding-window method (Calhoun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2014), we use Pearson correlation to calculate FC matrices within non-overlapping windows of length 1 < W ≤ 490 along the temporal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' At every window, we create an edge set of a unweighted and undirected graph with no self-edges by thresholding the top 1 ≤ ϵ < 100 percentile values of the lower triangle of the FC matrix (excluding the principal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='ukbiobank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='uk 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='humanconnectome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='org 17 Campbell Spasov Toschi Li`o diagonal) as connected following Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For both datasets, we choose W = 30 and ϵ = 5 resulting in T = ⌊490/30⌋ = 16 graph snapshots each with E(s, t) = ⌊(360(360 − 1)/2)(5/100)⌋ = 3, 231 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Baselines We compare DBGDGM against a range of static and dynamic unsupervised graph repre- sentation learning baseline models, all with publicly available code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In particular, we focus on baselines that are generative and can quantify uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We leave comparisons to popular deterministic baselines such as DynamicTriad (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2018), DySAT (Sankar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020), and DynNode2Vec (Mahdavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2018) for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Furthermore, since all of the baselines were originally designed to model large single-graph datasets, we had to adapt each implementation to work with smaller multi-graph datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Variational graph auto encoder3 (VGAE) (Kipf and Welling, 2016b) An extension of the variational autoencoder (Kingma and Welling, 2013) (VAE) for graph structured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Specifically, VGAE uses a graph convolutional network (GCN) (Kipf and Welling, 2016a) to learn a distribution over node embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Originally designed for static graphs, we train VGAE on each dynamic graph snapshot independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Overlapping stochastic block model4 (OSBM) (Mehta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019) A deep gener- ative version of the overlapping stochastic block model (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In particular, OSBM places a stick-breaking prior over the number of communities which allows the model to automatically infer the optimal number of communities from the data during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Similar to VGAE, OSBM uses a GCN to parameterize the distribution over node embed- dings and is designed for static graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Variational graph RNN5 (VGRNN) (Hajiramezanali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019) An extension of VGAE for dynamic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Using a modified graph RNN architecture, VGRNN is able to learn dependencies between and within changing graph topology over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Similar to DBGDGM, the prior distribution over node embeddings is parameterized using hidden states from previous timepoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Evolving latent space model6 (ELSM) (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019) A generative model for dynamic graphs that learns node embeddings and performs community detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In par- ticular, node embeddings are initially sampled from a Gaussian mixture model over com- munities and then evolved over time using an LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Unlike the previous baselines, ELSM does not use a GNNs to parameterize model distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' vGraph7 (VGRAPH) (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019) Similar to DBGDGM, VGRAPH simultane- ously learns node embeddings and community assignments by modeling nodes as being 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='com/tkipf/gae 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='com/nikhil-dce/SBM-meet-GNN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='com/VGraphRNN/VGRNN 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='com/sh-gupta/ELSM 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='com/fanyun-sun/vGraph 18 DBGDGM: Dynamic Brain Graph Deep Generative Model generated from a mixture of communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The generative process of VGRAPH also re- lies on edge information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Since VGRAPH only models static graphs, we train it on each dynamic graph snapshot independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Common neighbors (CMN) In light of recent work demonstrating that heuristic meth- ods are able to outperform deep-learning based models on dynamic link prediction tasks (Skard- ing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Poursafaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2022), we include our own heuristic-based generative model baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' More formally, let π(t) vi ∈ [0, 1]V denote a vector of Jaccard index scores for node v(t) i ∈ V with all other nodes v(t) j ∈ V for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The Jaccard index between two nodes v(t) i , v(t) j ∈ V is defined |Γ(v(t) i ) ∩ Γ(v(t) j )|/|Γ(v(t) i ) ∪ Γ(v(t) j )| where Γ(v(t) i ) denotes the set of neighbors of node v(t) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We define the probability of node v(t) i having a linked neighbor v(t) j at snapshot t as p(v(t) j |v(t) i ) = Categorical(π(t−1) vi ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (15) This simple generative model captures the intuition that nodes are more likely to form links if they had common neighbors in a previous snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Implementation details Software and hardware All models are developed in Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='7 (Python Core Team, 2019) using scikit-learn 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='1 (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2011), PyTorch(Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019), and numpy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='1 (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Statistical significance tests are carried out using deep- significance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='1 (Ulmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Experiments are performed on a Linux server (Debian 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='113-1) with a NVIDIA RTX A6000 GPU with 48 GB memory and 16 CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Training and testing All baselines are implemented as per the original paper and/or code repository given in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For the static graph baselines VGAE, OSBM, VGRAPH we train on each snapshot independently and use the node and/or community embeddings at the last training snapshot to make predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Hyperparameter optimization We use model and training hyperparameter values de- scribed in the original implementation of each baseline as a starting point for tuning on the validation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Since searching for optional values for each hyperparameter con- figuration was outside the scope of this paper, we focus mainly on tuning the dimensions of hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For DBGDGM, we use a learning rate of 1e-4 with a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We choose the number of communities K ∈ {3, 6, 8, 12, 16, 24} based on lowest average validation NLL (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In the generative model, we fix the temporal smoothness hyperparameters σφ = σψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In the inference network, we fix the number of layers for all NNs to Lφ = Lψ = Lz = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For the Gumbel-softmax reparameterization trick we anneal the softmax temperature parameter starting from a maximum of 1 to a minimum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='05 at a rate of 3e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Finally, we train all models for 1, 000 epochs using early-stopping with a patience of 15 based on the lowest validation NLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Interpretability analysis Using DBGDGM, for each community we average the node distributions across subjects and timepoints and take the top 10% most probable nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We use these high probability 19 Campbell Spasov Toschi Li`o 3 6 8 12 16 24 Number of communities 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='65 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='80 Validation nll hcp 3 6 8 12 16 24 Number of communities 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='65 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='75 Validation nll ukb Figure 4: Elbow plot for finding the optimal number of communities K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' nodes to calculate overlap between each community and the brain regions that comprise each functional network from Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' More specifically, the coloured proportions in Figure 3 represent the proportion of top nodes in each community, which belong to a given functional network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Abbreviation Functional network AUD Auditory network CON Cingulo-opercular network DAN Dorsal-attention network DMN Default mode network FPN Frontoparietal network LAN Language network ORA Orbito-affective network PMM Posterior-multimodal network SMN Somatomotor network VIS1 Visual network 1 VIS2 Visual network 2 VMM Ventral-multimodal network Table 2: Functional connectivity networks (FCNs) from Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (2019) 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'}