Consistent Latent Position Estimation and Vertex Classification for Random Dot Product Graphs

Sussman, Daniel L.; Tang, Minh; Priebe, Carey E.

doi:10.1109/tpami.2013.135

Cited by 90 publications

(104 citation statements)

References 23 publications

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“…It is proved in [14,41] that the adjacency spectral embedding provides a consistent estimate of the true latent positions in random dot product graphs. The key to this result is a tight concentration, in Frobenius norm, of the adjacency spectral embedding, X, about the true latent positions X.…”

Section: Definition 4 Given An Adjacency Matrixmentioning

confidence: 99%

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Community Detection and Classification in Hierarchical Stochastic Blockmodels

Lyzinski

Tang

Athreya

et al. 2017

IEEE Trans. Netw. Sci. Eng.

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131

111

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Abstract-In disciplines as diverse as social network analysis and neuroscience, many large graphs are believed to be composed of loosely connected smaller graph primitives, whose structure is more amenable to analysis We propose a robust, scalable, integrated methodology for community detection and community comparison in graphs. In our procedure, we first embed a graph into an appropriate Euclidean space to obtain a low-dimensional representation, and then cluster the vertices into communities. We next employ nonparametric graph inference techniques to identify structural similarity among these communities. These two steps are then applied recursively on the communities, allowing us to detect more fine-grained structure. We describe a hierarchical stochastic blockmodel-namely, a stochastic blockmodel with a natural hierarchical structure-and establish conditions under which our algorithm yields consistent estimates of model parameters and motifs, which we define to be stochastically similar groups of subgraphs. Finally, we demonstrate the effectiveness of our algorithm in both simulated and real data. Specifically, we address the problem of locating similar sub-communities in a partially reconstructed Drosophila connectome and in the social network Friendster.

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Section: Definition 4 Given An Adjacency Matrixmentioning

confidence: 99%

“…Therefore, each entry of U P (A−P )U P is of order O(log n) asymptotically almost surely, and as a consequence, √ nρ n ) (see [41]).…”

Section: Proof Of Theorem 15mentioning

confidence: 99%

Community Detection and Classification in Hierarchical Stochastic Blockmodels

Lyzinski

Tang

Athreya

et al. 2017

IEEE Trans. Netw. Sci. Eng.

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131

111

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“…A further generalization asserts that each node is in its own group, and therefore has a "latent position" that characterizes its probability of connecting with other nodes (homologous to latent variable models in neural coding) [88]. A particularly popular version of these models assumes that the probability of connections between a pair of nodes is equal to the dot product between the nodes' latent positions [89][90][91]. In these models, an extensive set of theoretical investigations have established the kinds of claims we desire when using a statistical model to make inferences about our data [92,93], as well as a number of extensions, including a generalized random dot product [94], a random dot product with node-wise covariates [95], and a latent structure model [96] (for review, see [97]).…”

Section: Statistical Models Of Connectomesmentioning

confidence: 99%

Connectal Coding: Discovering the Structures Linking Cognitive Phenotypes to Individual Histories

Vogelstein

Bridgeford

Pedigo

et al. 2019

Preprint

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Cognitive phenotypes characterize our memories, beliefs, skills, and preferences, and arise from our ancestral, developmental, and experiential histories. These histories are written into our brain structure through the building and modification of various brain circuits. Connectal coding, by way of analogy with neural coding, is the art, study, and practice of identifying the network structures that link cognitive phenomena to individual histories. We propose a formal statistical framework for connectal coding and demonstrate its utility in several applications spanning experimental modalities and phylogeny.Scientific explanations are characterized by modeling mechanisms of phenomena that describe (i) the constituent parts, (ii) the properties of those parts, and (iii) the interactions among them [1]. What we measure and model is often limited by technology.In brain sciences, 20 th century innovations enabled studying the parts and their properties, while studying interactions was limited [2] and laborious [3]. For example, nanoscale electron microscopy (EM) enabled studies of ultrastructure [4], microscale light microscopy and physiology enabled studies of single cells [5], and macroscale magnetic resonance imaging (MRI) enabling studies of brain regions [6,7]. 21 st century innovations include serial EM for measuring subcellular interactions [8], improved LM [9][10][11][12][13][14], and MRI [15-18] to estimate interactions within and across brain regions.These 21 st technologies now enable modeling brain connectivity, in a complementary fashion to models of brain activity that were developed in the 20th century [19,20]. Models of brain activity, typically referred to as neural coding, link patterns of brain activity to past, ongoing, and future events. By contrast, models of brain connectivity, which we refer to as connectal coding, link patterns of brain connectivity to past, ongoing, and future events. The nature of the patterns, events, and links change by virtue of switching focus from activity to connectivity. Moreover, the statistical models one can leverage to learn those links from the data must also change.The goal of this manuscript is to introduce in clear terms, motivate from first principles, and formalize this emerging approach to studying the brain. While neural activity coding is well established and widely accepted as a (possibly the) legitimate framework for studying the brain, connectal coding remains in its infancy. Below we outline our rationale for why connectal codes are not just valuable, but required to unify 20th and 21st century mentalities to model the parts, their properties, and interactions among them together to infer improved scientific explanations of cognitive phenomena. Modeling Brains as Networks

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“…First, eigen‐decomposition of the adjacency matrix of random dot product graph G ′ leads to consistent latent positions, where vertex nomination performance via k ‐nearest‐neighbor of these latent positions converges to Bayes optimal as | V | = n → ∞ and k / m → 0 (where the size of the graph increases, but the relative proportion of vertex classes and of interesting vertices does not). Empirical results on random, induced subgraphs of the Wikipedia graph validate the application of this technique to real data . The proofs in Ref depend on the generative graph model being a random dot product graph, which is relaxed to an unknown link function by Tang et al Taken together, this pair of papers begins to bridge the gap between theoretical results and naturally occurring data on this topic.…”

Section: Literature Reviewmentioning

confidence: 99%

Vertex nomination

Coppersmith

2014

WIREs Computational Stats

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Vertex nomination is a subclass of recommender systems operating on attributed graphs. Attributed graphs are an attractive way to represent data from a diverse set of natural and manmade phenomena. Frequently, these data have latent attributes or class memberships that are of interest and uncovering them is of some intrinsic value. So‐called recommender systems address the ‘more like this’ problem: given a subset of the data labeled as ‘interesting’, find unlabeled examples that are ‘similarly interesting’, often with the assumption that they are from the same latent class or possess similar latent attributes. Unsurprisingly, recommender systems operating on attributed graphs have generated an interesting body of research, detailing both theoretical and practical advances for a range of applications. This advanced review examines the relevant literature, particularly focused on the importance and inclusion of edge‐ and vertex attributes, used in conjunction with the graph structure. We include example applications from human language technology, biology, and neuroscience for concreteness, although the algorithms discussed are widely applicable. WIREs Comput Stat 2014, 6:144–153. doi: 10.1002/wics.1294 This article is categorized under: Algorithms and Computational Methods > Quadratic and Nonlinear Programming

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Consistent Latent Position Estimation and Vertex Classification for Random Dot Product Graphs

Cited by 90 publications

References 23 publications

Community Detection and Classification in Hierarchical Stochastic Blockmodels

Community Detection and Classification in Hierarchical Stochastic Blockmodels

Connectal Coding: Discovering the Structures Linking Cognitive Phenotypes to Individual Histories

Vertex nomination

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