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However, many domains are comprised of a single, partially-labeled network. Thus, relational versions of Expectation Maximization i. Although R-EM methods can significantly improve predictive performance in networks that are densely labeled, they do not achieve the same gains in sparsely labeled networks and can perform worse than RML methods. We have shown the fixed-point methods that R-EM uses for approximate learning and inference result in errors that prevent convergence in sparsely labeled networks. To address this, we propose two methods that do not experience this problem. Then we develop a Relational Data Augmentation R-DA method, which integrates over a range of stochastic parameter values for inference.

We note that existing relational machine learning RML approaches have limitations that prevent their application in large scale domains. First, semi-supervised methods for RML do not fully utilize all the unlabeled instances in the network.

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Second, the collective inference procedures necessary to jointly infer the missing labels are generally viewed as too expensive to apply in large scale domains. In our recent work, we address each of these limitations. We analyze the effect of full semi-supervised RML and find that collective inference methods can introduce considerable bias into predictions. We correct this by implementing a maximum entropy constraint on the inference step, forcing the predictions to have the same distribution as the observed labels.

Next, we outline a massively scalable variational inference algorithm for large scale relational network domains. We extend this inference algorithm to incorporate the maximum entropy constraint, proving that it only requires a constant amount of overhead while remaining massively parallel. Generative Models of Networks: Research in statistical relational learning focuses on methods to exploit correlations among the attributes of linked nodes to predict user characteristics with greater accuracy. Concurrently, research on generative graph models has primarily focused on modeling network structure without attributes, producing several models that are able to replicate structural characteristics of networks such as power law degree distributions or community structure.

However, there has been little work on how to generate networks with real-world structural properties and correlated attributes. AGM then combines the attribute correlations with the structural probabilities to sample networks conditioned on attribute values, while keeping the expected edge probabilities and degrees of the input graph model. Pfeiffer III, S.

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Moreno, T. La Fond, J. Neville, and B. Attributed Graph Models: Modeling network structure with correlated attributes. In addition, we have investigated scalable generative graph models that focus on modeling distributions of unattributed graphs that match real world network properties and scale to large datasets. In network analysis, the "assortativity" statistic, defined as the correlation between the degrees of linked nodes in the network, is often used to summarize the joint degree distribution. The measure can distinguish between types of networkssocial networks commonly exhibit positive assortativity, in contrast to biological or technological networks that are typically disassortative.

Despite this, little work has focused on scalable graph models that capture assortativity in networks. In our recent work, extending the ideas above, we developed a generative graph model that explicitly estimates and models the joint degree distribution. Our Binned Chung Lu method accurately captures both the joint degree distribution and assortativity, while still matching characteristics such as the degree distribution and clustering coefficients.

Further, our method has subquadratic learning and sampling methods that enable scaling to large, real world networks. Moore, S. Mussmann, J. Pfeiffer III, and J. Hypothesis Testing Across Networks: The recent interest in networks has fueled a great deal of research on the analysis and modeling of graphs. However, much of this work has focused on analyzing the structure of a single large network drawn from a specific domain e. Although some of the work has compared the structure of networks from various domains e.

The lack of statistical hypothesis testing methods to compare across networks is because previous work has focused on assessing the significance of patterns within a network e. Here we exploit recent work on mixed Kronecker Product Graph Models mKPGMs —which accurately capture the structural characteristics of real world networks and their natural variation—to develop a principled approach for hypothesis testing across networks.

Each sample is left out once, and the model is retrained every time on the rest of the data. The variance was This outlier was unsurprisingly the worst prediction, and the closest prediction was within 0. The r 2 score for the regression without any holdouts is 0. The variance of these predictions was This was the worst prediction, and the best prediction was within 0. So, what does all this mean? There's much room here for improvement, but both models are performing well above the expectations of random guesses, or even if we made random guesses bounded by the date range of our data.

There are outlier predictions in both models, and variance is high. That said, we are getting information about the year a network represents using only statistics that measure each network's overall structure. To isolate the effect of each predictor, we can turn to the coefficients. With the authors-as-nodes network, density is by far the largest positive predictor.

Each time the density score increases by one point, the predicted date will increase by about years. Density in my model was expressed as a value between 0. The next most informative coefficient is average clustering, followed by average path distance, which reduces the predicted date as it increases. Average degree, triadic closure, and radius provide less predictive information, and diameter is effectively irrelevant to our model.

Over time, more authors share at least one periodical between them.

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The average node and its neighbors are more likely to form a complete graph. Meanwhile, the average path distance—or the average number of steps it takes to connect two nodes following the shortest path between them for all possible pairs of network nodes—is decreasing.

With the periodicals-as-nodes network, density remains important, but not nearly as important as it was in the first model. Average path distance is the second largest coefficient, and in remains a negative predictor of date. A network's radius is the third largest coefficient, and diameter remains the least valuable, but now, all seven network measures are at least a partial factor in the predictions. Over time, as with author networks, density is increasing.

Periodicals are more likely to share at least one author between them, and the average number of steps it takes to connect two nodes is decreasing. Further, as the date increases, a given periodical is marginally more likely to form a complete network with its immediate neighbors. If nodes A and B are connected and nodes B and C are connected, nodes A and C are more likely to be connected triadic closure. Since the radius of periodical networks is also increasing over time, we can say that the network is becoming more eccentric.

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Throughout the data, there is substantial variability or noise foregrounding the trends or signal , but a signal seems to be present. This preliminary result suggests that the periodicals-as-nodes networks provide more predictive capacity than authors-as-nodes networks. To return, then, to hierarchies of taste, I see a lot of potential. A more developed experiment could build on this work to address several questions.

Is "who gets reviewed" consolidating over time? Or, are sets of periodicals becoming more cliquish? Is homophily is changing over time along genre lines, by periodical type, by author gender, etc. Are there patterns to authors garnering single-work reviews vs. Banf and Rhee developed a novel GRN inference strategy called GRACE Gene Regulatory network inference ACcuracy Enhancement 11 , which generates GRNs through multiple steps to integrate various knowledge related to the regulation of gene expression: initial network prediction from gene expression data using a random forest regression model and integrating information related to gene regulation, subsequent network module extraction by meta-network construction based on information of functionally related genes, and further selection of regulatory links using ensembles of Markov Random Fields Banf and Rhee, To infer the developmental GRN in Arabidopsis , the authors incorporated conserved sequence information in its promoter regions and experimentally determined cis -motifs for TFs, together with gene expression data from 83 tissues and stages, and obtained an initial GRN containing regulators, 4, targets, and 10, links.

To enhance confidence of the initially predicted GRN, the authors integrated knowledge from various information resources such as AraNet 12 , ATRM Arabidopsis Transcriptional Regulatory Map 13 , SUBA3 14 , and AraCyc 15 , and demonstrated its potential to produce high-confidence regulatory networks, thereby suggesting a benefit of integration of multiple clues from various information resources to improve accuracy of the GRNs.

TABLE 2. Examples of combined approaches for GRN inference in plants and other species. In terms of recent advances in both resolution and throughput to acquire genome and transcriptome datasets Reuter et al.

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Here, we highlight emerging applications of these approaches, through GRN reconstruction, from these three specific aspects. Population-scale transcriptome sequencing enables us to shed light on molecular consequences of regulatory variations in complex traits. Through transcriptome sequencing across mapping populations, eQTL analysis has been widely used to identify cis - and trans -QTLs, and reconstruct regulatory networks to mine genetic factors that determine various traits, including agronomic traits of crop species Albert et al.

Moreover, a transcriptome-wide association study TWAS was proposed to identify associations between gene expression and traits Gusev et al. For example, integrating genome and transcriptome data of whole blood RNA-Seq samples across 3, unrelated individuals, Luijk et al. Moreover, population-scale transcriptome sequencing across multiple tissue types, have been applied to reconstruct GRNs through integration with other resources on molecular networks, such as PPI and TF motifs, to reveal tissue-specific gene regulation Sonawane et al.

High-throughput sequencing applications at single-cell level have rapidly emerged, and enabled us to decipher GRNs underlying cellular heterogeneity Liu and Trapnell, ; Libault et al. For GRN inference from single-cell transcriptome datasets, several computational algorithms have recently been developed. Chan et al. Although, till date, there are only a small number of scRNA-Seq datasets from higher plant species Perroud et al.

Through a longitudinal transcriptome analysis of short-lived killifish, Nothobranchius furzeri , Baumgart et al. For crop improvements, trajectories of physiological states, resulting from interaction between genetic and environmental factors, often influence the phenotypes of eventual agronomic traits; longitudinal study of cellular networks provides clues to identify gene-environment interactions associated with the phenotypic changes in crops Mochida et al.

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Through construction of an integrated atlas of gene expression and regulatory networks in developing maize, Walley et al. In tropical rice, as introduced in the previous sections, integrating time-series datasets of transcriptome, nucleosome-free chromatin from ATAC-seq, and known cis -motifs for TFs from five tropical rice cultivars under controlled and agricultural field conditions, Wilkins et al. These examples from staple crops illuminate that combinatorial use of multiple omics data is a promising approach to improve the performance of GRN inference, as well as to mine better clues to improve agronomically important traits of crops under field conditions.

In the last few years, approaches to reconstruct GRNs have advanced by synergistic innovation of high-throughput sequencing and computational techniques; GRNs have played crucial roles to elucidate cellular systems and identify key genes that manipulate cellular functions. A lot of statistical- and ML-based approaches have been proposed and applied to infer GRNs based on transcriptome datasets; these have contributed to identify regulatory relationships of genes involved in various biological phenomena in plants.

Integration of GRNs and other networks, such as epigenetic, PPI, and metabolic networks, provides clues to identify molecular relations that function as interfaces, and will provide new insights into trans -omics networks across multiple omics layers Yugi et al. ML has provided algorithms to find useful patterns from large and heterogeneous unstructured data, acquired through multiple high-throughput techniques Ma et al.

Recently, ML-based approaches have been applied to extract features associated with cellular states and responses from high-throughput data, including transcriptomic and epigenomic data, and develop computational models that classify the cellular states and responses in applications such as precision oncology and drug development Aliper et al.

In plant science, ML-based integrative analysis of large-scale data from multiple omics spectra, such as genomic variations and molecular networks, as well as high-throughput phenomics, will enable us to decipher complex cellular systems and figure out molecular features associated with quantitative traits in plants and crops, and apply the results to design traits through optimizing GRNs in crop breeding. From the perspective of ML in GRN study, it will offer us algorithms not only for GRN inference but also for feature extraction across multi-dimensional datasets from various high-throughput experimental techniques.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Albert, E. Allele specific expression and genetic determinants of transcriptomic variations in response to mild water deficit in tomato. Plant J. Aliper, A. Deep learning applications for predicting pharmacological properties of drugs and drug repurposing using transcriptomic data.

Banf, M. Enhancing gene regulatory network inference through data integration with markov random fields. Barabasi, A. Bargmann, B. Plant 6, — Basnet, R. A systems genetics approach identifies gene regulatory networks associated with fatty acid composition in brassica rapa seed. Plant Physiol. Baumgart, M. Longitudinal RNA-seq analysis of vertebrate aging identifies mitochondrial complex i as a small-molecule-sensitive modifier of lifespan. Cell Syst. Blais, A. Constructing transcriptional regulatory networks.

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Genes Dev. Blum, C. Experimental noise cutoff boosts inferability of transcriptional networks in large-scale gene-deletion studies. Calabrese, G. Camacho, D. Next-Generation machine learning for biological networks. Cell , — Chan, T. Gene regulatory network inference from single-cell data using multivariate information measures. Dasgupta, S. Single-cell RNA sequencing: a new window into cell scale dynamics. Davie, K. A single-cell transcriptome atlas of the aging drosophila brain.

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Cell Predicting gene regulatory networks by combining spatial and temporal gene expression data in Arabidopsis root stem cells. Desai, J.

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Improving gene regulatory network inference by incorporating rates of transcriptional changes. Dewey, G. Google Scholar. Efroni, I. The potential of single-cell profiling in plants. Genome Biol. Faith, J. Large-scale mapping and validation of Escherichia coli transcriptional regulation from a compendium of expression profiles.

PLoS Biol. Fiers, M. Mapping gene regulatory networks from single-cell omics data. Brief Funct. Genomics 17, — Foo, M. A framework for engineering stress resilient plants using genetic feedback control and regulatory network rewiring. ACS Synth. Fuxman Bass, J.