A Tensor-Based Formulation of Hetero-Functional Graph Theory

Recently, hetero-functional graph theory (HFGT) has developed as a means to mathematically model the structure of large-scale complex flexible engineering systems. It does so by fusing concepts from network science and model-based systems engineering (MBSE). For the former, it utilizes multiple graph-based data structures to support a matrix-based quantitative analysis. For the latter, HFGT inherits the heterogeneity of conceptual and ontological constructs found in model-based systems engineering including system form, system function, and system concept. These diverse conceptual constructs indicate multi-dimensional rather than two-dimensional relationships. This paper provides the first tensor-based treatment of hetero-functional graph theory. In particular, it addresses the “system concept” and the hetero-functional adjacency matrix from the perspective of tensors and introduces the hetero-functional incidence tensor as a new data structure. The tensor-based formulation described in this work makes a stronger tie between HFGT and its ontological foundations in MBSE. Finally, the tensor-based formulation facilitates several analytical results that provide an understanding of the relationships between HFGT and multi-layer networks.

A Comparison Study between Autism Spectrum Disorder and Typically Control in Graph Frequency Bands Using Graph and Triadic Interaction Metrics

Many researchers using many different approaches have attempted to find features discriminating between autism spectrum disorder (ASD) and typically control (TC) subjects. In this study, this attempt has been continued by analyzing global metrics of functional graphs and metrics of functional triadic interactions of the brain in the low, middle, and high-frequency bands (LFB, MFB, and HFB) of the structural graph. The graph signal processing (GSP) provided the combinatorial usage of the functional graph of resting-state fMRI and structural graph of DTI. In comparison to TCs, ASDs had significantly higher clustering coefficients in the MFB, higher efficiencies and strengths in the MFB and HFB, and lower small-world propensity in the HFB. These results show over-connectivity, more global integration, and probably decreased local specialization in ASDs compared to TCs. Triadic analysis showed that the numbers of unbalanced triads were significantly lower for ASDs in the MFB. This finding may show the reason for restricted and repetitive behavior in ASDs. Also, in the MFB and HFB, the numbers of balanced triads and the energies of triadic interactions were significantly higher and lower for ASDs, respectively. These findings may reflect the disruption of the optimum balance between functional integration and specialization. All of these results demonstrated that the significant differences between ASDs and TCs existed in the MFB and HFB of the structural graph when analyzing the global metrics of the functional graph and triadic interaction metrics. In conclusion, the results demonstrate the promising perspective of GSP for attaining discriminative features and new knowledge, especially in the case of ASD.

Identification of Autism Spectrum Disorder With Functional Graph Discriminative Network

Autism spectrum disorder (ASD) is a specific brain disease that causes communication impairments and restricted interests. Functional connectivity analysis methodology is widely used in neuroscience research and shows much potential in discriminating ASD patients from healthy controls. However, due to heterogeneity of ASD patients, the performance of conventional functional connectivity classification methods is relatively poor. Graph neural network is an effective graph representation method to model structured data like functional connectivity. In this paper, we proposed a functional graph discriminative network (FGDN) for ASD classification. On the basis of pre-built graph templates, the proposed FGDN is able to effectively distinguish ASD patient from health controls. Moreover, we studied the size of training set for effective training, inter-site predictions, and discriminative brain regions. Discriminative brain regions were determined by the proposed model to investigate its applicability and biomarkers for ASD identification. For functional connectivity classification and analysis, FGDN is not only an effective tool for ASD identification but also a potential technique in neuroscience research.

Hydraulic parameters in the area of the main dam intake structure of the river

The article presents the study results of changes in flow hydraulic parameters and morphometric elements of the river in the area of the main dam intake structure. A model for studying the direction of flow and other parameters of the flow in the area of the main structure for obtaining water from the Amudarya without a dam has been developed. A functional graph of the depth dependence of the expenditure on the study object was obtained. Graphs of the functional dependence of the diurnal cross-section of the riverbed on the flow depth and the variation of the flow velocity depending on the flow depth were constructed. The direction and distribution of the flow in the area of the main dam intake structure were studied.

On the Structure of Graphs of Markoff Triples

We sharpen the bounds of J. Bourgain, A. Gamburd and P. Sarnak (2016) on the possible number of nodes outside the ‘giant component’ and on the size of individual connected components in the suitably defined functional graph of Markoff triples modulo $p$. This is a step towards the conjecture that there are no such nodes at all.