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京都大学 / 新制・課程博士 / 博士(医学) / 甲第25192号 / 医博第5078号 / 新制||医||1072(附属図書館) / 京都大学大学院医学研究科医学専攻 / (主査)教授 村川 泰裕, 教授 斎藤 通紀, 教授 李 聖林 / 学位規則第4条第1項該当 / Doctor of Agricultural Science / Kyoto University / DFAM

Real-world large-scale complex networks such as the Internet, social networks and biological networks have increasingly attracted the interest of researchers from many areas. Accurate modelling of the statistical regularities of these large-scale networks is critical to understand their global evolving structures and local dynamical patterns. Traditionally, the Erdos and Renyi random graph model has helped the investigation of various homogeneous networks. During the past decade, a special computational methodology has emerged to study complex networks, the outcome of which is identified by two models: the Watts and Strogatz small-world model and the Barabasi-Albert scale-free model. At the core of the complex network modelling process is the extraction of characteristics of real-world networks. I have developed computer simulation algorithms for study of the properties of current theoretical models as well as for the measurement of two real-world complex networks, which lead to the isolation of three complex network modelling essentials. The main contribution of the thesis is the introduction and study of a new General Two-Stage growth model (GTS Model), which aims to describe and analyze many common-featured real-world complex networks. The tools we use to create the model and later perform many measurements on it consist of computer simulations, numerical analysis and mathematical derivations. In particular, two major cases of this GTS model have been studied. One is named the U-P model, which employs a new functional form of the network growth rule: a linear combination of preferential attachment and uniform attachment. The degree distribution of the model is first studied by computer simulation, while the exact solution is also obtained analytically. Two other important properties of complex networks: the characteristic path length and the clustering coefficient are also extensively investigated, obtaining either analytically derived solutions or numerical results by computer simulations. Furthermore, I demonstrate that the hub-hub interaction behaves in effect as the link between a network's topology and resilience property. The other is called the Hybrid model, which incorporates two stages of growth and studies the transition behaviour between the Erdos and Renyi random graph model and the Barabasi-Albert scale-free model. The Hybrid model is measured by extensive numerical simulations focusing on its degree distribution, characteristic path length and clustering coefficient. Although either of the two cases serves as a new approach to modelling real-world large-scale complex networks, perhaps more importantly, the general two-stage model provides a new theoretical framework for complex network modelling, which can be extended in many ways besides the two studied in this thesis.

The interaction between the study of geometric and analytic aspects of Riemannian manifolds and that of graphs is a very amazing subject. The study of synthetic curvature notions on graphs adds new contributions to this topic. In this thesis, we mainly study two kinds of synthetic curvature notions: the Ollivier-Ricci cuvature on locally finite graphs and the combinatorial curvature on infinite semiplanar graphs. In the first part, we study the Ollivier-Ricci curvature. As known in Riemannian geometry, a lower Ricci curvature bound prevents geodesics from diverging too fast on average. We translate this Riemannian idea into a combinatorial setting using the Olliver-Ricci curvature notion. Note that on a graph, the analogue of geodesics starting in different directions, but eventually approaching each other again, would be a triangle. We derive lower and upper Ollivier-Ricci curvature bounds on graphs in terms of number of triangles, which is sharp for instance for complete graphs. We then describe the relation between Ollivier-Ricci curvature and the local clustering coefficient, which is an important concept in network analysis introduced by Watts-Strogatz. Furthermore, positive lower boundedness of Ollivier-Ricci curvature for neighboring vertices imply the existence of at least one triangle. It turns out that the existence of triangles can also improve Lin-Yau\'s curvature dimension inequality on graphs and then produce an implication from Ollivier-Ricci curvature lower boundedness to the curvature dimension inequality. The existence of triangles prevents a graph from being bipartite. A finite graph is bipartite if and only if its largest eigenvalue equals 2. Therefore it is natural that Ollivier-Ricci curvature is closely related to the largest eigenvalue estimates. We combine Ollivier-Ricci curvature notion with the neighborhood graph method developed by Bauer-Jost to study the spectrum estimates of a finite graph. We can always obtain nontrivial estimates on a non-bipartite graph even if its curvature is nonpositive. This answers one of Ollivier\'s open problem in the finite graph setting. In the second part of this thesis, we study systematically infinite semiplanar graphs with nonnegative combinatorial curvature. Unlike the previous Gauss-Bonnet formula approach, we explore an Alexandrov approach based on the observation that the nonnegative combinatorial curvature on a semiplanar graph is equivalent to nonnegative Alexandrov curvature on the surface obtained by replacing each face by a regular polygon of side length one with the same facial degree and gluing the polygons along common edges. Applying Cheeger-Gromoll splitting theorem on the surface, we give a metric classification of infinite semiplanar graphs with nonnegative curvature. We also construct the graphs embedded into the projective plane minus one point. Those constructions answer a question proposed by Chen. We further prove the volume doubling property and Poincare inequality which make the running of Nash-Moser iteration possible. We in particular explore the volume growth behavior on Archimedean tilings on a plane and prove that they satisfy a weak version of relative volume comparison with constant 1. With the above two basic inequalities in hand, we study the geometric function theory of infinite semiplanar graphs with nonnegative curvature. We obtain the Liouville type theorem for positive harmonic functions, the parabolicity. We also prove a dimension estimate for polynomial growth harmonic functions, which is an extension of the solution of Colding-Minicozzi of a conjecture of Yau in Riemannian geometry.

The aim of the thesis is to define the dependency of the classification of patients affected by neurodegenerative diseases on the choice of the parcellation atlas. Part of this thesis is the application of the functional connectivity analysis and the calculation of graph metrics according to the method published by Olaf Sporns and Mikail Rubinov [1] on fMRI data measured at CEITEC MU. The application is preceded by the theoretical research of parcellation atlases for brain segmentation from fMRI frames and the research of mathematical methods for classification as well as classifiers of neurodegenerative diseases. The first chapters of the thesis brings a theoretical basis of knowledge from the field of magnetic and functional magnetic resonance imaging. The physical principles of the method, the conditions and the course of acquisition of image data are defined. The third chapter summarizes the graph metrics used in the diploma thesis for analyzing and classifying graphs. The paper presents a brief overview of the brain segmentation methods, with the focuse on the atlas-based segmentation. After a theoretical research of functional connectivity methods and mathematical classification methods, the findings were used for segmentation, calculation of graph metrics and for classification of fMRI images obtained from 96 subjects into the one of two classes using Binary classifications by support vector machines and linear discriminatory analysis. The data classified in this study was measured on patiens with Parkinson’s disease (PD), Alzheimer’s disease (AD), Mild cognitive impairment (MCI), a combination of PD and MCI and subjects belonging to the control group of healthy individuals. For pre-processing and analysis, the MATLAB environment, the SPM12 toolbox and The Brain Connectivity Toolbox were used.

The interaction between the study of geometric and analytic aspects of Riemannian manifolds and that of graphs is a very amazing subject. The study of synthetic curvature notions on graphs adds new contributions to this topic. In this thesis, we mainly study two kinds of synthetic curvature notions: the Ollivier-Ricci cuvature on locally finite graphs and the combinatorial curvature on infinite semiplanar graphs. In the first part, we study the Ollivier-Ricci curvature. As known in Riemannian geometry, a lower Ricci curvature bound prevents geodesics from diverging too fast on average. We translate this Riemannian idea into a combinatorial setting using the Olliver-Ricci curvature notion. Note that on a graph, the analogue of geodesics starting in different directions, but eventually approaching each other again, would be a triangle. We derive lower and upper Ollivier-Ricci curvature bounds on graphs in terms of number of triangles, which is sharp for instance for complete graphs. We then describe the relation between Ollivier-Ricci curvature and the local clustering coefficient, which is an important concept in network analysis introduced by Watts-Strogatz. Furthermore, positive lower boundedness of Ollivier-Ricci curvature for neighboring vertices imply the existence of at least one triangle. It turns out that the existence of triangles can also improve Lin-Yau\''s curvature dimension inequality on graphs and then produce an implication from Ollivier-Ricci curvature lower boundedness to the curvature dimension inequality. The existence of triangles prevents a graph from being bipartite. A finite graph is bipartite if and only if its largest eigenvalue equals 2. Therefore it is natural that Ollivier-Ricci curvature is closely related to the largest eigenvalue estimates. We combine Ollivier-Ricci curvature notion with the neighborhood graph method developed by Bauer-Jost to study the spectrum estimates of a finite graph. We can always obtain nontrivial estimates on a non-bipartite graph even if its curvature is nonpositive. This answers one of Ollivier\''s open problem in the finite graph setting. In the second part of this thesis, we study systematically infinite semiplanar graphs with nonnegative combinatorial curvature. Unlike the previous Gauss-Bonnet formula approach, we explore an Alexandrov approach based on the observation that the nonnegative combinatorial curvature on a semiplanar graph is equivalent to nonnegative Alexandrov curvature on the surface obtained by replacing each face by a regular polygon of side length one with the same facial degree and gluing the polygons along common edges. Applying Cheeger-Gromoll splitting theorem on the surface, we give a metric classification of infinite semiplanar graphs with nonnegative curvature. We also construct the graphs embedded into the projective plane minus one point. Those constructions answer a question proposed by Chen. We further prove the volume doubling property and Poincare inequality which make the running of Nash-Moser iteration possible. We in particular explore the volume growth behavior on Archimedean tilings on a plane and prove that they satisfy a weak version of relative volume comparison with constant 1. With the above two basic inequalities in hand, we study the geometric function theory of infinite semiplanar graphs with nonnegative curvature. We obtain the Liouville type theorem for positive harmonic functions, the parabolicity. We also prove a dimension estimate for polynomial growth harmonic functions, which is an extension of the solution of Colding-Minicozzi of a conjecture of Yau in Riemannian geometry.

A prominent challenge in the healthcare system today is the limitation of resources in combi- nation with an increasing need for healthcare services. The pressure on healthcare is already extremely high and increasing due to a larger number of people seeking care as well as an aging population with an increased need for care. Therefore, it becomes more important to distribute resources effectively within healthcare to ensure high-quality care for everyone. Still, research shows that overcrowding of emergency departments and hospital wards is increasing affecting patient safety negatively with several negative implications including higher rates of medical errors and higher mortality. The problem is that healthcare is a complex system with many components that are interrelated and therefore hard to study with traditional approaches. Despite the huge quantity of studies on the overcrowding problem, there is yet to find a solution that could solve the problem. Thus, this thesis aims to design a data-driven pipeline to explore the clinical and logistical complexity of Emergency medicine patients admitted to hospital wards adopting a complex graph approach. Complex network theory provides a suitable tool to investigate complex networks by breaking complex systems down into smaller graphs with objects (nodes) and studying the relationship between these through various analysis tools. In this thesis, five complex networks were constructed representing co-morbidities in the car- diac, medicine, surgery, stroke, and orthopedic wards of the Academic Hospital of Uppsala, a hospital suffering from overcrowding. These networks were analyzed using degree distribution, centrality metrics, clustering coefficient, and community detection to reveal structural and clin- ical patterns. A comprehensive network of all hospital co-morbidities was also created and an- alyzed to compare it with the ward structures. Additionally, a network mapping patient flow from the emergency department based on chief complaints and ICD codes to wards was created and analyzed to identify admission patterns. The analysis of the co-morbidity networks revealed that there was an indication of structure between the wards. This was based on the visualization of nodes and edges of the networks, identified communities, and community comparisons between the wards. Further, it showed that there was a big overlap of common co-morbidities which could indicate the contrary. But it was also revealed that in terms of community structure, the wards were considerably different from each other indicating a good separation of diseases. The results of this research show that complex network theory could be used to increase the understanding of the complexity of healthcare wards in terms of the structure of diseases as well as clinical variability and allow for a discussion regarding if this is related to clinical or logistical factors. It also shows the potential of using complex network theory to increase the understanding of the path patients take from the emergency department to the wards based on the community detection analysis showing that there is a structure of where patient ends up based on the assigned ICD code and chief complaint in the emergency department. Previous studies have typically focused on specific diseases or patient flow within a single ward or the emergency department. This approach offers a tool to examine patient logistics across multiple wards alongside their clinical characteristics. The insights gained could help improve hospital structure by more efficiently distributing patients between wards, thereby enhancing resource use and hospital operations. Further research using complex network theory could deepen understanding of overcrowding issues and identify potential solutions. / En stor utmaning inom sjukvårdssystemet idag är begräsningen av resurser i kombination med ett ökat vårdbehov. Trycket på sjukvården är redan högt och ökar till följd av ett ökat antal personer som söker vård samt en åldrande befolkning med ett ökat vårdbehov. Därav blir det viktigare att fördela resurser inom sjukvården på ett effektivt sätt för att säkerställa en högkva- litativ vård till alla. Forskning visar dock att överbeläggningar på akutvårdsavdelningar och sjukvårdsavdelningar ökar vilket påverkar patientsäkerheten negativt med flera negativa kon- sekvenser däribland en högre andel medicinska misstag och en högre mortalitet. Problemet är att sjukvården är ett komplext system med många komponenter som samverkar och det är därav svårt att studera med traditionella tillvägagångssätt. Trots det höga antalet studier på överbeläggningar inom sjukvården behöver man fortfarande hitta en lösning på problemet. Därav är målet med denna avhandling att designa en datadriven pipeline för att undersöka den kliniska och logistiska komplexiteten hos patienter inlagda från akutvårdsavdelningen med hjälp av en komplex grafmetodik. Komplex nätverksteori är ett lämpligt verktyg för att studera komplexa nätverk genom att bryta ned det i mindre komponen- ter och undersöka sambanden mellan dem med hjälp av olika analysverktyg. I denna avhandling skapades 5 komplexa nätverk som representerade komorbiditeter utifrån tilldelad ICD-10-kod på hjärt-, medicin-, kirurgi-, stroke- och ortopediska avdelningen vid det akademiska sjukhuset i Uppsala, ett sjukhus som för närvarande lider av överbeläggningar. Nätverken analyserades med hjälp av gradfördelning, olika centralitetsmått, klusterkoefficient och samhällsdetektering för att identifiera skillnader eller likheter när det gäller struktur och klinisk variation. Ett heltäckande komplext nätverk skapades där alla komorbiditeter på hela sjukhuset inkluderades för att möjliggöra en jämförelse med strukturen på avdelningarna. Utö- ver detta, skapades och analyserades ett nätverk för att kartlägga patientflödet från akuten till sjukvårdsavdelningarna baserat på huvudorsak till patientens akutbesök och ICD kod. Analysen av samhällsstrukturen visade att det fanns en indikation av struktur mellan avdelning- arna. Detta baserat på visualisering av noder och kopplingar i nätverken, identifierade sam- hällen samt jämförelser av samhällen mellan avdelningarna. Vidare visade det dock att det fanns ett stort överlapp av vanliga komorbiditeter vilket kunde indikera motsatsen. Det visades dock att även när det gäller samhällsstruktur var avdelningarna väldigt olika vilket indikerade en god separering av sjukdomar. Resultaten av denna forskning visar att komplex nätverksteori kan användas för att öka förstå- elsen för komplexiteten på sjukvårdsavdelningarna gällande strukturen mellan sjukdomar såväl som klinisk variationen och öppnar upp för en diskussion om dessa är relaterade till kliniska eller logistiska faktorer. Det visar också potentialen att använda komplex nätverksteori för att öka förståelsen för den väg som patienterna tar från akutvårdsavdelningen till avdelningarna baserat på samhällsdetekteringsanalysen som visar att det finns en struktur av var patienten hamnar baserat på den tilldelade ICD-koden och huvudklagomål från akutvårdsavdelningen. Tidigare studier som har använt detta tillvägagångssätt har i huvudsak undersökt specifika sjuk- domar eller flöden på en specifik avdelning eller akutvårdsavdelning. Det här tillvägagångssät- tet ger ett verktyg för att utforska logistiken för patienters rutter till olika avdelningar samtidigt som deras kliniska egenskaper beaktas. Resultaten genom denna pipeline kan ge en grund för att öka förståelsen för hur man bättre kan strukturera sjukhuset genom att dela patienter mellanvavdelningar och genom detta effektivisera användningen av resurser och potentiellt förbättra rutiner på sjukhuset. Genom vidare studier, kan komplex nätverksteori användas för att öka förståelsen kring faktorer relaterade till problemet med överbeläggningar och hitta potentiella lösningar på problemet.