The field of generative graph models has seen increased popularity during recent years as it allows us to model the underlying distribution of a network and thus recreate it. From allowing anonymization of sensitive information in social networks to data augmentation of rare diseases in the brain, the ability to generate synthetic data has multiple applications in various domains. However, most current methods face the bottleneck of trying to generate the entire adjacency matrix and are thus limited to graphs with less than tens of thousands of nodes. In contrast, large real-world graphs like social networks or transaction graphs can extend significantly beyond these boundaries. Furthermore, the current scalable approaches are predominantly based on stochasticity and do not capture local structures and communities. In this paper, we propose Graphwave Edge-Linking CELL or GELCELL, a novel three-step architecture for generating graphs at scale. First, instead of constructing the entire network, GELCELL partitions the data and generates each cluster separately, allowing for efficient and parallelizable training. Then, by encoding the nodes, it trains a classifier to predict the edges between the partitions to patch them together, creating a synthetic version of the original large graph. Although it does suffer from some limitations due to necessary constraints on the cluster sizes, the results showed that GELCELL, given optimized parameters, can produce graphs with reasonable accuracy on all data tested, with the largest having 400 000 nodes and 1 000 000 edges. / Generativa grafmodeller har sett ökad popularitet under de senaste åren eftersom det möjliggör modellering av grafens underliggande distribution, och vi kan på så sätt återskapa liknande kopior. Förmågan att generera syntetisk data har ett flertal applikationsområden i en mängd av områden, allt från att möjligöra anonymisering av känslig data i sociala nätverk till att utöka mängden tillgänglig data av ovanliga hjärnsjukdomar. Dagens metoder har länge varit begränsade till grafer med under tiotusental noder, då dessa inte är tillräckligt skalbara, men grafer som sociala nätverk eller transaktionsgrafer kan sträcka sig långt utöver dessa gränser. Dessutom är de nuvarande skalbara tillvägagångssätten till största delen baserade på stokasticitet och fångar inte lokala strukturer och kluster. I denna rapport föreslår vi ”Graphwave EdgeLinking CELL” eller GELCELL, en trestegsarkitektur för att generera grafer i större skala. Istället för att återskapa hela grafen direkt så partitionerar GELCELL all datat och genererar varje kluster separat, vilket möjliggör både effektiv och parallelliserbar träning. Vi kan sedan koppla samman grafen genom att koda noderna och träna en modell för att prediktera länkarna mellan kluster och återskapa en syntetisk version av originalet. Metoden kräver vissa antaganden gällande max-storleken på dess kluster men är flexibel och kan rymma domänkännedom om en specifik graf i form av informerad parameterinställning. Trots detta visar resultaten på varierade träningsdata att GELCELL, givet optimerade parametrar, är kapabel att genera grafer med godtycklig precision upp till den största beprövade grafen med 400 000 noder och 1 000 000 länkar.

The Turán problem is a fundamental problem in extremal graph theory. It asks what the maximum number of edges a given graph G can have, not containing some forbidden graph H, and is solved using the Turán number ex(n,H), density π(H) and graph Tr(n). Turán's theorem tells us that the Turán graph Tr(n) is the largest Kr+1-free simple graph on n vertices. This paper is an overview of Turán problems for cliques Kn, hypercubes Qn and Hamming graphs H(s,d). We end it by proving a new result we call "the layer theorem", solving the Hamming-Turán problem using a method of creating layers of vertices in a graph. This theorem gives a lower bound for the Hamming-relative Turán density as follows: <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cpi_%7Bs,d%7D(%5Cmathcal%7BH%7D_%7Bs,d%7D,F)%20%5Cgeq%201%20-%20%5Cdfrac%7Bf+g%7D%7B%7C%7CH(s,d)%7C%7C%7D" /> where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f%20=%20%5Cbinom%7Bs%7D%7B2%7D%5Cleft(1-%5Cdfrac%7Br-2%7D%7Br-1%7D%5Cright)ds%5E%7Bd-1%7D%20%5Ctext%7B%20and%20%7D%20g%20=%20%5Csum_%7Bi=1%7D%5E%7Bn/(t-1)%7D%20(d-i(t-1))(s-1)%5E%7Bi(t-1)+1%7D%5Cbinom%7Bd%7D%7Bi(t-1)%7D" /> for the forbidden graph F stretching over t layers and r = χ(F). / Turán-problemet är det fundamentala problemet inom extremal grafteori. Det ställer frågan vad det maximala antalet kanter en given graf G kan ha utan att innehålla någon förbjuden graf H, och löses med hjälp av Turán-talet ex(n,H), -densiteten π(H) and -grafen Tr(n). Turáns sats säger oss att Turán-grafen Tr(n) är den största Kr+1-fria enkla grafen på n hörn. Denna uppsats är en överblick av Turán-problem i klickar Kn, hyperkuber Qn och Hamming-grafer H(s,d). Vi avslutar den med att bevisa ett nytt resultat som vi kallar "lagersatsen", vilket löser Hamming-Turán-problemet med hjälp av en metod som skapar lager av hörnen i en graf. Lagersatsen ger en undre gräns för den Hamming-relativa Turán-densiteten enligt följande: <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cpi_%7Bs,d%7D(%5Cmathcal%7BH%7D_%7Bs,d%7D,F)%20%5Cgeq%201%20-%20%5Cdfrac%7Bf+g%7D%7B%7C%7CH(s,d)%7C%7C%7D" /> där <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?f%20=%20%5Cbinom%7Bs%7D%7B2%7D%5Cleft(1-%5Cdfrac%7Br-2%7D%7Br-1%7D%5Cright)ds%5E%7Bd-1%7D%20%5Ctext%7B%20and%20%7D%20g%20=%20%5Csum_%7Bi=1%7D%5E%7Bn/(t-1)%7D%20(d-i(t-1))(s-1)%5E%7Bi(t-1)+1%7D%5Cbinom%7Bd%7D%7Bi(t-1)%7D" /> för den förbjudna grafen F som sträcker sig över t lager samt r = χ(F).

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Zugl.: Marburg, Univ., Diss., 2007

Title: Parameterized Complexity Author: Ondřej Suchý Department: Department of Applied Mathematics Advisor: Prof. RNDr. Jan Kratochvíl, CSc. Advisor's e-mail address: honza@kam.mff.cuni.cz Abstract: This thesis deals with the parameterized complexity of NP-hard graph problems. We explore the complexity of the problems in various scenarios, with respect to miscellaneous parameters and their combina- tions. Our aim is rather to classify in this multivariate manner whether the particular parameters make the problem fixed-parameter tractable or intractable than to present the algorithm achieving the best running time. In the questions we study typically the first-choice parameter is unsuccessful, in which case we propose to use less standard ones. The first family of problems investigated provides a common general- ization of many well known and studied domination and independence problems. Here we suggest using the dual parameterization and show that, in contrast to the standard solution-size, it can confine the in- evitable combinatorial explosion. Further studied problems are ana- logues of the Steiner problem in directed graphs. Here the parameter- ization by the number of terminals to be connected seems to be previ- ously unexplored in the directed setting. Unfortunately, the problems are shown to be...

Title: Computer Modeling of Tissue Development Author: Vojtěch Bednář Department: Department of applied mathematics Supervisor: Doc. RNDr. Zdeněk Hedrlín, CSc. Abstract: This thesis describes hybrid individual cell-based approach to modeling of systems of biological cells. In the first part reaction-diffusion model of environment is introduced together with vax equilibrium and model of a cell based on zygotic graph and cummulative states. Further, simulations modeling three biologically motivated situations are introduced: Lumen formation, tumor growth, and cellular migration in chronic inflammation. The first model shows a scenario of hollow structure formation based on directional division and cellular migration. The second model is concerned with the growth of a progeny of a slightly damaged cell. The resulting tumor exhibits three stages of malign transformation. Further, emergence of an aggressive tumor without detectable precursor is observed on one hand and a continual transformation of a benign neoplasm into a malign one is seen on the other hand. Each of these cases is a consequence of different parametrization of the model situation. The last model analyses the role of membrane enzymatic activity in migrating cells of the immune system in chronic inflammation. In this model it is observed that...

The aim of this thesis was to perform inter-company comparison by using different tools for a set of five construction companies. For comparison were selected five joint stock companies, which are in terms of market share, the largest. To the resulting comparison has been used inter-comparison methods, bankruptcy and credibility models and it was created credit rating companies AGR (Aspect Global Rating). Based on the chosen methodology, we compare the companies among themselves in terms of their economic situation.

Development of housing construction in the context of regional population development in Czechia Abstract The main aim of the thesis is to analyze and describe the development of housing construction in the context of regional population development in the Czech Republic in the years 2000- 2019. Spatial autocorrelation (LISA) and Webb's graph, which is used in analyzes of regional population development, were used for data analysis. Areas with above-average and, on the contrary, below-average housing construction are observed in four five-year periods. These areas are mutually compared in terms of their population development. The main result of the thesis is the confirmation of the assumption that in most cases, areas with above-average housing construction are also characterized by above-average population growth and are located in the hinterland of large cities (especially Prague) as part of the suburbanization process. Keywords: population development, housing construction, Webb's graph, spatial autocorrelation, Czechia

There is a wide range of different algorithms for insolvency prediction. The complex concept of insolvency proceedings from the point of view of both parties (debtor versus creditor) and from the point of view of the macroeconomics in this dissertation is new. It is often very difficult to generate forecasts using numerical quantifiers and traditional statistical methods. The reason is the lack of input data. Therefore, the work uses trend analysis tools based on the least information intensive quantifiers, ie trends, increasing, constant, and decreasing. A trend model solution is a set of scenarios where a set of variables is quantified by these trends. All possible transitions between the scenarios are generated and plotted in transition graphs. The oriented transition graph has as a node a set of scenarios, and as a branch the transitions between the scenarios. The given path through the transition graph describes any possible future and past behavior of the insolvency system being investigated. The Transition graph is a complete list of trend-based forecasts. The heuristics for determination of the payoff values from the insolvency proceedings applicable to the decision tree tools and the generated transition graphs from trend analyzes are also presented and used in the thesis. A nine-dimensional model serves as a case study. Vague variables are used in models that may have a major impact on the entire insolvency process, eg greed level and political situation.

This masters'sthesis deals with graph theory and utilization of this theory for design and optimization of data network structures. Introduction chapter describes graph theory in general view, i.e. fundamental terms used for graph description, graph distinguishing, etc. Next part describes graph algorithms, for example a shortest path finding. After this I write about actual routing protocols where the graph algorithms are used. Last but one part deals with queuing theory and final part describes practical presentation of using graph theory for design and optimization of data network structure in Matlab programme environment.