We give a strongly polynomial time combinatorial algorithm to minimise the largest eigenvalue of the weighted Laplacian of a bipartite graph. This is accomplished by solving the dual graph embedding problem which arises from a semidefinite programming formulation. In particular, the problem for trees can be solved in time cubic in the number of vertices.

The established methodologies for studying computational complexity can be applied to the new problems posed by very large-scale integrated (VLSI) circuits. This thesis develops a ''VLSI model of computation'' and derives upper and lower bounds on the silicon area and time required to solve the problems of sorting and discrete Fourier transformation. In particular, the area A and time T taken by any VLSI chip using any algorithm to perform an N-point Fourier transform must satisfy AT2 ≥ c N2 log2 N, for some fixed c > 0. A more general result for both sorting and Fourier transformation is that AT2x = Ω(N1 + x log2x N) for any x in the range 0 < x < 1. Also, the energy dissipated by a VLSI chip during the solution of either of these problems is at least Ω(N3/2 log N). The tightness of these bounds is demonstrated by the existence of nearly optimal circuits for both sorting and Fourier transformation. The circuits based on the shuffle-exchange interconnection pattern are fast but large: T = O(log2 N) for Fourier transformation, T = O(log3 N) for sorting; both have area A of at most O(N2 / log1/2 N). The circuits based on the mesh interconnection pattern are slow but small: T = O(N1/2 loglog N), A = O(N log2 N).

The availability and use of knowledge graphs have become commonplace as a compact storage of information and for lookup of facts. However, the discrete representation makes the knowledge graph unavailable for tasks that need a continuous representation, such as predicting relationships between entities, where the most probable relationship needs to be found. The need for a continuous representation has spurred the development of knowledge graph embeddings. The idea is to position the entities of the graph relative to each other in a continuous low-dimensional vector space, so that their relationships are preserved, and ideally leading to clusters of entities with similar characteristics. Several methods to produce knowledge graph embeddings have been created, from simple models that minimize the distance between related entities to complex neural models. Almost all of these embedding methods attempt to create an accurate static representation of each entity and relation. However, as with words in natural language, both entities and relations in a knowledge graph hold different meanings in different local contexts.  With the recent development of Transformer models, and their success in creating contextual representations of natural language, work has been done to apply them to graphs. Initial results show great promise, but there are significant differences in archi- tecture design across papers. There is no clear direction on how Transformer models can be best applied to create contextual knowledge graph embeddings. Two of the main differences in previous work is how the attention mask is applied in the model and what input graph structures the model is trained on.  This report explores how different attention masking methods and graph inputs affect a Transformer model (in this report, BERT) on a link prediction task for triples. Models are trained with five different attention masking methods, which to varying degrees restrict attention, and on three different input graph structures (triples, paths, and interconnected triples).  The results indicate that a Transformer model trained with a masked language model objective has the strongest performance on the link prediction task when there are no restrictions on how attention is directed, and when it is trained on graph structures that are sequential. This is similar to how models like BERT learn sentence structure after being exposed to a large number of training samples. For more complex graph structures it is beneficial to encode information of the graph structure through how the attention mask is applied. There also seems to be some indications that the input graph structure affects the models’ capabilities to learn underlying characteristics in the knowledge graph that is trained upon.

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We give a strongly polynomial time combinatorial algorithm to minimise the largest eigenvalue of the weighted Laplacian of a bipartite graph. This is accomplished by solving the dual graph embedding problem which arises from a semidefinite programming formulation. In particular, the problem for trees can be solved in time cubic in the number of vertices.

The flight search engine is a technology used in the air travel industry. It allows the traveler to search and book for the best flight options, such as the combination of flights while keeping the best services, options, and price. The computation for a flight search query can be very intensive given its parameters and complexity. The project goal is to predict the flight search queries computation cost for a new flight search engine product when dealing with parameters change and optimizations. The problem of flight search cost prediction is a regression problem. We propose to solve the problem by delimiting the problem based on its business logic and meaning. Our problem has data defined as a graph, which is why we have chosen Graph Neural Network. We have investigated multiple pretraining strategies for the evaluation of node embedding concerning a realworld regression task, including using a line graph for the training. The embeddings are used for downstream regression tasks. Our work is based on some stateoftheart Machine Learning, Deep Learning, and Graph Neural Network methods. We conclude that for some business use cases, the predictions are suitable for production use. In addition, the prediction of tree ensemble boosting methods produces negatives predictions which further degrade the R2 score by 4% because of the business meaning. The Deep Neural Network outperformed the most performing Machine Learning methods by 8% to 12% of R2 score. The Deep Neural Network also outperformed Deep Neural Network with pretrained node embedding from the Graph Neural Network methods by 11% to 17% R2 score. The Deep Neural Network achieved 93%, 81%, and 63% R2 score for each task with increasing difficulty. The training time range from 1 hour for Machine Learning models, 2 to 10 hours for Deep Learning models, and 8 to 24 hours for Deep Learning model for tabular data trained end to end with Graph Neural Network layers. The inference time is around 15 minutes. Finally, we found that using Graph Neural Network for the node regression task does not outperform Deep Neural Network. / Flygsökmotor är en teknik som används inom flygresebranschen. Den gör det möjligt för resenären att söka och boka de bästa flygalternativen, t.ex. kombinationer av flygningar med bästa service, alternativ och pris. Beräkningen av en flygsökning kan vara mycket intensiv med tanke på dess parametrar och komplexitet. Projektets mål är att förutsäga beräkningskostnaden för flygsökfrågor för en ny produkt för flygsökmotor när parametrar ändras och optimeringar görs. Problemet med att förutsäga kostnaderna för flygsökning är ett regressionsproblem. Vi föreslår att man löser problemet genom att avgränsa det utifrån dess affärslogik och innebörd. Vårt problem har data som definieras som en graf, vilket är anledningen till att vi har valt Graph Neural Network. Vi har undersökt flera förträningsstrategier för utvärdering av nodinbäddning när det gäller en regressionsuppgift från den verkliga världen, bland annat genom att använda ett linjediagram för träningen. Inbäddningarna används för regressionsuppgifter i efterföljande led. Vårt arbete bygger på några toppmoderna metoder för maskininlärning, djupinlärning och grafiska neurala nätverk. Vi drar slutsatsen att förutsägelserna är lämpliga för produktionsanvändning i vissa Vi drar slutsatsen att förutsägelserna är lämpliga för produktionsanvändning i vissa fall. Dessutom ger förutsägelserna från trädens ensemble av boostingmetoder negativa förutsägelser som ytterligare försämrar R2poängen med 4% på grund av affärsmässiga betydelser. Deep Neural Network överträffade de mest effektiva metoderna för maskininlärning med 812% av R2poängen. Det djupa neurala nätverket överträffade också det djupa neurala nätverket med förtränad node embedding från metoderna för grafiska neurala nätverk med 11 till 17% av R2poängen. Deep Neural Network uppnådde 93, 81 och 63% R2poäng för varje uppgift med stigande svårighetsgrad. Träningstiden varierar från 1 timme för maskininlärningsmodeller, 2 till 10 timmar för djupinlärningsmodeller och 8 till 24 timmar för djupinlärningsmodeller för tabelldata som tränats från början till slut med grafiska neurala nätverkslager. Inferenstiden är cirka 15 minuter. Slutligen fann vi  att användningen av Graph Neural Network för uppgiften om regression av noder inte överträffar Deep Neural Network.

Cette thèse aborde le problème du manque de performance des outils exploitant des représentationsà base de graphes en reconnaissance des formes. Nous proposons de contribuer aux nouvellesméthodes proposant de tirer partie, à la fois, de la richesse des méthodes structurelles et de la rapidité des méthodes de reconnaissance de formes statistiques. Deux principales contributions sontprésentées dans ce manuscrit. La première correspond à la proposition d'une nouvelle méthode deprojection explicite de graphes procédant par analyse multi-facettes des graphes. Cette méthodeeffectue une caractérisation des graphes suivant différents niveaux qui correspondent, selon nous,aux point-clés des représentations à base de graphes. Il s'agit de capturer l'information portéepar un graphe au niveau global, au niveau structure et au niveau local ou élémentaire. Ces informationscapturées sont encapsulés dans un vecteur de caractéristiques numériques employantdes histogrammes flous. La méthode proposée utilise, de plus, un mécanisme d'apprentissage nonsupervisée pour adapter automatiquement ses paramètres en fonction de la base de graphes àtraiter sans nécessité de phase d'apprentissage préalable. La deuxième contribution correspondà la mise en place d'une architecture pour l'indexation de masses de graphes afin de permettre,par la suite, la recherche de sous-graphes présents dans cette base. Cette architecture utilise laméthode précédente de projection explicite de graphes appliquée sur toutes les cliques d'ordre 2pouvant être extraites des graphes présents dans la base à indexer afin de pouvoir les classifier.Cette classification permet de constituer l'index qui sert de base à la description des graphes etdonc à leur indexation en ne nécessitant aucune base d'apprentissage pré-étiquetées. La méthodeproposée est applicable à de nombreux domaines, apportant la souplesse d'un système de requêtepar l'exemple et la granularité des techniques d'extraction ciblée (focused retrieval). / This thesis addresses the problem of lack of efficient computational tools for graph based structural pattern recognition approaches and proposes to exploit computational strength of statistical pattern recognition. It has two fold contributions. The first contribution is a new method of explicit graph embedding. The proposed graph embedding method exploits multilevel analysis of graph for extracting graph level information, structural level information and elementary level information from graphs. It embeds this information into a numeric feature vector. The method employs fuzzy overlapping trapezoidal intervals for addressing the noise sensitivity of graph representations and for minimizing the information loss while mapping from continuous graph space to discrete vector space. The method has unsupervised learning abilities and is capable of automatically adapting its parameters to underlying graph dataset. The second contribution is a framework for automatic indexing of graph repositories for graph retrieval and subgraph spotting. This framework exploits explicit graph embedding for representing the cliques of order 2 by numeric feature vectors, together with classification and clustering tools for automatically indexing a graph repository. It does not require a labeled learning set and can be easily deployed to a range of application domains, offering ease of query by example (QBE) and granularity of focused retrieval.

The internet is the home for massive volumes of valuable data constantly being created, making it difficult for users to find information relevant to them. In recent times, online users have been relying on the recommendations made by websites to narrow down the options. Online reviews have also become an increasingly important factor in the final choice of a customer. Unfortunately, attackers have found ways to manipulate both reviews and recommendations to mislead users. A Recommendation System is a special type of information filtering system adapted by online vendors to provide suggestions to their customers based on their requirements. Collaborative filtering is one of the most widely used recommendation systems; unfortunately, it is prone to shilling/profile injection attacks. Such attacks alter the recommendation process to promote or demote a particular product. On the other hand, many spammers write deceptive reviews to change the credibility of a product/service. This work aims to address these issues by treating the review manipulation and shilling attack scenarios independently. For the shilling attacks, we build an efficient Reinforcement Learning-based shilling attack method. This method reduces the uncertainty associated with the item selection process and finds the most optimal items to enhance attack reach while treating the recommender system as a black box. Such practical online attacks open new avenues for research in building more robust recommender systems. When it comes to review manipulations, we introduce a method to use a deep structure embedding approach that preserves highly nonlinear structural information and the dynamic aspects of user reviews to identify and cluster the spam users. It is worth mentioning that, in the experiment with real datasets, our method captures about 92\% of all spam reviewers using an unsupervised learning approach.<br>