Densest Subgraph in Dynamic Graph Streams

In this paper, we consider the problem of approximating the densest subgraph in the dynamic graph stream model. In this model of computation, the input graph is defined by an arbitrary sequence of edge insertions and deletions and the goal is to analyze properties of the resulting graph given memory that is sub-linear in the size of the stream. We present a single-pass algorithm that returns a $(1+\epsilon)$ approximation of the maximum density with high probability; the algorithm uses O(\epsilon^{-2} n \polylog n) space, processes each stream update in \polylog (n) time, and uses \poly(n) post-processing time where $n$ is the number of nodes. The space used by our algorithm matches the lower bound of Bahmani et al.~(PVLDB 2012) up to a poly-logarithmic factor for constant $\epsilon$. The best existing results for this problem were established recently by Bhattacharya et al.~(STOC 2015). They presented a $(2+\epsilon)$ approximation algorithm using similar space and another algorithm that both processed each update and maintained a $(4+\epsilon)$ approximation of the current maximum density in \polylog (n) time per-update.Comment: To appear in MFCS 201

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008

Algorithms for Computing Maximum Cliques in Hyperbolic Random Graphs

In this paper, we study the maximum clique problem on hyperbolic random graphs. A hyperbolic random graph is a mathematical model for analyzing scale-free networks since it effectively explains the power-law degree distribution of scale-free networks. We propose a simple algorithm for finding a maximum clique in hyperbolic random graph. We first analyze the running time of our algorithm theoretically. We can compute a maximum clique on a hyperbolic random graph G in O(m + n^{4.5(1-?)}) expected time if a geometric representation is given or in O(m + n^{6(1-?)}) expected time if a geometric representation is not given, where n and m denote the numbers of vertices and edges of G, respectively, and ? denotes a parameter controlling the power-law exponent of the degree distribution of G. Also, we implemented and evaluated our algorithm empirically. Our algorithm outperforms the previous algorithm [BFK18] practically and theoretically. Beyond the hyperbolic random graphs, we have experiment on real-world networks. For most of instances, we get large cliques close to the optimum solutions efficiently

Algorithms for Computing Maximum Cliques in Hyperbolic Random Graphs

In this paper, we study the maximum clique problem on hyperbolic random graphs. A hyperbolic random graph is a mathematical model for analyzing scale-free networks since it effectively explains the power-law degree distribution of scale-free networks. We propose a simple algorithm for finding a maximum clique in hyperbolic random graph. We first analyze the running time of our algorithm theoretically. We can compute a maximum clique on a hyperbolic random graph $G$ in $O(m + n^{4.5(1-\alpha)})$ expected time if a geometric representation is given or in $O(m + n^{6(1-\alpha)})$ expected time if a geometric representation is not given, where $n$ and $m$ denote the numbers of vertices and edges of $G$, respectively, and $\alpha$ denotes a parameter controlling the power-law exponent of the degree distribution of $G$. Also, we implemented and evaluated our algorithm empirically. Our algorithm outperforms the previous algorithm [BFK18] practically and theoretically. Beyond the hyperbolic random graphs, we have experiment on real-world networks. For most of instances, we get large cliques close to the optimum solutions efficiently.Comment: Accepted in ESA 202

Burning a binary tree and its generalization

Graph burning is a graph process that models the spread of social contagion. Initially, all the vertices of a graph $G$ are unburnt. At each step, an unburnt vertex is put on fire and the fire from burnt vertices of the previous step spreads to their adjacent unburnt vertices. This process continues till all the vertices are burnt. The burning number $b(G)$ of the graph $G$ is the minimum number of steps required to burn all the vertices in the graph. The burning number conjecture by Bonato et al. states that for a connected graph $G$ of order $n$, its burning number $b(G) \leq \lceil \sqrt{n} \rceil$. It is easy to observe that in order to burn a graph it is enough to burn its spanning tree. Hence it suffices to prove that for any tree $T$ of order $n$, its burning number $b(T) \leq \lceil \sqrt{n} \rceil$ where $T$ is the spanning tree of $G$. It was proved in 2018 that $b(T) \leq \lceil \sqrt{n + n_2 + 1/4} +1/2 \rceil$ for a tree $T$ where $n_2$ is the number of degree $2$ vertices in $T$. In this paper, we provide an algorithm to burn a tree and we improve the existing bound using this algorithm. We prove that $b(T)\leq \lceil \sqrt{n + n_2 + 8}\rceil -1$ which is an improved bound for $n\geq 50$. We also provide an algorithm to burn some subclasses of the binary tree and prove the burning number conjecture for the same

Enhancing explainability and scrutability of recommender systems

Our increasing reliance on complex algorithms for recommendations calls for models and methods for explainable, scrutable, and trustworthy AI. While explainability is required for understanding the relationships between model inputs and outputs, a scrutable system allows us to modify its behavior as desired. These properties help bridge the gap between our expectations and the algorithm’s behavior and accordingly boost our trust in AI. Aiming to cope with information overload, recommender systems play a crucial role in filtering content (such as products, news, songs, and movies) and shaping a personalized experience for their users. Consequently, there has been a growing demand from the information consumers to receive proper explanations for their personalized recommendations. These explanations aim at helping users understand why certain items are recommended to them and how their previous inputs to the system relate to the generation of such recommendations. Besides, in the event of receiving undesirable content, explanations could possibly contain valuable information as to how the system’s behavior can be modified accordingly. In this thesis, we present our contributions towards explainability and scrutability of recommender systems: • We introduce a user-centric framework, FAIRY, for discovering and ranking post-hoc explanations for the social feeds generated by black-box platforms. These explanations reveal relationships between users’ profiles and their feed items and are extracted from the local interaction graphs of users. FAIRY employs a learning-to-rank (LTR) method to score candidate explanations based on their relevance and surprisal. • We propose a method, PRINCE, to facilitate provider-side explainability in graph-based recommender systems that use personalized PageRank at their core. PRINCE explanations are comprehensible for users, because they present subsets of the user’s prior actions responsible for the received recommendations. PRINCE operates in a counterfactual setup and builds on a polynomial-time algorithm for finding the smallest counterfactual explanations. • We propose a human-in-the-loop framework, ELIXIR, for enhancing scrutability and subsequently the recommendation models by leveraging user feedback on explanations. ELIXIR enables recommender systems to collect user feedback on pairs of recommendations and explanations. The feedback is incorporated into the model by imposing a soft constraint for learning user-specific item representations. We evaluate all proposed models and methods with real user studies and demonstrate their benefits at achieving explainability and scrutability in recommender systems.Unsere zunehmende Abhängigkeit von komplexen Algorithmen für maschinelle Empfehlungen erfordert Modelle und Methoden für erklärbare, nachvollziehbare und vertrauenswürdige KI. Zum Verstehen der Beziehungen zwischen Modellein- und ausgaben muss KI erklärbar sein. Möchten wir das Verhalten des Systems hingegen nach unseren Vorstellungen ändern, muss dessen Entscheidungsprozess nachvollziehbar sein. Erklärbarkeit und Nachvollziehbarkeit von KI helfen uns dabei, die Lücke zwischen dem von uns erwarteten und dem tatsächlichen Verhalten der Algorithmen zu schließen und unser Vertrauen in KI-Systeme entsprechend zu stärken. Um ein Übermaß an Informationen zu verhindern, spielen Empfehlungsdienste eine entscheidende Rolle um Inhalte (z.B. Produkten, Nachrichten, Musik und Filmen) zu filtern und deren Benutzern eine personalisierte Erfahrung zu bieten. Infolgedessen erheben immer mehr In- formationskonsumenten Anspruch auf angemessene Erklärungen für deren personalisierte Empfehlungen. Diese Erklärungen sollen den Benutzern helfen zu verstehen, warum ihnen bestimmte Dinge empfohlen wurden und wie sich ihre früheren Eingaben in das System auf die Generierung solcher Empfehlungen auswirken. Außerdem können Erklärungen für den Fall, dass unerwünschte Inhalte empfohlen werden, wertvolle Informationen darüber enthalten, wie das Verhalten des Systems entsprechend geändert werden kann. In dieser Dissertation stellen wir unsere Beiträge zu Erklärbarkeit und Nachvollziehbarkeit von Empfehlungsdiensten vor. • Mit FAIRY stellen wir ein benutzerzentriertes Framework vor, mit dem post-hoc Erklärungen für die von Black-Box-Plattformen generierten sozialen Feeds entdeckt und bewertet werden können. Diese Erklärungen zeigen Beziehungen zwischen Benutzerprofilen und deren Feeds auf und werden aus den lokalen Interaktionsgraphen der Benutzer extrahiert. FAIRY verwendet eine LTR-Methode (Learning-to-Rank), um die Erklärungen anhand ihrer Relevanz und ihres Grads unerwarteter Empfehlungen zu bewerten. • Mit der PRINCE-Methode erleichtern wir das anbieterseitige Generieren von Erklärungen für PageRank-basierte Empfehlungsdienste. PRINCE-Erklärungen sind für Benutzer verständlich, da sie Teilmengen früherer Nutzerinteraktionen darstellen, die für die erhaltenen Empfehlungen verantwortlich sind. PRINCE-Erklärungen sind somit kausaler Natur und werden von einem Algorithmus mit polynomieller Laufzeit erzeugt , um präzise Erklärungen zu finden. • Wir präsentieren ein Human-in-the-Loop-Framework, ELIXIR, um die Nachvollziehbarkeit der Empfehlungsmodelle und die Qualität der Empfehlungen zu verbessern. Mit ELIXIR können Empfehlungsdienste Benutzerfeedback zu Empfehlungen und Erklärungen sammeln. Das Feedback wird in das Modell einbezogen, indem benutzerspezifischer Einbettungen von Objekten gelernt werden. Wir evaluieren alle Modelle und Methoden in Benutzerstudien und demonstrieren ihren Nutzen hinsichtlich Erklärbarkeit und Nachvollziehbarkeit von Empfehlungsdiensten

Twitter as an innovation process with damping effect

Understanding the innovation process, that is the underlying mechanisms through which novelties emerge, diffuse and trigger further novelties is undoubtedly of fundamental importance in many areas (biology, linguistics, social science and others). The models introduced so far satisfy the Heaps' law, regarding the rate at which novelties appear, and the Zipf's law, that states a power law behavior for the frequency distribution of the elements. However, there are empirical cases far from showing a pure power law behavior and such a deviation is present for elements with high frequencies. We explain this phenomenon by means of a suitable "damping" effect in the probability of a repetition of an old element. While the proposed model is extremely general and may be also employed in other contexts, it has been tested on some Twitter data sets and demonstrated great performances with respect to Heaps' law and, above all, with respect to the fitting of the frequency-rank plots for low and high frequencies