Clique Decompositions in Random Graphs via Refined Absorption

We prove that if $p\ge n^{-\frac{1}{3}+\beta}$ for some $\beta > 0$, then asymptotically almost surely the binomial random graph $G(n,p)$ has a $K_3$-packing containing all but at most $n + O(1)$ edges. Similarly, we prove that if $d \ge n^{\frac{2}{3}+\beta}$ for some $\beta > 0$ and $d$ is even, then asymptotically almost surely the random $d$-regular graph $G_{n,d}$ has a triangle decomposition provided $3 \mid d \cdot n$. We also show that $G(n,p)$ admits a fractional $K_3$-decomposition for such a value of $p$. We prove analogous versions for a $K_q$-packing of $G(n,p)$ with $p\ge n^{-\frac{1}{q+0.5}+\beta}$ and leave of $(q-2)n+O(1)$ edges, for $K_q$-decompositions of $G_{n,d}$ with $(q-1)~|~d$ and $d\ge n^{1-\frac{1}{q+0.5}+\beta}$ provided $q\mid d\cdot n$, and for fractional $K_q$-decompositions.Comment: 49 page

Robust expansion and hamiltonicity

This thesis contains four results in extremal graph theory relating to the recent notion of robust expansion, and the classical notion of Hamiltonicity. In Chapter 2 we prove that every sufficiently large ‘robustly expanding’ digraph which is dense and regular has an approximate Hamilton decomposition. This provides a common generalisation of several previous results and in turn was a crucial tool in Kühn and Osthus’s proof that in fact these conditions guarantee a Hamilton decomposition, thereby proving a conjecture of Kelly from 1968 on regular tournaments. In Chapters 3 and 4, we prove that every sufficiently large 3-connected $D$-regular graph on $n$ vertices with $D$ ≥ n/4 contains a Hamilton cycle. This answers a problem of Bollobás and Häggkvist from the 1970s. Along the way, we prove a general result about the structure of dense regular graphs, and consider other applications of this. Chapter 5 is devoted to a degree sequence analogue of the famous Pósa conjecture. Our main result is the following: if the $i$$^{th}$ largest degree in a sufficiently large graph $G$ on n vertices is at least a little larger than $n$/3 + $i$ for $i$ ≤ $n$/3, then $G$ contains the square of a Hamilton cycle

A bandwidth theorem for approximate decompositions

We provide a degree condition on a regular $n$-vertex graph $G$ which ensures the existence of a near optimal packing of any family $\mathcal H$ of bounded degree $n$-vertex $k$-chromatic separable graphs into $G$. In general, this degree condition is best possible. Here a graph is separable if it has a sublinear separator whose removal results in a set of components of sublinear size. Equivalently, the separability condition can be replaced by that of having small bandwidth. Thus our result can be viewed as a version of the bandwidth theorem of B\"ottcher, Schacht and Taraz in the setting of approximate decompositions. More precisely, let $\delta_k$ be the infimum over all $\delta\ge 1/2$ ensuring an approximate $K_k$-decomposition of any sufficiently large regular $n$-vertex graph $G$ of degree at least $\delta n$. Now suppose that $G$ is an $n$-vertex graph which is close to $r$-regular for some $r \ge (\delta_k+o(1))n$ and suppose that $H_1,\dots,H_t$ is a sequence of bounded degree $n$-vertex $k$-chromatic separable graphs with $\sum_i e(H_i) \le (1-o(1))e(G)$. We show that there is an edge-disjoint packing of $H_1,\dots,H_t$ into $G$. If the $H_i$ are bipartite, then $r\geq (1/2+o(1))n$ is sufficient. In particular, this yields an approximate version of the tree packing conjecture in the setting of regular host graphs $G$ of high degree. Similarly, our result implies approximate versions of the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree.Comment: Final version, to appear in the Proceedings of the London Mathematical Societ

Packing and embedding large subgraphs

This thesis contains several embedding results for graphs in both random and non random settings. Most notably, we resolve a long standing conjecture that the threshold probability for Hamiltonicity in the random binomial subgraph of the hypercube equals $1/2$. %posed e.g.~by Bollob\'as, In Chapter 2 we obtain the following perturbation result regarding the hypercube \cQ^n: if H\subseteq\cQ^n satisfies $\delta(H)\geq\alpha n$ with $\alpha>0$ fixed and we consider a random binomial subgraph \cQ^n_p of \cQ^n with $p\in(0,1]$ fixed, then with high probability H\cup\cQ^n_p contains $k$ edge-disjoint Hamilton cycles, for any fixed $k\in\mathbb{N}$. This result is part of a larger volume of work where we also prove the corresponding hitting time result for Hamiltonicity. In Chapter 3 we move to a non random setting. %to a deterministic one. %Instead of embedding a single Hamilton cycle our result concerns packing more general families of graphs into a fixed host graph. Rather than pack a small number of Hamilton cycles into a fixed host graph, our aim is to achieve optimally sized packings of more general families of graphs. More specifically, we provide a degree condition on a regular $n$-vertex graph $G$ which ensures the existence of a near optimal packing of any family $\mathcal H$ of bounded degree $n$-vertex $k$-chromatic separable graphs into $G$. %In general, this degree condition is best possible. %In particular, this yields an approximate version of the tree packing conjecture %in the setting of regular host graphs $G$ of high degree. %Similarly, our result implies approximate versions of the Oberwolfach problem, %the Alspach problem and the existence of resolvable designs in the setting of %regular host graphs of high degree. In particular, this yields approximate versions of the the tree packing conjecture, the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree

Behavioral analysis in cybersecurity using machine learning: a study based on graph representation, class imbalance and temporal dissection

The main goal of this thesis is to improve behavioral cybersecurity analysis using machine learning, exploiting graph structures, temporal dissection, and addressing imbalance problems.This main objective is divided into four specific goals: OBJ1: To study the influence of the temporal resolution on highlighting micro-dynamics in the entity behavior classification problem. In real use cases, time-series information could be not enough for describing the entity behavior classification. For this reason, we plan to exploit graph structures for integrating both structured and unstructured data in a representation of entities and their relationships. In this way, it will be possible to appreciate not only the single temporal communication but the whole behavior of these entities. Nevertheless, entity behaviors evolve over time and therefore, a static graph may not be enoughto describe all these changes. For this reason, we propose to use a temporal dissection for creating temporal subgraphs and therefore, analyze the influence of the temporal resolution on the graph creation and the entity behaviors within. Furthermore, we propose to study how the temporal granularity should be used for highlighting network micro-dynamics and short-term behavioral changes which can be a hint of suspicious activities. OBJ2: To develop novel sampling methods that work with disconnected graphs for addressing imbalanced problems avoiding component topology changes. Graph imbalance problem is a very common and challenging task and traditional graph sampling techniques that work directly on these structures cannot be used without modifying the graph’s intrinsic information or introducing bias. Furthermore, existing techniques have shown to be limited when disconnected graphs are used. For this reason, novel resampling methods for balancing the number of nodes that can be directly applied over disconnected graphs, without altering component topologies, need to be introduced. In particular, we propose to take advantage of the existence of disconnected graphs to detect and replicate the most relevant graph components without changing their topology, while considering traditional data-level strategies for handling the entity behaviors within. OBJ3: To study the usefulness of the generative adversarial networks for addressing the class imbalance problem in cybersecurity applications. Although traditional data-level pre-processing techniques have shown to be effective for addressing class imbalance problems, they have also shown downside effects when highly variable datasets are used, as it happens in cybersecurity. For this reason, new techniques that can exploit the overall data distribution for learning highly variable behaviors should be investigated. In this sense, GANs have shown promising results in the image and video domain, however, their extension to tabular data is not trivial. For this reason, we propose to adapt GANs for working with cybersecurity data and exploit their ability in learning and reproducing the input distribution for addressing the class imbalance problem (as an oversampling technique). Furthermore, since it is not possible to find a unique GAN solution that works for every scenario, we propose to study several GAN architectures with several training configurations to detect which is the best option for a cybersecurity application. OBJ4: To analyze temporal data trends and performance drift for enhancing cyber threat analysis. Temporal dynamics and incoming new data can affect the quality of the predictions compromising the model reliability. This phenomenon makes models get outdated without noticing. In this sense, it is very important to be able to extract more insightful information from the application domain analyzing data trends, learning processes, and performance drifts over time. For this reason, we propose to develop a systematic approach for analyzing how the data quality and their amount affect the learning process. Moreover, in the contextof CTI, we propose to study the relations between temporal performance drifts and the input data distribution for detecting possible model limitations, enhancing cyber threat analysis.Programa de Doctorado en Ciencias y Tecnologías Industriales (RD 99/2011) Industria Zientzietako eta Teknologietako Doktoretza Programa (ED 99/2011