To save content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about saving content to .
To save content items to your Kindle, first ensure no-reply@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
There are many cases in which we put weights on the edges of a graph or digraph and ask for the minimum or maximum weight object. The optimization questions that arise from this are the backbone of Combinatorial optimization. When the weights are random variables, we can ask for properties of the optimum value, which will be also a random variable. In this chapter, we consider three of the most basic optimization problems: minimum weight spanning trees, shortest paths, and minimum weight matchings.
In this chapter, we describe the main goal of the book, its organization, course outline, and suggestions for instructions and self-study. The textbook material is aimed for a one-semester undergraduate/graduate course for mathematics and computer science students. The course might also be recommended for students of physics, interested in networks and the evolution of large systems, as well as engineering students, specializing in telecommunication. Our textbook aims to give a gentle introduction to the mathematical foundations of random graphs and to build a platform to understand the nature of real-life networks. The text is divided into three parts and presents the basic elements of the theory of random graphs and networks. To help the reader navigate through the text, we have decided to start with describing in the preliminary part (Part I) the main technical tools used throughout the text. Part II of the text is devoted to the classic Erdős–Rényi–Gilbert uniform and binomial random graphs. Part III concentrates on generalizations of the Erdős–Rényi–Gilbert models of random graphs whose features better reflect some characteristic properties of real-world networks.
Whether a graph is connected, i.e., there is a path between any two of its vertices, is of particular importance. Therefore, in this chapter, we first establish the threshold for the connectivity of a random graph. We then view this property in terms of the graph process and show that w.h.p. the random graph becomes connected at precisely the time when the last isolated vertex joins the giant component. This “hitting time” result is the precursor to several similar results. After this, we deal with k-connectivity, i.e., the parameter that measures the strength of connectivity of a graph. We show that the threshold for this property is the same as for the existence of vertices of degree k in a random graph.
The previous chapter dealt with the existence of small subgraphs of a fixed size. In this chapter, we concern ourselves with the existence of large subgraphs, most notably perfect matchings and Hamilton cycles. Having dealt with perfect matchings, we turn our attention to long paths in sparse random graphs, i.e., in those where we expect a linear number of edges. We next study one of the most celebrated and difficult problems of random graphs: the existence of a Hamilton cycle in a random graph. In the last section of this chapter, we consider the general problem of the existence of arbitrary spanning subgraphs in a random graph
In this chapter, we mainly explore how the typical component structure evolves as the number of edges m increases. The following statements should be qualified with the caveat, w.h.p. The evolution of Erdős–Rényi–Gilbert type random graphs has clearly distinguishable phases. The first phase, at the beginning of the evolution, can be described as a period when a random graph is a collection of small components which are mostly trees. Next, a random graph passes through a phase transition phase when a giant component, of order comparable with the order of random graph, starts to emerge.
Large real-world networks although being globally sparse, in terms of the number of edges, have their nodes/vertices connected by relatively short paths. In addition, such networks are locally dense, i.e., vertices lying in a small neighborhood of a given vertex are connected by many edges. This observation is called the “small-world” phenomenon, and it has generated many attempts, both theoretical and experimental, to build and study appropriate models of small-world networks. The first attempt to explain this phenomenon and to build a more realistic model was introduced by Watts and Strogatz in 1998 followed by the publication of an alternative approach by Kleinberg in 2000. The current chapter is devoted to the presentation of both models.
In this chapter, we look first at the diameter of random graphs, i.e., the extreme value of the shortest distance between a pair of vertices. Then we look at the size of the largest independent set and the related value of the chromatic number. One interesting feature of these parameters is that they are often highly concentrated.
The current study is an approximate replication of Gray and DiLoreto’s (2016) study, which proposed a model predicting that course structure, learner interaction and instructor presence would influence students’ perceived learning and satisfaction in online learning, with student engagement acting as a mediator between two of the predictors and the outcome variables. Using mixed methods, the current study investigated whether Gray and DiLoreto’s model would be able to explain the relationships among the same variables in a computer-assisted language learning environment. A mediation analysis was conducted using survey responses from a sample of 215 college-level students, and qualitative analysis was conducted on the survey responses from a subsample of 50 students. Similar to Gray and DiLoreto’s study, positive correlational relationships emerged between the variables. However, the model proposed by Gray and DiLoreto did not fit our data well, leading us to suggest alternative path-analytic models with both student engagement and learner interaction as mediators. These models showed that the role of course organization and instructor presence were pivotal in explaining the variation in students’ perceived learning and satisfaction both directly and indirectly via student engagement and learner interaction. Moreover, qualitative analysis of students’ responses to open-ended questions suggested that from students’ perspectives, course structure was the most salient factor affecting their experiences within online language learning contexts, followed by learner interaction, and then by instructor presence.
Two-part framework and the Tweedie generalized linear model (GLM) have traditionally been used to model loss costs for short-term insurance contracts. For most portfolios of insurance claims, there is typically a large proportion of zero claims that leads to imbalances, resulting in lower prediction accuracy of these traditional approaches. In this article, we propose the use of tree-based methods with a hybrid structure that involves a two-step algorithm as an alternative approach. For example, the first step is the construction of a classification tree to build the probability model for claim frequency. The second step is the application of elastic net regression models at each terminal node from the classification tree to build the distribution models for claim severity. This hybrid structure captures the benefits of tuning hyperparameters at each step of the algorithm; this allows for improved prediction accuracy, and tuning can be performed to meet specific business objectives. An obvious major advantage of this hybrid structure is improved model interpretability. We examine and compare the predictive performance of this hybrid structure relative to the traditional Tweedie GLM using both simulated and real datasets. Our empirical results show that these hybrid tree-based methods produce more accurate and informative predictions.
Multilayer networks are in the focus of the current complex network study. In such networks, multiple types of links may exist as well as many attributes for nodes. To fully use multilayer—and other types of complex networks in applications, the merging of various data with topological information renders a powerful analysis. First, we suggest a simple way of representing network data in a data matrix where rows correspond to the nodes and columns correspond to the data items. The number of columns is allowed to be arbitrary, so that the data matrix can be easily expanded by adding columns. The data matrix can be chosen according to targets of the analysis and may vary a lot from case to case. Next, we partition the rows of the data matrix into communities using a method which allows maximal compression of the data matrix. For compressing a data matrix, we suggest to extend so-called regular decomposition method for non-square matrices. We illustrate our method for several types of data matrices, in particular, distance matrices, and matrices obtained by augmenting a distance matrix by a column of node degrees, or by concatenating several distance matrices corresponding to layers of a multilayer network. We illustrate our method with synthetic power-law graphs and two real networks: an Internet autonomous systems graph and a world airline graph. We compare the outputs of different community recovery methods on these graphs and discuss how incorporating node degrees as a separate column to the data matrix leads our method to identify community structures well-aligned with tiered hierarchical structures commonly encountered in complex scale-free networks.
We investigate artificial intelligence and machine learning methods for optimizing the adversarial behavior of agents in cybersecurity simulations. Our cybersecurity simulations integrate the modeling of agents launching Advanced Persistent Threats (APTs) with the modeling of agents using detection and mitigation mechanisms against APTs. This simulates the phenomenon of how attacks and defenses coevolve. The simulations and machine learning are used to search for optimal agent behaviors. The central question is: under what circumstances, is one training method more advantageous than another? We adapt and compare a variety of deep reinforcement learning (DRL), evolutionary strategies (ES) and Monte Carlo Tree Search methods within Connect 4, a baseline game environment, and on both a simulation supporting a simple APT threat model, SNAPT, as well as CyberBattleSim, an open-source cybersecurity simulation. Our results show that when attackers are trained by DRL and ES algorithms, as well as when they are trained with both algorithms being used in alternation, they are able to effectively choose complex exploits that thwart a defense. The algorithm that combines DRL and ES achieves the best comparative performance when attackers and defenders are simultaneously trained, rather than when each is trained against its non-learning counterpart.
In this paper, we present a model of the spread of the COVID-19 pandemic simulated by a multi-agent system (MAS) based on demographic data and medical knowledge. Demographic data are linked to the distribution of the population according to age and to an index of socioeconomic fragility with regard to the elderly. Medical knowledge are related to two risk factors: age and obesity. The contributions of this approach are as follows. Firstly, the two aggravating risk factors are introduced into the MAS using fuzzy sets. Secondly, the worsening of disease caused by these risk factors is modeled by fuzzy aggregation operators. The appearance of virus variants is also introduced into the simulation through a simplified modeling of their contagiousness. Using real data from inhabitants of an island in the Antilles (Guadeloupe, FWI), we model the rate of the population at risk which could be critical cases, if neither social distancing nor barrier gestures are respected by the entire population. The results show that hospital capacities are exceeded. The results show that hospital capacities are exceeded. The socioeconomic fragility index is used to assess mortality and also shows that the number of deaths can be significant.