Narrow Results

Software has emerged as an indispensable part of everyday existence in the increasingly digital age, impacting everything from healthcare to finance to transportation to industries. Given the increasing reliance on software, ensuring it is reliable and safe is crucial. Discovering and fixing vulnerabilities is crucial for sustaining the security and dependability of software. Numerous issues, including illegal accessibility, data breaches, and service interruptions, may result from these vulnerabilities. Software updates are applied by patching to address vulnerabilities, increase functionality, or improve performance. Adopting best practices for patching and vulnerability management can significantly lower security risks and improve the overall resilience of software environments despite certain limitations. Very few attempts have been made to model vulnerability Patch models (VPMs), even though Vulnerability Discovery Modeling (VDM) has been modeled based on the impact of vulnerabilities discovered over time, which help software vendors identify security trends, forecast security investments, and plan patches. The proposed modeling framework, which includes vulnerability discovery modeling and vulnerability Patch modeling, is designed to provide a comprehensive understanding of the patch management workflow. This paper also introduces a new modeling framework in the context of Patch. A statistical analysis has been conducted utilizing patch datasets to demonstrate the proposed systematic layout, providing a solid foundation for the research findings.

Software vulnerabilities can be defined as potential threats to an existing software structure. With the advent of networks and the internet to far and wide places, the software has become vulnerable to attacks by elements irrespective of their geographical locations. It would need consistent efforts and constant upgradation of vulnerability detection methods to keep up with the attacks. Machine learning methods can be used to detect vulnerabilities in less time. The proposed work’s main aim is to find the best possible methods to improve vulnerability detection. Classical statistical methods as well as hyper-tuned machine learning methods have been used to analyze vulnerability detection. The proposed work has employed the grid search method to improve the accuracy of SVM and neural networks. This study aims to move toward machine learning methods of vulnerability detection from classical statistical techniques with the help of empirical results, charts and graphs.

AND/OR graphs and minimum-cost solution graphs have been studied extensively in artificial intelligence (see, e.g., Nilsson [14]). Generally, the AND/OR graphs are used to model problem solving processes. The minimum-cost solution graph can be used to attack the problem with the least resource. However, in many cases we want to solve the problem within the shortest time period and we assume that we have as many concurrent resources as we need to run all concurrent processes. In this paper, we will study this problem and present an algorithm for finding the minimum-time-cost solution graph in an AND/OR graph. We will also study the following problems which often appear in industry when using AND/OR graphs to model manufacturing processes or to model problem solving processes: finding maximum (additive and non-additive) flows and critical vertices in an AND/OR graph. A detailed study of these problems provide insight into the vulnerability of complex systems such as cyber-infrastructures and energy infrastructures (these infrastructures could be modeled with AND/OR graphs). For an infrastructure modeled by an AND/OR graph, the protection of critical vertices should have highest priority since terrorists could defeat the whole infrastructure with the least effort by destroying these critical points. Though there are well known polynomial time algorithms for the corresponding problems in the traditional graph theory, we will show that generally it is NP-hard to find a non-additive maximum flow in an AND/OR graph, and it is both NP-hard and coNP-hard to find a set of critical vertices in an AND/OR graph. We will also present a polynomial time algorithm for finding a maximum additive flow in an AND/OR graph, and discuss the relative complexity of these problems.

The rupture degree of an incomplete connected graph G is defined by

In a communication network, the vulnerability measures are essential to guide the designer in choosing an appropriate topology. They measure the stability of the network to disruption of operation after the failure of certain stations or communication links. If a station or operative is captured in a spy network, then the adjacent stations will be betrayed and are therefore useless in the whole network. In this sense, Margaret B. Cozzens and Shu-Shih Y. Wu modeled a spy network as a graph and then defined the neighbor integrity of a graph to obtain the vulnerability of a spy network [10]. The neighbor integrity of a graph G, is defined to be , where S is any vertex subversion strategy of G and c(G/S) is the maximum order of the components of G/S. In this paper, we investigate the transformation graphs G^-+-, G^+--, G^++-, G^---, G^+-+, G^-++, G^--+ and G⁺⁺⁺ of a graph G, and determine their neighbor integrity.

The rupture degree of an incomplete connected graph G is defined by

The vulnerability shows the endurance of the network until the communication collapse after the breakdown of certain stations or communication links. If a spy or a station is invaded in a spy network, then the adjacent stations are treacherous. A vulnerability parameter the neighbor rupture degree can be used to obtain the vulnerability of a spy network. The neighbor rupture degree of a noncomplete connected graph G is defined to be

The binding number of a graph G is defined to be the minimum of $|N(S)|/\left|S\right|$ taken over all nonempty $S\subseteq V(G)$ such that $N(S)\ne V(G)$. Binding number, one indicator to better understand graph, is an important characteristic quantity of a graph. In this paper, the relationships between the binding number and some other graph vulnerability parameters, namely the toughness, integrity, rupture degree and scattering number, are established. Exact values for the binding numbers of wheel related graphs namely gear, helm, sunflower and friendship graph are obtained.

A vulnerability parameter the neighbor rupture degree can be used to obtain the vulnerability of a spy network. The neighbor rupture degree of a noncomplete connected graph $G$ is defined to be

Link residual closeness is reported as a new graph vulnerability measure, a graph-based approach to network vulnerability analysis, and more sensitive than some other existing vulnerability measures. Residual closeness is of great theoretical and practical significance to network design and optimization. In this paper, how some of the graph types perform when they suffer a link failure is discussed. Vulnerability of graphs to the failure of individual links is computed via link residual closeness which provides a much fuller characterization of the network.

The vulnerability measure of a graph or a network depends on robustness of the remained graph, after being exposed to any intervention or attack. In this paper, we consider two edge vulnerability parameters that are the edge neighbor rupture degree and the edge scattering number. The values of these parameters of some specific graphs and their graph operations are calculated. Thus, we analyze and compare which parameter is distinctive for the different type of graphs by using tables.

Let $G$ be a graph and $S\subseteq V(G)$. We define by $〈S〉$ the subgraph of $G$ induced by $S$. For each vertex $u\in S$ and for each vertex $v\in S\setminus \{u\}$, $d_{(G,S\setminus \{u\})}(u,v)$ is the length of the shortest path in $〈V(G)-((S-\{u\})-\{v\})〉$ between $u$ and $v$ if such a path exists, and $\infty$ otherwise. For a vertex $u\in S$, let ${\omega }_{(G,S\setminus \{u\})}(u)={\sum }_{v\in S\setminus \{u\}}{(\frac{1}{2})}^{d_{(G,S\setminus \{u\})}(u,v)-1}$ where $\left(\right.\frac{1}{2}{\left)\right.}^{\infty }=0$. Jäger and Rautenbach [27] define a set $S$ of vertices to be exponential independent if ${\omega }_{(G,S\setminus \{u\})}(u)<1$ for every vertex $u$ in $S$. The exponential independence number ${\alpha }_e(G)$ of $G$ is the maximum order of an exponential independent set. In this paper, we give a general theorem and we examine exponential independence number of some tree graphs and thorn graph of some graphs.

Link residual closeness is a novel graph based network vulnerability parameter. In this model, nodes are perfectly reliable and the links fail independently of each other. In this paper, the behavior of composite network types to the failure of individual links are observed and analyzed via link residual closeness which provides a much fuller characterization of the network.

The robustness evaluates the capability of networks in resisting failures or attacks on some parts of networks. The concept of vulnerability is very important in network analysis. Isolated rupture degree is a novel graph-theoretic concept defined as a measure of network vulnerability. In this paper, the relationships between isolated rupture degree and some other vulnerability parameters such as isolated scattering number and isolated toughness are established. Exact values for isolated rupture degree of thorny networks are obtained.

In several different applications and contexts, networks are essential frameworks and appear.Vulnerability value is a measure of the network’s durability in the face of damage that may lead to a reduction or complete loss of the network’s particular functionality. The domination number and it’s types can be used network vulnerability parameters. Recently, the disjunctive total domination number has been defined by Henning and Naicker. In this paper, the disjunctive total domination numbers of the transformation graph $G^{++z}$ when $z=\{+,-\}$ of some graphs $G$ have been obtained. Furthermore, some new general results have been given for the parameter mentioned above.

Computer networks are prone to targeted attacks and random failures. Robustness is a measure of an ability of a network to continue functioning when part of the network is either naturally damaged or targeted for attack. The study of network robustness is a critical tool in the characterization and understanding of complex interconnected systems. There are several proposed graph metrics that predicates network resilience against such attacks. Isolated rupture degree is a novel graph-theoretic concept defined as a measure of network vulnerability. Isolated rupture degree is argued as an appropriate measure for modelling the robustness of network topologies in the face of possible node destruction. In this paper, the relationships between isolated rupture degree and some other graph parameters such as connectivity, covering number, minimum vertex degree are established. The isolated rupture degrees of $2K_2$-free graphs, middle graphs, corona graphs of a middle graph and a complete graph $K_2$ on two vertices are evaluated, then compared and the more stable graph types are reported. A sharp upper bound for the isolated rupture degree of middle graphs is established.

Networks are known to be prone to node or link failures. A central issue in the analysis of networks is the assessment of their stability and reliability. A central concept that is used to assess stability and robustness of the performance of a network under failures is that of vulnerability. Node and link residual closeness are novel sensitive graph based characteristics for network vulnerability analysis. Node and link residual closeness measure the vulnerability even when the removal of nodes or links does not disconnect the network. Node and link residual closeness are of great theoretical and practical significance to network design and optimization. In this paper, vulnerabilities of multipartite network type topologies to the failure of individual nodes and links are computed via node and link residual closeness which provides a much fuller characterization of the network. Then, how multipartite network type topologies perform when they suffer a node or a link failure is analyzed.

We study the structural robustness of the scale free network against the cascading failure induced by overload. In this paper, a failure mechanism based on betweenness-degree ratio distribution is proposed. In the cascading failure model we built the initial load of an edge which is proportional to the node betweenness of its ends. During the edge random deletion, we find a phase transition. Then based on the phase transition, we divide the process of the cascading failure into two parts: the robust area and the vulnerable area, and define the corresponding indicator to measure the performance of the networks in both areas. From derivation, we find that the vulnerability of the network is determined by the distribution of betweenness-degree ratio. After that we use the connection between the node ability coefficient and distribution of betweenness-degree ratio to explain the cascading failure mechanism. In simulations, we verify the correctness of our derivations. By changing connecting preferences, we find scale free networks with a slight assortativity, which performs better both in robust area and vulnerable area.

Link residual closeness (LRC) is a new sensitive characteristic for network vulnerability analysis. LRC measures the vulnerability even when the removal of links does not disconnect the network. In this paper, the robustness of wheel type network topologies is modeled and analyzed via LRC in the face of possible link destruction.

Motivated by problems in telecommunication satellites, we investigate rearrangeable permutation networks made of binary switches. A simple counting argument shows that the number of switches necessary to build a n × n rearrangeable networks (i.e. capable of realizing all one-to-one mappings of its n inputs to its n outputs) is at least ⌈ log₂ (n!) ⌉ = n log₂ n - n log₂ e + o(n) as n → ∞. For n = 2^r, the r-dimensional Beneš network gives a solution using switches. Waksman, and independently Goldstein and Leibholz, improved these networks using n log₂ n - n + 1 switches. We provide an extension of this result to arbitrary values of n, using switches. Finally the fault-tolerance issue of these networks is discussed.