Fractal behaviours of networks induced on infinite tree structures by random walks

Tree graphs such as Cayley trees provide a stage to support the self-organization of fractal networks by the flow of walkers from the root vertex to the outermost shell of the tree graph. This network model is a typical example that demonstrates the ability of a random process on a network to generate fractality. However, the finite scale of the tree structure assumed in the model restricts the size of fractal networks. In this study, we removed the restriction on the size of the trees by introducing a lifetime τ (number of steps of random walks) of walkers. As a result, we successfully induced a size-independent fractal structure on a tree graph without a boundary. Our numerical results show that the mean number of offspring d b of the original tree structure determines the value of the fractal box dimension db through the relation d b — 1 = (n b — 1) -θ . The lifetime τ controls the presence or absence of small-world and scale-free properties. The ideal fractal behaviour can be maintained by selecting an appropriate value of τ. The numerical results contribute to the development of a systematic method for generating fractal small-world and scale-free networks while controlling the value of the fractal box dimension. Unlike other models that use recursive rules to generate self-similar structures, this model specifically produces small-world fractal networks with scale-free properties.

Incentives against Max-Min Fairness in a Centralized Resource System

Resource allocating mechanisms draw much attention from various areas, and exploring the truthfulness of these mechanisms is a very hot topic. In this paper, we focus on the max-min fair allocation in a centralized resource system and explore whether the allocation is truthful when a node behaves strategically. The max-min fair allocation enables nodes receive appropriate resources, and we introduce an efficient algorithm to find out the allocation. To explore whether the allocation is truthful, we analyze how the allocation varies when a new node is added to the system, and we discuss whether the node can gain more resources if it misreports its resource demands. Surprisingly, if a node misrepresents itself by creating several fictitious nodes but keeps the sum of these nodes’ resource demands the same, the node can achieve more resources evidently. We further present some illustrative examples to verify the results, and we show that a node can achieve 1.83 times resource if it misrepresents itself as two nodes. Finally, we discuss the influence of node’s misrepresenting behavior in tree graph: some child nodes gain fewer resources even if their parent node gains more resources by creating two fictitious nodes.

Methodological approach to spacecraft development cost calculation

The aim of this work is to develop a methodological approach to the development of a unified, common for all developers, computer-aided branch procedure for spacecraft development cost estimation based on a spacecraft cost model and practical recommendations on computer-aided branch procedure development. The cost parameters of the development, manufacturing, and operation of new spacecraft and their technical level (perfection) are the determining factors in competitiveness assessment. To decide on the advisability of starting or resuming the development of a new spacecraft, one has to correctly estimate the development cost. Using the standardized calculation method in estimating the development cost for new space hardware is unacceptable for lack of bug-free design and production documentation, which is the end product of any development activity. Parametric methods (the basic methods used to estimate the spacecraft development cost in the USA and Europe) cannot be used in the development of a high-quality procedure for spacecraft development cost estimation for lack of a branch statistical database on spacecraft development labor intensiveness and materials consumption at the State Space Agency of Ukraine. This calls for a nonstandard cost model of spacecraft development. The authors’ cost model is based on a method of componentwise analogy for simple spacecraft components, moving (up and down) along the edges of a weighted oriented tree graph that models the spacecraft technical structure, and fuzzy analysis methods. The tree graph Gi(V(С),D) models the spacecraft technical structure (V, C, and D are the sets of graph vertices and edges and spacecraft components, respectively; to each graph vertex there corresponds a spacecraft component). The paper presents a nonstandard cost model of spacecraft development, which in its essence is close to a nonlinear parametric cost model, and a scientific methodology for the development of an advanced branch procedure for spacecraft development cost calculation with component and stage detailing.

Disjoint Genus-0 Surfaces in Extremal Graph Theory and Set Theory Lead To a Novel Topological Theorem

When an edge is removed, a cycle graph Cn becomes a n-1 tree graph. This observation from extremal set theory leads us to the realm of set theory, in which a topological manifold of genus-1 turns out to be of genus-0. Starting from these premises, we prove a theorem suggesting that a manifold with disjoint points must be of genus-0, while a manifold of genus-1 cannot encompass disjoint points.

Image Copy Detection and Evolution Visualisation Using Tree Graphs

Image copy detection is an important problem for several applications such as detecting forgery to enforce copyright protection and intellectual property. One of the important problems following copy detection, however, is the assessment of the type of modifications undergone by an original image to form its copies. In this work, we propose a method for quantifying some of these modifications when multiple copies of the same image are available. We also propose an algorithm to estimate temporal precedence between images (i.e., the order of creation of the copies). Using the estimated relations, a tree graph is then built to visualise the history of evolution of the original image into its copies. Our work is important for ensuring better interpretation of image copies after their detection. It also lays a new ground for enhancing image indexing and search on the Web.

On Harmonic Index and Diameter of Quasi-Tree Graphs

The harmonic index of a graph G ( H G ) is defined as the sum of the weights 2 / d u + d v for all edges u v of G , where d u is the degree of a vertex u in G . In this paper, we show that H G ≥ D G + 5 / 3 − n / 2 and H G ≥ 1 / 2 + 2 / 3 n − 2 D G , where G is a quasi-tree graph of order n and diameter D G . Indeed, we show that both lower bounds are tight and identify all quasi-tree graphs reaching these two lower bounds.