ELASTICITY: Topological Characterization of Robustness in Complex Networks

Just as a herd of animals relies on its robust social structure to survive in the wild, similarly robustness is a crucial characteristic for the survival of a complex network under attack. The capacity to measure robustness in complex networks defines the resolve of a network to maintain functionality in the advent of classical component failures and at the onset of cryptic malicious attacks. To date, robustness metrics are deficient and unfortunately the following dilemmas exist: accurate models necessitate complex analysis while conversely, simple models lack applicability to our definition of robustness. In this paper, we define robustness and present a novel metric, elasticity- a bridge between accuracy and complexity-a link in the chain of network robustness. Additionally, we explore the performance of elasticity on Internet topologies and online social networks, and articulate results

The Weak Hyperedge Tenacity of the Hypercycles

Graphs play an important role in our daily life. For example, the urban transport network can be represented by a graph, as the intersections are the vertices and the streets are the edges of the graph. Suppose that some edges of the graph are removed, the question arises how damaged the graph is. There are some criteria for measuring the vulnerability of graph; the tenacity is the best criteria for measuring it. Since the hypergraph generalize the standard graph by defining any edge between multiple vertices instead of only two vertices, the above question is about the hypergraph. When a hyperedge is omitted from hypergraph, we have two kinds of deletion: strong deletion and weak deletion. Weak hyperedge deletion just deletes the connection between the vertices in the hyperedge and the vertices became in the hypergraph. In this paper, we obtain the tenacity of hypercycles by weak hyperedge deletion

Inverse domination integrity of graphs

With the growing demand for information transport, networks and network architecture have grown increasingly vital. Nodes and the connections that connect them make up a communication network. When the communication network’s nodes or links are destroyed, the network’s efficiency reduces. If a network is modeled by a graph, then there are various graph theoretical parameters used to express the vulnerability of communication networks such as connectivity, integrity, weak integrity, neighbor integrity, hub integrity, domination integrity, toughness, tenacity etc. In this paper, we introduce a new vulnerability parameter known as an inverse domination integrity which is defined as IDI(G) = min S⊆V (G) {|S| + m(G − S)}, where S is an inverse dominating set and m(G − S) denotes the order of largest component of G − S. We derive few bounds of an inverse domination integrity of graphs. Also, we determine an inverse domination integrity of some families of graphs. Finally, we compute different types of measures of vulnerabilities of probabilistic neural network which are useful in classification and pattern recognition problems.Publisher's Versio

Toughness of the corona of two graphs

The toughness of a non-complete graph G = (V , E) is defined as τ (G) = min{|S|/ω(G − S)}, where the minimum is taken over all cutsets S of vertices of G and ω(G − S) denotes the number of components of the resultant graph G − S by deletion of S. The corona of two graphs G and H , written as G ◦ H , is the graph obtained by taking one copy of G and |V (G)| copies of H , and then joining the ith vertex of G to every vertex in the ith copy of H . In this paper, we investigate the toughness of this kind of graphs and obtain the exact value for the corona of two graphs belonging to some families as paths, cycles, stars, wheels or complete graphs.Ministerio de Educación y Ciencia MTM2008-06620-C03-02Generalitat de Cataluña 1298 SGR2009Junta de Andalucía P06-FQM-0164

Neighbor Isolated Tenacity of Graphs

The tenacity of a graph is a measure of the vulnerability of a graph. In this paper we investigate a refinement that involves the neighbor isolated version of this parameter. The neighbor isolated tenacity of a noncomplete connected graph G is defined to be NIT(G) = min {|X|+ c(G/X) / i(G/X), i(G/X) ≥ 1} where the minimum is taken over all X, the cut strategy of G , i(G/X)is the number of components which are isolated vertices of G/X and c(G/X) is the maximum order of the components of G/X. Next, the relations between neighbor isolated tenacity and other parameters are determined and the neighbor isolated tenacity of some special graphs are obtained. Moreover, some results about the neighbor isolated tenacity of graphs obtained by graph operations are given

Domination Integrity of Some Path Related Graphs

The stability of a communication network is one of the important parameters for network designers and users. A communication network can be considered to be highly vulnerable if the destruction of a few elements cause large damage and only few members are able to communicate. In a communication network several vulnerability measures like binding number, toughness, scattering number, integrity, tenacity, edge tenacity and rupture degree are used to determine the resistance of network to the disruption after the failure of certain nodes (vertices) or communication links (edges). Domination theory also provides a model to measure the vulnerability of a graph network. The domination integrity of a simple connected graph is one such measure. Here we determine the domination integrity of square graph of path as well as the graphs obtained by composition (lexicographic product) of two paths