On Harary index of graphs

Xu, Kexiang; Das, Kinkar Ch.

doi:10.1016/j.dam.2011.06.003

Cited by 64 publications

(29 citation statements)

References 21 publications

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“…The class of connectivity/cohesiveness-based metrics comprises objectives like maximizing the number of disconnected components (fragmentation) in the interdicted graph, minimizing the number of pairwise-connected nodes, or minimizing the size of the largest remaining component. The distance-based metrics consider the length of the shortest path between two nodes in the pre-and post-interdicted graphs and consist of such metrics as graph efficiency and Harary index [46]. Using these metrics, a CNP can be formulated as a mixed-integer program [2,37,41,42].…”

Section: Metrics and Exact Solutionsmentioning

confidence: 99%

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Robustness of solutions to critical node detection problems with imperfect data: a computational study

Gillen

Veremyev

Prokopyev

et al. 2016

Optimization Methods and Software

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A class of critical node detection problems based upon the metric of communication efficiency is considered. While both exact integer programming and heuristic centrality-based methods exist for the solution of these problems, previous work has been mostly focused on the case where perfect information about the network is available. In this paper we suppose that some level of misinformation about nodes or edges has been inflicted on the observer's perception of the network, that is, there are hidden elements or fake additional elements. An extensive computational study is conducted to ascertain whether the exact integer-programming-based solutions perform better under imperfect information than heuristic methods. For large networks, exact methods cannot produce a solution in a reasonable amount of time, hence an approximation of the exact method is also considered for such instances. The obtained approximate solutions are again compared to centrality-based heuristics under the presence of imperfect data.

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Section: Metrics and Exact Solutionsmentioning

confidence: 99%

“…In particular, two common distance-based metrics that capture different aspects of communication efficiency were chosen for this study: graph efficiency and an exponential decay function. Graph efficiency-essentially a scaled Harary index [46]-is defined as Graph efficiency:…”

Section: Metrics and Exact Solutionsmentioning

confidence: 99%

Robustness of solutions to critical node detection problems with imperfect data: a computational study

Gillen

Veremyev

Prokopyev

et al. 2016

Optimization Methods and Software

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“…Details on the applications of the Harary index can be found in the survey [7], whereas we can find details on its mathematical properties in [5,10] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Harary index of the k-th power of a graph

Xiong

Gutman

2013

APPL ANAL DISCRETE M

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The k-th power of a graph G, denoted by G k , is a graph with the same set of vertices as G, such that two vertices are adjacent in G k if and only if their distance in G is at most k. The Harary index H is the sum of the reciprocal distances of all pairs of vertices of the underlying graph. Lower and upper bounds on H(G k ) are obtained. A Nordhaus-Gaddum type inequality for H(G k ) is also established.

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“…They have revolved around the idea of considering the sum of reciprocal shortest-path distances in graphs. It is worth mentioning that the sum of all the reciprocal shortest-path distances in a graph is known in graph theory as the Harary index, which was introduced by Plavšić et al in 1993 [38] and studied elsewhere [46,45]. The so-called efficiency index introduced by Latora and Marchiori [30] (defined below in Eq.…”

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confidence: 99%

Communicability Angle and the Spatial Efficiency of Networks

Estrada¹,

Hatano

2016

SIAM Rev.

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Abstract. We introduce the concept of communicability angle between a pair of nodes in a graph. We provide strong analytical and empirical evidence that the average communicability angle for a given network accounts for its spatial efficiency on the basis of the communications among the nodes in a network. We determine characteristics of the spatial efficiency of more than a hundred real-world complex networks that represent complex systems arising in a diverse set of scenarios. In particular, we find that the communicability angle correlates very well with the experimentally measured value of the relative packing efficiency of proteins that are represented as residue networks. We finally show how we can modulate the spatial efficiency of a network by tuning the weights of the edges of the networks. This allows us to predict effects of external stresses on the spatial efficiency of a network as well as to design strategies to improve important parameters in real-world complex systems.distance; graph planarity; Euclidean distance 1. Introduction. Graphs are frequently used to represent discrete objects both in abstract mathematics and computer sciences as well as in applications, such as theoretical physics, biology, ecology and social sciences [26,14]. In the particular case of representing the networked skeleton of complex systems, graphs receive the denomination of complex networks; we will hereafter use graphs and networks interchangeably.Complex networks are ubiquitous in many real-world scenarios, ranging from the biomolecular -those representing gene transcription, protein interactions, and metabolic reactions -to the social and infrastructural organization of modern society [11,36,9]. In many of these networks, nodes and edges are used to represent physically embedded objects [4], namely spatial networks. In urban street networks, for instance, the nodes describe the intersection of streets, which are represented by the edges of the graph. These streets and their intersections are embedded in the twodimensional space representing the surface occupied by the corresponding city [28]. Thus, these networks are planar graphs in the sense that we can draw them in a plane without edge intersections, except for the few bridges and overpasses present in a city. Another spatial network is the brain network, in which the nodes account for brain regions embedded in the three-dimensional space occupied by the brain, while the edges represent the communication or physical connections among these regions [7]. We can also capture the three-dimensional structure of proteins by means of the residue networks in which nodes describe amino acids and the edges represent physical interactions among them. Other examples include the following: infrastructures, such as the Internet, transportation networks, water and electricity supply networks, etc. [4]; anatomical networks, such as vascular and organ/tissue networks; the networks of channels in fractured rocks; the networks representing the corridors and galleries in animal nest...

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On Harary index of graphs

Cited by 64 publications

References 21 publications

Robustness of solutions to critical node detection problems with imperfect data: a computational study

Robustness of solutions to critical node detection problems with imperfect data: a computational study

Harary index of the k-th power of a graph

Communicability Angle and the Spatial Efficiency of Networks

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