GTI-space : the space of generalized topological indices

A new extension of the generalized topological indices (GTI) approach is carried out torepresent 'simple' and 'composite' topological indices (TIs) in an unified way. Thisapproach defines a GTI-space from which both simple and composite TIs represent particular subspaces. Accordingly, simple TIs such as Wiener, Balaban, Zagreb, Harary and Randićconnectivity indices are expressed by means of the same GTI representation introduced for composite TIs such as hyper-Wiener, molecular topological index (MTI), Gutman index andreverse MTI. Using GTI-space approach we easily identify mathematical relations between some composite and simple indices, such as the relationship between hyper-Wiener and Wiener index and the relation between MTI and first Zagreb index. The relation of the GTI space with the sub-structural cluster expansion of property/activity is also analysed and some routes for the applications of this approach to QSPR/QSAR are also given

"Clumpiness" Mixing in Complex Networks

Three measures of clumpiness of complex networks are introduced. The measures quantify how most central nodes of a network are clumped together. The assortativity coefficient defined in a previous study measures a similar characteristic, but accounts only for the clumpiness of the central nodes that are directly connected to each other. The clumpiness coefficient defined in the present paper also takes into account the cases where central nodes are separated by a few links. The definition is based on the node degrees and the distances between pairs of nodes. The clumpiness coefficient together with the assortativity coefficient can define four classes of network. Numerical calculations demonstrate that the classification scheme successfully categorizes 30 real-world networks into the four classes: clumped assortative, clumped disassortative, loose assortative and loose disassortative networks. The clumpiness coefficient also differentiates the Erdos-Renyi model from the Barabasi-Albert model, which the assortativity coefficient could not differentiate. In addition, the bounds of the clumpiness coefficient as well as the relationships between the three measures of clumpiness are discussed.Comment: 47 pages, 11 figure

"Clumpiness" Mixing in Complex Networks

Three measures of clumpiness of complex networks are introduced. The measures quantify how most central nodes of a network are clumped together. The assortativity coefficient defined in a previous study measures a similar characteristic, but accounts only for the clumpiness of the central nodes that are directly connected to each other. The clumpiness coefficient defined in the present paper also takes into account the cases where central nodes are separated by a few links. The definition is based on the node degrees and the distances between pairs of nodes. The clumpiness coefficient together with the assortativity coefficient can define four classes of network. Numerical calculations demonstrate that the classification scheme successfully categorizes 30 real-world networks into the four classes: clumped assortative, clumped disassortative, loose assortative and loose disassortative networks. The clumpiness coefficient also differentiates the Erdos-Renyi model from the Barabasi-Albert model, which the assortativity coefficient could not differentiate. In addition, the bounds of the clumpiness coefficient as well as the relationships between the three measures of clumpiness are discussed.Comment: 47 pages, 11 figure

Spectral Scaling in Complex Networks

A complex network is said to show topological isotropy if the topological structure around a particular node looks the same in all directions of the whole network. Topologically anisotropic networks are those where the local neighborhood around a node is not reproduced at large scale for the whole network. The existence of topological isotropy is investigated by the existence of a power-law scaling between a local and a global topological characteristic of complex networks obtained from graph spectra. We investigate this structural characteristic of complex networks and its consequences for 32 real-world networks representing informational, technological, biological, social and ecological systems.Comment: 9 pages, 3 figure

How the parts organize in the whole : a top-downview of molecular descriptors and properties for QSARand drug design

Sometimes the complexity of a system, or the properties derived from it, do depend neither on the individual characteristics of the components of the system nor on the nature of the physical forces that hold them together. In such cases the properties derived from the 'organization' of the system given by the connectivity of its elements can be determinant for explaining the structure of such systems. Here we explore the necessity of accounting for these structural characteristics in the molecular descriptors. We show that graph theory is the most appropriate mathematical theory to account for such molecular features. We review a method (TOPS-MODE) that is able to transform simple molecular descriptors, such as logP, polar surface area, molar refraction, charges, etc., into series of descriptors that account for the distribution of these characteristics (hydrophobicity, polarity, steric effects, etc) across the molecule. We explain the mathematical and physical principles of the TOPS-MODE method and develop three examples covering the description and interpretation of skin sensitisation of chemicals, chromosome aberration produced by organic molecules and drug binding to human serum albumin