Ernesto Estrada

Subgraph centrality in complex networks

We introduce a new centrality measure that characterizes the participation of each node in all subgraphs in a network. Smaller subgraphs are given more weight than larger ones, which makes this measure appropriate for characterizing network motifs. We show that the subgraph centrality [C(S)(i)] can be obtained mathematically from the spectra of the adjacency matrix of the network. This measure is better able to discriminate the nodes of a network than alternate measures such as degree, closeness, betweenness, and eigenvector centralities. We study eight real-world networks for which C(S)(i) displays useful and desirable properties, such as clear ranking of nodes and scale-free characteristics. Compared with the number of links per node, the ranking introduced by C(S)(i) (for the nodes in the protein interaction network of S. cereviciae) is more highly correlated with the lethality of individual proteins removed from the proteome.

Recent advances on the role of topological indices in drug discovery research

The role of topological indices in drug development research is updated. A series of definitions in the fields of topological indices and drug discovery technologies are introduced. In all cases where it is possible the IUPAC recommendations for terms used in medicinal chemistry and in computational drug design are used. Recent advances on the use of topological indices in the lead discovery process are reviewed making emphasis on two approaches: combined use of connectivity and charge indices and TOSS-MODE approach. Studies of similarity/dissimilarity and rational combinatorial library design are also updated. The use of these descriptors in lead optimization process is critically analyzed. Topological indices QSAR, the problem of 2D QSAR versus 3D QSAR, strategies of orthogonalization and the use of linear combination and semiempirical connectivity indices are also described. The main directions of progress for these indices in QSAR and drug research are analyzed with examples of application of novel statistical techniques, such as artificial neural networks, genetic algorithms and partial least squares. Future outlooks of development in this area of research are also given.

Virtual identification of essential proteins within the protein interaction network of yeast

Topological analysis of large scale protein‐protein interaction networks (PINs) is important for understanding the organizational and functional principles of individual proteins. The number of interactions that a protein has in a PIN has been observed to be correlated with its indispensability. Essential proteins generally have more interactions than the nonessential ones. We show here that the lethality associated with removal of a protein from the yeast proteome correlates with different centrality measures of the nodes in the PIN, such as the closeness of a protein to many other proteins, or the number of pairs of proteins which need a specific protein as an intermediary in their communications, or the participation of a protein in different protein clusters in the PIN. These measures are significantly better than random selection in identifying essential proteins in a PIN. Centrality measures based on graph spectral properties of the network, in particular the subgraph centrality, show the best performance in identifying essential proteins in the yeast PIN. Subgraph centrality gives important structural information about the role of individual proteins, and permits the selection of possible targets for rational drug discovery through the identification of essential proteins in the PIN.

Spectral measures of bipartivity in complex networks

We introduce a quantitative measure of network bipartivity as a proportion of even to total number of closed walks in the network. Spectral graph theory is used to quantify how close to bipartite a network is and the extent to which individual nodes and edges contribute to the global network bipartivity. It is shown that the bipartivity characterizes the network structure and can be related to the efficiency of semantic or communication networks, trophic interactions in food webs, construction principles in metabolic networks, or communities in social networks.

SPECTRAL MOMENTS OF THE EDGE ADJACENCY MATRIX IN MOLECULAR GRAPHS. 1. DEFINITION AND APPLICATIONS TO THE PREDICTION OF PHYSICAL PROPERTIES OF ALKANES

A novel graph theoretical invariant based on the spectral moments of the edge adjacency matrix (E) is proposed. Spectral moments of the E matrix are used to describe seven physical properties of alkanes. All the regression models found are very significant from the statistical point of view. The spectral moments are expressed as linear combinations of the different structural fragments of the molecular graph. The use of the substructural approach for the description of seven physical properties of alkanes is also proved. The results obtained are interpreted in term of structural features of molecules.

Using network centrality measures to manage landscape connectivity

We use a graph-theoretical landscape modeling approach to investigate how to identify central patches in the landscape as well as how these central patches influence (1) organism movement within the local neighborhood and (2) the dispersal of organisms beyond the local neighborhood. Organism movements were theoretically estimated based on the spatial configuration of the habitat patches in the studied landscape. We find that centrality depends on the way the graph-theoretical model of habitat patches is constructed, although even the simplest network representation, not taking strength and directionality of potential organisms flows into account, still provides a coarse-grained assessment of the most important patches according to their contribution to landscape connectivity. Moreover, we identify (at least) two general classes of centrality. One accounts for the local flow of organisms in the neighborhood of a patch, and the other accounts for the ability to maintain connectivity beyond the scale of the local neighborhood. Finally, we study how habitat patches with high scores on different network centrality measures are distributed in a fragmented agricultural landscape in Madagascar. Results show that patches with high degree and betweenness centrality are widely spread, while patches with high subgraph and closeness centrality are clumped together in dense clusters. This finding may enable multispecies analyses of single-species network models.

Edge Adjacency Relationships and a Novel Topological Index Related to Molecular Volume

published in Advance ACS Absrracfs, June 15, 1994. where v,, is the index of refraction and W t h e molar volume. 0095-2338/95/1635-0031

Characterization of the folding degree of proteins

09.00/0

Network Properties Revealed through Matrix Functions

MOTIVATION The characterization of the folding degree of chains is central to the elucidation of structure--function relationships in proteins. Here we present a new index for characterizing the folding degree of a (protein) chain. This index shows a range of features that are desirable for the study of the relation between structure and function in proteins. RESULTS A novel index characterizing the folding degree of (protein) chains is developed based on the spectral moments of a matrix representing the dihedral angles (phi, omega and epsilon) of the protein main chain. The proposed index is normalized to the chain size, is not correlated to the gyration radius of the backbone chain and is able to distinguish between structures for which the sum of the main-chain dihedral angles is identical. The index is well correlated to the percentages of helix and strand in proteins, shows a linear dependence with temperature changes, and is able to differentiate among protein families. AVAILABILITY On request from the author.

Spectral Moments of the Edge-Adjacency Matrix of Molecular Graphs. 2. Molecules Containing Heteroatoms and QSAR Applications

The emerging field of network science deals with the tasks of modeling, comparing, and summarizing large data sets that describe complex interactions. Because pairwise affinity data can be stored in a two-dimensional array, graph theory and applied linear algebra provide extremely useful tools. Here, we focus on the general concepts of centrality, communicability, and betweenness, each of which quantifies important features in a network. Some recent work in the mathematical physics literature has shown that the exponential of a networks adjacency matrix can be used as the basis for defining and computing specific versions of these measures. We introduce here a general class of measures based on matrix functions, and show that a particular case involving a matrix resolvent arises naturally from graph-theoretic arguments. We also point out connections between these measures and the quantities typically computed when spectral methods are used for data mining tasks such as clustering and ordering. We finish with computational examples showing the new matrix resolvent version applied to real networks.