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84
Networks, Dynamics, and the SmallWorld Phenomenon
 American Journal of Sociology
, 1999
"... The smallworld phenomenon formalized in this article as the coincidence of high local clustering and short global separation, is shown to be a general feature of sparse, decentralized networks that are neither completely ordered nor completely random. Networks of this kind have received little atte ..."
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Cited by 106 (1 self)
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The smallworld phenomenon formalized in this article as the coincidence of high local clustering and short global separation, is shown to be a general feature of sparse, decentralized networks that are neither completely ordered nor completely random. Networks of this kind have received little attention, yet they appear to be widespread in the social and natural sciences, as is indicated here by three distinct examples. Furthermore, small admixtures of randomness to an otherwise ordered network can have a dramatic impact on its dynamical, as well as structural, properties—a feature illustrated by a simple model of disease transmission.
The Wiener Index Of Simply Generated Random Trees
 Random Struct. Alg
, 2003
"... Asymptotics are obtained for the mean, variance and higher moments as well as for the distribution of the Wiener index of a random tree from a simply generated family (or, equivalently, a critical Galton Watson tree). We also establish a joint asymptotic distribution of the Wiener index and the in ..."
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Cited by 37 (16 self)
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Asymptotics are obtained for the mean, variance and higher moments as well as for the distribution of the Wiener index of a random tree from a simply generated family (or, equivalently, a critical Galton Watson tree). We also establish a joint asymptotic distribution of the Wiener index and the internal path length, as well as asymptotics for the covariance and other mixed moments. The limit laws are described using functionals of a Brownian excursion. The methods include both Aldous' theory of the continuum random tree and analysis of generating functions. 1.
Graph Kernels
, 2007
"... We present a unified framework to study graph kernels, special cases of which include the random walk (Gärtner et al., 2003; Borgwardt et al., 2005) and marginalized (Kashima et al., 2003, 2004; Mahé et al., 2004) graph kernels. Through reduction to a Sylvester equation we improve the time complexit ..."
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Cited by 37 (4 self)
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We present a unified framework to study graph kernels, special cases of which include the random walk (Gärtner et al., 2003; Borgwardt et al., 2005) and marginalized (Kashima et al., 2003, 2004; Mahé et al., 2004) graph kernels. Through reduction to a Sylvester equation we improve the time complexity of kernel computation between unlabeled graphs with n vertices from O(n 6) to O(n 3). We find a spectral decomposition approach even more efficient when computing entire kernel matrices. For labeled graphs we develop conjugate gradient and fixedpoint methods that take O(dn 3) time per iteration, where d is the size of the label set. By extending the necessary linear algebra to Reproducing Kernel Hilbert Spaces (RKHS) we obtain the same result for ddimensional edge kernels, and O(n 4) in the infinitedimensional case; on sparse graphs these algorithms only take O(n 2) time per iteration in all cases. Experiments on graphs from bioinformatics and other application domains show that these techniques can speed up computation of the kernel by an order of magnitude or more. We also show that certain rational kernels (Cortes et al., 2002, 2003, 2004) when specialized to graphs reduce to our random walk graph kernel. Finally, we relate our framework to Rconvolution kernels (Haussler, 1999) and provide a kernel that is close to the optimal assignment kernel of Fröhlich et al. (2006) yet provably positive semidefinite.
The Wiener index and the Szeged index of benzenoid systems in linear time
 J. Chem. Inf. Comput. Sci
, 1997
"... A linear time algorithm is presented which, for a given benzenoid system G, computes the Wiener index of G. The algorithm is based on an isometric embedding of G into the Cartesian product of three trees, combined with the notion of the Wiener index of vertexweighted graphs. An analogous approach y ..."
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Cited by 26 (6 self)
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A linear time algorithm is presented which, for a given benzenoid system G, computes the Wiener index of G. The algorithm is based on an isometric embedding of G into the Cartesian product of three trees, combined with the notion of the Wiener index of vertexweighted graphs. An analogous approach yields also a linear algorithm for computing the Szeged index of benzenoid systems. 1.
Faces of generalized permutohedra
"... Abstract. The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f, h and γvectors. These polytopes include permutohedra, associahedra, graphassociahedra, simple graphic zonotopes, nestohedra, and other interesting polytopes. We give several explici ..."
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Cited by 18 (3 self)
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Abstract. The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f, h and γvectors. These polytopes include permutohedra, associahedra, graphassociahedra, simple graphic zonotopes, nestohedra, and other interesting polytopes. We give several explicit formulas for hvectors and γvectors involving descent statistics. This includes a combinatorial interpretation for γvectors of a large class of generalized permutohedra which are flag simple polytopes, and confirms for them Gal’s conjecture on nonnegativity of γvectors. We calculate explicit generating functions and formulae for hpolynomials of various families of graphassociahedra, including those corresponding to all Dynkin diagrams of finite and affine types. We also discuss relations with Narayana numbers and with Simon Newcomb’s problem. We give (and conjecture) upper and lower bounds for f, h, and γvectors within several classes of generalized permutohedra. An appendix discusses the equivalence of various notions of deformations of simple polytopes.
Centers of complex networks
 J Theor Biol
, 2003
"... Abstract. The central vertices in complex networks are of particular interest because they might play the role of organizational hubs. Here, we consider three different geometric centrality measures, eccentricity, status, and centroid value, that were originally used in the context of resource place ..."
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Cited by 14 (0 self)
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Abstract. The central vertices in complex networks are of particular interest because they might play the role of organizational hubs. Here, we consider three different geometric centrality measures, eccentricity, status, and centroid value, that were originally used in the context of resource placement problems. We show that these quantities lead to useful descriptions of the centers of biological networks which often, but not always, correlate with a purely local notion of centrality such as the vertex degree. We introduce the notion of local centers as local optima of a centrality value “landscape ” on a network and discuss briefly their role. 1 S. Wuchty, P.F. Stadler: Centers of Complex Networks 2
An integrated som fuzzy artmap neural system for the evaluation of toxicity
 J. Chem. Inf. Comput. Sci
, 2002
"... Selforganized maps (SOM) have been applied to analyze the similarities of chemical compounds and to select from a given pool of descriptors the smallest and more relevant subset needed to build robust QSAR models based on fuzzy ARTMAP. First, the category maps for each molecular descriptor and for ..."
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Cited by 9 (1 self)
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Selforganized maps (SOM) have been applied to analyze the similarities of chemical compounds and to select from a given pool of descriptors the smallest and more relevant subset needed to build robust QSAR models based on fuzzy ARTMAP. First, the category maps for each molecular descriptor and for the target activity variable were created with SOM and then classified on the basis of topology and nonlinear distribution. The best subset of descriptors was obtained by choosing from each cluster the index with the highest correlation with the target variable and then in order of decreasing correlation. This process was terminated when a dissimilarity measure increased, indicating that the inclusion of more molecular indices would not add supplementary information. The optimal subset of descriptors was used as input to a fuzzy ARTMAP architecture modified to effect predictive capabilities. The performance of the integrated SOMfuzzy ARTMAP approach was evaluated with the prediction of the acute toxicity LC50 of a homogeneous set of 69 benzene derivatives in the fathead minnow and the oral rat toxicity LD50 of a heterogeneous set of 155 organic compounds. The proposed methodology minimized the problem of misclassification of similar compounds and significantly enhanced the predictive capabilities of a properly trained fuzzy ARTMAP network.
The Wiener polynomial of a graph
, 2008
"... The Wiener index is a graphical invariant that has found extensive application in chemistry. We define a generating function, which we call the Wiener polynomial, whose derivative is a qanalog of the Wiener index. We study some of the elementary properties of this polynomial and compute it for some ..."
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Cited by 7 (3 self)
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The Wiener index is a graphical invariant that has found extensive application in chemistry. We define a generating function, which we call the Wiener polynomial, whose derivative is a qanalog of the Wiener index. We study some of the elementary properties of this polynomial and compute it for some common graphs. We then find a formula for the Wiener polynomial of a dendrimer, a certain highly regular tree of interest to chemists, and show that it is unimodal. Finally, we point out a connection with the Poincaré polynomial of a finite Coxeter group.
MOLecular Structure GENeration with MOLGEN, new features and future developments
 Fresenius J. Anal. Chem
, 1997
"... MOLGEN is a computer program system which is designed for generating molecular graphs fast, redundancy free and exhaustively. In the present paper we describe its basic features, new features of the current release MOLGEN 3.5, and future developments which provide considerable improvements and ex ..."
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Cited by 6 (4 self)
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MOLGEN is a computer program system which is designed for generating molecular graphs fast, redundancy free and exhaustively. In the present paper we describe its basic features, new features of the current release MOLGEN 3.5, and future developments which provide considerable improvements and extensions. 1 Introduction MOLGEN [17] is a generator for molecular graphs (=connectivity isomers or constitutional formulae) allowing to generate all isomers that correspond to a given molecular formula and (optional) further conditions like prescribed and forbidden substructures, ring sizes etc. The input consists of ffl the empirical formula, together with ffl an optional list of macroatoms, which means prescribed substructures that must not overlap, ffl an optional goodlist, that consists of prescribed substructures which may overlap, ffl an optional badlist, containing forbidden substructures, ffl an optional interval for the minimal and maximal size of rings, ffl an optional num...