Clustering coefficient

SEPPA Concept of unit patch of residue triangle with spatial clustering coefficient http //lifecenter.sgst.cn/seppa/ Sun et al. (39)... 

As the first example, we shall discuss two notions of modules often used in topological models of networks (Milo et al. 2002 Ravasz et al. 2002 Itzkovitz et al. 2003 Clauset et al. 2004 Newman 2006). The first notion is that of a cluster on the basis of the nodes within the cluster having more interactions among themselves than with nodes outside of the cluster. Clusters can be quantitatively defined on the basis of a cluster coefficient. Escherichia coli has been shown to be clustered in a hierarchical fashion (Ravasz et al. 2002). Another definition is that of a network motif (Mho et al. 2002 Shen-Orr et al. 2002 Kashtan et al. 2004 Yeger-Lotem et al. 2004). These authors... 

In addition, the determinantal form of the individual terms in this expansion implies antisymmetrization of the cluster coefficients, such that Iff- = -tf-j =... 

The present work details the derivation of a full coupled-cluster model, including single, double, and triple excitation operators. Second quantization and time-independent diagrams are used to facilitate the derivation the treatment of (diagram) degeneracy and permutational symmetry is adapted from time-dependent methods. Implicit formulas are presented in terms of products of one- and two-electron integrals, over (molecular) spin-orbitals and cluster coefficients. Final formulas are obtained that restrict random access requirements to rank 2 modified integrals and sequential access requirements to the rank 3 cluster coefficients. 

Also, projection onto the reference determinant allows AE, and hence the correlated energy, to be calculated once the cluster coefficients are known ... 

It is clear that the necessary and sufficient number of equations in the coupled set [Eq. (78)] is equal to the number of unique cluster coefficients, provided that a solution exists. Since the coupled-cluster equations are non-Hermitian and nonlinear, the existence of solutions and the reality of eigenvalues corresponding to solutions are not guaranteed. However, Zivkovic and Monkhorst3536 have recently shown, using analytic continuations of solutions to the Cl problem, that for physically reasonable cases both the existence of solutions and the reality of eigenvalues are assured. 

The specific equations used to determine the cluster coefficients in the CCSDT model will now be given. Projection of the Schrodinger equation [cf. Eq. (77)] onto the singly excited space gives... 

The equations for the cluster coefficients and the correlated energy in a CCSDT model were given in operator form in Section III [cf. Eqs. (80)— (83)] this form is, of course, not amenable to calculations. In Section V the time-independent techniques discussed in Section IV are applied to evaluate the requisite matrix elements in terms of cluster coefficients and one- and two-electron integrals over the spin-orbital basis. 

The expressions for the matrix elements obtained in the preceding section, together with Eqs. (80)-(83), enable us to write implicit equations determining the cluster coefficients and the correlated energy in terms of the cluster coefficients and the one- and two-electron integrals over the spin-orbital basis. We may write Eq. (80), the projection of the Schro-dinger equation for the CCSDT wave function on the singly excited space, as... 

Individual cluster coefficients, as well as integrals, cannot in general be freely reordered in expressions for matrix elements. In particular, the rearrangement of coefficients can be seen to correspond to a topological deformation of a diagram that alters the sequence of topologically distinct... 

We now show that the projection of the Schrodinger equation for the CCSDT wave function on the triply excited space [cf. Eq. (106)] can be written in terms of (at worst) products of unmodified rank 3 cluster coefficients and modified rank 2 integrals. Tensor notation with repeated index summation convention will be used, except, of course, for the permutation operator. 

The fourth and fifth terms, [XII] and [XIII], are anomalous in that modified cluster coefficients (e.g., rank 2) are used in the final tensor products. The particular modified cluster coefficients involved, t2, have already been used in several intermediate tensor products [cf. Eqs. (145), (157), (160), (169), and (174)]. 

The algebraic expressions obtained from a diagrammatic evaluation of the coupled-cluster equations for a CCSDT model are resolvable into products of unmodified cluster coefficients (or trivially modified in the case of t2) and modified one- and two-electron integrals. At no stage of the calculation are tensors of rank greater than 2 required, except for the initial contraction and final expansion of the rank 3 triples cluster coefficients. 

The model ensures a set of simple non-dominated networks (Pareto optimal) while minimizing the numbers of indirect links in the networks. The total number of indirect links is minimized to keep the network sparse (i.e., to keep the network simple). The model also maximizes the percentage difference between clustering coefficients to maintain one of the small world network (SWN) characteristic and minimizes the sum of squared errors to fit the experimental data. One of the constraints is included to ensure the minimum connectivity in all non dominated networks. The other constraint is included to make the L(avgshortestpath) nearly equal to log -if so that SWN characteristic can be ensured. Mathematically, the model (Ganta, 2007) is ... 

Step 5 For the updated population of directed networks with random links and time delay links, calculate the characteristic parameters i.e. average clustering coefficient avgcc) and average shortest path length (avgshortestpath). Calculate similar parameters, namely, Crandom (Cron) and Lrandom (Lran), for the equivalent random networks i.e. random networks with same number of nodes and links (Newman et al (2000)). 

The clustering coefficient of a vertex, denoted as Ci, is a local vertex invariant derived from the adjacencymatrixbyconsideringthe first-neighbor interconnectivity [Bonchev and Buck, 2007]. It was proposed as a measure of the clustered structure of a graph around a vertex and is defined as... 

Muralidhar, R., and Ramkrishna, D., Inverse Problems of Agglomeration Kinetics, 2 Binary Clustering Coefficients from Self-Preserving Spectra, J. Colloid Interface Set 131 (2) (1989) 503-513. 

Quadruply excited configuration Coefficient from 37-configuration wave function by full C.I. Unlinked four-cluster coefficient calculated from pair correlation coefficients ... 

The differences between the full C.I. coefficients and the unlinked cluster coefficients are due to the true four-electron correlations, L/1234. They are very small. 

The clustering coefficient around a node (v) of degree k [CC (r)] is defined as... 

The coefficients are the irreducible cluster coefficients [4] constructed from yk/k. The correction to the second virial coefficient is due to the existence of binary junctions (k = 2), and should vanish if y2 in A2 were made to vanish. Thus association does not affect the second virial coefficient if there is no binary cross-linking. The frequently... 

Definition 3 The local clustering coefficient (Watts, 1999), Cv, of a vertex v with K neighbors measures the density of the links in the neighborhood of v. 

The clustering coefficient of a random graph with mean degree 2 is... 

Clustering coefficients Watts and Strogatz proposed a measure of clustering (Watts, 1999) and defined it as a measure of local vertices density, thus for each node v, the local clustering around its neighborhood is defined in the following way ... 

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