On two recent geometrical characterizations of hyperellipticity

Costa, Antonio F.; Porto, Ana M.

doi:10.5209/rev_rema.2004.v17.n1.16778

Cited by 5 publications

(23 citation statements)

References 5 publications

(7 reference statements)

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“…At the same time, as will become clear along the remainder of this article, the more connected the network is, the less hierarchical levels it tends to have. It should be also observed that algorithmic implementation of hierarchy identification, such as those reported in [9] and [10] (see also [35]), are typically more computationally efficient than the use of the matrix arithmetic presented in this Section.…”

Section: Virtual Edges and Hierarchiesmentioning

confidence: 99%

“…Observe that the hierarchies d provide a radial reference frame or coordinate system which can be used to partially identify nodes and edges with respect to the reference node i. The concept o hierarchy in a complex network is also related to the concept of roles [34] and the distance transform [29,30] of the nodes in the original network Γ with respect to the reference node [10].…”

Section: Virtual Edges and Hierarchiesmentioning

confidence: 99%

“…The set obtained from the above vector has also been called [10] the ring of radius d centered at i, being henceforth represented as R d (i). Observe that the ring of radius 2 centered at i = 8 in Figure 2 is R 2 (8) = {1, 3, 4, 6, 10, 11, 13}.…”

Section: Virtual Edges and Hierarchiesmentioning

confidence: 99%

“…Although such distributions emphasize important properties of the analyzed networks, further valuable topological information can be gathered not only by considering the clustering coefficient, but also by analyzing such features along the hierarchical levels of the networks [9,10]. While some attention has been focused on the relevant issue of hierarchy in complex networks (e.g.…”

Section: Introductionmentioning

confidence: 99%

“…[11,12,13,17,18,19,20,21,22,23,24,28,26,27]), and hierarchical extensions of the node degree and clustering coefficient were only more recently formalized in [9,10] by using concepts derived from mathematical morphology [25,29,30] including dilations and distance transforms in graphs. Despite their recent introduction, such concepts have already yielded valuable results when applied to essentiality of protein-protein interaction networks [37], bone structure characterization [38], and community finding [32,33].…”

Section: Introductionmentioning

confidence: 99%

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Hierarchical Characterization of Complex Networks

Costa

Silva

2006

J Stat Phys

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While the majority of approaches to the characterization of complex networks has relied on measurements considering only the immediate neighborhood of each network node, valuable information about the network topological properties can be obtained by considering further neighborhoods. The current work discusses on how the concepts of hierarchical node degree and hierarchical clustering coefficient (introduced in cond-mat/0408076), complemented by new hierarchical measurements, can be used in order to obtain a powerful set of topological features of complex networks. The interpretation of such measurements is discussed, including an analytical study of the hierarchical node degree for random networks, and the potential of the suggested measurements for the characterization of complex networks is illustrated with respect to simulations of random, scale-free and regular network models as well as real data (airports, proteins and word associations). The enhanced characterization of the connectivity provided by the set of hierarchical measurements also allows the use of agglomerative clustering methods in order to obtain taxonomies of relationships between nodes in a network, a possibility which is also illustrated in the current article.

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Section: Virtual Edges and Hierarchiesmentioning

confidence: 99%

Section: Virtual Edges and Hierarchiesmentioning

confidence: 99%

Section: Virtual Edges and Hierarchiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hierarchical Characterization of Complex Networks

Costa

Silva

2006

J Stat Phys

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The fundamental inequality for cocompact Fuchsian groups

Kosenko

Tiozzo

2022

Forum of Mathematics, Sigma

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Sznajd Complex Networks

Costa

2005

Int. J. Mod. Phys. C

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The Sznajd cellular automata corresponds to one of the simplest and yet most interesting models of complex systems. While the traditional two-dimensional Sznajd model tends to a consensus state (pro or cons), the assignment of the contrary to the dominant opinion to some of its cells during the system evolution is known to provide stabilizing feedback implying the overall system state to oscillate around null magnetization. The current article presents a novel type of geographic complex network model whose connections follow an associated feedbacked Sznajd model, i.e. the Sznajd dynamics is run over the network edges. Only connections not exceeding a maximum Euclidean distance D are considered, and any two nodes within such a distance are randomly selected and, in case they are connected, all network nodes which are no further than D are connected to them. In case they are not connected, all nodes within that distance are disconnected from them. Pairs of nodes are then randomly selected and assigned to the contrary of the dominant connectivity. The topology of the complex networks obtained by such a simple growth scheme, which are typically characterized by patches of connected communities, is analyzed both at global and individual levels in terms of a set of hierarchical measurements introduced recently. A series of interesting properties are identified and discussed comparatively to random and scale-free models with the same number of nodes and similar connectivity.

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On two recent geometrical characterizations of hyperellipticity

Abstract: We obtain short and unified new proofs of two recent characterizations of hyperellipticity given in [4] and [6], as well as a way of establishing a relation between them.

Cited by 5 publications

References 5 publications

Hierarchical Characterization of Complex Networks

Hierarchical Characterization of Complex Networks

The fundamental inequality for cocompact Fuchsian groups

Sznajd Complex Networks

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