Spanning Trees With Edge Conflicts and Wireless Connectivity

We introduce the problem of finding a spanning tree along with a partition of the tree edges into fewest number of feasible sets, where constraints on the edges define feasibility. The motivation comes from wireless networking, where we seek to model the irregularities seen in actual wireless environments. Not all node pairs may be able to communicate, even if geographically close - thus, the available pairs are specified with a link graph {L}=(V,E). Also, signal attenuation need not follow a nice geometric formula - hence, interference is modeled by a conflict (hyper)graph {C}=(E,F) on the links. The objective is to maximize the efficiency of the communication, or equivalently, to minimize the length of a schedule of the tree edges in the form of a coloring. We find that in spite of all this generality, the problem can be approximated linearly in terms of a versatile parameter, the inductive independence of the interference graph. Specifically, we give a simple algorithm that attains a O(rho log n)-approximation, where n is the number of nodes and rho is the inductive independence, and show that near-linear dependence on rho is also necessary. We also treat an extension to Steiner trees, modeling multicasting, and obtain a comparable result. Our results suggest that several canonical assumptions of geometry, regularity and "niceness" in wireless settings can sometimes be relaxed without a significant hit in algorithm performance

A new method to measure complexity in binary or weighted networks and applications to functional connectivity in the human brain

BACKGROUND: Networks or graphs play an important role in the biological sciences. Protein interaction networks and metabolic networks support the understanding of basic cellular mechanisms. In the human brain, networks of functional or structural connectivity model the information-flow between cortex regions. In this context, measures of network properties are needed. We propose a new measure, Ndim, estimating the complexity of arbitrary networks. This measure is based on a fractal dimension, which is similar to recently introduced box-covering dimensions. However, box-covering dimensions are only applicable to fractal networks. The construction of these network-dimensions relies on concepts proposed to measure fractality or complexity of irregular sets in [Formula: see text]. RESULTS: The network measure Ndim grows with the proliferation of increasing network connectivity and is essentially determined by the cardinality of a maximum k-clique, where k is the characteristic path length of the network. Numerical applications to lattice-graphs and to fractal and non-fractal graph models, together with formal proofs show, that Ndim estimates a dimension of complexity for arbitrary graphs. Box-covering dimensions for fractal graphs rely on a linear log-log plot of minimum numbers of covering subgraph boxes versus the box sizes. We demonstrate the affinity between Ndim and the fractal box-covering dimensions but also that Ndim extends the concept of a fractal dimension to networks with non-linear log-log plots. Comparisons of Ndim with topological measures of complexity (cost and efficiency) show that Ndim has larger informative power. Three different methods to apply Ndim to weighted networks are finally presented and exemplified by comparisons of functional brain connectivity of healthy and depressed subjects. CONCLUSION: We introduce a new measure of complexity for networks. We show that Ndim has the properties of a dimension and overcomes several limitations of presently used topological and fractal complexity-measures. It allows the comparison of the complexity of networks of different type, e.g., between fractal graphs characterized by hub repulsion and small world graphs with strong hub attraction. The large informative power and a convenient computational CPU-time for moderately sized networks may make Ndim a valuable tool for the analysis of biological networks

On irregular total labellings

Two new graph characteristics, the total vertex irregularity strength and the total edge irregularity strength, are introduced. Estimations on these parameters are obtained. For some families of graphs the precise values of these parameters are proved. (c) 2006 Elsevier B.V. All rights reserved.C

Analysis and synthesis of iris images

Of all the physiological traits of the human body that help in personal identification, the iris is probably the most robust and accurate. Although numerous iris recognition algorithms have been proposed, the underlying processes that define the texture of irises have not been extensively studied. In this thesis, multiple pair-wise pixel interactions have been used to describe the textural content of the iris image thereby resulting in a Markov Random Field (MRF) model for the iris image. This information is expected to be useful for the development of user-specific models for iris images, i.e. the matcher could be tuned to accommodate the characteristics of each user\u27s iris image in order to improve matching performance. We also use MRF modeling to construct synthetic irises based on iris primitive extracted from real iris images. The synthesis procedure is deterministic and avoids the sampling of a probability distribution making it computationally simple. We demonstrate that iris textures in general are significantly different from other irregular textural patterns. Clustering experiments indicate that the synthetic irises generated using the proposed technique are similar in textural content to real iris images