Local Tomography of Large Networks under the Low-Observability Regime

This article studies the problem of reconstructing the topology of a network of interacting agents via observations of the state-evolution of the agents. We focus on the large-scale network setting with the additional constraint of $partial$ observations, where only a small fraction of the agents can be feasibly observed. The goal is to infer the underlying subnetwork of interactions and we refer to this problem as $local$ $tomography$. In order to study the large-scale setting, we adopt a proper stochastic formulation where the unobserved part of the network is modeled as an Erd\"{o}s-R\'enyi random graph, while the observable subnetwork is left arbitrary. The main result of this work is establishing that, under this setting, local tomography is actually possible with high probability, provided that certain conditions on the network model are met (such as stability and symmetry of the network combination matrix). Remarkably, such conclusion is established under the $low$-$observability$ $regime$, where the cardinality of the observable subnetwork is fixed, while the size of the overall network scales to infinity.Comment: To appear in IEEE Transactions on Information Theor

Oriented Matroids -- Combinatorial Structures Underlying Loop Quantum Gravity

We analyze combinatorial structures which play a central role in determining spectral properties of the volume operator in loop quantum gravity (LQG). These structures encode geometrical information of the embedding of arbitrary valence vertices of a graph in 3-dimensional Riemannian space, and can be represented by sign strings containing relative orientations of embedded edges. We demonstrate that these signature factors are a special representation of the general mathematical concept of an oriented matroid. Moreover, we show that oriented matroids can also be used to describe the topology (connectedness) of directed graphs. Hence the mathematical methods developed for oriented matroids can be applied to the difficult combinatorics of embedded graphs underlying the construction of LQG. As a first application we revisit the analysis of [4-5], and find that enumeration of all possible sign configurations used there is equivalent to enumerating all realizable oriented matroids of rank 3, and thus can be greatly simplified. We find that for 7-valent vertices having no coplanar triples of edge tangents, the smallest non-zero eigenvalue of the volume spectrum does not grow as one increases the maximum spin \jmax at the vertex, for any orientation of the edge tangents. This indicates that, in contrast to the area operator, considering large \jmax does not necessarily imply large volume eigenvalues. In addition we give an outlook to possible starting points for rewriting the combinatorics of LQG in terms of oriented matroids.Comment: 43 pages, 26 figures, LaTeX. Version published in CQG. Typos corrected, presentation slightly extende

A Complete Grammar for Decomposing a Family of Graphs into 3-connected Components

Tutte has described in the book "Connectivity in graphs" a canonical decomposition of any graph into 3-connected components. In this article we translate (using the language of symbolic combinatorics) Tutte's decomposition into a general grammar expressing any family of graphs (with some stability conditions) in terms of the 3-connected subfamily. A key ingredient we use is an extension of the so-called dissymmetry theorem, which yields negative signs in the grammar. As a main application we recover in a purely combinatorial way the analytic expression found by Gim\'enez and Noy for the series counting labelled planar graphs (such an expression is crucial to do asymptotic enumeration and to obtain limit laws of various parameters on random planar graphs). Besides the grammar, an important ingredient of our method is a recent bijective construction of planar maps by Bouttier, Di Francesco and Guitter.Comment: 39 page

Networks, (K)nots, Nucleotides, and Nanostructures

Designing self-assembling DNA nanostructures often requires the identification of a route for a scaffolding strand of DNA through the target structure. When the target structure is modeled as a graph, these scaffolding routes correspond to Eulerian circuits subject to turning restrictions imposed by physical constraints on the strands of DNA. Existence of such Eulerian circuits is an NP-hard problem, which can be approached by adapting solutions to a version of the Traveling Salesperson Problem. However, the author and collaborators have demonstrated that even Eulerian circuits obeying these turning restrictions are not necessarily feasible as scaffolding routes by giving examples of nontrivially knotted circuits which cannot be traced by the unknotted scaffolding strand. Often, targets of DNA nanostructure self-assembly are modeled as graphs embedded on surfaces in space. In this case, Eulerian circuits obeying the turning restrictions correspond to A-trails, circuits which turn immediately left or right at each vertex. In any graph embedded on the sphere, all A-trails are unknotted regardless of the embedding of the sphere in space. We show that this does not hold in general for graphs on the torus. However, we show this property does hold for checkerboard-colorable graphs on the torus, that is, those graphs whose faces can be properly 2-colored, and provide a partial converse to this result. As a consequence, we characterize (with one exceptional family) regular triangulations of the torus containing unknotted A-trails. By developing a theory of sums of A-trails, we lift constructions from the torus to arbitrary n-tori, and by generalizing our work on A-trails to smooth circuit decompositions, we construct all torus links and certain sums of torus links from circuit decompositions of rectangular torus grids. Graphs embedded on surfaces are equivalent to ribbon graphs, which are particularly well-suited to modeling DNA nanostructures, as their boundary components correspond to strands of DNA and their twisted ribbons correspond to double-helices. Every ribbon graph has a corresponding delta-matroid, a combinatorial object encoding the structure of the ribbon-graph\u27s spanning quasi-trees (substructures having exactly one boundary component). We show that interlacement with respect to quasi-trees can be generalized to delta-matroids, and use the resulting structure on delta-matroids to provide feasible-set expansions for a family of delta-matroid polynomials, both recovering well-known expansions of this type (such as the spanning-tree expansion of the Tutte polynnomial) as well as providing several previously unknown expansions. Among these are expansions for the transition polynomial, a version of which has been used to study DNA nanostructure self-assembly, and the interlace polynomial, which solves a problem in DNA recombination

Algorithms, haplotypes and phylogenetic networks

Preface. Before I started my PhD in computational biology in 2005, I had never even heard of this term. Now, almost four years later, I think I have some idea of what is meant by it. One of the goals of my PhD was to explore different topics within computational biology and to see where the biggest opportunities for discrete/combinatorial mathematicians could be found. Roughly speaking, the first two years of my PhD I focussed mainly on problems related to haplotyping and genome rearrangements and the last two years on phylogenetic networks. I must say I really enjoyed learning so much about both mathematics and biology. It was especially amazing to learn how exact, theoretical mathematics can be used to solve complex, practical problems from biology. The topics I studied clearly show how extremely useful mathematics can be for biology. But I also learned that there are many more interesting topics in computational biology than the ones that I could study so far. The number of opportunities for discrete mathematicians is absolutely immense. I did not include my studies on genome rearrangements in this thesis, because my most interesting results [Hur07a; Hur07b] are not directly related to biology. This work is nevertheless interesting to mathematicians and I recommend them to read it. I can certainly conclude that also in this field there is a vast number of opportunities for mathematicians and that the topic genome rearrangements provides numerous beautiful mathematical problems. I could never have written this thesis without a great amount of help from many different people. I want to thank my supervisors Leen Stougie and Judith Keijsper for guiding me, for helping me, for correcting my mistakes, for supplying ideas and for the enjoyable time I had while working with them. I also want to thank the Dutch BSIK/BRICKS project for funding my research and Gerhard Woeginger for giving me the opportunity to work in his group and being my second promotor. I want to thank Jens Stoye and Julia Zakotnik for the work we did together and for the great time I had in Bielefeld. I want to thank Ferry Hagen and Teun Boekhout for helping me to make my work relevant for "real" biology. I also want to thank John Tromp, Rudi Cilibrasi, Cor Hurkens and all others I worked with during my PhD. I want to thank Erik de Vink and Mike Steel for reading and commenting my thesis. I want to thank my colleagues from the Combinatorial Optimisation group at the Technische Universiteit Eindhoven for the pleasant working conditions and the fun things we did besides work. I especially want to thank Matthias Mnich, not only a great colleague but also a good friend, for all his ideas, his humour and our good and fruitful cooperation. I also want to thank Steven Kelk. I must say that I was very lucky to work with Steven during my PhD. He introduced me to problems, had an enormous amount of ideas, found the critical mistakes in my proofs and made my PhD a success both in terms of results and in terms of enjoying work. Finally, I want to thank Conno Hendriksen and Bas Heideveld for assisting me during my PhD defence and I want to thank them and all my other friends and family for helping me with everything in my life but research

Co-Design Strategies for Energy-Efficient UWB and UHF Wireless Systems

This paper reviews the most recent methods, combining nonlinear harmonic-balance-based analysis with electromagnetic (EM) simulation, for optimizing, at the circuit level, modern radiative RF/microwave systems. In order to maximize the system efficiency, each subsystem must be designed layoutwise, accounting for the presence of the others, that is, accounting for its actual terminations, rather than the ideal ones (50 Ω). In this way, the twofold goal of minimizing size and losses of the system is obtained by reducing intersystem matching networks. Indeed, terminations are complex, frequency-dispersive, and variable with the signal level, if active operations are concerned, and are responsible for performance degradation if not properly optimized. This approach is nowadays necessary, given the ever increased spread of pervasively distributed RF microsystems adopting miniaturized antennas, such as radio frequency identification (RFID) or wireless sensor networks, that must be low-cost, low-profile, low-power, and must simultaneously perform localization, identification, and sensing. For the design of a transmitter and a receiver connected with the respective antennas, suitable figures of merit are considered, encompassing radiation and nonlinear performance. Recent representative low-profile realizations, adopting ultra-wideband (UWB) excitations are used to highlight the benefit of the proposed nonlinear/EM approach for next generation energy autonomous microsystem, such as UWB-RFID tags

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic