Algebraic Connectivity and Degree Sequences of Trees

We investigate the structure of trees that have minimal algebraic connectivity among all trees with a given degree sequence. We show that such trees are caterpillars and that the vertex degrees are non-decreasing on every path on non-pendant vertices starting at the characteristic set of the Fiedler vector.Comment: 8 page

Sharp Asymptotics for the Truncated Two-Point Function of the Ising Model with a Positive Field

We prove that the correction to exponential decay of the truncated two points function in the homogeneous positive field Ising model is $c\|x\|^{-(d-1)/2}$. The proof is based on the development in the random current representation of a "modern" Ornstein-Zernike theory, as developed by Campanino, Ioffe and Velenik.Comment: 26 pages, 7 figures, details adde

The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane

We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on a version of the triangular lattice in the complex plane have unique scaling limits, which are invariant under rotations, scalings, and, in the case of the MST, also under translations. However, they are not expected to be conformally invariant. We also prove some geometric properties of the limiting MST. The topology of convergence is the space of spanning trees introduced by Aizenman, Burchard, Newman & Wilson (1999), and the proof relies on the existence and conformal covariance of the scaling limit of the near-critical percolation ensemble, established in our earlier works.Comment: 56 pages, 21 figures. A thoroughly revised versio

Advanced 30/20 GHz multiple-beam antennas for communications satellites

Design concepts under development utilize two separate spacecraft antenna systems, one uplink at 30 GHz and the other a downlink at 20 GHz, where each antenna provides multiple fixed and scanning beams. Two contractors completed configuration trade-off studies and breadboarding of critical technology components, and are fabricating and testing proof-of-concept (POC) models to demonstrate the technology feasibility. Technology developments required for the proposed systems are presented, along with each contractor's progress to date. The technology development areas discussed include: (1) offset Cassegrain and shaped reflector systems for narrow beams with low sidelobes and wideangle off-axis scan; (2) diplexed beam-forming networks for dual polarization, low sidelobes, and fixed and scan-beam operation; (3) fast switching networks for scanning beams; and (4) fabrication of precision feed components and large offset reflectors

Degree Correlations and Motifs in Technological Networks

Recent network research has sought to characterize complex systems with a number of statistical metrics, such as power law exponent (if any), clustering coefficient, community behavior, and degree correlation. A larger goal of such research is to obtain insight into the systems’ functions by means of these and similar analyses. In this paper we examine network models of mechanical assemblies. Such systems are well understood functionally. We show that they have both rich and varied community structure as well as negative degree correlations (disassortative mixing), and show that this can be explained by additional powerful constraints that arise from identifiable first principles. In addition, we note that their main “motif” is closed loops (as it is for electric and electronic circuits), a pattern that conventional network analysis does not detect but which is used by software designed to aid in the design of such systems. The implication is that functional understanding of complex systems requires considerable domain knowledge beyond what typical network analysis tools employ

More Than the Sum of Its Parts: Unlocking the Power of Network Structure for Understanding Organization and Function in Microbiomes

Plant and soil microbiomes are integral to the health and productivity of plants and ecosystems, yet researchers struggle to identify microbiome characteristics important for providing beneficial outcomes. Network analysis offers a shift in analytical framework beyond who is present to the organization or patterns of coexistence between microbes within the microbiome. Because microbial phenotypes are often significantly impacted by coexisting populations, patterns of coexistence within microbiomes are likely to be especially important in predicting functional outcomes. Here, we provide an overview of the how and why of network analysis in microbiome research, highlighting the ways in which network analyses have provided novel insights into microbiome organization and functional capacities, the diverse network roles of different microbial populations, and the eco-evolutionary dynamics of plant and soil microbiomes

The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane

We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on a version of the triangular lattice in the complex plane have unique scaling limits, which are invariant under rotations, scalings, and, in the case of the MST, also under translations. However, they are not expected to be conformally invariant. We also prove some geometric properties of the limiting MST. The topology of convergence is the space of spanning trees introduced by Aizenman, Burchard, Newman & Wilson (1999), and the proof relies on the existence and conformal covariance of the scaling limit of the near-critical percolation ensemble, established in our earlier works

Exploring countrywide spatial systems: Spatio-structural correlates at the regional and national scales

In this paper we take a step towards extending space syntax analysis into the countrywide scale, through the study of three very-large spatial systems in the UK, namely the top-tier road network of the entire country (170,007 nodes), the complete road network (1,208,674 nodes) of three contiguous NUTS1 regions (the East of England, South East of England and Greater London) and the complete road network of UK's mainland (2,031,971 nodes). We compare the results of our analysis with several types of functional and socio-economic data, finding clear statistical associations at the scale of the entire country between network structure, vehicular movement flows and the spatial distribution of several socio-economic variables. We conclude by arguing that space syntax models and analysis hold their value at very-large territorial scales, being highly robust and producing coherent results between datasets of different sources, themes and dimensionalities