Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)

Although the ``scale-free'' literature is large and growing, it gives neither a precise definition of scale-free graphs nor rigorous proofs of many of their claimed properties. In fact, it is easily shown that the existing theory has many inherent contradictions and verifiably false claims. In this paper, we propose a new, mathematically precise, and structural definition of the extent to which a graph is scale-free, and prove a series of results that recover many of the claimed properties while suggesting the potential for a rich and interesting theory. With this definition, scale-free (or its opposite, scale-rich) is closely related to other structural graph properties such as various notions of self-similarity (or respectively, self-dissimilarity). Scale-free graphs are also shown to be the likely outcome of random construction processes, consistent with the heuristic definitions implicit in existing random graph approaches. Our approach clarifies much of the confusion surrounding the sensational qualitative claims in the scale-free literature, and offers rigorous and quantitative alternatives.Comment: 44 pages, 16 figures. The primary version is to appear in Internet Mathematics (2005

Biomolecule-directed assembly of nanoscale building blocks studied via lattice Monte Carlo simulation

We perform lattice Monte Carlo simulations to study the self-assembly of functionalized inorganic nanoscale building blocks using recognitive biomolecule linkers. We develop a minimal coarse-grained lattice model for the nanoscale building block (NBB) and the recognitive linkers. Using this model, we explore the influence of the size ratio of linker length to NBB diameter on the assembly process and the structural properties of the resulting aggregates, including the spatial distribution of NBBs and aggregate topology. We find the constant-kernel Smoluchowski theory of diffusion-limited cluster–cluster aggregation describes the aggregation kinetics for certain size ratios. © 2004 American Institute of Physics.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70812/2/JCPSA6-121-8-3919-1.pd

High-Performance, Low-Complexity Deadlock Avoidance for Arbitrary Topologies/Routings

Recently, the use of graph-based network topologies has been proposed as an alternative to traditional networks such as tori or fat-trees due to their very good topological characteristics. However they pose practical implementation challenges such as the lack of deadlock avoidance strategies. Previous proposals are either exceedingly complex, underutilise network resources or lack flexibility. We propose- and prove formally- three generic, low-complexity dead-lock avoidance mechanisms that only require local information. The main strengths of our method are its topology- and routing- independence and that the virtual channel count is bounded by the length of the longest path. We evaluate our proposed mechanisms against previous proposals through an extensive simulation study to measure the impact on the performance using both synthetic and realistic traffic. First we compare against a well-known HPC mechanism for dragonfly and achieved similar performance level. Then we moved to Graph-based networks and show that our mechanisms can greatly outperform traditional, spanning-tree based mechanisms, even if these use a much larger number of virtual channels. Overall, we find that our proposal provides a simple, flexible and high performance deadlock-avoidance solution