Distance-Dependent Kronecker Graphs for Modeling Social Networks

This paper focuses on a generalization of stochastic Kronecker graphs, introducing a Kronecker-like operator and defining a family of generator matrices H dependent on distances between nodes in a specified graph embedding. We prove that any lattice-based network model with sufficiently small distance-dependent connection probability will have a Poisson degree distribution and provide a general framework to prove searchability for such a network. Using this framework, we focus on a specific example of an expanding hypercube and discuss the similarities and differences of such a model with recently proposed network models based on a hidden metric space. We also prove that a greedy forwarding algorithm can find very short paths of length O((log log n)^2) on the hypercube with n nodes, demonstrating that distance-dependent Kronecker graphs can generate searchable network models

Two-dimensional projections of an hypercube

We present a method to project a hypercube of arbitrary dimension on the plane, in such a way as to preserve, as well as possible, the distribution of distances between vertices. The method relies on a Montecarlo optimization procedure that minimizes the squared difference between distances in the plane and in the hypercube, appropriately weighted. The plane projections provide a convenient way of visualization for dynamical processes taking place on the hypercube.Comment: 4 pages, 3 figures, Revtex

Generalizing Kronecker graphs in order to model searchable networks

This paper describes an extension to stochastic Kronecker graphs that provides the special structure required for searchability, by defining a “distance”-dependent Kronecker operator. We show how this extension of Kronecker graphs can generate several existing social network models, such as the Watts-Strogatz small-world model and Kleinberg’s latticebased model. We focus on a specific example of an expanding hypercube, reminiscent of recently proposed social network models based on a hidden hyperbolic metric space, and prove that a greedy forwarding algorithm can find very short paths of length O((log log n)^2) for graphs with n nodes

Hitting time for quantum walks on the hypercube

Hitting times for discrete quantum walks on graphs give an average time before the walk reaches an ending condition. To be analogous to the hitting time for a classical walk, the quantum hitting time must involve repeated measurements as well as unitary evolution. We derive an expression for hitting time using superoperators, and numerically evaluate it for the discrete walk on the hypercube. The values found are compared to other analogues of hitting time suggested in earlier work. The dependence of hitting times on the type of unitary ``coin'' is examined, and we give an example of an initial state and coin which gives an infinite hitting time for a quantum walk. Such infinite hitting times require destructive interference, and are not observed classically. Finally, we look at distortions of the hypercube, and observe that a loss of symmetry in the hypercube increases the hitting time. Symmetry seems to play an important role in both dramatic speed-ups and slow-downs of quantum walks.Comment: 8 pages in RevTeX format, four figures in EPS forma

Orthogonal-Array based Design Methodology for Complex, Coupled Space Systems

The process of designing a complex system, formed by many elements and sub-elements interacting between each other, is usually completed at a system level and in the preliminary phases in two major steps: design-space exploration and optimization. In a classical approach, especially in a company environment, the two steps are usually performed together, by experts of the field inferring on major phenomena, making assumptions and doing some trial-and-error runs on the available mathematical models. To support designers and decision makers during the design phases of this kind of complex systems, and to enable early discovery of emergent behaviours arising from interactions between the various elements being designed, the authors implemented a parametric methodology for the design-space exploration and optimization. The parametric technique is based on the utilization of a particular type of matrix design of experiments, the orthogonal arrays. Through successive design iterations with orthogonal arrays, the optimal solution is reached with a reduced effort if compared to more computationally-intense techniques, providing sensitivity and robustness information. The paper describes the design methodology in detail providing an application example that is the design of a human mission to support a lunar base

Populations of models, Experimental Designs and coverage of parameter space by Latin Hypercube and Orthogonal Sampling

In this paper we have used simulations to make a conjecture about the coverage of a $t$ dimensional subspace of a $d$ dimensional parameter space of size $n$ when performing $k$ trials of Latin Hypercube sampling. This takes the form $P(k,n,d,t)=1-e^{-k/n^{t-1}}$. We suggest that this coverage formula is independent of $d$ and this allows us to make connections between building Populations of Models and Experimental Designs. We also show that Orthogonal sampling is superior to Latin Hypercube sampling in terms of allowing a more uniform coverage of the $t$ dimensional subspace at the sub-block size level.Comment: 9 pages, 5 figure