Locating the Source of Diffusion in Large-Scale Networks

How can we localize the source of diffusion in a complex network? Due to the tremendous size of many real networks--such as the Internet or the human social graph--it is usually infeasible to observe the state of all nodes in a network. We show that it is fundamentally possible to estimate the location of the source from measurements collected by sparsely-placed observers. We present a strategy that is optimal for arbitrary trees, achieving maximum probability of correct localization. We describe efficient implementations with complexity O(N^{\alpha}), where \alpha=1 for arbitrary trees, and \alpha=3 for arbitrary graphs. In the context of several case studies, we determine how localization accuracy is affected by various system parameters, including the structure of the network, the density of observers, and the number of observed cascades.Comment: To appear in Physical Review Letters. Includes pre-print of main paper, and supplementary materia

Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization

A versatile method is described for the practical computation of the discrete Fourier transforms (DFT) of a continuous function $g(t)$ given by its values $g_{j}$ at the points of a uniform grid $F_{N}$ generated by conjugacy classes of elements of finite adjoint order $N$ in the fundamental region $F$ of compact semisimple Lie groups. The present implementation of the method is for the groups SU(2), when $F$ is reduced to a one-dimensional segment, and for $SU(2)\times ... \times SU(2)$ in multidimensional cases. This simplest case turns out to result in a transform known as discrete cosine transform (DCT), which is often considered to be simply a specific type of the standard DFT. Here we show that the DCT is very different from the standard DFT when the properties of the continuous extensions of these two discrete transforms from the discrete grid points $t_j; j=0,1, ... N$ to all points $t \in F$ are considered. (A) Unlike the continuous extension of the DFT, the continuous extension of (the inverse) DCT, called CEDCT, closely approximates $g(t)$ between the grid points $t_j$. (B) For increasing $N$, the derivative of CEDCT converges to the derivative of $g(t)$. And (C), for CEDCT the principle of locality is valid. Finally, we use the continuous extension of 2-dimensional DCT to illustrate its potential for interpolation, as well as for the data compression of 2D images.Comment: submitted to JMP on April 3, 2003; still waiting for the referee's Repor

Source-Channel Communication in Sensor Networks

Sensors acquire data, and communicate this to an interested party. The arising coding problem is often split into two parts: First, the sensors compress their respective acquired signals, potentially applying the concepts of distributed source coding. Then, they communicate the compressed version to the interested party, the goal being not to make any errors. This coding paradigm is inspired by Shannon’s separation theorem for point-to-point communication, but it leads to suboptimal performance in general network topologies. The optimal performance for the general case is not known. In this paper, we propose an alternative coding paradigm based on joint source-channel coding. This coding paradigm permits to determine the optimal performance for a class of sensor networks, and shows how to achieve it. For sensor networks outside this class, we argue that the goal of the coding system could be to approach our condition for op- timal performance as closely as possible. This is supported by examples for which our coding paradigm significantly outperforms the traditional separation-based coding paradigm. In particular, for a Gaussian exam- ple considered in this paper, the distortion of the best coding scheme according to the separation paradigm decreases like 1/logM, while for our coding paradigm, it decreases like 1/M, where M is the total number of sensors

On the capacity of large Gaussian relay networks

The capacity of a particular large Gaussian relay network is determined in the limit as the number of relays tends to infinity. Upper bounds are derived from cut-set arguments, and lower bounds follow from an argument involving uncoded transmission. It is shown that in cases of interest, upper and lower bounds coincide in the limit as the number of relays tends to infinity. Hence, this paper provides a new example where a simple cut-set upper bound is achievable, and one more example where uncoded transmission achieves optimal performance. The findings are illustrated by geometric interpretations

A lower bound to the scaling behavior of sensor networks

For a class of sensor networks, the task is to monitor an underlying physical phenomenon over space and time through a noisy observation process. The sensors can communicate back to a data collector over a noisy channel. The key parameters in such a setting are the fidelity (or distortion) at which the underlying physical phenomenon can be estimated by a central data collector, and the cost of operating the communication networ

Gamow-Teller strength distributions for nuclei in pre-supernova stellar cores

Electron-capture and $\beta$-decay of nuclei in the core of massive stars play an important role in the stages leading to a type II supernova explosion. Nuclei in the f-p shell are particularly important for these reactions in the post Silicon-burning stage of a presupernova star. In this paper, we characterise the energy distribution of the Gamow-Teller Giant Resonance (GTGR) for mid-fp-shell nuclei in terms of a few shape parameters, using data obtained from high energy, forward scattering (p,n) and (n,p) reactions. The energy of the GTGR centroid $E_{GT}$ is further generalised as function of nuclear properties like mass number, isospin and other shell model properties of the nucleus. Since a large fraction of the GT strength lies in the GTGR region, and the GTGR is accessible for weak transitions taking place at energies relevant to the cores of presupernova and collapsing stars, our results are relevant to the study of important $e^-$-capture and $\beta$-decay rates of arbitrary, neutron-rich, f-p shell nuclei in stellar cores. Using the observed GTGR and Isobaric Analog States (IAS) energy systematics we compare the coupling coefficients in the Bohr-Mottelson two particle interaction Hamiltonian for different regions of the Isotope Table.Comment: Revtex, 28 pages +7 figures (PostScript Figures, uuencoded, filename: Sutfigs.uu). If you have difficulty printing the figures, please contact [email protected]. Accepted for publication in Phys. Rev. C, Nov 01, 199

Practical solution to the Monte Carlo sign problem: Realistic calculations of 54Fe

We present a practical solution to the "sign problem" in the auxiliary field Monte Carlo approach to the nuclear shell model. The method is based on extrapolation from a continuous family of problem-free Hamiltonians. To demonstrate the resultant ability to treat large shell-model problems, we present results for 54Fe in the full fp-shell basis using the Brown-Richter interaction. We find the Gamow-Teller beta^+ strength to be quenched by 58% relative to the single-particle estimate, in better agreement with experiment than previous estimates based on truncated bases.Comment: 11 pages + 2 figures (not included