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Register a new userCan one see the shape of a network?
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Added by
Melanie Weber
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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Traditionally, network analysis is based on local properties of vertices, like their degree or clustering, and their statistical behavior across the network in question. This paper develops an approach which is different in two respects. We investigate edge-based properties, and we define global characteristics of networks directly. The latter will provide our affirmative answer to the question raised in the title. More concretely, we start with Formans notion of the Ricci curvature of a graph, or more generally, a polyhedral complex. This will allow us to pass from a graph as representing a network to a polyhedral complex for instance by filling in triangles into connected triples of edges and to investigate the resulting effect on the curvature. This is insightful for two reasons: First, we can define a curvature flow in order to asymptotically simplify a network and reduce it to its essentials. Second, using a construction of Bloch, which yields a discrete Gauss-Bonnet theorem, we have the Euler characteristic of a network as a global characteristic. These two aspects beautifully merge in the sense that the asymptotic properties of the curvature flow are indicated by that Euler characteristic.
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We show that the spectrum of the Schrodinger operator on a finite, metric graph determines uniquely the connectivity matrix and the bond lengths, provided that the lengths are non-commensurate and the connectivity is simple (no parallel bonds between vertices and no loops connecting a vertex to itself). That is, one can hear the shape of the graph! We also consider a related inversion problem: A compact graph can be converted into a scattering system by attaching to its vertices leads to infinity. We show that the scattering phase determines uniquely the compact part of the graph, under similar conditions as above.
By amplifying photonic qubits it is possible to produce states that contain enough photons to be seen with a human eye, potentially bringing quantum effects to macroscopic scales [1]. In this paper we theoretically study quantum states obtained by amplifying one side of an entangled photon pair with different types of optical cloning machines for photonic qubits. We propose a detection scheme that involves lossy threshold detectors (such as human eye) on the amplified side and conventional photon detectors on the other side. We show that correlations obtained with such coarse-grained measurements prove the entanglement of the initial photon pair and do not prove the entanglement of the amplified state. We emphasize the importance of the detection loophole in Bell violation experiments by giving a simple preparation technique for separable states that violate a Bell inequality without closing this loophole. Finally we analyze the genuine entanglement of the amplified states and its robustness to losses before, during and after amplification.
Neural network applications have become popular in both enterprise and personal settings. Network solutions are tuned meticulously for each task, and designs that can robustly resolve queries end up in high demand. As the commercial value of accurate and performant machine learning models increases, so too does the demand to protect neural architectures as confidential investments. We explore the vulnerability of neural networks deployed as black boxes across accelerated hardware through electromagnetic side channels. We examine the magnetic flux emanating from a graphics processing units power cable, as acquired by a cheap $3 induction sensor, and find that this signal betrays the detailed topology and hyperparameters of a black-box neural network model. The attack acquires the magnetic signal for one query with unknown input values, but known input dimensions. The network reconstruction is possible due to the modular layer sequence in which deep neural networks are evaluated. We find that each layer components evaluation produces an identifiable magnetic signal signature, from which layer topology, width, function type, and sequence order can be inferred using a suitably trained classifier and a joint consistency optimization based on integer programming. We study the extent to which network specifications can be recovered, and consider metrics for comparing network similarity. We demonstrate the potential accuracy of this side channel attack in recovering the details for a broad range of network architectures, including random designs. We consider applications that may exploit this novel side channel exposure, such as adversarial transfer attacks. In response, we discuss countermeasures to protect against our method and other similar snooping techniques.
We show that finding a graph realization with the minimum Randic index for a given degree sequence is solvable in polynomial time by formulating the problem as a minimum weight perfect b-matching problem. However, the realization found via this reduction is not guaranteed to be connected. Approximating the minimum weight b-matching problem subject to a connectivity constraint is shown to be NP-Hard. For instances in which the optimal solution to the minimum Randic index problem is not connected, we describe a heuristic to connect the graph using pairwise edge exchanges that preserves the degree sequence. In our computational experiments, the heuristic performs well and the Randic index of the realization after our heuristic is within 3% of the unconstrained optimal value on average. Although we focus on minimizing the Randic index, our results extend to maximizing the Randic index as well. Applications of the Randic index to synchronization of neuronal networks controlling respiration in mammals and to normalizing cortical thickness networks in diagnosing individuals with dementia are provided.
It is widely believed that dark matter exists within galaxies and clusters of galaxies. Under the assumption that this dark matter is composed of the lightest, stable supersymmetric particle, assumed to be the neutralino, the feasibility of its indirect detection via observations of a diffuse gamma-ray signal due to neutralino annihilations within M31 is examined. To this end, first the dark matter halo of the close spiral galaxy M31 is modeled from observations, then the resultant gamma-ray flux is estimated within supersymmetric model configurations. We conclude that under favorable conditions such as the rapid accretion of neutralinos on the central black hole in M31 and/or the presence of many clumps inside its halo with $r^{-3/2}$ inner profiles, a neutralino annihilation gamma-ray signal is marginally detectable by the ongoing collaboration CELESTE.
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