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Electrical networks with prescribed current and applications to random walks on graphs

  • * Corresponding author: Amir Moradifam

    * Corresponding author: Amir Moradifam
The second author is supported by NSF grant DMS-1715850.
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  • In this paper we study Current Density Impedance Imaging (CDII) on Electrical Networks. The inverse problem is to determine the conductivity matrix of an electrical network from the prescribed knowledge of the magnitude of the induced current along the edges coupled with the imposed voltage or injected current on the boundary nodes. This problem leads to a weighted $ l^1 $ minimization problem for the corresponding voltage potential. We also investigate the problem of determining the transition probabilities of random walks on graphs from the prescribed expected net number of times the walker passes along the edges of the graph. Convergent numerical algorithms for solving such problems are also presented. Our results can be utilized in the design of electrical networks when certain current flow on the network is desired as well as the design of random walk models on graphs when the expected net number of the times the walker passes along the edges is prescribed. We also show that a mass preserving flow $ J = (J_{ij}) $ on a network can be uniquely recovered from the knowledge of $ |J| = (|J_{ij}|) $ and the flux of the flow on the boundary nodes, where $ J_{ij} $ is the flow from node $ i $ to node $ j $ and $ J_{ij} = -J_{ji} $, and discuss its potential application in cryptography.

    Mathematics Subject Classification: Primary: 35R30, 94C99; Secondary: 49N45, 05C81.


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  • Table 1.  Numerical errors for algorithm 1 on 100 node graph with 1121 edges

    Tolerance Relative L2 Error Number of Iterations Elapsed Time(s)
    $ 10^{-3} $ 1.2171$ \times 10^{-3} $ 16 0.069309
    $ 10^{-4} $ 1.3160$ \times 10^{-4} $ 22 0.102846
    $ 10^{-5} $ 1.4494$ \times 10^{-5} $ 92 0.358250
    $ 10^{-6} $ 1.3615$ \times 10^{-6} $ 133 0.405979
     | Show Table
    DownLoad: CSV

    Table 2.  Numerical errors for algorithm 2 on 100 node graph with 1121 edges

    Tolerance Relative L2 Error Number of Iterations Elapsed Time(s)
    $ 10^{-2} $ 1.3069$ \times 10^{-3} $ 7 0.055400
    $ 10^{-3} $ 1.3908$ \times 10^{-4} $ 9 0.071342
    $ 10^{-4} $ 1.0235$ \times 10^{-5} $ 12 0.086956
    $ 10^{-5} $ 1.1987$ \times 10^{-6} $ 24 0.147310
     | Show Table
    DownLoad: CSV

    Table 3.  Average Number of Iterations

    Tolerance Algorithm 1 Algorithm 2
    $ 10^{-3} $ 21.175 15.918
    $ 10^{-4} $ 46.097 18.905
    $ 10^{-5} $ 111.847 23.486
    $ 10^{-6} $ 227.624 32.846
     | Show Table
    DownLoad: CSV
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