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April  2019, 13(2): 353-375. doi: 10.3934/ipi.2019018

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

 University of California, Riverside, Riverside, CA 92501, USA

Received  June 2018 Revised  September 2018 Published  January 2019

Fund Project: The second author is supported by NSF grant DMS-1715850.

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.

Citation: Christina Knox, Amir Moradifam. Electrical networks with prescribed current and applications to random walks on graphs. Inverse Problems & Imaging, 2019, 13 (2) : 353-375. doi: 10.3934/ipi.2019018
##### References:
 [1] J.-F. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration, Multiscale Modeling & Simulation, 8 (2009), 337-369.  doi: 10.1137/090753504.  Google Scholar [2] R. M. Christley, G. Pinchbeck, R. Bowers, D. Clancy, N. French, R. Bennett and J. Turner, Infection in social networks: using network analysis to identify high-risk individuals, American Journal of Epidemiology, 162 (2005), 1024-1031.  doi: 10.1093/aje/kwi308.  Google Scholar [3] S.-Y. Chung and C. A. Berenstein, $\omega$-harmonic functions and inverse conductivity problems on networks, SIAM Journal on Applied Mathematics, 65 (2005), 1200-1226.  doi: 10.1137/S0036139903432743.  Google Scholar [4] C. Cooper, R. Elsasser, H. Ono and T. Radzik, Coalescing random walks and voting on connected graphs, SIAM Journal on Discrete Mathematics, 27 (2013), 1748-1758.  doi: 10.1137/120900368.  Google Scholar [5] E. B. Curtis and J. A. 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Fama, Random walks in stock market prices, Financial Analysts Journal, 51 (1995), 75-80.   Google Scholar [12] D. Gabay, Applications of the method of multipliers to variational inequalities, in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary Value Problems, Studies in Mathematics and its Applications (eds. F. M. and G. R.), vol. 15, 1983, chapter 9,299–331. Google Scholar [13] D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Computers & Mathematics with Applications, 2 (1976), 17-40.   Google Scholar [14] R. Glowinski and A. Marroco, Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problèmes de Dirichlet non linéaires, Revue Française D'automatique, Informatique, Recherche Opérationnelle. Analyse Numérique, 9 (1975), 41–76.  Google Scholar [15] T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343.  doi: 10.1137/080725891.  Google Scholar [16] P. R. Halmos, Finite-dimensional Vector Spaces, Springer-Verlag, New York-Heidelberg, 1974.  Google Scholar [17] K. F. Hasanov, A. W. Ma, R. S. Yoon, A. I. Nachman and M. Joy, A new approach to current density impedance imaging, in Engineering in Medicine and Biology Society, 2004. IEMBS'04. 26th Annual International Conference of the IEEE, vol. 1, IEEE, 2004, 1321–1324. Google Scholar [18] M. R. Henzinger, A. Heydon, M. Mitzenmacher and M. Najork, Measuring index quality using random walks on the web, Computer Networks, 31 (1999), 1291-1303.   Google Scholar [19] N. Hoell, A. Moradifam and A. Nachman, Current density impedance imaging of an anisotropic conductivity in a known conformal class, SIAM Journal on Mathematical Analysis, 46 (2014), 1820-1842.  doi: 10.1137/130911524.  Google Scholar [20] R. L. Jerrard, A. Moradifam and A. I. Nachman, Existence and uniqueness of minimizers of general least gradient problems, J. Reine Angew. Math., 734 (2018), 71-97.  doi: 10.1515/crelle-2014-0151.  Google Scholar [21] M. Joy, G. Scott and M. Henkelman, In vivo detection of applied electric currents by magnetic resonance imaging, Magnetic Resonance Imaging, 7 (1989), 89-94.   Google Scholar [22] S. Kim, O. Kwon, J. K. Seo and J.-R. Yoon, On a nonlinear partial differential equation arising in magnetic resonance electrical impedance tomography, SIAM Journal on Mathematical Analysis, 34 (2002), 511-526.   Google Scholar [23] Y. J. Kim, O. Kwon, J. K. Seo and E. J. Woo, Uniqueness and convergence of conductivity image reconstruction in magnetic resonance electrical impedance tomography, Inverse Problems, 19 (2003), 1213-1225.  doi: 10.1088/0266-5611/19/5/312.  Google Scholar [24] O. Kwon, J.-Y. Lee and J.-R. Yoon, Equipotential line method for magnetic resonance electrical impedance tomography, Inverse Problems, 18 (2002), 1089-1100.  doi: 10.1088/0266-5611/18/4/310.  Google Scholar [25] L. Lovász, Random walks on graphs, Combinatorics, Paul Erdos is Eighty, 2 (1993), 1-46.   Google Scholar [26] A. Moradifam, Existence and structure of minimizers of least gradient problems, Indiana University Mathematics Journal, 67 (2018), 1025-1037.  doi: 10.1512/iumj.2018.67.7360.  Google Scholar [27] A. Moradifam and A. Nachman, Convergence of the alternating split Bregman algorithm in infinite-dimensional Hilbert spaces. Google Scholar [28] A. Moradifam, A. Nachman and A. Tamasan, Conductivity imaging from one interior measurement in the presence of perfectly conducting and insulating inclusions, SIAM Journal on Mathematical Analysis, 44 (2012), 3969-3990.  doi: 10.1137/120866701.  Google Scholar [29] A. Moradifam, A. Nachman and A. Timonov, A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data, Inverse Problems, 28 (2012), 084003, 23PP. doi: 10.1088/0266-5611/28/8/084003.  Google Scholar [30] A. Nachman, A. Tamasan and A. Timonov, Current density impedance imaging, in Tomography and Inverse Transport Theory, vol. 559, Amer. Math. Soc. Providence, RI, 2011,135–149. doi: 10.1090/conm/559/11076.  Google Scholar [31] A. Nachman, A. Tamasan and A. Timonov, Conductivity imaging with a single measurement of boundary and interior data, Inverse Problems, 23 (2007), 2551-2563.  doi: 10.1088/0266-5611/23/6/017.  Google Scholar [32] A. Nachman, A. Tamasan and A. Timonov, Recovering the conductivity from a single measurement of interior data, Inverse Problems, 25 (2009), 035014, 16PP. doi: 10.1088/0266-5611/25/3/035014.  Google Scholar [33] A. Nachman, A. Tamasan and A. Timonov, Reconstruction of planar conductivities in subdomains from incomplete data, SIAM Journal on Applied Mathematics, 70 (2010), 3342-3362.  doi: 10.1137/10079241X.  Google Scholar [34] A. Nachman, A. Tamasan and J. Veras, A weighted minimum gradient problem with complete electrode model boundary conditions for conductivity imaging, SIAM Journal on Applied Mathematics, 76 (2016), 1321-1343.  doi: 10.1137/15M100897X.  Google Scholar [35] S. Ribas, B. Ribeiro-Neto, R. L. Santos, E. de Souza e Silva, A. Ueda and N. Ziviani, Random walks on the reputation graph, in Proceedings of the 2015 International Conference on The Theory of Information Retrieval, ACM, 2015,181–190. Google Scholar [36] P. Sarkar and A. W. Moore, Random walks in social networks and their applications: A survey, in Social Network Data Analytics, Springer, 2011, 43–77. doi: 10.1007/978-1-4419-8462-3_3.  Google Scholar [37] S. Setzer, Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage, in International Conference on Scale Space and Variational Methods in Computer Vision, Springer, 2009,464–476. Google Scholar [38] S. Setzer, Operator splittings, Bregman methods and frame shrinkage in image processing, International Journal of Computer Vision, 92 (2011), 265-280.  doi: 10.1007/s11263-010-0357-3.  Google Scholar [39] A. Skogseid and V. Fasano, Statistical Mechanics and Random Walks: Principles, Processes, and Applications, Nova Science Publishers, 2012. Google Scholar [40] M. E. Yildiz, R. Pagliari, A. Ozdaglar and A. Scaglione, Voting models in random networks, in Information Theory and Applications Workshop (ITA), 2010, Institute of Electrical and Electronics Engineers, 2010, 1–7. doi: 10.1109/ITA.2010.5454090.  Google Scholar

show all references

##### References:
 [1] J.-F. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration, Multiscale Modeling & Simulation, 8 (2009), 337-369.  doi: 10.1137/090753504.  Google Scholar [2] R. M. Christley, G. Pinchbeck, R. Bowers, D. Clancy, N. French, R. Bennett and J. Turner, Infection in social networks: using network analysis to identify high-risk individuals, American Journal of Epidemiology, 162 (2005), 1024-1031.  doi: 10.1093/aje/kwi308.  Google Scholar [3] S.-Y. Chung and C. A. Berenstein, $\omega$-harmonic functions and inverse conductivity problems on networks, SIAM Journal on Applied Mathematics, 65 (2005), 1200-1226.  doi: 10.1137/S0036139903432743.  Google Scholar [4] C. Cooper, R. Elsasser, H. Ono and T. Radzik, Coalescing random walks and voting on connected graphs, SIAM Journal on Discrete Mathematics, 27 (2013), 1748-1758.  doi: 10.1137/120900368.  Google Scholar [5] E. B. Curtis and J. A. Morrow, Inverse Problems for Electrical Networks, vol. 13, World Scientific, 2000. Google Scholar [6] P. G. Doyle and J. L. Snell, Random Walks and Electric Networks, Mathematical Association of America, 1984.  Google Scholar [7] M. Draief and A. Ganesh, A random walk model for infection on graphs: Spread of epidemics & rumours with mobile agents, Discrete Event Dynamic Systems, 21 (2011), 41-61.  doi: 10.1007/s10626-010-0092-5.  Google Scholar [8] J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, 55 (1992), 293-318.  doi: 10.1007/BF01581204.  Google Scholar [9] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, SIAM, 1999. doi: 10.1137/1.9781611971088.  Google Scholar [10] E. Esser, Applications of Lagrangian-based alternating direction methods and connections to split bregman, CAM report, 9 (2009), 31. Google Scholar [11] E. F. Fama, Random walks in stock market prices, Financial Analysts Journal, 51 (1995), 75-80.   Google Scholar [12] D. Gabay, Applications of the method of multipliers to variational inequalities, in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary Value Problems, Studies in Mathematics and its Applications (eds. F. M. and G. R.), vol. 15, 1983, chapter 9,299–331. Google Scholar [13] D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Computers & Mathematics with Applications, 2 (1976), 17-40.   Google Scholar [14] R. Glowinski and A. Marroco, Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problèmes de Dirichlet non linéaires, Revue Française D'automatique, Informatique, Recherche Opérationnelle. Analyse Numérique, 9 (1975), 41–76.  Google Scholar [15] T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343.  doi: 10.1137/080725891.  Google Scholar [16] P. R. Halmos, Finite-dimensional Vector Spaces, Springer-Verlag, New York-Heidelberg, 1974.  Google Scholar [17] K. F. Hasanov, A. W. Ma, R. S. Yoon, A. I. Nachman and M. Joy, A new approach to current density impedance imaging, in Engineering in Medicine and Biology Society, 2004. IEMBS'04. 26th Annual International Conference of the IEEE, vol. 1, IEEE, 2004, 1321–1324. Google Scholar [18] M. R. Henzinger, A. Heydon, M. Mitzenmacher and M. Najork, Measuring index quality using random walks on the web, Computer Networks, 31 (1999), 1291-1303.   Google Scholar [19] N. Hoell, A. Moradifam and A. Nachman, Current density impedance imaging of an anisotropic conductivity in a known conformal class, SIAM Journal on Mathematical Analysis, 46 (2014), 1820-1842.  doi: 10.1137/130911524.  Google Scholar [20] R. L. Jerrard, A. Moradifam and A. I. Nachman, Existence and uniqueness of minimizers of general least gradient problems, J. Reine Angew. Math., 734 (2018), 71-97.  doi: 10.1515/crelle-2014-0151.  Google Scholar [21] M. Joy, G. Scott and M. Henkelman, In vivo detection of applied electric currents by magnetic resonance imaging, Magnetic Resonance Imaging, 7 (1989), 89-94.   Google Scholar [22] S. Kim, O. Kwon, J. K. Seo and J.-R. Yoon, On a nonlinear partial differential equation arising in magnetic resonance electrical impedance tomography, SIAM Journal on Mathematical Analysis, 34 (2002), 511-526.   Google Scholar [23] Y. J. Kim, O. Kwon, J. K. Seo and E. J. Woo, Uniqueness and convergence of conductivity image reconstruction in magnetic resonance electrical impedance tomography, Inverse Problems, 19 (2003), 1213-1225.  doi: 10.1088/0266-5611/19/5/312.  Google Scholar [24] O. Kwon, J.-Y. Lee and J.-R. Yoon, Equipotential line method for magnetic resonance electrical impedance tomography, Inverse Problems, 18 (2002), 1089-1100.  doi: 10.1088/0266-5611/18/4/310.  Google Scholar [25] L. Lovász, Random walks on graphs, Combinatorics, Paul Erdos is Eighty, 2 (1993), 1-46.   Google Scholar [26] A. Moradifam, Existence and structure of minimizers of least gradient problems, Indiana University Mathematics Journal, 67 (2018), 1025-1037.  doi: 10.1512/iumj.2018.67.7360.  Google Scholar [27] A. Moradifam and A. Nachman, Convergence of the alternating split Bregman algorithm in infinite-dimensional Hilbert spaces. Google Scholar [28] A. Moradifam, A. Nachman and A. Tamasan, Conductivity imaging from one interior measurement in the presence of perfectly conducting and insulating inclusions, SIAM Journal on Mathematical Analysis, 44 (2012), 3969-3990.  doi: 10.1137/120866701.  Google Scholar [29] A. Moradifam, A. Nachman and A. Timonov, A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data, Inverse Problems, 28 (2012), 084003, 23PP. doi: 10.1088/0266-5611/28/8/084003.  Google Scholar [30] A. Nachman, A. Tamasan and A. Timonov, Current density impedance imaging, in Tomography and Inverse Transport Theory, vol. 559, Amer. Math. Soc. Providence, RI, 2011,135–149. doi: 10.1090/conm/559/11076.  Google Scholar [31] A. Nachman, A. Tamasan and A. Timonov, Conductivity imaging with a single measurement of boundary and interior data, Inverse Problems, 23 (2007), 2551-2563.  doi: 10.1088/0266-5611/23/6/017.  Google Scholar [32] A. Nachman, A. Tamasan and A. Timonov, Recovering the conductivity from a single measurement of interior data, Inverse Problems, 25 (2009), 035014, 16PP. doi: 10.1088/0266-5611/25/3/035014.  Google Scholar [33] A. Nachman, A. Tamasan and A. Timonov, Reconstruction of planar conductivities in subdomains from incomplete data, SIAM Journal on Applied Mathematics, 70 (2010), 3342-3362.  doi: 10.1137/10079241X.  Google Scholar [34] A. Nachman, A. Tamasan and J. Veras, A weighted minimum gradient problem with complete electrode model boundary conditions for conductivity imaging, SIAM Journal on Applied Mathematics, 76 (2016), 1321-1343.  doi: 10.1137/15M100897X.  Google Scholar [35] S. Ribas, B. Ribeiro-Neto, R. L. Santos, E. de Souza e Silva, A. Ueda and N. Ziviani, Random walks on the reputation graph, in Proceedings of the 2015 International Conference on The Theory of Information Retrieval, ACM, 2015,181–190. Google Scholar [36] P. Sarkar and A. W. Moore, Random walks in social networks and their applications: A survey, in Social Network Data Analytics, Springer, 2011, 43–77. doi: 10.1007/978-1-4419-8462-3_3.  Google Scholar [37] S. Setzer, Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage, in International Conference on Scale Space and Variational Methods in Computer Vision, Springer, 2009,464–476. Google Scholar [38] S. Setzer, Operator splittings, Bregman methods and frame shrinkage in image processing, International Journal of Computer Vision, 92 (2011), 265-280.  doi: 10.1007/s11263-010-0357-3.  Google Scholar [39] A. Skogseid and V. Fasano, Statistical Mechanics and Random Walks: Principles, Processes, and Applications, Nova Science Publishers, 2012. Google Scholar [40] M. E. Yildiz, R. Pagliari, A. Ozdaglar and A. Scaglione, Voting models in random networks, in Information Theory and Applications Workshop (ITA), 2010, Institute of Electrical and Electronics Engineers, 2010, 1–7. doi: 10.1109/ITA.2010.5454090.  Google Scholar
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
 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
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
 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
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
 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
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