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Modulus metrics on networks
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Abstract
The concept of $p$-modulus gives a way to measure the richness of a family of objects on a graph. In this paper, we investigate the families of connecting walks between two fixed nodes and show how to use $p$-modulus to form a parametrized family of graph metrics that generalize several well-known and widely-used metrics. We also investigate a characteristic of metrics called the "antisnowflaking exponent" and present some numerical findings supporting a conjecture about the new metrics. We end with explicit computations of the new metrics on some selected graphs.
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Keywords:
- p-modulus,
- graph metrics,
- snowflaking,
- mincut,
- shortest path,
- effective resistance.
Mathematics Subject Classification: 90C35.Citation: 
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Figure 1. The path graph
$P_3$ on three nodes
Figure 2. Antisnowflaking exponent for different
$p$ values
Figure 3. The cycle graph
$C_N$ and the extremal density$\rho^*$ for$\Gamma(a, c)$ and$\Gamma(a, b)$ 
Figure 4.
$K_6$ - Complete graph on 6 nodes
Figure 5. The complete graph
$K_N$ and the extremal density$\rho^*$ for$\Gamma(a, b)$ 
Figure 6. The square graph
Figure 7. Eigenvalues of
$M$ as$\beta$ varies, given$\alpha = 1$ 
Figure 8. Comparisons of times required to compute
$d_p$ distances on several square 2D grids for different values of$p$ -
References
[1] N. Albin, J. Clemens, N. Fernando and P. Poggi-Corradini, Blocking duality for p-modulus on networks and applications, arXiv: 1612.00435. 
[2] N. Albin, M. Brunner, R. Perez, P. Poggi-Corradini and N. Wiens, Modulus on graphs as a generalization of standard graph theoretic quantities, Conform. Geom. Dyn., 19 (2015), 298-317. doi: 10.1090/ecgd/287. 
[3] N. Albin and P. Poggi-Corradini, Minimal subfamilies and the probabilistic interpretation for modulus on graphs, J. Anal., 24 (2016), 183-208. doi: 10.1007/s41478-016-0002-9.
[4] N. Albin, F. D. Sahneh, M. Goering and P. Poggi-Corradini, Modulus of families of walks on graphs, in Complex Analysis and Dynamical Systems VII, vol. 699 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2017, 35–55. doi: 10.1090/conm/699/14080.
[5] G. Csardi and T. Nepusz, The igraph software package for complex network research, InterJournal, Complex Systems (2006), 1695, URL http://igraph.sf.net.
[6] S. Diamond and S. Boyd, CVXPY: A Python-embedded modeling language for convex optimization, Journal of Machine Learning Research, 17 (2016), Paper No. 83, 5 pp.
[7] J. Ding, J. R. Lee and Y. Peres, Cover times, blanket times, and majorizing measures, Ann. of Math., 175 (2012), 1409-1471. doi: 10.4007/annals.2012.175.3.8.
[8] P. G. Doyle and J. L. Snell, Random Walks and Electric Networks, vol. 22 of Carus Mathematical Monographs, Mathematical Association of America, Washington, DC, 1984.
[9] M. Goering, N. Albin, F. Sahneh, C. Scoglio and P. Poggi-Corradini, Numerical investigation of metrics for epidemic processes on graphs, in 2015 49th Asilomar Conference on Signals, Systems and Computers, 2015, 1317–1322. doi: 10.1109/ACSSC.2015.7421356.
[10] E. Jones, T. Oliphant and P. Peterson et al., SciPy: Open source scientific tools for Python, 2001–, URL http://www.scipy.org/, [Online; accessed 2/28/2018].
[11] D. A. Levin, Y. Peres and E. L. Wilmer, Markov Chains and Mixing Times, American Mathematical Society, Providence, RI, 2009, With a chapter by James G. Propp and David B. Wilson.
[12] D. A. Spielman, Graphs, vectors, and matrices, Bull. Amer. Math. Soc. (N.S.), 54 (2017), 45-61. doi: 10.1090/bull/1557.
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Figure 1. The path graph
$P_3$ on three nodes -
Figure 2. Antisnowflaking exponent for different
$p$ values -
Figure 3. The cycle graph
$C_N$ and the extremal density$\rho^*$ for$\Gamma(a, c)$ and$\Gamma(a, b)$ -
Figure 4.
$K_6$ - Complete graph on 6 nodes -
Figure 5. The complete graph
$K_N$ and the extremal density$\rho^*$ for$\Gamma(a, b)$ - Figure 6. The square graph
-
Figure 7. Eigenvalues of
$M$ as$\beta$ varies, given$\alpha = 1$ -
Figure 8. Comparisons of times required to compute
$d_p$ distances on several square 2D grids for different values of$p$