Figure 1.
The path graph $P_3$ on three nodes
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January
2019, 24(1): 1-17.
doi: 10.3934/dcdsb.2018161
Modulus metrics on networks
Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA |
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.
Mathematics Subject Classification:
90C35.
Citation:
Nathan Albin, Nethali Fernando, Pietro Poggi-Corradini. Modulus metrics on networks. Discrete & Continuous Dynamical Systems - B,
2019, 24
(1)
: 1-17.
doi: 10.3934/dcdsb.2018161
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References:
| [1] |
N. Albin, J. Clemens, N. Fernando and P. Poggi-Corradini, Blocking duality for p-modulus on networks and applications, arXiv: 1612.00435.Google Scholar |
| [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.Google Scholar |
| [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].Google Scholar |
| [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. |
show all references
References:
| [1] |
N. Albin, J. Clemens, N. Fernando and P. Poggi-Corradini, Blocking duality for p-modulus on networks and applications, arXiv: 1612.00435.Google Scholar |
| [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.Google Scholar |
| [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].Google Scholar |
| [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. |
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 6.
The square graph
Figure 8.
Comparisons of times required to compute $d_p$ distances on several square 2D grids for different values of $p$
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