Modulus metrics on networks

Nathan Albin; Nethali Fernando; Pietro Poggi-Corradini

doi:10.3934/dcdsb.2018161

Article Contents

2019, Volume 24, Issue 1: 1-17. Doi: 10.3934/dcdsb.2018161

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Modulus metrics on networks

Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA

* Corresponding author: Pietro Poggi-Corradini
* Corresponding author: Pietro Poggi-Corradini

Received: December 31, 2016

Revised: February 28, 2018

Early access: May 2018

Published: January 2019

The authors are supported by NSF n. 1515810.

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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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Mathematics Subject Classification: 90C35.

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Figure 1. The path graph $P_3$ on three nodes

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Figure 2. Antisnowflaking exponent for different $p$ values

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Figure 3. The cycle graph $C_N$ and the extremal density $\rho^*$ for $\Gamma(a, c)$ and $\Gamma(a, b)$

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Figure 4. $K_6$- Complete graph on 6 nodes

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Figure 5. The complete graph $K_N$ and the extremal density $\rho^*$ for $\Gamma(a, b)$

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Figure 6. The square graph

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Figure 7. Eigenvalues of $M$ as $\beta$ varies, given $\alpha = 1$

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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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References

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