# American Institute of Mathematical Sciences

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

Received  January 2017 Revised  March 2018 Published  June 2018

Fund Project: The authors are supported by NSF n. 1515810.

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.

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
##### References:

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##### References:
The path graph $P_3$ on three nodes
Antisnowflaking exponent for different $p$ values
The cycle graph $C_N$ and the extremal density $\rho^*$ for $\Gamma(a, c)$ and $\Gamma(a, b)$
$K_6$- Complete graph on 6 nodes
The complete graph $K_N$ and the extremal density $\rho^*$ for $\Gamma(a, b)$
The square graph
Eigenvalues of $M$ as $\beta$ varies, given $\alpha = 1$
Comparisons of times required to compute $d_p$ distances on several square 2D grids for different values of $p$
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