Article Contents
Article Contents

# Modulus metrics on networks

• * Corresponding author: Pietro Poggi-Corradini
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.

Mathematics Subject Classification: 90C35.

 Citation:

• 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 $p$

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