Metric and ultrametric inequalities for directed graphs

Consider an electrical network modeled by a digraph G each directed edge e of which is a semiconductor with a monomial conductance function y_e^* = f_e (y_e) = y_e^r / μ_e^s  if   y_e ≥0  and  y_e^* = 0  if   y_e < 0. Here e is a directed edge, y_e is the potential difference (voltage),  y_e^* ≥0  is  the current in e, and μ_e > 0 is the resistance of e; furthermore, r > 0 and s > 0 are two real parameters common for all edges. In particular, case r = s = 1  corresponds to the Ohm law, while  r = ½, s =1  may be interpreted as the square law of resistance, which is typical of hydraulics and gas dynamics.

We show that for every ordered pair of nodes a, b of the network, the effective resistance μ_a,b is well-defined. In other words, any two-pole network with poles a and b can be effectively replaced by two oppositely directed edges: from a to b of resistance μ_a,b and from b to a of resistance μ_b,a. Furthermore, for every three nodes a, b, c, the inequality μ_a,c^s/r + μ_c,b^s/r ≥ μ_a,b^s/r holds, in which the equality is achieved if and only if every directed path from a to b in G contains c.

In addition to case (i) s = r = 1, some limit values of parameters  s and r  are also of interest being related to classic triangle inequalities. Namely,
(ii) the length/time  of a shortest directed path,
(iii) the inverse width of a bottleneck path,
(iv) the inverse capacity (maximum flow per unit time)
from a terminal a to a terminal b correspond to:
(ii) r = s → ∞,

(iii)  r = 1, s → ∞,

(iv) r → 0, s = 1$, respectively.

These results generalize ones obtained in 1987 for monomial isotropic networks, which are modeled by undirected graphs. In this special case, effective resistances form a metric space  
when  s ≥ r and an ultrametric one when s/r  → ∞.
Of course, for anisotropic networks, which are modeled by directed graphs, the symmetry, μ_a,b=μ_b,a may fail. For the linear isotropic networks triangle inequality is known since 1967 and it was extended to the linear anisotropic case in 2016.