A "network of networks" (from history to algebra)

Abstract

Recall first the algebraic treatment of flows or tensions in a transportation network $N$, i.e. a connected antisymmetric 1-graph $G(X, U)$. Assume that, unusually, we take the values of flows (resp. tensions) in $\mathbb{C}$. So the algebraic lattices $\Gamma$ of flow (resp. tension) values associated to $G(X, U)$ are lattices of $\mathbb{C}$. These lattices are congruent modulo the action of the special linear group SL($2, \mathbb{C}$). Then, it is well known one can define a lattice function $G_{k}(\Gamma)$, as a modular function of weight $2k$, on the set $\mathcal{R}$ of all lattices of $\mathbb{C}$. Let now $N_{1}, N_{2}, ..., N_{p}$ be connected antisymmetric 1-graphs and $C_{n}$, the set of hermitian symmetric matrices $n \times n$. Let also $\mathcal{R'} $ be the set of all the lattices of $C_{n}$. The previous structure can be transposed to any $ n \times n $ symmetric hermitian matrices of flow (or tension) values of the $G_{i}$. In this case, the Siegel space $S_{n}= C_{n}$ replaces the Poincar\'{e} half-plane, and the symplectic group Sp$(2n, \mathbb{R})$ takes the place of the special linear group SL($2, \mathbb{C}$). We get now the new lattice function as a function of all the lattices of $S_{n}$, i.e. a model of the "network of networks" $\mathcal{R'}$. In the end, we study the tree of minimal length of $\mathcal{R'}$.