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

TL;DR

This paper explores an algebraic framework for flows and tensions in transportation networks using complex and hermitian matrices, connecting classical network theory with advanced algebraic and geometric structures.

Contribution

It introduces a novel algebraic model of networks using lattices of complex and hermitian matrices, extending classical network concepts to the Siegel space and symplectic groups.

Findings

Lattices of flow/tension values form congruence classes under SL(2,C) and Sp(2n,R).

A new lattice function is defined on the set of all lattices in Siegel space.

The minimal length tree of the network of networks is studied.

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 $\mathcal{R'}$.