Search arXivSearch

arXiv · 2408.06344

The Signatures of Ideal Flow Networks

Abstract

An Ideal Flow Network (IFN) is a strongly connected network where relative flows are preserved (irreducible premagic matrix). IFN can be decomposed into canonical cycles to form a string code called network signature. A network signature can be composed back into an IFN by assignment and merging operations. Using string manipulations on network signatures, we can derive total flow, link values, sum of rows and columns, and probability matrices and test for irreducibility.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kardi Teknomo. 2024-07-21. The Signatures of Ideal Flow Networks. https://arxiv.org/abs/2408.06344

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

Flip Dynamics for Sampling Colorings: Improving $(11/6-ε)$ Using a Simple Metric

We present improved bounds for randomly sampling $k$-colorings of graphs with maximum degree $Δ$; our results hold without any further structural assumptions on the graph. The Glauber dynamics is a simple single-site update Markov chain. Jerrum (1995) proved an optimal $O(n\log{n})$ mixing-time bound for Glauber dynamics whenever $k>2Δ$ where $Δ$ is the maximum degree of the input graph. This bound was improved by Vigoda (1999) to $k>(11/6)Δ$ using a "flip" dynamics which recolors (small) maximal two-colored components in each step. Vigoda's result was the best known for general graphs for 20 years until Chen et al. (2019) established optimal mixing of the flip dynamics for $k>(11/6-\varepsilon)Δ$ where $\varepsilon\approx 10^{-5}$. We present the first substantial improvement over these results. We prove an optimal mixing-time bound of $O(n\log{n})$ for the flip dynamics when $Δ\geq125$ and $k\geq1.809Δ$. This yields, through recent spectral independence results, an optimal $O(n\log{n})$ mixing time for the Glauber dynamics for every fixed $Δ\geq125$ in the same range of $k/Δ$. Our proof utilizes path coupling with a simple weighted Hamming distance for "unblocked" neighbors.

cs.DM

Factorisability of Low Dimensional Non-Negative Integer Matrices

We consider the problem of determining if a given two-dimensional nonnegative integer matrix $M$ is the product of two such matrices, excluding trivial units. A matrix $M$ with no such factorisation is called prime and therefore belongs to the minimal (infinite rank) generator of $2 \times 2$ matrices over the natural numbers, otherwise it is called composite. We also consider the problem of finding a (non-unique) factorisation of a composite matrix. Our results have applications in computational group theory and the theory of codes, where such matrices are called incidence matrices. We analyse the complexity of primality and finding a factorisation for a composite matrix, providing a first efficient algorithm.

cs.DM