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Uniform Sampling of Negative Edge Weights in Shortest Path Networks

Lukas Geis, Daniel Allendorf, Thomas Bläsius, Alexander Leonhardt, Ulrich Meyer, Manuel Penschuck, Hung Tran

TL;DR

A maximum entropy edge weight model for shortest path networks that allows for negative weights and empirically study the performance characteristics of an implementation of the novel sampling algorithm as well as the output produced by the model.

Abstract

We consider a maximum entropy edge weight model for shortest path networks that allows for negative weights. Given a graph $G$ and possible weights $\mathcal{W}$ typically consisting of positive and negative values, the model selects edge weights $w \in \mathcal{W}^m$ uniformly at random from all weights that do not introduce a negative cycle. We propose an MCMC process and show that (i) it converges to the required distribution and (ii) that the mixing time on the cycle graph is polynomial. We then engineer an implementation of the process using a dynamic version of Johnson's algorithm in connection with a bidirectional Dijkstra search. We empirically study the performance characteristics of an implementation of the novel sampling algorithm as well as the output produced by the model.

Uniform Sampling of Negative Edge Weights in Shortest Path Networks

TL;DR

A maximum entropy edge weight model for shortest path networks that allows for negative weights and empirically study the performance characteristics of an implementation of the novel sampling algorithm as well as the output produced by the model.

Abstract

We consider a maximum entropy edge weight model for shortest path networks that allows for negative weights. Given a graph and possible weights typically consisting of positive and negative values, the model selects edge weights uniformly at random from all weights that do not introduce a negative cycle. We propose an MCMC process and show that (i) it converges to the required distribution and (ii) that the mixing time on the cycle graph is polynomial. We then engineer an implementation of the process using a dynamic version of Johnson's algorithm in connection with a bidirectional Dijkstra search. We empirically study the performance characteristics of an implementation of the novel sampling algorithm as well as the output produced by the model.

Paper Structure

This paper contains 25 sections, 13 theorems, 6 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Related work
  4. Our contribution
  5. The Negative Edge Weight Model
  6. The MCMC sampler
  7. The sampler is approximately uniform
  8. Rapid mixing for the $n$-cycle
  9. Algorithmic insights
  10. An online version of Johnson's algorithm
  11. Maintaining a feasible potential
  12. Bidirectional search
  13. Empirical evaluation
  14. Acceptance Rate
  15. Queue-Insertions
  16. ...and 10 more sections

Key Result

Theorem 4

Let $G = (V, E)$ be a directed graph and $\mathcal{W}$ a weight interval. Then, the MCMC process converges to a uniform distribution on $\mathbb S\xspace_{G, \mathcal{W}}$.

Figures (9)

  • Figure 1: Potential updates on a cycle graph. The number above an edge $(a,b)$ is $w(a,b)$ and below $w_{\phi}(a,b) = w(a,b) + \phi(b) - \phi(a)$; green nodes indicate the original potential (as in top panel). In the top panel, update $w(u,v) {\gets} -2$ broke edge $(u,v)$, i. e., we need to increase the gradient at least $\mathcal{B}_{uv}\xspace = 2$ to make $\phi$ feasible again.
  • Figure 2: Acceptance-Rate of MCMC over time for different initial weight functions and varying average degrees for different graph classes.
  • Figure 3: Queue-Insertions per step by SSSP algorithms on $\mathcal{GNP}\xspace(n = 10000,\overline{d}\xspace = 10)$.
  • Figure 4: Average weight, fraction of negative edges, and average runtime per step over time for $\mathcal{GNP}\xspace, \mathcal{RHG}\xspace, \mathcal{DSF}\xspace$ graphs with $10000$ nodes and average degree $10$.
  • Figure 5: Runs to see $99\%$ of $\mathbb S\xspace_{C_n, \mathcal{W}}$ on $n$-cycle for $\mathcal{W} = \left\{-1, 0, 1\right\}$ as function of MCMC steps.
  • ...and 4 more figures

Theorems & Definitions (16)

  • Definition 1
  • Theorem 4
  • Theorem 5
  • Lemma 7: based on DBLP:journals/jacm/Johnson77
  • Lemma 8
  • Lemma 9
  • Corollary 10
  • Lemma 11: Corollary 3.3 of DBLP:journals/cpc/MartinR06
  • Lemma 12: Proposition 6 of GeomBounds
  • Lemma 13
  • ...and 6 more