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Graph Generation Methods under Partial Information

Tong Sun, Jianshu Hao, Michael C. Fu, Guangxin Jiang

Abstract

We study the problem of generating graphs with prescribed degree sequences for bipartite, directed, and undirected networks. We first propose a sequential method for bipartite graph generation and establish a necessary and sufficient interval condition that characterizes the admissible number of connections at each step, thereby guaranteeing global feasibility. Based on this result, we develop bipartite graph enumeration and sampling algorithms suitable for different problem sizes. We then extend these bipartite graph algorithms to the directed and undirected cases by incorporating additional connection constraints, as well as feasibility verification and symmetric connection steps, while preserving the same algorithmic principles. Finally, numerical experiments demonstrate the performance of the proposed algorithms, particularly their scalability to large instances where existing methods become computationally prohibitive.

Graph Generation Methods under Partial Information

Abstract

We study the problem of generating graphs with prescribed degree sequences for bipartite, directed, and undirected networks. We first propose a sequential method for bipartite graph generation and establish a necessary and sufficient interval condition that characterizes the admissible number of connections at each step, thereby guaranteeing global feasibility. Based on this result, we develop bipartite graph enumeration and sampling algorithms suitable for different problem sizes. We then extend these bipartite graph algorithms to the directed and undirected cases by incorporating additional connection constraints, as well as feasibility verification and symmetric connection steps, while preserving the same algorithmic principles. Finally, numerical experiments demonstrate the performance of the proposed algorithms, particularly their scalability to large instances where existing methods become computationally prohibitive.
Paper Structure (39 sections, 9 theorems, 78 equations, 10 figures, 10 tables, 12 algorithms)

This paper contains 39 sections, 9 theorems, 78 equations, 10 figures, 10 tables, 12 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Bipartite Graph Algorithms
  4. Node Connection Interval
  5. Bipartite Graph Enumeration Algorithm
  6. Bipartite Graph Sampling Algorithms
  7. Uniform Bipartite Graph Sampling Algorithm.
  8. Efficient Bipartite Graph Sampling Algorithm.
  9. Extended Algorithms
  10. Directed Graph Algorithms
  11. Directed Graph Enumeration Algorithm
  12. Directed Graph Sampling Algorithms
  13. Undirected Graph Algorithms
  14. Undirected Graph Enumeration Algorithm
  15. Undirected Graph Sampling Algorithms
  16. ...and 24 more sections

Key Result

Lemma 1

For the prescribed degree sequences $\mathbf{a}$ and $\mathbf{b}$, a bipartite graph exists if and only if $\sum_{i=1}^{m} \left [ \mathbf{a} \right ]_i = \sum_{j=1}^{n} \left [ \mathbf{b} \right ]_j$, and the following condition holds for all $r\in \left\{1, 2, \dots, n\right\}$:

Figures (10)

  • Figure 1: Five bipartite graphs, one directed graph, and one undirected graph consistent with $\mathbf{a} = \mathbf{b} = (2, 1, 1)$.
  • Figure 2: Connection process for node $\left[\mathbf{b}\right]_1$ in Example \ref{['example1']} with degree sequences $\mathbf{a} = \mathbf{b} = (2,\,1,\,1)$.
  • Figure 3: Enumeration process for node $\left[\mathbf{b}\right]_1$ in Example \ref{['example1']} with degree sequences $\mathbf{a} = \mathbf{b} = (2,\,1,\,1)$.
  • Figure 4: All the bipartite graphs in Example \ref{['example1']} with degree sequences $\mathbf{a} = \mathbf{b} = (2,\,1,\,1)$.
  • Figure 5: Runtime as a function of the number of nodes for bipartite graphs with degree sequences $\mathbf{a} = \mathbf{b} = (n-1,\,n-1,\,n-2,\,n-3,\,\ldots,\,2,\,1)$ averaged over 20 replications.
  • ...and 5 more figures

Theorems & Definitions (21)

  • Example 1
  • Definition 1
  • Lemma 1
  • Remark 1
  • Definition 2
  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Proposition 2
  • Lemma A.1: Gale--Ryser Theorem gale1957theorem
  • ...and 11 more