Exploring the space of graphs with fixed discrete curvatures

TL;DR

This work tackles the inverse problem of constructing graphs with prescribed discrete edge curvatures, focusing on Forman--Ricci curvature. It develops an approximate reconstruction method using an evolutionary MCMC to sample graphs whose curvature sequences approximate a target, and complements it with an exact reconstruction framework based on Markov bases that shows the space of graphs with fixed curvature and degree sequences is connected via transpositions and Markov moves. Empirical experiments on synthetic targets (ER, SBM, BA) and a real metabolic network reveal that curvature constraints alone are local and may not reproduce global network properties, suggesting the benefit of incorporating additional graph statistics into the loss or constraints. Overall, the paper lays groundwork for curvature-constrained graph generation and provides a principled algebraic mechanism to navigate the space of graphs with fixed curvature features through Markov-basis moves.

Abstract

Discrete curvatures are quantities associated to the nodes and edges of a graph that reflect the local geometry around them. These curvatures have a rich mathematical theory and they have recently found success as a tool to analyze networks across a wide range of domains. In this work, we consider the problem of constructing graphs with a prescribed set of discrete edge curvatures, and explore the space of such graphs. We address this problem in two ways: first, we develop an evolutionary algorithm to sample graphs with discrete curvatures close to a given set. We use this algorithm to explore how other network statistics vary when constrained by the discrete curvatures in the network. Second, we solve the exact reconstruction problem for the specific case of Forman-Ricci curvature. By leveraging the theory of Markov bases, we obtain a finite set of rewiring moves that connects the space of all graphs with a fixed discrete curvature.

Figures (10)

• Figure 1: The Forman--Ricci curvatures $F(e)$ and augmented Forman--Ricci curvatures $F^{\#}(e)$ in a graph. The differences due to the presence of a triangle are highlighted in red.
• Figure 2: Evolution of curvature sequences throughout the sampling algorithm. Top left: Histogram of the target sequence $\mathcal{C}^\star$. This is the Forman--Ricci curvature sequence of the C. elegans metabolic network (see Section \ref{['subsubsection: target sequences']}). Bottom left: Evolution of the mean squared error MSE$_t$ between the target sequence and the Forman--Ricci curvature sequences. Highlight box: Histogram of Forman--Ricci curvatures $\mathcal{C}(G_t)$ at four points during the algorithm: (A) the initial graph (B) after $5.0e4$ accepted mutations, (C) after $11.0e4$ accepted mutations and (D) at the end of the algorithm, after $16.0e4$ accepted mutations and $T=2.0e6$ total iterations.
• Figure 3: Histograms of the Forman--Ricci curvature of an Erdős--Rényi, stochastic block model and Barabási--Albert random graph and the C. elegans metabolic network.
• Figure 4: Network statistics for graphs sampled using the algorithm from Section \ref{['subsection: description of algorithm']}, based on four target Forman--Ricci curvature sequences (ER, SBM, BA, C. elegans). The histograms of the observed statistics in the sampled graph ensemble are shown in blue and the red line indicates the statistic of the target graph together with its standard score.
• Figure 5: Network statistics for graphs sampled using the algorithm from Section \ref{['subsection: description of algorithm']}, based on four target augmented Forman--Ricci curvature sequences (ER, SBM, BA, C. elegans). The histograms of the observed statistics in the sampled graph ensemble are shown in blue and the red line indicates the statistic of the target graph together with its standard score.

Theorems & Definitions (15)

• Example 4.1
• Theorem 4.2: stanton_2012_constructing
• Theorem 4.3: stanton_2012_constructing
• Proposition 4.4
• Definition 4.5
• Theorem 4.6: Fundamental theorem of Markov bases MarkovBases
• Example 4.7
• Example 4.8: $\Delta=4$
• Proposition 4.9
• Proposition 4.10