A Geometric Chung Lu model and the Drosophila Medulla connectome

TL;DR

The paper tackles the mismatch between purely random or purely geometric network models and real-world spatial networks by introducing a generic geometric Chung-Lu model that combines a distance-aware connection function with node heterogeneity. It formalizes the model on a $d$-dimensional torus, extends it to bounded regions, and provides parameter estimation procedures from observed data. Applying the framework to the Drosophila Medulla connectome, the authors demonstrate that the synthetic graphs reproduce key spectral and centrality properties, while highlighting boundary effects and limitations in capturing certain local structures. The work offers a flexible approach for generating synthetic spatial networks with realistic topology, potentially benefiting studies of neural connectomics and other spatial networks.

Abstract

Many real world graphs have edges correlated to the distance between them, but, in an inhomogeneous manner. While the Chung-Lu model and the geometric random graph models both are elegant in their simplicity, they are insufficient to capture the complexity of these networks. In this paper, we develop a generalized geometric random graph model that preserves many graph theoretic aspects of these real world networks. We test the validity of this model on a graphical representation of the Drosophila Medulla connectome.

Figures (13)

• Figure 1: Three views of the graph of the connectome of the Drosophila Medula, with the vertex locations computed as the centroid of the synapse locations. Note that the dimensions of the bounding box of the connectome is $7290 \; \mu m \times 8685 \; \mu m \times 1212 \; \mu m$. Note that $98\%$ of the data lies in a bounding box of dimensions $4300 \; \mu m \times 4005 \; \mu m \times 1130 \; \mu m$.
• Figure 2: The connectome of the of the Drosophila Medula connectome
• Figure 3: Note that while the diameter of the largest connected component in Figure \ref{['fig:graphofconnectome']} has diameter 6, the neighborhoods of the vertices with the highest, $100^{th}$ highest and $200^{th}$ highest valencies are 927, 37 and 23 respectively, suggesting a correlation between the valency of a vertex and is average distance from neighbors.
• Figure 4: The empirical cumulative distribution of edge distances have similar 'S' shaped in spite of its valence.
• Figure 5: The empirical probabilities (blue) for $F_1$ (top) and $F_2$ (bottom) against their modeled probabilities (orange) ($\hat{F}_1$ and $\hat{F}_2$ respectively.). The fits are good, particularly for longer edges.

Theorems & Definitions (21)

• Definition 3.1: Parameters of the Model
• Definition 3.2: Additional Constraints and Assumptions
• Lemma 3.3
• proof
• Remark 3.4
• Theorem 3.5
• proof
• Lemma 3.6
• proof
• Definition 3.7