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Optimal network geometry detection for weak geometry

R. Michielan, C. Stegehuis

TL;DR

The paper tackles the problem of detecting latent geometric structure in networks when only edge data are available, focusing on weak geometry regimes. It introduces a mixed-integer linear programming (MILP) framework to optimize edge-based and degree-based statistics that distinguish geometric from non-geometric graphs within the Geometric Inhomogeneous Random Graph (GIRG) family, including hyperbolic-inspired models. A key finding is that weak geometry is detectable for all $1/T>0$, with plain subgraph counts having a fundamental limit, while incorporating degree information via weighted subgraphs (notably weighted cliques) drastically expands detectability across regimes. The results yield practical tools for geometry detection in network data and highlight directions for extending MILP-based statistics to broader local and induced-subgraph observables.

Abstract

Network geometry, characterized by nodes with associated latent variables, is a fundamental feature of real-world networks. Still, when only the network edges are given, it may be difficult to assess whether the network contains an underlying source of geometry. This paper investigates the limits of geometry detection in networks in a wide class of models that contain geometry and power-law degrees, which include the popular hyperbolic random graph model. We specifically focus on the regime in which the geometric signal is weak, characterized by the inverse temperature $1/T<1$. We show that the dependencies between edges can be tackled through Mixed-Integer Linear Problems, which lift the non-linear nature of network analysis into an exponential space in which simple linear optimization techniques can be employed. This approach allows us to investigate which subgraph and degree-based statistic is most effective at detecting the presence of an underlying geometric space. Interestingly, we show that even when the geometric effect is extremely weak, our Mixed-Integer programming identifies a network statistic that efficiently distinguishes geometric and non-geometric networks.

Optimal network geometry detection for weak geometry

TL;DR

The paper tackles the problem of detecting latent geometric structure in networks when only edge data are available, focusing on weak geometry regimes. It introduces a mixed-integer linear programming (MILP) framework to optimize edge-based and degree-based statistics that distinguish geometric from non-geometric graphs within the Geometric Inhomogeneous Random Graph (GIRG) family, including hyperbolic-inspired models. A key finding is that weak geometry is detectable for all , with plain subgraph counts having a fundamental limit, while incorporating degree information via weighted subgraphs (notably weighted cliques) drastically expands detectability across regimes. The results yield practical tools for geometry detection in network data and highlight directions for extending MILP-based statistics to broader local and induced-subgraph observables.

Abstract

Network geometry, characterized by nodes with associated latent variables, is a fundamental feature of real-world networks. Still, when only the network edges are given, it may be difficult to assess whether the network contains an underlying source of geometry. This paper investigates the limits of geometry detection in networks in a wide class of models that contain geometry and power-law degrees, which include the popular hyperbolic random graph model. We specifically focus on the regime in which the geometric signal is weak, characterized by the inverse temperature . We show that the dependencies between edges can be tackled through Mixed-Integer Linear Problems, which lift the non-linear nature of network analysis into an exponential space in which simple linear optimization techniques can be employed. This approach allows us to investigate which subgraph and degree-based statistic is most effective at detecting the presence of an underlying geometric space. Interestingly, we show that even when the geometric effect is extremely weak, our Mixed-Integer programming identifies a network statistic that efficiently distinguishes geometric and non-geometric networks.

Paper Structure

This paper contains 11 sections, 36 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Motif optimization: $1/T>1$
  4. Motif optimization: $1/T<1$
  5. Optimal subgraph to detect geometry
  6. Most powerful subgraph: mountains subgraph.
  7. Cliques.
  8. Most powerful degree-based subgraph
  9. Optimal weighted statistic: weighted cliques
  10. Conclusion
  11. Mixed Integer Linear Programming formulations

Figures (5)

  • Figure 1: $B(H)$ for all subgraphs of size 4, 5, 6 with $B(H)>0$ for $\tau=2.4, 1/T>1$. The colors of the nodes and edges denote the optimal $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ values of \ref{['eq:optproblem']}.
  • Figure 2: $B(H)$ for all subgraphs of size 4, 5, 6 with $B(H)>0$ for $\tau=2.4, 1/T=0.95$. The colors of the nodes and edges denote the optimal $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ values of \ref{['eq:optproblemhightemp']}.
  • Figure 3: $B_W(H)$ for all subgraphs of size 4, 5, 6 with $B_W(H)>0$ for $\tau=2.4, 1/T=0.8$. The colors of the nodes and edges denote the optimal $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ values of \ref{['eq:optproblemweighted']}.
  • Figure 4: $B_W(H)$ for the 12 subgraphs of size 4, 5, 6 with the largest values of $B_W(H)>0$ for $\tau=2.2, 1/T=0.6$. The colors of the nodes and edges denote the optimal $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ values of \ref{['eq:optproblemweighted']}, for which the legend is as in Figure \ref{['fig:tau24highweightedgamma08']}.
  • Figure 5: $B_W(H)$ for the 12 subgraphs of size 4, 5, 6 with the largest values of $B_W(H)>0$ for $\tau=2.2, 1/T=0.4$.The colors of the nodes and edges denote the optimal $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ values of \ref{['eq:optproblemweighted']}, for which the legend is as in Figure \ref{['fig:tau24highweightedgamma08']}.