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Graph GOSPA metric: a metric to measure the discrepancy between graphs of different sizes

Jinhao Gu, Ángel F. García-Fernández, Robert E. Firth, Lennart Svensson

TL;DR

The paper introduces the graph GOSPA metric, a principled distance between graphs of different sizes that incorporates both node attributes and edge structure by optimizing node assignments and penalising edge mismatches. It extends the generalised optimal sub-pattern assignment framework from sets to graphs and provides a polynomial-time computable LP lower bound, while handling undirected, weighted, and directed graphs. A decomposable objective separates localisation, missed, and false-node costs from the edge-mismatch term, enabling interpretable diagnostics. Validation on simulated graphs and molecular datasets demonstrates the metric’s interpretability, computational feasibility, and competitive performance against existing graph distances such as GED, MCS, and GCD.

Abstract

This paper proposes a metric to measure the dissimilarity between graphs that may have a different number of nodes. The proposed metric extends the generalised optimal subpattern assignment (GOSPA) metric, which is a metric for sets, to graphs. The proposed graph GOSPA metric includes costs associated with node attribute errors for properly assigned nodes, missed and false nodes and edge mismatches between graphs. The computation of this metric is based on finding the optimal assignments between nodes in the two graphs, with the possibility of leaving some of the nodes unassigned. We also propose a lower bound for the metric, which is also a metric for graphs and is computable in polynomial time using linear programming. The metric is first derived for undirected unweighted graphs and it is then extended to directed and weighted graphs. The properties of the metric are demonstrated via simulated and empirical datasets.

Graph GOSPA metric: a metric to measure the discrepancy between graphs of different sizes

TL;DR

The paper introduces the graph GOSPA metric, a principled distance between graphs of different sizes that incorporates both node attributes and edge structure by optimizing node assignments and penalising edge mismatches. It extends the generalised optimal sub-pattern assignment framework from sets to graphs and provides a polynomial-time computable LP lower bound, while handling undirected, weighted, and directed graphs. A decomposable objective separates localisation, missed, and false-node costs from the edge-mismatch term, enabling interpretable diagnostics. Validation on simulated graphs and molecular datasets demonstrates the metric’s interpretability, computational feasibility, and competitive performance against existing graph distances such as GED, MCS, and GCD.

Abstract

This paper proposes a metric to measure the dissimilarity between graphs that may have a different number of nodes. The proposed metric extends the generalised optimal subpattern assignment (GOSPA) metric, which is a metric for sets, to graphs. The proposed graph GOSPA metric includes costs associated with node attribute errors for properly assigned nodes, missed and false nodes and edge mismatches between graphs. The computation of this metric is based on finding the optimal assignments between nodes in the two graphs, with the possibility of leaving some of the nodes unassigned. We also propose a lower bound for the metric, which is also a metric for graphs and is computable in polynomial time using linear programming. The metric is first derived for undirected unweighted graphs and it is then extended to directed and weighted graphs. The properties of the metric are demonstrated via simulated and empirical datasets.
Paper Structure (26 sections, 6 theorems, 45 equations, 20 figures, 6 tables)

This paper contains 26 sections, 6 theorems, 45 equations, 20 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Problem formulation and background
  3. A metric for graphs
  4. GOSPA metric
  5. Graph metric based on optimal assignments
  6. Graph GOSPA metric
  7. Examples
  8. LP Graph GOSPA metric
  9. Integer linear programming formulation
  10. LP relaxation
  11. Metric decomposition
  12. Extensions to other types of graphs
  13. Graph GOSPA metric for weighted undirected graphs
  14. Graph GOSPA metric for directed graphs
  15. Graphs without node attributes
  16. ...and 11 more sections

Key Result

Lemma 1

For $1<p<\infty$, a scalar $c > 0$, edge mismatch penalty $\epsilon>0$ and base metric $d(\cdot,\cdot)$, the graph GOSPA metric $d^{(c,\epsilon)}(\cdot,\cdot)$ in eq:graph_metric between two graphs $X$ and $Y$ can be written as where and $W_{1:n_{X},1:n_{Y}}$ is the matrix formed by the first $n_{X}$ rows and the first $n_{Y}$ columns of matrix $W$ (e.g., removing the last row and column of $W$)

Figures (20)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5: Example to illustrate the node and edge mismatch costs for the same ground truth graph $X$, and different estimated graphs $Y$ ($\Delta \ll c$): (a) three properly assigned nodes and no edge mismatch; (b) three properly assigned nodes and one missed edge; (c) two properly assigned nodes, one missed node and two half-edge mismatch penalties; (d) Two properly assigned nodes and two unassigned nodes ($\delta \gg c$), three half-edge mismatch penalties.
  • ...and 15 more figures

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Proposition 2
  • Proposition 3
  • Lemma 4
  • Lemma 5
  • Lemma 6