Constellation Reporting Framework

• Constellation Reporting Framework is a unified system that segments, analyzes, and reports structured configurations across astronomical, network, and relational data.
• It integrates advanced methods such as binary tree decomposition, quadtree filtering, MILP optimization, and time-expanded graph scheduling for scalable performance.
• The framework enables efficient classification, robust KPI monitoring, and dynamic optimization, supporting practical applications from celestial mapping to satellite network management.

The Constellation Reporting Framework encompasses a variety of computational strategies, mathematical models, and workflow architectures for the efficient segmentation, analysis, and reporting of structured configurations—termed "constellations"—across astronomical, network, and relational data domains. Practical instantiations range from celestial region membership determination, big data pattern matching, performance evaluation of satellite networks, and the optimal design of constellation configurations, each leveraging specialized algorithms and formulation strategies.

1. Binary Tree Decomposition of the Celestial Sphere

The foundational approach for astronomical constellation membership, as articulated in (Glaschke, 2010), segments the IAU-defined celestial sphere into regions using a binary tree. Boundary segments are recursively selected as candidate splits, extended over the sphere, and used to partition the boundaries into two subsets. The criterion for split selection is to minimize the largest subset size, thus:

$$\underset{S}{\text{minimize}}\;\; \max(|P_{\text{left}}(S)|, |P_{\text{right}}(S)|)$$

where $P_{\text{left}}(S)$ and $P_{\text{right}}(S)$ represent the boundary segments on either side of split $S$. The recursive decomposition yields a binary tree that partitions all sky coordinates into rectangles or rectangular-like cells, each uniquely associated with an IAU constellation. Membership queries are thus reduced to a traversal of the tree, typically resolved in 9–11 decisions. This enables high-throughput classification in both real-time and batch contexts, presenting marked efficiency gains over exhaustive boundary checks and rectilinear meshes. Limitations arise in global tree balance and the potential for suboptimal partitioning in regions with convoluted boundaries.

2. Scalable Constellation Query Processing on Big Data

Spatial constellation queries, defined as the matching of point sets to geometric patterns based on pairwise or relative distances, are formalized in (Porto et al., 2017). Pure constellation queries require:

$$| \text{dist}(s_i, s_j) - \text{dist}(q_i, q_j) | \leq \varepsilon$$

for all $i, j$, with $\varepsilon$ denoting an additive tolerance. The framework integrates:

• Quadtree Filtering: Prunes candidate sets in high-dimensional data by leveraging spatial partitioning and error tolerances.
• Composition Algorithms:
  • Bucket_NL (bucket nested-loop) for sparse candidate buckets.
  • Matrix Multiplication (MM/MMM-NL) for dense scenarios, where Boolean matrices encode candidate compatibility, rapidly compressing combinatorial search spaces.
• Scale-invariant Search: General constellations employ a continuous-to-discrete transformation, defining scale factors for pattern matching with modified error bounds, ensuring tractability without loss of completeness.

Threshold regimes for algorithm choice are data-dependent: Bucket_NL excels for $\varepsilon < 0.003$, MM/MMM-NL for higher tolerances. The system’s architecture supports distributed implementation (Apache Spark), with linear memory growth in data size and substantial speedup over baseline methods. Primary applications include astronomical pattern recognition, seismic event clustering, and geospatial analytics.

3. Performance Evaluation and KPI Frameworks for Satellite Networks

A systematic Key Performance Indicator (KPI) framework for LEO mega-constellation networks is detailed in (Wang et al., 22 Jan 2024). KPIs are categorized as:

• Constellation KPIs: N-asset coverage, area traffic capacity (Kbps/km$^2$), service availability.
• Radio Interface Technology (RIT) KPIs: Peak/user data rates, unmet capacity, latency measures, access/handover statistics.

Methodologically, the reporting system partitions the service area using hexagonal cells (H3 geospatial indexing), simulates dynamic network topology via snapshot-based intervals, and restricts interference analysis to a finite "interfering area" containing the target satellites and their immediate neighbors. KPIs are statistically synthesized across all time steps:

KPI	Value/Range	Notes
Area traffic capacity	~4 Kbps/km²	Achieved for reference constellation
Service availability	0.36–0.39	Probability per cell
Access success prob.	>95% (nearest scheme)	CDF/heatmap shows coverage
Handover failure rate	~10% avg., up to 40%	Resource bottleneck hotspots

Relevant system equations include received power $P_{(b,l)}^u(t)$, path loss $PL(d)$, and SINR $\gamma_{(b,l)}^{u,c}(t)$, incorporating beam geometry and atmospheric factors. Simulation methodology supports reproducibility and calibration against ITU/3GPP standards. The comprehensiveness of this framework enables dynamic, scalable performance reporting, facilitating real-world monitoring and optimization.

4. Optimization-Based Constellation Design

A unified mixed-integer linear programming (MILP) framework for constellation configuration is articulated in (Rogers et al., 14 Jul 2025), with five core formulations:

• SCLP: Min-Cost continuous coverage,

$$\min \sum_j c_j x_j, \;\; \sum_j V_{tjp} x_j \geq r_{tp} \;\; \forall t, \forall p$$

• PSCLP: Partial coverage via binary time-target indicators, enforcing percent criteria $D_p$.
• MCLP: Maximal observation reward, fixed satellite count $N$.
• MMRT/MART: Minimize max/average revisit times with auxiliary constraints and cardinality set.

Extensibility is demonstrated via add-ons:

• Inter-Satellite Link (ISL) robustness, ensuring Hamiltonian cycles (inspired by Dirac’s theorem).
• Hard revisit-time constraints, deployment cost parametrization, and dual-target accommodation.

Comparative analyses quantify trade-offs among coverage, satellite count, revisit-time minimization, and cost. Solutions leverage commercial MILP solvers, enabling guarantees of global optimality provided discrete candidate orbital slots and visibility matrices.

5. Collaborative Data Downloading in Mega-Constellations

The Hurry framework (Luo et al., 5 Jun 2024) provides a dynamic, collaborative model for data downloading in mega-constellation networks, using a Time-Expanded Graph as the basis for scheduling:

• Nodes: Represent satellites and ground stations per time slot.
• Edges: Encode possible data transfers, bandwidth constraints, and buffer capacities.
• Algorithm: Adaptive Min-Cost Max-Flow, with horizon determination via doubling strategy.

Performance metrics demonstrate 100% completion of fixed data volume downloads, throughput improvements (11–66%) over the prior art CoDld, and resilience to Proximal Station Bias Degradation (PSBD). Satellite Queue Deviation Index (SQDI) is actively monitored to trigger plan updates under topology changes.

6. Visualization and Temporal Reporting of Relational Constellations

In data repository contexts (Reitz, 2010), ego-centered and time-aware visualization frameworks segment complex relation networks into small, navigable graphs:

• Entities (nodes): Focal “ego” and alters, selected via rating functions.
• Relations (edges): Temporal segmentation using time-color and intensity views, mapping quantitative metrics to visual attributes (e.g., node fill $d \in [0,1]$).
• Data interface: Abstracts repository schema; configuration files enable flexible adaptation.

Applications to DBLP and Wikipedia demonstrate web-based exploration and quantifiable reporting of relation dynamics.

7. Community Engagement and Policy Reporting for Satellite Constellations

The IAU CPS Community Engagement Hub (Barentine et al., 2023) integrates reporting on the societal, cultural, and observational impacts of satellite constellations. The framework, informed by international conferences (SATCON1/2, DQS-I/II), foregrounds stakeholder participation, documentation, and the translation of technical parameters—such as time-averaged brightness and orbital path predictions—into accessible reports for diverse audiences. While direct technical formulas are not a focal point, the ecosystem encourages the bridging of scientific rigor with community-relevant policy frameworks.

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Collectively, the Constellation Reporting Framework is characterized by computational efficiency, modular extensibility, and robust handling of multidimensional metrics. It enables scalable classification, optimal configuration, resource-aware performance monitoring, and transparent reporting across astronomical, network, and sociotechnical domains.