# Defining Quantitative Metrics for Emergent Structures in IO Simulations ## 1. Objective Following the identification of a promising dynamic regime in Run 15 [[releases/archive/Information Ontology 1/0127_IO_Simulation_Run15]], a key next step is to develop quantitative measures that can characterize the emergent structures and dynamics more precisely. This will allow for: * Objective comparison of different parameter regimes. * Identification of phase transitions or critical points. * A more rigorous assessment of the model's ability to generate targeted emergent phenomena. This node defines a set of metrics to be implemented and calculated in future simulation analysis. ## 2. Target Phenomena and Desired Metrics Based on the hypothetical outcomes outlined in [[releases/archive/Information Ontology 1/0105_IO_Simulation_Outcomes_v2]] and the initial observations from Run 15, we want to quantify: 1. **Domain Size:** The typical size or spatial extent of the relatively uniform regions (domains) in the `φ` field. 2. **Boundary Sharpness:** How sharply defined are the interfaces between domains? 3. **Boundary Dynamics:** How do the domain boundaries move, interact, and change over time? 4. **Oscillation Frequencies:** What are the dominant frequencies of oscillation within the system, both globally (global field) and locally (within domains)? 5. **"Particle" Compactness (If Applicable):** If localized, particle-like structures emerge, how can we quantify their spatial compactness or density? ## 3. Proposed Metrics Here are specific metrics to quantify these phenomena in the 1D continuous-state network model: 1. **Domain Size (Average Domain Length):** * *Algorithm:* 1. Identify "domains" as contiguous regions where `φ(i, S)` is within a certain threshold `δ` of the average value within that region. 2. For each node `i`, find the longest contiguous segment (domain) containing `i` where `|φ(i, S) - <φ>_domain| < δ`, where `<φ>_domain` is the average `φ` within that segment. 3. Calculate the length of each domain. 4. Average these lengths across all nodes to get the average domain length at time `S`. 5. Track this average domain length over time. * *Parameters:* `δ` (domain definition threshold). 2. **Boundary Sharpness (Average Gradient Magnitude):** * *Algorithm:* 1. Calculate the spatial gradient of `φ` at each node: `grad(i, S) = |φ(i+1, S) - φ(i-1, S)| / 2` (using periodic boundaries). 2. Calculate the average gradient magnitude across the entire system at time `S`: `<|grad(φ)|>_S = (1/N) * Σ_i grad(i, S)`. 3. Track this average gradient over time. Higher values indicate sharper boundaries. 3. **Boundary Dynamics (Boundary Density and Velocity):** * *Algorithm:* 1. Define a "boundary" as a location where `grad(i, S)` exceeds a threshold `grad_threshold`. 2. Calculate the "boundary density" as the number of such points per unit length (or fraction of nodes that are boundaries). 3. Track the positions of these boundaries over time. 4. Calculate the average boundary velocity (magnitude of displacement per time step). * *Parameters:* `grad_threshold`. 4. **Oscillation Frequencies (Power Spectral Density):** * *Algorithm:* 1. For a subset of nodes (or all nodes), record the time series of `φ(i, t)` over the simulation duration. 2. Calculate the Power Spectral Density (PSD) of each time series using a Fast Fourier Transform (FFT). 3. Average the PSDs across the selected nodes to obtain a representative frequency spectrum for the system. 4. Identify the dominant frequencies (peaks in the PSD). * *Parameters:* Number of nodes for PSD calculation, FFT window size. 5. **"Particle" Compactness (If Localized Structures Emerge):** * *Algorithm:* (This is more speculative, contingent on observing localized structures) 1. Define a "particle" as a localized region where `|φ(i, S) - <φ>_background| > amplitude_threshold` (a region where the state deviates significantly from the average background). 2. Calculate the "compactness" as the ratio of the "particle's" peak amplitude to its spatial extent (e.g., the full width at half maximum of the localized profile). Higher compactness means a more localized, high-amplitude structure. 3. Track the number of "particles," their average compactness, and their lifetimes. * *Parameters:* `amplitude_threshold`, method for defining spatial extent. ## 4. Implementation Notes * These metrics should be implemented as Python functions that take the simulation results (e.g., `phi_history`, `theta_history`) as input and return the calculated metrics. * The code should be designed to efficiently calculate these metrics across multiple time steps and for different parameter sets. * The parameters for the metrics themselves (e.g., `δ`, `grad_threshold`, `amplitude_threshold`) will need to be tuned based on visual inspection of the simulation data and the specific emergent phenomena observed. ## 5. Conclusion: A Quantitative Toolkit for IO Exploration Defining these quantitative metrics provides the necessary tools for a more rigorous analysis of the IO continuous-state network model. By implementing these measures and applying them to future simulation runs, we can move beyond qualitative descriptions and begin to map the parameter space, identify key transitions, and assess the framework's ability to generate targeted emergent behaviors. This is a crucial step towards validating the potential of Information Dynamics as a fundamental theory. The next step is to implement these metrics in code and apply them to the existing Run 15 data and subsequent simulations.