Calculate Cascade Metrics From a Network Edge List

Performs the core cascade analysis on an edge list returned by build_network(). This function computes the topology score, layer-level role scores, node-level metrics, and the overall cascade balance score.

Usage

calculate_cascade(network_df, alpha_parameter = 0.9)

Arguments

network_df

A data frame representing the edge list. Required columns:

from: Source node identifier.
to: Target node identifier.
layer: Integer (1-3). The "degree" of the interaction (1 = Core, 2 = Community, 3 = Distant).

alpha_parameter

Numeric. Damping factor for Alpha Centrality (defaults to 0.9).

Value

An object of class cascade_analysis containing:

cascade_score: The global Balance Score ($S_c$, 0-1).
summary: A summary table aggregating roles by Layer (Degree).
node_data: Detailed metrics for every node.
topology_score: Baseline topological health score.

Details

Theoretical Foundation: Unlike general connectivity metrics, this method resembles the work of Long, Cunningham, and Braithwaite (2013), examining participants' roles based on the structure of the network formed through the research. It assesses how influence ripples outward from the core team (Layer 1) to the broader community (Layer 3+).

Operational Definitions: The function maps Social Network Analysis (SNA) metrics to four key influence roles:

1. Knitting (Cohesion & Bonding): Measures how well the network strengthens internal bonds within a specific group.

Metrics: Community detection (Walktrap) + Eigenvector Centrality.
Interpretation: High scores indicate a tight-knit, resilient core.

2. Bridging (Connection & Spanning): Measures the ability to connect otherwise disconnected groups (filling "structural holes").

Metrics: Structural Holes (Constraint) + Degree Centrality on the inter-layer graph.
Interpretation: High scores indicate key "brokers" connecting silos.

3. Channeling (Flow & Transmission): Measures the efficiency of information flow and resource distribution.

Metrics: PageRank (local) + Harmonic Centrality (global) on the inter-layer graph.
Interpretation: High scores indicate effective pipelines for moving resources.

4. Reaching (Access & Inclusion): Measures the extent of the network's periphery and accessibility.

Metrics: Clustering Coefficient + Communicability (matrix exponential) on the inter-layer graph.
Interpretation: High scores indicate an inclusive network with reduced barriers.

The Scoring Process:

Layer (Degree) Scoring: For each network layer (degree of separation), influence is calculated by combining local cohesion (Knitting + Bridging), global flow (Channeling), and peripheral access (Reaching): $$s_{\text{layer}} = \gamma(\alpha L + \beta G) + \lambda T$$ where $L$ represents combined Knitting and Bridging scores, $G$ represents Channeling score, $T$ represents Reaching score, and weights are:
- $\alpha = 0.4$ (local cohesion weight)
- $\beta = 0.3$ (global flow weight)
- $\lambda = 0.3$ (peripheral access weight)
- $\gamma$ varies by layer: 0.9 (Layer 1), 0.5 (Layer 2), 0.45 (Layer 3)
Cascade Balance Score: Calculated based on the equality of layer scores. $$S_{c} = 1 - \operatorname{Gini}(\{s_{\text{layer},k}\})$$ where $\{s_{\text{layer},k}\}$ represents the set of all layer influence scores.

Cascade Balance Interpretation:

$S_c < 0.50$: Very Low Balance (Core-dominated)
$0.50 \le S_c < 0.59$: Low Balance
$0.60 \le S_c < 0.69$: Moderate Balance
$0.70 \le S_c \le 0.79$: High Balance
$S_c \ge 0.80$: Very High Balance (Equitable distribution)

Pipeline

For typical use, call analyze_cascade rather than calculate_cascade directly. The pipeline accepts raw cascade survey parameters, automatically routes to a full or scaled analysis based on expected edge count, and averages results across multiple stochastic runs when scaling is required.

References

Christakis, N. A., & Fowler, J. H. (2009). Connected: The Surprising Power of Our Social Networks and How They Shape Our Lives. Little, Brown Spark.

Haddad, C. N., et al. (2024). The World Bank's New Inequality Indicator. World Bank. doi:10.1596/41687

Long, J. C., Cunningham, F. C., & Braithwaite, J. (2013). Bridges, brokers and boundary spanners in collaborative networks: a systematic review. BMC Health Services Research, 13, 158. doi:10.1186/1472-6963-13-158

Price, J. F. (2024). CEnTR*IMPACT: Community Engaged and Transformative Research - Inclusive Measurement of Projects & Community Transformation. CUMU.

Wang, H.-Y., et al. (2020). Comparison of Ferguson's delta and the Gini coefficient used for measuring the inequality of data. Health and Quality of Life Outcomes, 18(1), 111.