What formal properties or criteria determine whether a structural element is task-critical (e.g., causal dependence, information-theoretic necessity, task-graph reachability)? (a sub-question that arises while solving: What is the best way to display concept mapping and diagram trees in this app (a software application) for the concepts being shown?)

Question

Accepted Answer

# Analysis: Two Distinct Problems Conflated

You got the core issue right. Let’s pull these apart.

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## Problem 1: Formal Definition of Task-Criticality
**What makes a structural element task-critical?**

Here are candidate formal criteria from the literature:

| Criterion | Definition | Limitation |
|-----------|-----------|-----------|
| **Causal dependence** | Element E is task-critical if task outcome depends on E's state/properties | Doesn't distinguish *degree* of dependence; bidirectional causality unclear |
| **Information-theoretic necessity** | E is critical if removing E increases task-completion entropy/uncertainty | Requires quantifying information loss; domain-specific |
| **Graph reachability** | E is critical if all/most goal nodes are reachable *only* through E | Works for DAGs; fails for cyclic or probabilistic dependencies |
| **Failure impact** | E is critical if its failure prevents task completion | Practical but post-hoc; doesn't define criticality *a priori* |

**Missing from your question:** What task model are you using? (workflow, decision tree, resource dependency graph, something else?)

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## Problem 2: UI Display for Concept Maps/Diagram Trees
**How should critical elements be visualized?**

This depends on **user research**, not theory:
- What mental model do users already have?
- Do they confuse criticality with *importance* or *frequency of use*?
- Should criticality be persistent (color/shape) or contextual (on-demand highlighting)?

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## What I Need From You

1. **What task model** governs your domain? (e.g., workflow, knowledge hierarchy, dependency graph)
2. **What criticality metric**