s4h-network-centrality

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# Network: Centrality

In almost every network — social, organizational, digital, biological — a small number of nodes exert disproportionate influence. They are not necessarily the most visible or the most senior. They are the most structurally positioned. Albert-László Barabási's research on scale-free networks established that real-world networks follow power-law degree distributions: a few hubs have vastly more connections than average, while most nodes have very few. These hubs emerge through preferential attachment — new nodes connect to already-well-connected nodes — which means structural inequality in networks is self-reinforcing.

Centrality analysis unpacks what "important" means in a specific network. There are four distinct types of centrality, and they identify different kinds of importance. A node can be a hub (high degree), a broker (high betweenness), an efficient spreader (high closeness), or influential because it is connected to other influential nodes (high eigenvector). These often coincide — but when they diverge, the divergence is analytically rich.

The key implication: targeting or protecting the most central nodes has outsized effect. Removing a high-betweenness broker can fragment a network; reaching a high-eigenvector hub can cascade influence throughout it.

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## Your Process

**Step 1: Define the Network**
Name the nodes (what entities?) and the edges (what relationship?). Be precise about direction — is the relationship directional (A influences B but not vice versa) or undirected (mutual connection)? Name the time horizon and state what question you are trying to answer with centrality analysis.

**Framing check:** Confirm the network definition and the question before continuing. State what you've identified in one sentence, then use `AskUserQuestion`:
- **Question:** "I'm reading this as: [your one-sentence description of the nodes, the relationship, and the question to answer]. Is that right?"
- **Header:** "Framing"
- **Options:**
  - **Yes — proceed** — framing is correct
  - **Adjust** — one element is off; user will correct it before you continue
  - **Reframe** — different situation than read; incorporate the correction before proceeding

**Step 2: Enumerate the Nodes and Edges**
List all known nodes. For each pair, identify whether an edge exists and its weight or frequency if relevant. If working from description rather than data, reason from available information — who interacts with whom, who reports to whom, who collaborates, who is referenced.

**Step 3: Calculate Centrality by Type**

For each node, assess across four dimensions:

- **Degree centrality:** How many direct connections does this node have? (Raw count or normalised as a fraction of all possible connections.) High degree = a hub.
- **Betweenness centrality:** How often does this node sit on the shortest path between two other nodes? High betweenness = a broker or bottleneck — removing them disrupts information or resource flow.
- **Closeness centrality:** How short are this node's average distances to all other nodes? High closeness = fastest to reach everyone; efficient at spreading or receiving information.
- **Eigenvector / prestige centrality:** Is this node connected to other highly-connected nodes? High eigenvector = influence through association; the node's power is amplified by the power of its neighbours.

**Before narrowing:** Show the full centrality picture first. Use `AskUserQuestion`:
- **Question:** "I've mapped [N] nodes across all four centrality types. Before I identify the top-priority nodes, is there a specific type of centrality — hubs, brokers, spreaders, or prestige — you most care about for this question?"
- **Header:** "Priority"
- **Options:**
  - **Rank all four** — give a complete picture across all centrality types
  - **Focus on brokers** — betweenness centrality; who controls flow?
  - **Focus on hubs** — degree centrality; who is most connected?
  - **Focus on prestige** — eigenvector; who matters because of who they know?

**Step 4: Identify Structural Features**
Flag nodes with unusual profiles:
- **High betweenness, low degree:** A narrow broker — few connections but sits on critical paths. High leverage, high vulnerability.
- **High eigenvector, low degree:** Connected to powerful nodes but not widely. A "backdoor" to influence.
- **Isolates:** Nodes with no connections. Excluded from the network — may be by design or a gap.
- **Cut points / articulation nodes:** Nodes whose removal would disconnect parts of the network.

**Step 5: Apply to the Question**
Translate structural findings into actionable answers for the original question. Different questions call for different centrality types: targeting for influence → eigenvector and degree; identifying single points of failure → betweenness; seeding information efficiently → closeness.

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## Human Check-in

Before proceeding, use the `AskUserQuestion` tool. State your interpretation of the network and the question in 1–2 sentences, then ask:

- **Question:** "My read: [your 1–2 sentence interpretation]. How do you want to proceed?"
- **Header:** "Scope"
- **Options:**
  - **Full analysis** — Complete all steps, reasoning shown throughout
  - **Key findings only** — Bottom-line output: top nodes by type and the structural implications
  - **Specific focus** — Zoom in on one centrality type or one subset of nodes
  - **Reframe** — The read is off; correct it and the analysis will follow the corrected framing

Proceed based on their selection. If the user reframes, incorporate the correction before running any analysis.

## Output Format

**Network Definition**
[Nodes, edges, direction, time horizon, question being answered]

**Centrality Rankings**

| Node | Degree | Betweenness | Closeness | Eigenvector | Overall Assessment |
|------|--------|-------------|-----------|-------------|-------------------|
| | | | | | |

**Structural Features**
[Cut points, isolates, anomalous profiles, and what each means]

**Answer to the Original Question**
[Direct application of centrality findings to the specific question — who to target, protect, connect, or disconnect]

**Caveats**
[Where the analysis is limited by missing edges, assumed relationships, or a changing network structure]

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## Notes

Centrality analysis assumes the network is reasonably stable and that the edges you've mapped reflect real relationships. In dynamic networks — where ties form and dissolve quickly — centrality scores can be misleading snapshots. Pair with `/s4h-network-contagion` when the goal is to model spread, since centrality tells you *who* matters but not *how* something moves through the network. Pair with `/s4h-social-power-mapping` when the network is organizational and the edges are influence or authority rather than communication.

The most important structural insight is often not the top-ranked node — it is the node with surprisingly high betweenness relative to its degree. These "hidden brokers" are frequently invisible to conventional analysis because they are not the loudest, most senior, or most connected — they just happen to sit on the only path between two otherwise-disconnected parts of the network.

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## What's Next

After delivering this output, use `AskUserQuestion` to offer the next move:

- **Question:** "Centrality mapped. What's next?"
- **Header:** "Next"
- **Options:**
  - `/s4h-network-contagion` — Model how something spreads through this network, using the identified hubs and brokers as seed points or bottlenecks
  - `/s4h-network-weak-ties` — Identify the structural holes in this network and the bridging connections that span them
  - `/s4h-social-power-mapping` — Translate the structural centrality findings into an organizational power map
  - **Done** — Wrap up and synthesise what we have so far

From human-avatar/skills-for-humanity, MIT.

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