ECC3841 Chap.1 Network Foundations and Centrality
Network Foundations and Centrality
Network foundations turn a relationship into a graph. Begin by stating the nodes, what a link means and whether direction matters. Paths and components describe reachability; degree, closeness, betweenness and eigenvector centrality answer different questions about local contact, access, brokerage and recursive importance.
Network-level statistics such as density, clustering, average path length and diameter describe the structure as a whole. The examinable skill is choosing a statistic from the mechanism, showing its denominator and interpreting the result without claiming causation.
What this chapter covers
- 01
Nodes, links and directed versus undirected adjacency
- 02
Paths, geodesics, components and the giant component
- 03
Degree and degree distributions
- 04
Closeness as reciprocal distance
- 05
Betweenness as shortest-path brokerage
- 06
Eigenvector centrality and important neighbors
- 07
Density, clustering, diameter and average path length
- 08
Why the same degree distribution can hide different structures
Worked example · free
Compare degree and closeness on a path
- +1Degrees are two for B and C because each has two direct neighbors.
- +1Distances from B to A,C,D are one, one and two, sum four.
- +1Normalized closeness for B is three divided by four.
- +1C is symmetric and also has closeness three divided by four.
- +1The measures tie because the two interior nodes occupy symmetric positions.
Key terms
- Node
- An actor or object represented as a vertex in the network.
- Link
- The specified relationship joining a pair of nodes, with direction retained when relevant.
- Component
- A maximal set of nodes connected by paths.
- Geodesic
- A shortest path between two nodes.
- Network density
- The share of possible links that are present.
- Clustering coefficient
- A measure of how often a nodes neighbors are linked with one another.
Network Foundations and Centrality FAQ
Can two nodes lead different centrality rankings?
Yes. Degree values immediate contacts, closeness values short access, betweenness values brokerage and eigenvector centrality values important neighbors. A bridge can have modest degree and high betweenness. In calculations, write the graph convention, denominator and economic interpretation before giving the final ranking.
Why define the link before calculating?
Direction, weight and relationship type determine the adjacency matrix and every statistic built from it. A nomination network and a mutual-friendship network cannot be treated as the same graph. In calculations, write the graph convention, denominator and economic interpretation before giving the final ranking.
Does high density imply short paths?
Density often helps shorten routes, but placement matters. Two networks with the same number of links can have different clustering, component structure and average path length. In calculations, write the graph convention, denominator and economic interpretation before giving the final ranking.
What earns method credit?
State the formula or denominator, show path or link counts and connect the number to the economic mechanism. A ranking without its reason is incomplete. In calculations, write the graph convention, denominator and economic interpretation before giving the final ranking.
Exam move
Redraw small graphs and compute one measure at a time. Begin each sheet with a sentence defining the nodes, the relationship and whether an arrow has direction. Convert the picture to an adjacency matrix and check one row against the drawing before doing anything else. On an undirected graph, add the degrees and verify that the result equals twice the number of links.
If it does not, repair the representation rather than adjusting a later answer. Keep node-level and network-level statistics in separate columns. Degree, closeness, betweenness and eigenvector centrality rank nodes, while density, clustering, average path length and diameter describe the graph. For every centrality, write the question it answers and one case where it can disagree with degree.
Practise shortest paths by listing them, not by relying on visual distance. For betweenness, identify every pair that excludes the candidate node and split credit when several shortest routes exist. For closeness, state the normalization and explain how disconnected nodes are handled. For eigenvector centrality, first check that the equation rewards ties to already important neighbors.
Build comparison pairs: a star center versus a path interior, a bridge between cliques versus a dense-clique member, and two graphs with the same degree distribution but different clustering. After each calculation, write one economic sentence about access, brokerage, exposure or prestige. Avoid treating a large score as causal influence. Use symmetry as a check: structurally equivalent nodes should receive equal values.
On the final revision pass, cover the answers and reconstruct the denominator, path count and interpretation from the graph alone. Finish by changing one link and predicting which measures move before recalculating; that exercise exposes whether you understand the mechanism or only the formula.
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