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Defining clusters, part three
Our definition of cluster informally draws upon our formal definitions of connected components and cliques. In an undirected graph G=(V,E) a cluster is a subgraph induced by node set S⊆V (i.e., GS =...
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Defining clusters, part two: cliques
In an undirected graph G=(V,E) a clique is a subgraph induced by node set S⊆V (i.e., GS = (S,ES)) such that the density of GS is 1. Put another way, in a clique, every pair of nodes is adjacent....
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Defining clusters, part one: connected components
The following sections present a "semi-formal" definition of clusters in three parts:First, we formally define connected componentSecond, we introduce and formally define cliqueThird, we informally...
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Clusters
Informally speaking, a cluster is a group of nodes in a graph that are more connected to each other than they are to the rest of the graph. For example, the red and yellow regions below are clusters:
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Hubs
A hub is a node in a graph with much higher degree than average. For example, nodes 1 and 2 are both hubs in the figure below:
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Connected graphs and connected components
A connected graph (as defined above) is said to consist of a single connected component. If a graph is not connected, then it consists of two or more connected components. Six Degrees offers this...
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Earlier we stated that A path connects its origin and destination nodesTwo nodes are connected if there is at least one path that connects them A graph is connected if each of its nodes is connected to...
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Induced Subgraphs
It is often useful to consider only part of a graph. Induced subgraphs are one particularly convenient way to define a specific sub-part of a larger graph. Given a graph G=(V,E), an induced subgraph is...
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"Connected": a word of many meanings
The word "connected" speaks to the most basic structural properties of networks. It is arguably both the most important and the most overused term in the network vocabulary. Important and proper uses...
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