What is Highly Connected graph?
In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of an arbitrary directed graph form a partition into subgraphs that are themselves strongly connected.
What is strongly connected graph in data structure?
(definition) Definition: A directed graph that has a path from each vertex to every other vertex. Formal Definition: A directed graph D=(V, E) such that for all pairs of vertices u, v ∈ V, there is a path from u to v and from v to u.
Is a complete graph always connected?
A connected graph is a graph in which it’s possible to get from every vertex in the graph to every other vertex through a series of edges, called a path. By definition, every complete graph is a connected graph, but not every connected graph is a complete graph.
How do you know if a graph is strongly connected?
A simple solution is to perform Depth–first search (DFS) or Breadth–first search (BFS) starting from every vertex in the graph. If each DFS/BFS call visits every other vertex in the graph, then the graph is strongly connected.
How does the HCS clustering algorithm find clusters?
HCS clustering algorithm finds all the subgraphs with n vertices such that the minimum cut of those subgraphs contain more than n/2 edges, and identifies them as clusters. Such a subgraph is called a Highly Connected Subgraph (HCS). Single vertices are not considered clusters and are grouped into a singletons set S.
How is a similarity graph used in clustering?
Similarity graph is one of the models to represent the similarity between elements, and in turn facilitate generating of clusters. To construct a similarity graph from similarity data, represent elements as vertices, and elicit edges between vertices when the similarity value between them is above some threshold.
How can you tell if G is highly connected?
If G has a vertex x with degree <= n/2, then G has a minimum cut (that isolates x) with edges <= n/2, so G is not highly connected. So if G is highly connected, every vertex has degree >= n/2.
What is the goal of a cluster analysis?
The goal of cluster analysis is to group elements into disjoint subsets, or clusters, based on similarity between elements, so that elements in the same cluster are highly similar to each other (homogeneity), while elements from different clusters have low similarity to each other (separation).