* Paths
- geodesic paths
- necessarily self-avoiding walks
- need not be unique for a given pair of vertices
- important for small-world effect, information flow, the Internet,
transportation etc
- diameter of a network
- Eulerian
- a path that traverses each edge in a network exactly once
- at most two odd vertices => exactly two or zero
- this is only necessary condition, not sufficient
- Hamiltonian paths
- a path that visits each vertex in a network exactly once
- is hard! no non-trivial necessary condition is known
- garbage collection, job sequencing, travelling salesman etc
* Components
- Undirected graph
- a subset of vertices in a network s.t. there exists a path from
every vertex to every other vertex, and no other vertex can be added
to this set preserving this property. (Maximal subset)
- every vertex belongs to exactly one component
- connected network, single isolated vertex
- adjacency matrix is block diagonal
- Directed graph
- weakly connected components
- strongly connected components (SCCs)
- single vertex also possible
- every vertex is a part of at least one cycle
- every vertex belongs to exactly one SCC
- out-component
- set of all vertices reachable from a given vertex
- property of the network + the starting vertex
- a vertex can belong to more than two out-components
- all members of SCC to which the vertex belongs are also part of
its out-component
- all vertices reachable from the vertex are reachable from other
vertices in SCC also => out-components of all the vertices in SCC
are same.
- SCC of the vertex is subset of its out-component
- out-components actually belong to strongly connected components
- in-component
- set of all vertices from which a particular vertex can be
reached
- property of network + the vertex in question
- a vertex can belong to more than one in-component
- SCC of the vertex is subset of its in-component
- in-components actually belong to SCCs
- SCC is an intersection of out-component and an in-component