Difference between revisions of "MCSN Tuesday, 25-Oct-11"
Line 25: | Line 25: | ||
*** components cover the network | *** components cover the network | ||
*** components therefore define a partition | *** components therefore define a partition | ||
+ | *** Giant component formation | ||
+ | **** Play with Pajek: 100 vertices, vary the number of arcs. How many components are there? How big are they? You'll notice with more than 100 arcs that a large component arises. | ||
+ | **** Try this [http://ccl.northwestern.edu/netlogo/models/GiantComponent beautiful simulation] which illustrates giant component formation in a random graph. | ||
** K-cores | ** K-cores | ||
*** The k-core is a maximal subnet in which every vertex is degree k or greater. | *** The k-core is a maximal subnet in which every vertex is degree k or greater. | ||
Line 33: | Line 36: | ||
*** find the k-core by locating a vertex of degree k or more, then checking if at least k neighbors could be included in the k-core. | *** find the k-core by locating a vertex of degree k or more, then checking if at least k neighbors could be included in the k-core. | ||
*** OR: find the k-core by deleting vertices of degree less than k, then repeating. | *** OR: find the k-core by deleting vertices of degree less than k, then repeating. | ||
− | ** Cliques | + | ** Complete networks, triads and cliques |
− | *** | + | *** Complete networks |
+ | **** A complete network is 100% dense | ||
+ | **** A complete triad is a triangular complete network ("bone") | ||
+ | **** Complete triads may overlap | ||
+ | **** The set of complete triads ("skeleton") may have several components | ||
+ | **** Each component is considered a "social circle" | ||
+ | **** Complete triads do not form a partition | ||
+ | **** A partition can be generated by counting, for each vertex, how many triads include it | ||
+ | *** Cliques | ||
+ | *** A clique is a maximal complete subnetwork | ||
+ | *** Cliques may overlap | ||
+ | *** Cliques need not cover the network | ||
+ | *** Cliques generally do not form a partition |
Revision as of 09:58, 25 October 2011
- Review of quiz and concepts:
- Density
- # lines divided by # possible lines
- Cannot exceed 100%
- Degree
- indegree and outdegree
- total degree: sum over all vertices
- average degree...
- ...and a neat trick for computing total and average degree (e.g. if all you know is there are 8 vertices and 8 lines...)
- Components
- criteria: (a) violate no one-way streets; (b) do not visit any vertex twice.
- path (a,b)
- semipath (not a, b)
- walk (a, not b)
- semiwalk (not a, not b)
- weak
- a subnet is weakly connected if each pair of vertices is connected by a semipath
- a weak component is a maximal weakly connected subnet
- find it by tracing all the vertices semipath-reachable from a particular vertex V
- strong
- a subnet is strongly connected if each pair of vertices is connected by a path
- a strong component is a maximal strongly connected subnet
- find it by tracing all the vertices path-reachable from a particular vertex V, which can also reach V
- components never overlap
- components cover the network
- components therefore define a partition
- Giant component formation
- Play with Pajek: 100 vertices, vary the number of arcs. How many components are there? How big are they? You'll notice with more than 100 arcs that a large component arises.
- Try this beautiful simulation which illustrates giant component formation in a random graph.
- criteria: (a) violate no one-way streets; (b) do not visit any vertex twice.
- K-cores
- The k-core is a maximal subnet in which every vertex is degree k or greater.
- k-cores "nest": the 0-core contains the 1-core, which contains the 2-core, which contains the 3-core
- k-cores are not necessarily connected
- a partition is defined by assigning each vertex to its maximum k-core.
- this partition cannot be computed from degree information alone!
- find the k-core by locating a vertex of degree k or more, then checking if at least k neighbors could be included in the k-core.
- OR: find the k-core by deleting vertices of degree less than k, then repeating.
- Complete networks, triads and cliques
- Complete networks
- A complete network is 100% dense
- A complete triad is a triangular complete network ("bone")
- Complete triads may overlap
- The set of complete triads ("skeleton") may have several components
- Each component is considered a "social circle"
- Complete triads do not form a partition
- A partition can be generated by counting, for each vertex, how many triads include it
- Cliques
- A clique is a maximal complete subnetwork
- Cliques may overlap
- Cliques need not cover the network
- Cliques generally do not form a partition
- Complete networks
- Density