MCSN Tuesday, 25-Oct-11
- Note:
- Repeat quiz on Thursday, either to replace or average with last quiz (your choice)
- Draft proposals are also due Thursday
- 4.8 due today
- Open Quiz#2.paj
- Review of quiz and chapter 3 concepts. It's very important that you understand this material! Please see me if you do not.
- Density
- # lines divided by # possible lines
- Cannot exceed 100%
- Undirected simple network on N vertices contains N*(N-1)/2 possible lines
- Directed simple network on N vertices contains N*N possible lines (if loops are allowed) or N*(N-1) possible lines (if loops are expressly disallowed)
- Degree
- degree (undirected network); indegree and outdegree (directed network)
- total degree: sum over all vertices
- average degree: divide that sum by the number of vertices
- There's a neat trick for computing total and average degree (especially if all you know is the number of vertices and the number of lines, but don't know how they're connected)
- Connectivity and 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
- k-cores themselves do not form a partition
- a partition can be 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
- Complete triads
- 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 thus generally do not form a partition
- Density
- Balance and cluster theory (ch 4)