Generating Networks Small World Network and Preferential - - PowerPoint PPT Presentation

Generating Networks

Small World Network and Preferential Attachment

f

Some more terms

• Diameter

Find all of the shortest paths between all nodes

• f the network, the Diameter is the longest

such path

• Average Path Length

Find the shortest path between all nodes and sum them up and divide by the number of paths

Small World Network Most nodes are not neighbors but there are very short paths between most nodes. That is that the average path length is small. Note the diameter

• f the network can still be large.

Erdős–Rényi model

All possible edges in the network have an equal probability of being added. This probability is a parameter of the construction of the network. Alternatively of all of the graphs with N nodes and M edges one is chosen with uniform probability.

Watts and Strogatz model

Start with a uniform ring network where each node n is connected to k nodes immediately next to it. Next for each edge let there be a random chance

• f probability β that this edge be removed and

replaced by another edge that does not exist and is not a self link.

Barabási–Albert model Start with a seed network of at least 2 nodes that are connected. Now add a node making k connections to existing nodes, choosing the nodes such that the probability of connection is biased by the number of existing connections to that node. This is called preferential attachment

• r “rich get richer”.

Thanks wikipedia http://en.wikipedia.org/wiki/Small-world_network http://en.wikipedia.org/wiki/Average_path_length http://en.wikipedia.org/wiki/Random_graph http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_mo http://en.wikipedia.org/wiki/Watts_and_Strogatz_model http://en.wikipedia.org/wiki/BA_model