Flashcards
Lecture 3, 4, 5
Cards in this deck (38)
- 1Lean ModelFind a system that contains only the minimal elements, if unsure about an element ignore it and add it later if needed when the solution to a simpler problem found
- 2Adjacency matrixA matrix showing the connections between nodes i and j
- 3Qualities of the adjacency matrix for a Undirected Graph Simple graphSymmetrical
Contains only 1s and 0s - 4Degree of a nodethe degree is the
number of times the node
appears as an endpoint of a link. - 5In degree vs Out degree
- 6Eulerian WalkA walk where every link is used exactly once
- 7Konigsberg Node of degree 1 Nodes of odd degreeNeed to have a walk where 2 or less nodes with degree 1. Because degree 1 nodes need to be start or end point
For same reason, cant have more than 2 nodes of odd degree - 8Konigsberg Node of degree 2Fine unless it is the starting node, in which case it must also be the end point (a circuit)
- 9Solution to konigsbergFind all odd degree nodes. Connect the until two or less odd degree nodes left
- 10Chinese PostmanShortest walk that uses every link at least once.
Issue when circular tour needed, can move odd node problem by walking corridor twice. Move odd nodes together to pair them up. - 11Heimholze AlgorithmStart in correct node and go on walk until fail. Then choose a node that you pass through and do a seperate walk, completing the circuit. Then when you do the first walk again when you pass through this node do the second walk before finishing the first
- 12Hamiltonian Path Hamiltonian CycleA path that visits every node on a network exactly once
The same except the first and last point are the same (ie one node is visited twice) - 13Local searchSort nodes into clusters then find hamiltonian path in between clusters
- 14Distance between node pair Diameter of a networkThe shortest route between two nodes
The longest distance between any two nodes
ie. the longest shortest path - 15Alternative definition of diameterThe average distance between all pairs of nodes
Not just the average between longest nodes but all nodes - 16Degree distributionThe distribution of the degree of the nodes. If a random node is taken in the graph the probability that the node has degree k is p(k)
- 17Mean degree (z) for G(N,K) for G(N,p)The expectation for the degree of the degree distribution
z = 2K/N
z = (N-1)p ~ Np - 18Excess degree distribution qkThe probability of finding a node that has k additional links if a random link is followed.
The key is that it is you find a node via a random link, not by starting at a random node
- 19Mean excess degreeThe expected value for the excess degree distribution
- 20ER GraphG(N,p) where N nodes are connected with the probability of a connection occuring being p
Or G(N,K) where K links are formed with the probability of the end points being any node being uniformly distributed - 21ER Graph mean degree and excess degree distributionz = (N − 1)p ≈ N p which is a binomial distribution and can be approximated using the poisson distribution
Mean degree equals the excess degree
pk = qk
z = q - 22Approximation for the number of people that can be reached in d stepsn(d) = q^d
- 23Diameter (quick estimate formula)D = ln(N)/ln(q)
its a very slowly rising function - 242 reasons diameter estimate works wellCanceling of two effects:
lots of cycles exist: reducing the effective degree
For real networks q>z - 25robustness of diameter estimatesAn error in q only minorly effects the estimates of the diameter
- 26Closeness centralityto find the most important node in a network.
Run dijkstras algorithm for every pair of nodes in a network
Find the node with the lowest average distance to other nodes - 27Rows and Columns matrix
- 28Travelling Salesman - some approachesTrial and error - very quick, non optimal, can give good insight
Brute force - slow, safe, optimal, easy to implement. trick is to enumerate efficiently.
Branch & Bound - Exact solution relatively fact, Hard to implement, becomes fast if you can bound effectively
Greedy - very fast, easy to implement on a computer, risky - 29Christophides algorithmConstruct a minimal spanning tree, connect the nodes with odd degree until an eulerian circuit appears
- 30Exponential Seriesexp(z) =
- 31mean degree distribution p_k
- 32Differentiation formulak x ^ (k-1) = d/dx x ^ k
- 33Kronecker delta
- 34calculating the: mean degree z mean excess degree q
- 35Closeness centrality formulaNode with highest C_i is most central
- 36Sanity check for degree distributionsMake sure p_k and q_k add up to 1
- 37G(N,K)Create N nodes
Create K links
Choose the endpoints of the links randomly with uniform probability among the nodes. Reject and redraw any links
that would make the graph non-simple. - 38G(N,p) How is it made, advantagesCreate N nodes
For every pair of nodes (i,j) create a link between these nodes with probability p
Gives completely random network
Each link is statistically independent
Information
- Description
- 2 Aerospace (Dynamics of Networks) Flashcards on Lecture 3, 4, 5, created by Oliver Boom on 04/03/2018.
- Created by
- Oliver Boom
- Views
- 0
- Rating