Let G be a graph which is connected (so you can go from any vertex to...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Let G be a graph which is connected (so you can go from any vertex to any other along the edges). Assume that G has no double edges (there is at most one edge between any two vertices) and no edge connects a vertex back to itself (each edge must connect two distinct vertices). The adjacency matrix, A, of graph G is defined by ai.j 1 {} if there is an edge between vertices i and j, 0 otherwise. The transition matrix P of the random walk on G is defined by ai,j D₁ Pi,j:= -Σaij. where D₁ := Prove that P is symmetrizable and that E;T; = D/D; where D:=E, D₁. [6] Hint: You are indirectly given T₁ (how?), so it is sufficient to 'guess' a solution and verify that it satisfies the detailed balance equations... (b) The crazy knight! Prove that a knight on a chess-board making legal chess moves at random and starting at a corner square will take on average 168 moves to return there. Hint: think of the graph associated with this random walk and count the num- ber of edges leaving each site, then use above results... Let G be a graph which is connected (so you can go from any vertex to any other along the edges). Assume that G has no double edges (there is at most one edge between any two vertices) and no edge connects a vertex back to itself (each edge must connect two distinct vertices). The adjacency matrix, A, of graph G is defined by ai.j 1 {} if there is an edge between vertices i and j, 0 otherwise. The transition matrix P of the random walk on G is defined by ai,j D₁ Pi,j:= -Σaij. where D₁ := Prove that P is symmetrizable and that E;T; = D/D; where D:=E, D₁. [6] Hint: You are indirectly given T₁ (how?), so it is sufficient to 'guess' a solution and verify that it satisfies the detailed balance equations... (b) The crazy knight! Prove that a knight on a chess-board making legal chess moves at random and starting at a corner square will take on average 168 moves to return there. Hint: think of the graph associated with this random walk and count the num- ber of edges leaving each site, then use above results...
Expert Answer:
Answer rating: 100% (QA)
To prove that the transition matrix P of the random walk on graph G is symmetrizable we need to show that there exists a diagonal matrix D such that D... View the full answer
Related Book For
Data Structures and Algorithm Analysis in Java
ISBN: 978-0132576277
3rd edition
Authors: Mark A. Weiss
Posted Date:
Students also viewed these mathematics questions
-
Let G be a connected graph. Show that if T is a spanning tree of G constructed using depth-first search, then an edge of G not in T must be a back edge, that is, it must connect a vertex to one of...
-
Let G be a directed graph on n vertices. If the associated undirected graph for G is Kn, prove that vV[od(v)]2 = vV [id(v)]2.
-
Let G be a group where a2 = e for all a G. Prove that G is abelian.
-
Question 4 Tic-tac-toe (also known as noughts and crosses) is a game for two players, X and O, who take turns marking the spaces in a 3x3 grid. The player who succeeds in placing three of their marks...
-
New tire retreading equipment, acquired at a cost of $160,000 at the beginning of a fiscal year, has an estimated useful life of four years and an estimated residual value of $16,000. The manager...
-
1. Creditor (lender) has an agreement with Debtor whereby Debtor has provided assets (real or personal property) as collateral in return for a loan. 2. Creditor registers the security agreement...
-
Show how the minimum wage creates unemployment in a competitive market.
-
A consumer electronics company is comparing the brightness of two different types of picture tubes for use in its television sets. Tube type A has mean brightness of 100 and standard deviation of 16,...
-
On June 1 , Year 1 , Williams Inc. paid $ 4 , 8 0 0 for a two - year insurance policy with coverage beginning that day. How much insurance expense will Williams Inc. report in its Year 1 income...
-
Get It Right, CPAs, has been retained to review its client's corporate formation calculations for 20XX. Maria, Roger, and Novak created Grassroots Tennis, Inc. (GTI), which began operations on March...
-
During a collision, the airbag in a car will deploy and fill up with nitrogen gas. In order for this to occur, solid sodium azide (NaN 3) is ignited and decomposes to produce solid sodium (Na) and...
-
Van Duzer explains that one of the consequences of the Fall on the nature of work is that people are working harder at a faster pace, not being able to take a pause, under the influence of increasing...
-
1. What is tension? 2. What does tension do to materials? 3. What is compression? 4. What does compression do to materials?
-
You decide to accept a quote from a store specializing in office remodeling. They offer to pay in quarterly installments due over 3 years, giving a footing of $600,000. of $600,000. The estimate as...
-
Corporate) taxes play an important role in making investment decisions. Taxes influence the incremental cash flows needed to assess the feasibility of an investment. Explain why and how taxes are...
-
Understanding how DRIPs work Under DRIP, the company gives any cash dividends that investors would have received to a bank, which acts as a trustee. The bank then uses the money to repurchase the...
-
An insurance company estimates that its customers have 2% probability of getting.Into an accident which will result in a medical bill of $52876. What is the actuarially fair premium for full...
-
Imagine that your best friend knows you are taking a psychology course and wonders what psychology is all about. How would you define psychology for your friend? Write an essay on the discipline of...
-
Show the optimal binary search tree for the following words, where the frequency of occurrence is in parentheses: a (0.18), and (0.19), I (0.23), it (0.21), or (0.19).
-
Two binary trees are similar if they are both empty or both nonempty and have similar left and right subtrees. Write a method to decide whether two binary trees are similar. What is the running time...
-
a. Prove that in a round robin tournament it is always possible to arrange the players in an order pi1 , pi2 , . . . , piN such that for all 1 j < N, pij has won the match against pij+1. b. Give an...
-
Describe how partial and circular reasoning can be helpful or harmful in resolving ethical dilemmas.
-
Describe the various legal risks for nurses.
-
Describe the various roles of advanced practice nurses.
Study smarter with the SolutionInn App