Math 366 Problem Set 10 Due Thursday, March 19, 2015 ✓ Solved

Math 366 Problem Set 10due Thursday March 19 20151 Let G Be The

Let G be the graph shown below, which is the cycle graph C7 with an added chord. Consider a random walk on G in which the probability pij of moving from vertex i to vertex j is given by pij = 1 di where di is the degree of vertex i. (a) Give the transition matrix P for this random walk. (b) Give the 2-step transition matrix P2. (c) What is the probability of being in state 6 after 2 steps, if we start in state 6? (d) What is the smallest value of n for which all entries of Pn are positive? How would you interpret the significance of this value for n? (e) For increasingly large values of n, at approximately what value of n do all rows of the matrix Pn first appear equal (to three decimal places)? What is the significance of this, and what are the individual probabilities of being in each state? 2. Consider the transition matrix P =

[a) Draw a planar graph consistent with P, including bidirectional edges with transition probabilities shown as edge weights. (b) What three-dimensional geometric figure could this graph correspond to? (c) Now make the edges one-way only by allowing the move i → j if and only if i

Sample Paper For Above instruction

Introduction

This paper discusses the properties of a random walk on a specific graph, particularly a cycle graph C7 with an added chord, and examines the transition matrix associated with this Markov process. It further explores the concepts of matrix powers, state probabilities, and the transition matrix's convergence to equilibrium. The second part analyzes a different transition matrix P, its graphical representation, and possible geometric interpretations, including modifications to make the graph directed and upper triangular. The goal is to understand the probabilistic and structural properties of Markov chains and their graphical representations.

Part 1: Random Walk on the Cycle Graph C7 with an Added Chord

Transition Matrix P

The graph G consists of 7 vertices arranged in a cycle with an additional chord connecting two non-adjacent vertices. Each vertex's degree, di, determines the transition probabilities, where pij = 1/di. The degrees are calculated based on the number of edges incident to each vertex, considering the cycle and the chord. For example, vertices on the cycle have degree 2, while the vertices connected by the chord may have degree 3.

Constructing the transition matrix P involves setting each row corresponding to a vertex, with probabilities distributed equally among its neighbors. The matrix P is thus a 7x7 matrix, with entries of 1/di for adjacent vertices and zeros elsewhere. For example, if vertex 1 is connected to vertices 2, 7, and 4 (because of the chord), its row in P would assign probabilities 1/3 to each of these vertices.

Two-step Transition Matrix P²

The matrix P^2 is obtained by matrix multiplication of P with itself, representing the probabilities of transitioning between vertices in two steps. It provides insights into the likelihood of reaching certain vertices after two moves, considering all possible paths.

Probability of Being in State 6 After Two Steps

Assuming the process starts at vertex 6, the probability of being in state 6 after two steps can be computed by examining the sixth row of P^2. This involves summing the probabilities of all paths of length 2 starting from 6 and ending at 6, which accounts for repeated transitions and path dependencies inherent in Markov chains.

Smallest n for All Entries of Pⁿ to be Positive

The matrix P^n becomes strictly positive when the Markov chain is irreducible and aperiodic, ensuring the existence of a unique stationary distribution. The smallest such n is known as the mixing time, which signifies the number of steps needed for the Markov process to reach a state close to equilibrium, regardless of the initial state.

Convergence of Rows in Pⁿ

As n increases, all rows of P^n tend to converge to the same distribution—the stationary distribution. In practice, this convergence occurs at a finite n, which can be approximated by examining the rows for equality within three decimal places. This indicates the chain's stabilization and loss of memory of initial states.

Part 2: Transition Matrix P and Its Geometric Interpretations

Graphical Representation of P

The matrix P, defined explicitly with transition probabilities, can be represented as a planar graph with nodes and weighted edges. Each node corresponds to a state, and edges between nodes are directed and weighted according to transition probabilities. Drawing this graph emphasizes the structure of transitions, whether bidirectional or unidirectional.

Geometric Figure Correspondence

The structure of the transition graph can be associated with certain geometric objects based on the connectivity and symmetry. For instance, a tetrahedral or pyramidal shape may correspond to a graph with hierarchical or layered transitions, reflecting the geometric nature of the probabilities and transitions in three dimensions.

Directed Graph with i

Adjusting the graph to allow transitions only from state i to j with i

Graph and Matrix Redraw

Redrawing the graph involves removing all edges where i ≥ j and recalculating transition probabilities proportionally among remaining edges. The matrix P would then be restructured as an upper triangular matrix, with zeros below the main diagonal, representing a unidirectional, hierarchical transition process.

Conclusion

This analysis highlights fundamental concepts in Markov chain theory, including transition matrices, chain convergence, and geometric interpretations. The process of modifying the transition structure emphasizes the importance of matrix consistency and the effects of directionality on the chain's behavior. These insights are crucial in applications ranging from stochastic modeling to probabilistic graphical models.

References

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• Durrett, R. (2010). Probability: Theory and Examples. Cambridge University Press.
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