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Introduction to Random Walk
- 2. What is Random Walk? A drunk man will find his way home, but a drunk bird may get lost forever.
- 3. What is Random Walk? • Random walk is a process, a model or a rule to generate path sequence of random motion. • A random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers.
- 4. What is Random Walk? • Many natural phenomena can be modelled as random walk • the path traced by a molecule as it travels in a liquid or a gas, • the search path of a foraging animal, • the price of a fluctuating stock and • the financial status of a gambler • PageRank • Recommender Systems • Investment theory of stock market • Generate fractal images • Even though they may not be truly random in reality
- 5. Simulation of Normally Distributed Random Walk • We start with initial location 100 and generate the random walk based on normal probability distribution.
- 6. Simulation on Higher Dimensions • In higher dimensions, the set of randomly walked points has interesting geometric properties. In fact, one gets a discrete fractal.
- 8. Simple random walks on graphs Let G=(V, E) be a connected graph, |V|=n and |E|=m • Given an initial vertex 𝑣0, select “at random” an adjacent vertex 𝑣1, and move to this neighbour • Then select “at random” a neighbour 𝑣2 of 𝑣1, and move to it. • etc.
- 14. Simple random walks on graphs The sequence of vertices 𝑣0, 𝑣1, 𝑣2, … , 𝑣 𝑘, … selected in this way is a simple random walk on G. At each step 𝑘, we have a random variable 𝑋 𝑘 taking values on 𝑉. Hence, the random sequence 𝑋0, 𝑋1, 𝑋2, … , 𝑋 𝑘, … Is a discrete time stochastic process defined on the state space 𝑉.
- 15. Simple random walks on graphs What does “at random” mean? If at time 𝑘 we are at vertex 𝑖, choose uniformly an adjacent vertex 𝑗 to move to. Let d(i) denote the degree of vertex i. These transition probabilities do not depend on “time” 𝑘
- 16. Simple random walks on graphs
- 17. Random Walk and Markov chain Correspondence between terminology of random walks and Markov chains
- 18. Random Walk and Markov chain • The Markov property holds: conditional on the present, the future is independent of the past • The random sequence of vertices visited by the walk, 𝑋0, 𝑋1, 𝑋2, … , 𝑋 𝑘, … • Is a Markov chain with state space V and matrix of transition probabilities 𝑃 = 𝑝𝑖𝑗 𝑖, 𝑗 ∈ 𝑉
- 19. Random Walk and Markov chain • A Markov chain describes a stochastic process over a set of states according to a transition probability matrix • Markov chains are memoryless • Random walks correspond to Markov chains: • The set of states is the set of nodes in the graph. • The elements of the transition probability matrix are the probabilities to follow and edge from one node to another.
- 20. Random Walk and Markov chain Stochastic matrix
- 21. Random Walk and Markov chain 𝑗∈𝑉 𝑝𝑖𝑗 = 1 Let D be the diagonal matrix with 𝐷 𝑖𝑗 = 1/𝑑(𝑖) and A be the adjacency matrix of G. Then P = DA In particular , if G is d-regular, 𝑃 = 1 𝑑 𝐴
- 22. Random Walk and Markov chain Let 𝜌 𝑘 be the row vector giving the probability distribution of 𝑋 𝑘, 𝜌 𝑘 𝑖 = 𝑃 𝑋 𝑘 = 𝑖 , 𝑖 ∈ 𝑉 The rule of the walk is expressed by the simple equation 𝜌 𝑘+1 = 𝜌 𝑘 𝑃 That is, if 𝜌0 is the initial distribution from which the starting vertex 𝑣0 is drawn, 𝜌 𝑘 = 𝜌0 𝑃 𝑘, 𝑘 ≥ 0
- 23. Random Walk and Markov chain • Since G is connected, the random walk on G corresponds to an irreducible Markov chain. Irreducible: There is a path from every node to every other node. • The Perron-Frobenius theorem for nonnegative matrices implies the existence of a unique probability distribution π, which is a positive left eigenvector of P associated to its dominant eigenvalue λ = 1
- 24. Random Walk and Markov chain If the initial vertex of the walk is drawn from 𝜋 , then the probability distribution at time k is 𝜌 𝑘 = 𝜋𝑃 𝑘 = π Hence, for all time k >=0 , 𝑃 𝑋 𝑘 = 𝑖 = 𝜋 𝑖 , 𝑖 ∈ 𝑉 The random walk is a stationary stochastic process 𝜋 is the stationary distribution
- 25. Random Walk and Markov chain Does a stationary distribution always exist? Is it unique? Yes, if the graph is “well-behaved”. Irreducible and Aperiodic, a directed graph is said to be aperiodic if there is no integer k > 1 that divides the length of every cycle of the graph
- 26. Several concepts How fast will the random surfer approach this stationary distribution? Mixing Time! The access time or hitting time H(i,j) is the expected number of step before node j is visited, starting from node i. In general 𝐻 𝑖, 𝑗 ≠ 𝐻 𝑗, 𝑖 The commute time is 𝐻 𝑖, 𝑗 + 𝐻 𝑗, 𝑖 The cover time (starting from a given distribution) is the expected number of steps to reach every node.
- 27. Random Walk with Recommender Systems A B C D a b c d e
- 28. Random Walk with Recommender Systems A B C D a b c d e A to c
- 29. Random Walk with Recommender Systems A B C D a b c d e A to e A B C b c d e a D
- 30. Random Walk for Recommendation TrustWalker: A Random Walk Model for Combining Trust-based and Item-based Recommendation
- 32. Random Walk algorithms A subset of Vc of a set genes V have “a prori” known property C Can we rank the other genes in the set VVc w.r.t their likelihood to belong to Vc?
- 33. Random Walk in DeepWalk DeepWalk, learning latent representations of vertices in a network
- 34. Random Walk in DeepWalk DeepWalk, learning latent representations of vertices in a network
- 35. Reference • http://people.revoledu.com/kardi/tutorial/StochasticProcess/Rando mWalk/index.html • http://www.mit.edu/~kardar/teaching/projects/chemotaxis(AndreaS chmidt)/random.htm • https://www.math.uchicago.edu/~lawler/srwbook.pdf • https://www.cs.cmu.edu/~avrim/598/chap5only.pdf • http://www.lirmm.fr/~sau/JCALM/Josep.pdf • http://homes.di.unimi.it/valentini/MB201213/slide/RandomWalksGr aphs.pdf • https://arxiv.org/pdf/1403.6652.pdf
- 36. Weekly Report • Released the DeepRec • Revised the sequential recommendation paper • I implemented a similar idea for rating prediction-> does not work • Prepare for the learning group Next Week: • Some review work • Revision
Editor's Notes
- #4: can all be approximated by random walk models, even though they may not be truly random in reality.
- #5: can all be approximated by random walk models, even though they may not be truly random in reality.
- #22: a regular graph is a graph where each vertex has the same number of neighbors
- #23: a regular graph is a graph where each vertex has the same number of neighbors
- #24: a regular graph is a graph where each vertex has the same number of neighbors
- #25: a regular graph is a graph where each vertex has the same number of neighbors
- #26: a regular graph is a graph where each vertex has the same number of neighbors
- #27: a regular graph is a graph where each vertex has the same number of neighbors
- #31: a regular graph is a graph where each vertex has the same number of neighbors
- #33: a regular graph is a graph where each vertex has the same number of neighbors
- #34: a regular graph is a graph where each vertex has the same number of neighbors
- #35: a regular graph is a graph where each vertex has the same number of neighbors