2. Let P be a k k stochastic matrix. Show that ₁ P = 1 for all 1 ≤ i ≤ k. Here j=1 Pr denotes the entry in the i-th row and j-th column of the n-th power of matrix P.
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- 12. A stochastic matrix Pis called doubly stochastic if Σi Pij = 1 for all j. It is called sub-stochastic if Ei Pij ≤ 1 for all j. Show that, if P is stochastic (respectively, doubly stochastic, sub-stochastic), then Pn is stochastic (respectively, doubly stochastic, sub-stochastic) for all n.Consider the following model to grow simple networks. At time t = 1 we start with a complete network with no = 6 nodes. At each time step t> 1 a new node is added to the network. The node arrives together with m = 2 new links, which are connected to m = 2 different nodes already present in the network. The probability II, that a new link is connected to node i is: N(t-1) II₁ = ki - 1 Z == with Z = (k; -1) j=1 where k; is the degree of node i, and N(t - 1) is the number of nodes in the network at timet - 1. (e) Write down the master equation of the model, i.e. the equation that describes the evolution of the average number N(t) of nodes that at time t have degree k.If Q is 3 x 2 matrix with orthonormal 3 then 4 columns and x = a. ||Qx|| = 1 b. ||Qx|| = 25 c. ||Qx|| = 0 С. d. ||Qx|| = 5 %3D
- Consider the following model to grow simple networks. At time t = 1 we start with a complete network with no = 6 nodes. At each time step t > 1 a new node is added to the network. The node arrives together with m = 2 new links, which are connected to m = 2 different nodes already present in the network. The probability II, that a new link is connected to node i is: N(t-1) II¿ = ki - 1 Ꮓ with Z=(k-1) j=1 where ki is the degree of node i, and N(t - 1) is the number of nodes in the network at timet - 1.Consider the following model to grow simple networks. At time t = 1 we start with a complete network with no = 6 nodes. At each time step > 1 a new node is added to the network. The node arrives together with m = 2 new links, which are connected to m = 2 different nodes already present in the network. The probability II, that a new link is connected to node i is: N(t-1) II¿ = ki - 1 Ꮓ = with Z(k-1) j=1 where k; is the degree of node i, and N(t -1) is the number of nodes in the network at time t-1. (c) Write down the differential equation governing the time evolution of the degree ki of node i fort >>1 in the mean-field approximation. Solve this equation with the initial condition ki(t) = m, where t; is the time of arrival of node i.Consider the following model to grow simple networks. At time t = 1 we start with a complete network with no = 6 nodes. At each time step t> 1 a new node is added to the network. The node arrives together with m = 2 new links, which are connected to m = 2 different nodes already present in the network. The probability II, that a new link is connected to node i is: ki II¿ = Z - 1 N(t-1) with Z(k-1) - j=1 where k, is the degree of node i, and N(t 1) is the number of nodes in the network at time t-1. (b) What is the average node degree (k) at time t? What is the average node degree in the limit t→ ∞o?
- If x(t) is ensemble member of an input ran- dom process X(t) and Y(t) is the ensemble member of an output random process of the LTI system, obtain the relationships for y(t) and Y(t).Given a 1000-by-1000 matrix Y, let I Y A = (6 ?). I What is the maximum number of steps that GMRES would require to converge? (Please show steps and explain.)= Consider the following model to grow simple networks. At time t = 1 we start with a complete network with no 6 nodes. At each time step t> 1 a new node is added to the network. The node arrives together with m = 2 new links, which are connected to m = 2 different nodes already present in the network. The probability II; that a new link is connected to node i is: ki – 1 II¿ Z N(t-1) with Z = (k; - 1) j=1 where k¿ is the degree of node i, and N(t − 1) is the number of nodes in the network at time t - 1. (a) Find an expression for the number of nodes, N(t), and the number of links, L(t), in the network as a function of time t. Find an expression for the value of Z as a function of time t. (b) What is the average node degree (k) at time t? What is the average node degree in the limit t→ ∞? (c) Write down the differential equation governing the time evolution of the degree ki of node i for t≫ 1 in the mean-field approximation. Solve this equation with the initial condition k¿(t;) = m,…
- Find the sıze of the matrix 0. 7.Let A € Rmxn, B € Rmxp be any matrix. Solve the following problem _min_ ||AX – B||2/17. XERnxp Hint: Write X = [X₁,..., Xp] and B = [b₁,..., bp]. Then AX – B = [Ax₁ – b₁,..., Axp – bp]. Use this to reduce the prob- lem accordingly.Q9. As usual, let In denote the n x n identity matrix. 1 (a) Find a 2 x 2 matrix such that A? = –I2 -1 (b) Show there is no 3 × 3 matrix such that A² = –I3. |