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- In the Erdös-Rényi random network model, suppose N=101 and p=1/20, that is, there are 101 vertices, and every pair of vertices has a probability of 1/20 of being connected by an edge. For the network model given what is the probability that a network generated with those parameters has exactly 400 edges? No need to give the decimal value, the mathematical expression will sufficea. Build an adjacency matrix ? for this map. b. How many paths of length 2 from V5 to V1 exist? c. How many paths of length 3 from V5 to V1 exist?Consider the Omega network and Butterfly network from p nodes in the leftmost column to p nodes in the rightmost column for some p=2^k. The Omega network is defined in Chapter 2 of the text book such that Si is connected to element S j if j=2i for or j=2i+1-p for See Chapter 2 in text book for its definition. The Butterfly network is an interconnection network composed of log p levels (as the omega network). In a Butterfly network, each switching node i at a level l is connected to the identically numbered element at level l + 1 and to a switching node whose number differs from itself only at the lth most significant bit. Therefore, switching node Si is connected to element S j at level l if j = i or j . Prove that for each node Si in the leftmost column and a node Sj in the rightmost column, there is a path from Si to Sj in the Omega network. Prove that for each node Si in the leftmost and a node Sj in the rightmost, there is a path from Si to Sj in the Butterfly network.
- Consider the following network: (a) Which of the following most accurately describes the connectedness of this network? Strongly Weakly Disconnected None of the above (b) When discussing path lengths on a weighted graph, one must first define how the weights are related to the length of a path between two nodes is then the sum of the distances of the links in that path. Consider the previous network and assume that the link weights represent distances. Using this distance metric, what is the shortest path between nodes 1 and 6? (c) A common way to define the distance between two nodes is the inverse (or reciprocal) of the link weight. Consider the previous network and assume that the distance between two adjacent nodes is defined as the reciprocal of the link weight. Using this distance metric, what is the shortest path between node 1 and 6?What are the three prerequisites for a network to function optimally? Let's break them down and examine them separately.Let A, B, C, D be the vertices of a square with side length 100. If we want to create a minimum-weight spanning tree to connect these four vertices, clearly this spanning tree would have total weight 300 (e.g. we can connect AB, BC, and CD). But what if we are able to add extra vertices inside the square, and use these additional vertices in constructing our spanning tree? Would the minimum-weight spanning tree have total weight less than 300? And if so, where should these additional vertices be placed to minimize the total weight? Let G be a graph with the vertices A, B, C, D, and possibly one or more additional vertices that can be placed anywhere you want on the (two-dimensional) plane containing the four vertices of the square. Determine the smallest total weight for the minimum-weight spanning tree of G. Round your answer to the nearest integer. Note: I encourage you to add n additional points (for n=1, 2, 3) to your graph and see if you can figure out where these point(s) need to…
- Let A, B, C, D be the vertices of a square with side length 100. If we want to create a minimum-weight spanning tree to connect these four vertices, clearly this spanning tree would have total weight 300 (e.g. we can connect AB, BC, and CD). But what if we are able to add extra vertices inside the square, and use these additional vertices in constructing our spanning tree? Would the minimum-weight spanning tree have total weight less than 300? And if so, where should these additional vertices be placed to minimize the total weight? Let G be a graph with the vertices A, B, C, D, and possibly one or more additional vertices that can be placed anywhere you want on the (two-dimensional) plane containing the four vertices of the square. Determine the smallest total weight for the minimum-weight spanning tree of G. Round your answer to the nearest integer. Attention: Please don't just copy these two following answers, which are not correct at all. Thank you.…Consider the following flow network. The figure describes a flow ? and the capacity of the edges: if (?, ?) appears next to an edge ?, then the capacity of the edge ?ₑ is ?, and the flow ?ₑ that goes through ? in ? is ?. For example, if ? = (?, ?₁), then ?ₑ = 2 and ?ₑ = 1. 1. Draw the residual network ? & of the above flow ?. [Draw a graph containing all the nodes, edges, and the values on the edges]. 2. Find an augmenting path that will increase the flow by 1. You only need to list the vertices in the path and indicate the resulting flow in the following figure (using the same notation as the above figure). 3. Find a minimum ?-? cut in the graph (where the weight of an edge is its capacity). Briefly justify why the cut you found is a minimum cut.Consider the network in the following figure. Edges that are not pictured have a length of ∞. Image attatched (a) What is the optimal TSP tour on this network, and what is its total length, z*? What is the optimal 1-tree Tˆ* on this network (“rooted” at node 1), and what is its total length, z(Tˆ*)? (b) Find the best Held-Karp lower bound you can. That is, find weights to add to one or more nodes so that z* and the right-hand side of the following formula are as close as possible. z* ≥ z(Tˆ*) + SUM{i∈N} λi(di(Tˆ*) − 2) Image attatched Here Tˆ* is the optimal 1-tree under the revised distance matrix, z(Tˆ*) is its cost and di(Tˆ*) is the degree of node i in 1-tree Tˆ*.
- Show the final flow that the Ford-Fulkerson Algorithm finds for this network, given that it proceeds to completion from the flow rates you have given in your answer to part, and augments flow along the edges (?,?1,?3,?) and (?,?2,?5,?). Identify a cut of the network that has a cut capacity equal to the maximum flow of the network.The details are in the attached image Your solution for both Uniform-Cost-Search and A* Search should include the following:- The status of the fringe (queue) at each step (i.e. show at each step the orderednodes and their costs in the queue). Provide the search tree for your solution, showing the order in which the nodes wereexpanded and the cost at each node. What is the shortest route found and its pathcost?Consider an unstructured overlay network in which each node randomly chooses c neighbors. If P and Q are both neighbors of R, what is the probability that they are also neighbors of each other?