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In this problem we '11 explore one f the applications of eigenvalues and eigenvectors google PageR- ank algorithm: Consider set of webpages hyperlinked by th...

Question

In this problem we '11 explore one f the applications of eigenvalues and eigenvectors google PageR- ank algorithm: Consider set of webpages hyperlinked by the given directed graphs in each diagrams(a)(6)For each diagram:Determine the corresponding transition mnatrix AsDetermine the google matrix G_ Does either graph have dangling node? If so which one(s)? Does either graph have disconnected node? If s0 which one(s)?Compute the PageRank of cach page in the set (Recall that the PageRank vec

In this problem we '11 explore one f the applications of eigenvalues and eigenvectors google PageR- ank algorithm: Consider set of webpages hyperlinked by the given directed graphs in each diagrams (a) (6) For each diagram: Determine the corresponding transition mnatrix As Determine the google matrix G_ Does either graph have dangling node? If so which one(s)? Does either graph have disconnected node? If s0 which one(s)? Compute the PageRank of cach page in the set (Recall that the PageRank vector for a web graph with transition matrix A. and damping factor p, is the unique probabilistic eigenvector of the matrix G, corresponding to the eigenvalue 1.)



Answers

Find the adjacency matrix of the graph or digraph.

Hello there were discussing problem number 13. Please don't mind this. I think I have a different edition of the book from where Called this question Discussing problem 13 off. Advanced answering mathematics, ninth edition, Chapter 23 Section one. So what this question asks us to do is that it answers to create an adjacency matrix for this graph. Now, what else? An adjacency matrix. It's a metrics for graph G. Where each entry A I G. Where I represents their own number and she represents the column number is assigned a value of one. If there exists an edge between the nodes I n Jane, the graph on zero. Otherwise, what does that mean? Let's have a look. So if you see the node one and the graph, their existence edge between one and four. One in three on one and two named e four e do and even respectively. So what we do, we take the first room That is equals one Onda. We check for all the values of J If there exists any No sorry. There exists any edge between that value I on that. Will you g So if j is four there it does exist. Er, there does exist an edge. So that would mean we fell a 14 with the value off one next, 12 on 13 So that would mean true. The second column, third column and the fourth column off First two are filled one and there does not exist an edge between one and itself, so that would mean you fill it with a zero. Similarly, we could do this for all the other three notes toe went for, as we can see, do not have any edge emanating from them. So that would mean we fill all the columns off the second row with zero and all the columns of the four through with seals finally looking at No. Three, there does exist one edge emanating from it leading towards one. So that would mean the third row first column would be filled with one. Well, there does not exist any other note any other edge. I'm sorry from three. So that would mean all the other places are filled with zero eso. That's it. This one's the metrics for this graph.

Do that. So we are given this graph on we have to draw it. Adjacency metrics. So what does an adjacency matrix mean? It means that will draw a metrics A corresponding to this graph. Where in each entry A i g. Off the graph where I represents the his own numbers and she represents the column numbers on a I G would be one If there exists an edge between the nodes I NJ that's starting from I and ending at G Andi, it will be zero after does not exist in age between I and change. So let's try to make metrics for the given graph. We see that the notes and the craft are numbered from 1 to 4. So that means I will be waiting from 1 to 4. And so will J. All right, so there exist. Are there emanate three edges from one, namely the food e to and e one. That means a 14 a 12 on a 13 would be filled with a one or the first truth. 2nd, 3rd and 4th columns would have won as their values, but there does not exist an edge starting from one point into itself. So that would mean we would fill it with a zero. Similarly, if you look at notes toe in Ford, there are no edges emanating from them. So it would fill all these columns with zero in case of row two. Andrew full finally looking at North three. There does exist in it starting from node three, ending at mood one, which is e three. That would mean a road three column one would have a one and all the other problems would have zero. Hence, Demetrick six, which is enclosed in this green figure, is the adjacency matrix off this craft. Yeah, Thank you.

Were given special graphs were asked to find an adjacency matrix For each of these graphs. Part a special graph is K n complete graph with Inverted sees So first noticed that this graph has in verse Eze, which I'll call. And we also have that this implies that the adjacency matrix it wouldn't be an n by N. Matrix. And moreover, we have that every vertex is connected to every I guess I should say, adjacent to every other Vertex. So it follows that the degree ever Text V is going to be an minus one and recall from a previous problem we have that he some off the rose. This is going to be the degree. I guess the sum of Roe I is that degree of be I minus the number of loops at the I. Now there are no loose at the I in this graph. This is a simple graph and so this is simply going to be equal to the degree of the I, which is and minus one. So we have this. Some of each row is going to be and minus one, and we have that the eye is not adjacent to the I main diagonal is just zeros and every other entry is going to be ones. So we obtained any tricks that looks like this where we have all zeros down the diagonal and one's everywhere else. In part B, we're given the graph C n. This is the cycle on Enver disease, Number of Vergis eases n and we have that the degree of each Vertex is going to be to and finally have that new loops are allowed. So by a previous problem, we have that the some of the roads. Well, instead of thinking of it that way, you have that the eye is adjacent to V J. If and only if I is equal to J plus or minus one mod. And so I draw the adjacency matrix have save you one b two all the way down to VN and V one V two already over to be in. And we have that for one breach. V. I have you guys not adjacent to itself. So we're going to have Ciro's down the mean diagonal and we have that he wants adjacent to be to likewise this is a symmetric matrix since its undirected it is also adjacent to view one. We have that the one is not adjacent to be three. We do have that V two of the Jasons to be three. And so we get sort of this checkerboard pattern of zeroes and ones. And so the form of the Matrix is actually going to depend on the value of N Well, no. So you know that the one is going to be Jason to the end. This will have ones in these corners and again there zeros down the main diagonal. Now, in part C, we're given the graph of u. N This is the wheel with, I guess cycle of Enver theses says the wheel on n plus one burgesses Technically and we have that overseas are V one through the end the n plus one lawyer the degree ah fi I is equal to Well, this is going to be three. Yes, I lies between one and and And in the case of the n plus one, the degree is going to be. And so the rules for this are yes, I allies between one end. Let me have that V I is adjacent to B J even only Yes, I is equal to J plus or minus one mod and or we have that J is equal to and plus one. So it follows that this craft is going to be very similar to the graph for the cycle CN except is going to end but plus one by n plus one graph and noticed that we have the first and entries of the first and rose each are the same as from the previous part we had that the main diagonal was still going to be all zeros. And for the last row and last column, yeah, this is gonna be adjacent to the Empress, warns Jason's to every Vertex except itself. Is this going to be a row of ones with zero at the end and a column of ones with a zero at the end and finally well and were given the graph k and then this is the back part. Tight complete graph with two Vertex since the one with Enver theses in V two with Enver Theses said the tumor reverse uses M plus n and we have this well, it'll liver disease. The one how long he to to the M and then also the M plus one all the way up to the M plus in we have that Vertex. The eye is adjacent to Vertex T J. If and only if we have that I lies between one m and J lies between M plus one and M plus n, or vice versa. And so it follows that adjacency matrix is going to be an M plus end by M plus n Matrix. It's gonna be a little bit hard to see, but I'll do my best to draw it. So we have that by our rules. Text one is adjacent to courtesies the and plus one through V M Plus in You Have That projects to was chasing to the M plus one through VN plus in and so on until we get to begin with the end minus one. This is going to be Jason TVM, plus warm through the M Plus n and finally, a VM begin. Half the VM is adjacent to V M, +13 VN plus in the M plus ones adjacent to view +13 PM The M plus and minus one is Jason to be one through the end and the M plus End is adjacent to the 1:30 p.m. Mrs Are adjacency matrix. Basically, it's the Matrix with for Sub Mitrice's. So we have is your matrix matrix called A and Matrix being matrix zero where a is an all one matrix. The B isn't all one matrix and in part E we're giving the graph Que en This is the end Hyper cube. Now, to find expression for the adjacency matrix is gonna be a little bit complicated, but I noticed that Q one is simply the line segment. Then points 01 And we had that for Kyu won. The adjacency matrix is there for simply 0110 and we have that for Q. Then suppose that Cure Ben is equal to you should do Que Tu as well. So we have the q of two. Well, now we have for possible bit strings to obtain this matrix. Thank you too. And recognize that this is the same as Q one and then I to I two and Cuban again. So I'm going to put forth that in general que en plus one has adjacency matrix, which is that makes him space here. Que en i n i and que en for n greater than or equal to zero where one, I should say. And then, of course, we have Q one up here.

In this video, we're gonna go through the answers. Question number 35 from chapter 9.5. So, in part, A were asked to find Ah, what's to show that the matrix A here has repeated I value our equals two minds Well, on dhe show that all the time it is. That's the form given in the question. Okay, so I start with the wagon values. So we need Thio. Find the determinant off the matrix A minus. Aw, times dead and see matrix. That's this. Determine in here. That's gonna be well minus ah times by minus three minus. Uh, because for that's gonna be able to u R squared. Ah, hoofs every, uh, minus. Ah, sets plus thio minus three plus four plus one. Okay, so this is easily fact arised as ah course one squared. If that's equal to zero, then we have repeated Eigen value eyes equal to minus one. Okay, cool. So now it's fine. The Eiken vector associating with that. So a plus the identity matrix is gonna be too two minus one for minus three plus one is minus two okay, times by infected, you go on is equal to zero So therefore, any item vector must be proportional to Okay, let's see if we let the first component be one second component. Must be two cases going breaks that fulfills. Ah, well, the quiet for a movements part B. Uh, so this is quite easy. So the an intravenous solution can be Rin as eat first I about you East the first time value, which was minus one times t times by dragon Vector 12 And it's very easy to check with us. Solution off the system, given you a question. Okay, But see, so I guess a little bit more tricky here for us to find a second mini independent solution. Eso we have only we had a repeat. I can value said can't just use the second night value because the repeated I could buy you only had one linearly independent. Uh, Ivan Vector. So it's Astra's use. The form next to is equal to t eat. The minus t does buy you one plus even my honesty times by you too. Okay, so we're gonna substitute that in. We're gonna need to find the first derivative using the what a fool on the first term is gonna be one minus t times e to the minus t Because it's the modesty. Different shades to minus eats. The modesty does buy you want because it's just a constant vector. Minus it reminds t you too, it Okay, so, saying, Ah, you two prime people to eight times you, too. So meaty. Prime waas one minus t Because by e to the minus t you want minus eight minus t you too. And then a times You too. Uh, well, that's just gonna be tee times e to the minus t times by eight times by Yeah, yes, I was by a you one. So say you one because you want is an aiken defector with Aiken value minus one and a you want he's gonna be equal to B minus one times You want. Just by the definition off, you won't be Knight director of the Matrix egg. Uh, we got close E to the minus t times bite matrix. A YouTube look. Okay, so let's have a look. What's going on here then? So this minus t eat my honesty? You want concussive with this modesty here? And then what we're left with is ah, eats the minus t You want minus you too equals eats the minus t a You too. So eat the virus Taken Never be zeros. We can cancel that. So this is gonna allow us to arrive at a plus. The identity matrix I times by e t equals you are. Okay, so now we console that. Let's have a look. At what a plus. I Yes, that's gonna be the matrix to minus one for minus two times by you, too. He calls when you want. Was the vector one too. So this means that you too. Okay. So if the first component is one, then we're gonna have second opponent is gonna be two times one minus one, which is just what? So that's, uh, a solution for you, too. Okay. Now, huh? Day as us girls just find what a close I Yeah, like squared times. The YouTube is okay. So this is just a plus I times by a close eye. That's just the definition of squaring. Something just most quiet by. You must buy twice by that by that thing. Okay, So then hey, they were just showing that April's I times you two is You want. It's that in this Caribbean is a close eye you want. But it was high time too. You want because that you wanted a infective. Ah, metrics. I with Ivan value minus one, that is just gonna be equal to zero. And that completes the


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