 Thank you very much for the invitation to this very nice place here, so I will try to give a kind of Yeah, quite elementary introduction to Google Mattress, not too elementary and especially about Spectre properties and Say after some yeah quite long introductory part I will speak about Concepts of reduced Google Mattress which takes it's not the main top Only topic of this talk, but it takes an important part this consults of work to learn in collaboration with Katja and Seedema And here, Katja will give a talk right and the first talk after the lunch now Here you see simple two Examples of the complex eigenvalue spectrum of two types of Google Mattress associated to the ISA Wikipedia Network from 2009 or the physical review citation network up to 2012 the red dots are what could we call? Corespace eigenvalues that means a fraction of them is a fraction which we could compute compute No, and the blue dots here are what called subspace eigenvalues, which are easier to come into the green one That's special case because here we used the quite new complicated method for computation Requiring high precision computations because of the matrix, which is numerically very tough But this is not the main topic of my talk, but I can give some some pride discussion I can give details So what is the main object we look at it's a kind of parent for beginners operators Imagine you have some discrete Markov process some some system the states with probabilities pj the time dependent on Dependent on the discrete time and some linear evolution where you have transition probabilities and of course this normalization of these probabilities gives you the column normalization of this matrix all problems be positive and This property implies that of course is a Vector of probabilities normalized it stays normalized over time. That's okay. It's important In general this matrix needs not to be symmetric in our case. That's important We therefore have complex eigenvalues and this property here allows to show that all Eigenvalues are inside the unit circle as they all all all on the border now We have one trivial left eigenvector, which is this unit eigenvector all entries one if you take that equation here Simply e times e transpose g is equal e transpose therefore you have one eigenvalue one And then you need also the as you said right eigenvector, which is less trivial which we call a page Frank In the context of Google matrices There's at least one such eigenvector. There may be many it may be degenerate which is quite complicated But in case if it's not degenerate and if you have a finite gap between the second and first eigenvalue You can show you start with an arbitrary initial distribution of probabilities You iterate it in time long enough you will converge to the page Frank No, and the rate of conversion goes like the second eigenvalue power t. It's an exponential Converter it depends how much smaller the lambda 2 is Compared to one that's a point. It may be small and you Must not have any degeneracy for this to work. No, this is called the power method It's an efficient way to compute a page Frank So now if you have a directed network, you know you construct the adjacency matrix Similar in a way, but now for the directed network We put a link if you have a link k2g You we put the matrix and a gk equal one. Otherwise you put it zero, but now the matrix is not symmetric. It's important We normalize then all Non-zero qualms of 8 1 we divided by their sum. This gives the matrix We call s 0. However, there may be empty qualms, which corresponds to what we call dangling nodes in that case replace The full column by one over n numbers as he is a vector containing unit numbers No, and this matrix we call s no This matrix s has already the properties of the Peron for Venus operator But at this stage the unit eigenvalue may be degenerate actually if you take the university networks for the web page Then we have may have degeneracy of three 3,000 times because of many invariant subspaces also for Wikipedia you may have degeneracies and therefore in order to obtain Unique page rank, but also for other reasons one applies typically a damping factor Taking this method one applies this factor alpha, which is smaller than one but rather close to one typically with 0.85 and Then we add some contribution of a projector here. This simply projector Which projects on these unit vector this matrix has the same symmetries as the s is the Peron for Venus operator but here One we keep one eigenvalue one and all other eigenvalues are scaled down with a factor alpha therefore, you have a gap and there's nice convergence and This lambda month here and after there's application of the damping factor The first n is no longer degenerate and the page rank algorithm to compute the leading page rank for this vector is Converges quite well. It is also possible to apply the same procedure to the inverted network Where you replace the adjacency matrix by its transpose call it a star this gives a matrix s star g star But here on the level of s this is no longer the transpose because we applied some normalization to the columns here This is called chairing and of course here green page is a famous paper where they introduced This expression essentially no and the idea Yes, some simple example know of a directed network five knots and second number of links adjacency matrix essentially the links they go from a call if you take a position unit column number to row number for example the one here is from Column one to row two now or Here's a this One for example from column two to row three et cetera et cetera Then you normalize on non empty column it gives you one over two one over three only the last column is empty The dangling node therefore we have to insert one over five here network sizes five. This gives the matrix s simple example and now here first application for university networks that That means the web the web pages of the University of Cambridge and Oxford from 2006 And we take of course all links which go outside so what's the only links inside the web page are kept no Links going out to other websites, of course move here. We have computed page rank and also chi rank for both cases sorry this Damping the usual damping factor now and it's a log log representation and in the horizontal axis We have the K rank that means after computation of the parent page hunk which has positive entries We order the nodes according to page hunk ordering. This gives what we call a K rank on for the case star rank for For the chair and case now therefore it's often monotonous code with the case and we see quite well There's a kind of typical power law localization with some Exponent close to two minus one no and this is I written here this eigenvalue formula It shows that the page rank at of some node i represents the importance of this nodes obtained of the sum of all other pages Pointing it but with the weight of their page hunk It's not only the sum of other links, but some times their own importance therefore You have a self-consisting equation. That's what makes a big difference between Google and Before that other search engines. That's the important thing and of course in search engines Problem is not finding matches when you search for a word or something It's but to order all the results in a reasonable order. That's the basic idea of Google Now Mention some numerical methods we used to study of course is this power method to compute page rank It's converges like well alpha 2. There's also a part from K and K star rank 2d rank which a combination of both obtained when we draw K and K star in Plane then we start with increasing squares from below each time Not enter the square. It's it's the next one in the 2d ranking. It's a combination of this The eigenvalues are complex now We can fully diagonalize matches say for size up to 10 power 4. It's not that big for typical Networks, but then we can also use an all the method to compute say 100 up to 10,000 eigenvalues largest eigenvalues for much bigger networks, you know And actually even but before we apply the anomaly method one has to take out what I call Invariant subspaces or network people call buckets You can determine in advance before the organization the buckets and this gives this Triangular this block structure here and this s s double s index for subspace Contains itself diagonal many diagonal block entries for each packet You can diagonalize them separately mostly exactly and that's has a advantage We take care of these degeneracies of the unit eigenvalue No, you can count it even and there's no problem numerical problem here And then we remain with a core space contribution which has actually typically leading eigenvalue below one because their Contribution from the core space core space nodes may enter the subspace nodes But not the other way around therefore the core space part. It says the same eigenvalues, but the leading eigenvalue is typically below one, but quite close to one no on main mention and certain for certain networks, especially Citation network of physical review or recently we looked also to Bitcoin network We have either a triangle and an extra structure. That means before we said if The fact that it's only approximately triangle because we have the dangling nodes You have some empty colliems said dangling nodes destroy the triangular structure But without them we would have a new potent matrix which has only zero eigenvalues But because of the dangling nodes it becomes Metals with the proper symmetries. However, they're not a lot. You're done blocks Which are numerically nightmare the poison that means even if you compute say for 10 minus 16 Double precision numbers you may have errors of all the points one Numerical errors and there are some tricks to take on some analytical analytical methods or high precision computations with much much more precision This was in particular important for the physical review citation Okay We applied it to these methods to the University of X that works as an example of Cambridge. I showed Oxford were 200,000 nodes and 2 million Links for a degree a typical average degree of 10 out in degree There's also a Linux kernel network. It's a nice directed record. We need to take the kernel functions as not Function typically may be called by other kernel functions No, and this gives you a directed network and functions may also call in Loops one function a may call be an PCA. So that's not a purely triangular network, but quite close It's also Wikipedia. Of course, this is I think four million nodes for English Wikipedia of 2013 and roughly 10 power Approximately 10 power It links if you take other language editions you have a little bit smaller networks Also, we'll trade network from the UN the data from 10 from the UN So you have a small now only 10 power 4 nodes But you have a more complicated structure because a mixture of countries and goods. It's about more correct more complicated problem so analyzed by Leo and Dima, no, we also studied we got data of Twitter 2009 At that time Twitter was still say modest in size There's 40 million nodes in 1.5 billion links But this was of course only anonymous data without any names and so on we had only Two columns no number link from to that was it nothing more Of course physical citation if you net it's only half a million papers up to 2012 and 5 yeah, maybe in 5 million links But okay, this is only citation links from inside for the gravity, you know each paper in our time sets much many more papers so So here I come back to Wikipedia. This is the old of Wikipedia 2009 You have your spectra for the also and also more recent Cambridge 2011 which is now bigger 2006 you see a red a coarse base. I knew blue subspace eigenvalues which are computed go much There it's easy as we can compute them essentially exactly without any problems at least most of them Here's the same as chair and here you have a big amount of subspace eigenvalues and here you see many subsets I can close of course in 2006 was even more extreme. We had blue dots on the full circle. It means We had so many complex eigenvalues on the circle itself that it was a could not even see in the green line here We have also computed some eigenvectors, of course a page rank at damping factor point eight five It's a black curves. We have also computed for some cases page hunk was a very small damping factor Say here one minus ten minus eight, which is numerically very very tricky required a particular Algorithm so here's a power mess a simple power method does not work It was a combination of power method and an all the methods switched one to the other two And then here we have also some cost base eigenvectors. Ah, sorry wanted to Here we have to the first red and green as of top Cospers and then blue and pink are two course by vectors with large imaginary partner angle Which I actually quite strongly localized if you say pink here test some big steps is a log log representation and the same also blue one here and That shows that these are vectors where you have maybe hundred or thousand knots, which are much more important than the other nuts And actually for many of these eigenvectors you can identify themes and Topics that means here I show again a part of the spectrum of Wikipedia No, and here's the eigenvalues and for each I may for example One eigenvector contains in top not for you the topic of England countries about priority football DNA Here mathematics these are pieces on the real axis, which are simply increased. Yes, the zoom So this was already a nice possibility to identify certain communities using eigenvectors and the question is how to analyze them in more detail and this will Give rise to the reduced Google metrics approach which I will attend to Two or three a few transparency later not right away, but in more detail So but before that I would want to give a brief mention in concerning Anderson localization Which is very well known in solid-state physics. You have written down the most simple. Ah, sorry Tight binding Hamiltonian eigenvalue equation. You have your hopping element You have a discrete lattice Cn is a wave function on that aside n and you have some hopping element t and some diagonal this order epsilon n No, and it can be shown and see this order is random and uniformly distributed in this interval to parameters w and t And it is known in that one dimension and even two dimension the eigenstates are always exponentially localized provided W is larger than zero In three dimension you have localization for critical This order buffer critical disorder, which is quite large and below you have diffusive dynamics Actual if you take this as a network if you put w to zero here, no take out system then we would have network take n points on a circle and Link only the neighbors one neighbor then you have very driven network in the Laplacian of that network would be this Hamiltonian here apart from some constant maybe Then but then you don't have localization But here in addition you have this this other term, which is not present of course in the network. So now if you go back to our Patron vectors typically for there we have a more power low localization with an exponent close to point nine Yeah However, if you look at these course base eigenvectors the leading eigenvector for which have eigenvalue close to one It turns out to set for certain university networks. We have this kind of picture means an exponential decay in this This is a K rank corresponding to the vector itself that again We order the nodes according to the modulus of the eigenvector components And you have a strong exponential up to and then subset some saturation at the constant value But it was extremely small 10 minus 20 10 minus 17 get 10 minus 8 and The corresponding course base eigenvalue. They're not exactly one first. I even I discovered numerically thought it's one I thought there's a bug in my program, but it's not true. It's there is a difference which is significant and Using appropriate algorithm can compute the small difference in a direct way. That's as possible And these are that means we have here what could be called quasi subspace That means we have a bucket which is not perfect It's linked only by a dangling node, but very late to the other nodes that means the probability to ruin the bucket is very Big, but not perfectly one. It's but it's very close to one In any cases gives a nice also exponential localization from some particular cases One other topic one can study is the issue of why factor For this you need to Try it to find the scaling you compute you take some cut off Number below one say maybe point nine point five or even point one and you count the number of eigenvalues Which models above this number which you call n lambda and then you see how does this number scale with a network size? The problem is to have such a scale you need many networks with different sizes. No, it's only And if you take the physical review citation that think it's quite easy You simply cut it at some you have it depends on time No, you cannot each time step you have a smaller network So you can you see here the network size of the physical review which in this period here doubles every 11 years doubled from 19 yeah 10 roughly to 2010 and Therefore you have you can have a time dependent Network which increases over time and we can apply the scaling and here you see the results If the network size is above say 10 power 4 a bit more then you have a nice power law Which is here and you can fit this exponent B Which turns out to be point five one if you take lambda C point five here None that different value which reduces somehow the exponent slightly and here we have a dependence Of this exponent for different values of third cut of Value of London, but here these values. Maybe these are numerically not perfectly stable because of certain problems now Similar thing can be done for the Linux kernel Network because here you have also different networks depending on time the different Linux versions You know, there are many and this was a time when the last Linux version was 2.6 something If you were doing it today, you would have even much more much bigger Scales here is also while similar exponent which is Point six five for these two cut of values which was studied in 2011 by Leo and Dima and Alex They will also both the talk of Stefan this afternoon. No, he will speak about This kind of exponents but in the context of my chaotic Maps if I understand correctly, there are also some Ulam maps which were also studied by Dima and Leo Which I did not mention it, but for which you can also compute this kind of exponents, no Ah, yeah, I had also some small part of random metrics Perron-Ferbenian matrices which we did also In this work here The point is people know very well as classical random metrics ensemble for Hermitian Hamiltonian Metrics very well since 1955 from Wigner And here everything is quite well known, especially The level statistics of close levels are uniform Universal and they apply to many complicated physical system like nuclear physics or In quantum chaos and so on you have a specialist in your levels space in this room as a Uniform shape and so on and the conditions Practically for this university are quite low. That means it's quite easy to enter in this region of universality And we tried similar kind of sample for Perron-Ferbenian, but it turns out there's no universality as you will see the idea simply you take a matrix or being the symmetries of some normalization and try Say every matrix element to be random with the same distribution some function p of g ij and The non-correct approximately non-correct approximately because you have to respect the some normalization You draw that means numerically draw them all and then you normalize which introduce a slight mini Correlations and the first thing you see because of the something you see the average must be 1 over n That means the average matrix is simply a projector matrix Which has one eigenvalue one and the other angle zero and the eigenvalue one is because the Flat correspond to flat page rank because the eigenvector everywhere one or one over n or one over square root n and other Eigen is zero this is only the average But then you have a spectrum one dot one and zero which is highly degenerate then you can think of Degenerate perturbation theory for the zero eigenvalue for the fluctuations, and then you can apply known results for non-symmetric Random matrices which gives you circle uniform Identity of states circuit eigenvalue density in the complex plane and from the variance of the area you can estimate The radius of of this circle is a known result And it turns out this is We have this kind of radius and then it depends what do you put for this variance and then you have different type of Models here for this is you can for example say You choose a uniform distribution between zero or two over one for every element which gives you a full matrix In that case the radius is one over Propulsion one over square root n it goes to zero for large networks You can also choose a sparse network if you'd say the probability has a high probability of zero value for g and a small Probability of some other values to respect this total average and variance and you get a different Parameter for sigma if you work it out you turn out that the radius scales like one over squared of q Use a number of non-zero elements per column I can also try a power law with an exponent here between two and three which is the interesting part below To it's not possible because you could not compute an average and above Three and turn back to the full case here now, and then you find the Radius which some power law which is in between yeah, if you take n here and here that means between square root or constant And if you try numerically to verify, it's quite easy. It works very well The only one realization was 400 eigenvalues No, I mean the network here's only the zoomed circle for which is very small because you see this Here if you have a sparse case, there are two versions uniform sparse Constantly means you put one or random numbers on the sparse elements, and they have a circles with Proportional one over square root of 20 here and all dots inside works Here's a one eigenvalue outside of course Then here's a power law you have also kind of circular is but the circle is not really uniform it There's a small problem. This model is a little bit more complicated You have a fit of these scale in theoretical curve because expo fitted exploring is a somewhat different, but okay Yes, this is a different version here. We put a triangular Random matrix that means the lower triangle is zero. This is not the model That means here you have different distributions in that case already the average has a non-trivial Angles which is Average is the blue squares and the fluctuations gives the red dots, okay But the different case important thing is what you find you find spectra which are nice circles No, there was some radius no here No, and the unit which is not at all what we see for realistic networks. So to speak there's no Universality actually actually the spectra of Wikipedia or university networks They have a subtle structure and it depends really on the networks There is a part for each net of particular information which gives you a particular eigenvalue spectrum now I turn to the issue of reduced Google metrics This is the first publications also together with Katya, but Katya will give also more further applications of this method in the afternoon Her talk but first some theoretical background the ideas are following now We take we have a directed network. We have a Google matrix for this matrix And now suppose we choose some sub network of a very small modest number of nodes and air Ah, sorry here a small number and air which provides a decomposition of reduced network and the other nodes Let's say called it scattering nodes. This gives the block structure of the matrix This is a small matrix. There's a bigger matrix here and also Corresponding to the page on vector and you can work out is quite easy algebra So eigenvalue equation for the page trunk you can write it as this such an eigenvalue equation There's an effective reduced matrix here. It's two or three lines of your matrix algebra This also is Follows the line of shoes formula if you take block inverses of matrices now It's the same ideas which are behind here's this one It's okay because we did it for the page trunk But imagine you would put here a lambda here for some other eigenvector Then you would also obtain lambda here But then the lambda would also appear here and G would be depend implicitly on the eigenvalue What we did here is we concentrated of course on the page we have one here now And then you have two type of conuses are direct links, no but here you have More complicated contribution which we call scattering Contributions which are the interesting new part and we can also work out as this matrix the same symmetry properties As Google matters such positive elements some number of column This is quite easy and I want only to mention. There is a nice analogy the issue of count code tick scattering With the s-metrics formula which was found by my own mind well all long time ago the context of nuclear physics of compound Nucleos scattering, but this formula especially with H being replaced by a big random matrix for some chaotic Say it's a classically chaotic system, but in a quantum version of the system and W Transition matrix elements from this system to scattering channels say open quantum dot as a classical example, no and This is a statistical object because H is random and many effort has been done to evaluate Statistical properties average of this matter of variances average conductance localization. You can also Take here more complicated. Hamiltonians, which are for one-dimensional Localization problems and so on. This is a model which has been exploited in the context of mesoscopic physics since At least 20 maybe 30 even more even more than 20 years Maybe 20 more than 25 years now and in context of nuclear physics even much longer than this But since important thing mathematically is quite relevant Quite the same it's the same mechanism of shoes formula, which is behind of deriving that we have some Say this is a scattering part and this is direct. Okay. Here's one normally There are some direct scratching phases in real physical models now Now the problem is a practical back here you see if you to evaluate if you need to compute this matrix inverse and This is normally numerically a very tough problem Because this part G is still a very big matrix comparable as the initial network and it has one eigenvalue close to one It means if you expand theoretically this converges, but the world right of course is very slow You had to have to add up millions of terms cannot do this But what you can do, you know if you take the initial G as a version with the damping factors And you know there's only one eigenvalue close to you one the other I was Say below alpha at most alpha and you can take out analytically its contribution by suitable projectors here PC is a projector on the eigen space to this leading eigenvalue. It's ideas. You compute first Using also patreon algorithm applied to this matrix left and right eigenvector leading eigenvector for this matrix GSS Compute its leading eigenvalue, which is lambda C which is very close to one Then you have a projector on this eigen space. You have a complemented Complementary projector you can write it as this and you have to exact analytic inverse and here you have some projected Matrix G which appears G bar and this G bar has a leading eigenvalue which is below alpha That means this series here converges fast in comparison to this one here and this can be numerically all this can be numerically Implemented in an algorithm. You can compute these projectors. Actually what you do you you have to apply this to some given vector and you have to Apply the successive matrix vector multiplication, but this can be all then and this converges quite quickly So as a result you have so three different components never remind direct links Then you come conclusion, which you call p p alpha projector Which is the contribution where we take out analytically the leading eigenvalue and okay We can rewrite this with some different vectors. This is a rank one matrix here. That's important thing and Then you have the real scattering Is a battery out No, the sound ah It's impression. That's okay. Then you have Yes, this is a real scattering Contributions which gives interesting indirectly it turns out that numerically the projector contribution is a dominant one in percentage It's maybe 95 or 96 percent of the normalization the direct links a few percent and also there's a few parents However, it's this contribution which has a more interesting structure and allows to identify is a indirect transfer lowers network and We have studied the first application was to certain leading politicians of different countries For example here I show Simply in what we call density in the log K log K star plane This represents somehow a picture of the networks now In yeah in K rank and the patron and she rank order to dimensions and the red dots are the political position of the politicians We chose now here for you know, we choose 20 leading politicians of us using English Wikipedia UK Here's a G 20. This is okay. So states lead of G 20 and here for German politician France, but here we choose the respective Wikipedia of Germany of German language French or Russian now So but I will present now here only some results for the G 20 and French case that means From here these and this one, but there are also other results You see the what you call K K plane first of the G 20 politicians You see Putin Obama Obama is leading in K rank in page time, but putting is leading in chair rank It's more communicative now, and then you have here other state leader came and I must say this from 2013 or 2000 that means it's not up to date now Trump does not appear here. No, he appears for the US in some place, but lower place Yes, we have for French politicians here as well. I think my actor presence not even in their year my corner. It's a cozy all on the Melanchon and so on This is the same if you choose 40 politicians, so here I show For the G 20 case the these reduced Google metrics in so if a color density That means red is for maximal matrix elements Blue for minimal and green Intermediate size. This is a full reduced Google metrics This is a contribution of the projector part, which is very good. You see numerically. It's essentially this This is a very simple structure. We don't see much here Here you have the direct links now and here you have the indirect links which turn up from this Power series which we have add up and also here you see most contribution come from diagonal terms That means in even in indirect links web page a politician We speak of the Wikipedia web page of a politician and they have some indirect links going out making something in coming back This gives a quite considerable number of contributions, which are not as interesting That is as well to see other contribution are more interesting So here I show the first direct contributions One corner of the previous but with the names now here if Obama Putin and what you see here is for example Obama is obtaining links from Cameron from Harper a strong rather strong. Yeah, I'm actually even stronger thing the strongest link is here coming from I'm not sure from Also gets it from some These are the direct links and now here we take indirect links But we're taking out the diagonal contribution which increases the other other dots and here We see for example Merkel becomes now a strong link from Oland I believe yeah, and There is no direct links in the form of a picture. There was a blue squared here if you see here Merkel any Olander Going to Merkel is nothing here, no But here there's a strong link that means the web page of Oland is for France going with citing other pages because French-German collaboration and then going back to Merkel and this is important that produces this kind of indirect links and here you can now for each politician we can Say we take the top three or four In each line that gives the top three or four followers or the other very round when he said somebody which we call Top three friends we can now Determine a fish with different network of friends or followers Simply taking the maximum. Ah, sorry taking the maximum values per line or per row for each case This gives you I will have a graphical representation for the French case Yeah, here we have the French case the bigger one. No, okay. Yes, I could hear Oland You see his followers and his friends of the first those who know French politicians can Try to understand things but again, I remind is speaking here of links of corresponding Wikipedia pages and Link may have May have positive may be positive or negative. No, you make you know, it has at this point doesn't have a Moral or other value as such is some it's important, no So and here you have this picture. So what we show here using this previous matrices We present here for example, then we take for the five say political groups a to right right extreme right left very left and The Green Party know we take Lydia leaders of these groups and Then we take two step maximum two steps of indirect either friends or followers No, these are for friends and followers and here using this full reduced Google metrics and you're using the The qr contribution of indirect links and if you take here for the friends Then we see we add to other this is a good in a royal and I think it's this raffa ra We add to other and then it stops. There are no other links. They remain quite inside But if you take these other matters, then you have much more interesting links here now and for the followers is a bit different here you have many links in both both cases, but sometimes they also strange structures, yeah Guys, yeah here in the corner of yeah from and so on okay, and there are many other applications not only politicians for example geopolitics or called terrorist networks or painters which Of which Katja will talk speak in the afternoon in all detail which even better nicer pictures in here So time, okay, then to close the only I want to speak some work in progress. We started recently which we call easing page Frank No, remind the easing model here well known Since long time ago here you have some spins with plus minus values classical spins No, you have some okay. You take news neighbor and some Ferromagnetic interactions in the field and people know there's a phase transition Ferromagnetic state depending on temperature and so on solved one in dementia case and to the may can be served analytically so on What we tried What we are trying is to apply a kind not exactly easing model But to double the number of nodes that means suppose we have some networks such as Wikipedia with nodes ij And we double the number we give them colors a red and blue can be for political affiliation That means more communists or left or conservative people. We double each node And then we say with a note given node i there's a probability WR this node is either prefer Provincial for red that means both versions linked to red one Or his preferential with a complementary probability. He's linking to the blue one In average Each node is it means is either this or this but then there's a distribution between different i nodes is then given by these probabilities, no This is one version one can do many other modulations of this And for the factor there's also this vector which appears either in the damping factor or in the dangling nodes You know this unit vector and we replace here They insert the probabilities for red and blue. This is some practical used to have Yeah, this is some practical meaning But what you can do now is we can compute the page rank of the increased network, no And you have components say red components of a node and blue components And now we can define the road quantity which counts the number of nodes where the red ones are Stronger that that's a patron for bear of red is stronger of blue and One must say that when you when one computes page rank there in the lower parts There are many identical values very often and therefore it equal even on numerical for double then even Exact equality is possible here. It's in this case. We add one or one over two Contribution to have some symmetric result and this gives a vote as a function So here's a probability of Preferential for red or blue and here's the vote of course extreme case if there's zero probability vote to zero and extreme cases But then you have here this Transition in between this is a pointer one should mention that here's a kind of gap. Okay. It's a green cover some nice fit No, but one should mention That okay here's the data points are not dense but here between this is a real gap This is because of this Choice of vector. It means there are certain number of nodes which essentially depend only From such contributions from the damping factor and then the nodes means if the W air crosses from point five to point five something bigger No, then you have a big jump in number here jump a big this step here. Okay You can also now study the effect of an elite. What does it mean? It means we take Say something small number say nl elite to notes for which we provide a different probability for the preference preference color preference or political preference and Wf for the other notes a big population and the elite to notes may be selected as to say top and EL nodes may be thousand now either You're using K rank or the k-star rank or 2d rank the different rankings No, and then we look how the vote of this curfew is modified is modified with respect to this elite and now and We did this for Wikipedia recent Wikipedia 2017 thousand elite and what you see here is this mode Modification here the probability of the elite is Zero that means perfect preference for blue the elite all elite people are only preferring blue color And then here this is the probability for the other notes And of course this is going down because that this gives you the vote for red notes But the maximum is about yeah the minimum here's about point five There's some strange pica, but this numeric if stable. I verified very carefully so many data points It's not a accident something that this is a green dots are for choosing top a little as chairing notes And here's a page rank or 2d rank notes Here is the same a different preference. That means this line here SW corresponds to the red points here That means here I take the data for K run K rank notes, but with different values of the elite probability going up here and you see And here in the color scale minus one corresponds to red and that's one to yell Yeah, yellow, but this is not sorry Yeah, this is more orange Should be orangutan not pure yellow. It's different to this doing this the yellow the point is the scale here minus One corresponds to this minimum value and this is really is never reached you have the strongest positive number is here that means the maximum Maybe for other values are a little bit above, but not that much no And you see of course if the elite of probability is close to point five and Here's a for other words is 0 5 and it's of course their positive effects here This line is a zero line and then he is positive and the other party is negative and here you see a strong Reduction due to the effect of elite. Okay, these are say preliminary results And there are also many questions, especially how to modernize this This is the first attempt to to modernize this type of Networks so conclusion on better summary I reminded some basic property for Google metrics, which are constructed from directed networks That web Wikipedia Twitter Linus and the citation and so on can efficiently compute page rank leading complex eigenvalues What's exploiting the structure of invariant subspaces, which is by the way I think believed strongly led to what Zergis told us yesterday about this the structure of networks with the G in G out, I think this in one sub is the G G out part I believe But it's another word Okay, then So something about localization Which quasi exponential localization? There are certain case where I can study while fractal scaling for certain networks typically networks where you have some way to Have the time dependent network with different network size. That's what you need And otherwise if you give a given network, you cannot scale off if you have only one network Then different simple models of random parameter do not really describe spectra of realistic Google metrics Then of course approach of reduced Google metrics For subnetworks gives this decomposition of gen three contribution allows to construct Friendful or network using either the full reduced Google or this third component And they're also a possibility to use different language editions of Wikipedia Which takes into accounts multi cultural aspect and finally some yeah new work beginning of work for what we call easing page rank and as the effect of Selected eliton notes on the world can can be studied So, thank you You speak of this here Maybe next yes, yeah, how I obtain this formula What I do is the following you have here a matrix And it's similar to what Pete told us the kind of we make a spectral decomposition of this matrix And you define a projector formally. That's one way of doing this a projector of the I can space but the point is since it's not symmetric the projector you need right and left eigenvectors This is for example a projector if you squared it's one if you multiply it with qc. It's zero Yeah, what you can do this you take this gss No to compute CLCL and same time you get lambda c the point It's like page rank algorithm But each time you also change the norm of the vector and the renormalization provides you automatically with the lambda c The eigenvalue you compute everything and if you test a verify that both lambda c is from left and right are actually the same That's then you compute CLCL lambda c No, when you have this you can apply PC to some vector You can of course apply to Qc which is 1-p also two vectors and then what I need to do It's not really told here if I go to this formula Take it here takes this. We have still contributions from other which are rectangular matrices here That means here you have a few number of columns, but very many lines That means you take one column That's only you take to compute as I do it column by column No, you take one column this gives you a big vector here. You multiply it to this and this is j bar You apply really literally is this formula as you see, okay You can economize if you multiply two g's you can economize one qc You apply this qc gcs qc and so on so on you add it up you test for convergence and so on and when you have done this Then you apply against here some This has a other way around many columns, but not small number of rows and then you get A row that means a column that can then you get a column of this matrix and then you do it again for the other columns And the nice thing is since this for column is completely in demand. It's very nice to make it in parallel You know you can do it If you do do open mp parallelization you can Compute it all parallel if you have a machine with 20 processors can and there's there's no loss in say scaling is perfect, which is very rare in numerical computations It's a bit tricky, but it's there's no essential problem But the assumption of course is that you do this for a matrix where you have applied the stamp And there's only one leading eigenvalue now Which is close Suppose you have you can we have formulated it theoretically if you have say two or three leading eigenvalues You can generalize it by Increasing this projector parts and instead of rank one you would fear would be more complicated, but not impossible Yes, the just one is what we have from the Google matrix because of the damping factor now this comes out Okay, you can say take now a model a network model the way you not with a damper, but suppose you have two eigenvectors which are close to one No Say different model not with damping factor and you want to compute reduced Metrics of this then it would be pertinent