 Thank you very much. First of all, thank you for the organizers for inviting me. And I am Leonardo Erman from Buenos Aires, Argentina, from the Conea, which is a nuclear energy agency. And OK, I will talk about the World Trade Network and the applications of the Google Matrix on this network. The idea of motivations is try to understand and try to analyze, model, or visualize the network that can be huge with a lot of information. For this work, we will focus on the Watering Network, which has the information of all the trades in the world for different products for the last 50 years, more or less. We have analyzed other kind of networks, like Wikipedia and universities coming from the World Wide Web, also others coming from software or from applications. Bitcoin, maybe I will mention some of that. But the main idea here is to focus on Watering Network. These are different views of networks. These are the Google Matrix that are ordered by page rank values for different networks. And we can see very different behaviors of very different kind of networks in this picture that I will explain later. OK, the database we will use comes from the World Trade United Nations Commodities Trade Network, ComTrade, which has the information from 1962 for all countries. The whole countries gives the information to ComTrade for all the imports and exports for all kind of products. We will use here the classification with this protocol, but we will choose 61 products in order to have a more good qualification of data on 227 countries for 2008. Number of countries are changing, and number of products we will fix in 61, the classification of products. The trade volume is given in dollars, so we will build networks that will have a dimension of 13, 227 times 61, more or less. It depends on the year. So this number of almost 14,000 nodes will have. OK, let's focus on the matrix we will have. We will do two different approaches. First approach will be to have the total money from one country to the other in dollars. And also, we can see the same kind of money matrix. We can build it from a specific for a given product. Then with that, we can build different kind of Google matrices. So yesterday, Klaus introduced a Google matrix. The idea is that we can build a page rank. We can compute page rank, which is a centrality measure, which is a spectral focused. And it's very good for directed network. It's very easy to compute. It depends on incoming links, the page rank. And then we can compute a chain rank. We will depend on outgoing links. And it has no local properties. You cannot reconstruct your page rank knowing only your neighbors. You need to know the whole matrix. It is like we can think, OK, page rank kind of infinite time limit of a random walk. So how we will build the matrix? Suppose we have this kind of matrix. In this case, the matrices will be weighted by the amount of money that suppose these are countries and these are the amount of money that it spends or buys from one to the other in a given year. So the recipe for building the Google matrix is the following. The money matrix of this network will be as follows. Suppose here the first and second node, they don't have outgoing links. So the columns are empty, 0 with 0s. Then suppose here the third column, it has 4,000 goings to 4. So we put 4,000 here and 2,800 to 6. So in this way, we build this matrix with each outgoing links is put in a column. So this matrix is positive and real. Then we make a matrix that is non-negative, but it's stochastic like a Markov process where the sum of all columns is normalized to 1. When we have 0s in a column, it's called a dangling node and we put 1 over n in each element of the column. And on the other elements, we normalize to 1. The good thing of this is that now we have a Peron-Fravenius theorem for non-negative matrices. And so we know that the largest eigenvalue is 1 but could have some degeneracies. So finally, we build the Google matrix with the dangling factor. And in this case, we have chosen 0.85. And the good thing of this matrix is that now we can use Peron-Fravenius theorem for positive matrix. So we know that we have at least one gap, which is at least 1 minus alpha. So we have a very good convergence of the system and so we can compute patient of the matrix, which is non-trivial for directed and weighted matrix. So with this money matrix, we build G Google matrix and we can build G star matrix, which is the same approach but using the transverse of the money matrix. So inverting the time. Here, of course, the information is there. You cannot cheat. Outgoing links are very important. It's not like the World Wide Web that you can increase your position adding some outgoing links. Here, it's not possible. These kind of things are. So the main idea is also to compare this page rank and chain rank to the simple volume ranking, like export rank and import rank, which measures the importance of a country by the volume that they import or they export. The total money they import or export. So this will go k bar here and this will be patient and chain rank. The good thing, now that we have these two definitions, we can use two dimensional ranking. OK, from data, we know these are the number of countries, how they evolve in time. And more countries are created in time from the 1960s. In red, you can see the distribution. OK, this continues, but we end up in 2010. But it continues. But data is not very reliable for the last years. So the amount of money here is in logarithmic scale, grows more or less exponentially. And the average number of links here also increases due to globalization. We want to have a democracy in countries, but not in product. I will explain this in some slides. So we will use two different approaches for the World Trade Network Google analysis. The first one would be to focus in a small matrix given by the number of countries. And then we can see for all commodities or we can see for a given product. And the other is to build the multiproduct World Trade Network, which will have these 61 products and the same amount of countries. So it will be a higher dimension. So if we want to analyze, for example, for a given product with this approach, it simply will have to compute only patient. With the other approach, OK, we use some personalized vector in order to have a democracy in countries, but not in products. But so we will work in a larger space with larger dimension. So if we want to compare these two rankings, we will have a list, suppose, of patient ranking this way. And if these are cars, suppose cars trade, here we have to take out only cars from a longer list. So but this is how Google works also. It gives a list and you take the position of a given, in this case, product or search. So these are how many matrix looks ordered by importance of the country here and here. So in the top it are more rich countries here. And these are how G and G star looks. And they are normalized each column to one. So this is why we say this is more democratic in countries because we give the same weight for all countries. And we see the distribution of countries of patient and chair rank and import rank and export rank. And they follow a C flow. But in export and imports, it's more like it has an exponential tail. But in patient and chair rank, it's a C flow. What about the spectrum? The spectrum of this Google matrix, we can see, for example, for all commodities, this is a spectrum in the complex plane. We have with alpha equal one. So we have one eigenvalue equal to one. And for all commodities, it's more symmetric matrix because usually the market is more symmetric in this way from imports and exports. So this is the whole spectrum. And it's close to the real. It's close to the real. When we see, for example, food, now it's getting more wider. And if you go inside for more specific kind of product like cereals, it's getting wider. And suppose barley, now the symmetry, the matrix is far too symmetric because usually when you buy from one country to the other, you buy barley, you don't say the same product. It's not balanced in a given specific product. So we can compute also this kind of information would be in the correlator. We can define a correlator between patient and chair rank. If the matrix is diagonal, this would be one. But it is symmetrical. And these are different networks. And the values of their correlators, usually networks coming from the worldwide web can have very large correlators. But other kind of networks, they have zero correlators. In our case, we'll be here in the watery network. So we'll be more similar to the cases of networks coming from the worldwide web. OK, this is an example of how suppose barley, how it looks the network, but in the KK Star plane. These are the networks only in this subset. OK, but these are for cars, the positions of the countries. The main thing is that, for example, here, Ukraine is the first country in exports of barley. But it goes to the sixth position in chair rank, which is related with export. But so what is the difference between export rank and chair rank is that you can have a lot of volume of exporting in a given product. But the chair rank measures not only the volume, but also the importance you have in the network. So for example, Ukraine has the largest volume, but it has not a central position in the network of this network. And on the other hand, USA is the opposite. It has the eighth position in volume and goes to the third position, because they can choose a better position in the network from the sales of barley. The same things can happen with cars, suppose. And now we can analyze how this ranking for all commodities evolves in time. This is the plane of K. So this K I haven't mentioned, but it's the position in page rank value. Page rank is a real value. K is the integral number that is the first K position is for the largest p-value. So here you have the most rich countries, in the sense that they have the largest page rank and chair rank values. And these are the positions. Sometimes we have this region that means that a position of chair rank is better than position of page rank. So these are a kind of good position in an economical point of view, because you are better exporter than importer in some way. On the other side, in these positions, you have the opposite countries that are better in importing than exporting. So this can be a dangerous region. So let's see, this is in 1962. Let's see how this is evolving in time. This is the year evolving. So we can see different things. One thing is that this part is more or less frozen. And here we have a very large movement of countries. We can see that some countries are always in this region and some others are moving. We can detect that here that we would have a lot of information. Suppose here you can see Greece always moving in this dangerous region. And OK, maybe we can mention that we can detect also some crisis due to spending a lot of time in this dangerous position. We can see a kind of solid state phase here and a kind of gas state due to mobility or velocity here. So we can go further and we can define velocity in this two-dimensional plane. Velocity square will be defined as the change in patient and the change in J-rank position, the changing from one year to the other to the next year. So we can define also the importance, which will be the sum over K and K star. So lower K plus K star will be most rich country or important country in this network. So we can see here that these are the density, the number of countries for a given importance or yes, in the network. And it's more or less uniform. OK, the case here, here in the last part of this importance, we have very, very small countries like Ireland and this kind of small countries. But I was saying that the most importance are more or less frozen here. These are relative comparison between countries. You could be in a crisis, but if the first country continues to be the first, it's not visible here. So when the country is this K plus K star grows, you can see that the velocity also grows. The points are the real data here. The red is the average for this data. And the blue one is a smooth average. So you can see that here it grows more or less linear then it's uniform. So here we are in the gas position and then decreases. But this is due to that we have very small countries that cannot go to very important positions. So if we analyze with more detail what is happening, suppose these are page rank, position and chain rank for these are here four countries and these are the five countries. But for different scale, it's the same. But with different scale, we can see that the frozen part of the solid state in a page rank is United States always in the first position. Then Germany and UK have some switch here. And then you have France, Japan. And you can see here that they are going up. In export, it's a little bit different what is happening in chain rank. United States lost the first position here. Also we have Germany going to third position and Japan fourth. And these are related with exports is chain rank. And so someone is entering here, which is China. China moved to the first position. China is this Xi'an cube and was very far away, like for 18th position, something like this, entered to the first one. So we can see that China is a kind of deposition in these two states. And on the other side, we see that UK goes from second position in the 60s to maybe now more than out of the top 10 positions. So maybe it's a case of sublimation. And one interesting thing we can say here, if we see the trajectory of Argentina, suppose I'm from Argentina, but it's a good example to show the position in exports in the last 20 years is more or less uniform stable. But you can see here a peak in Peshland, which is related with imports. This peak is located in the next year after the crisis that we have in 2001. So the main idea is that when you have a crisis, the idea you can export in the same way. But imports, you cannot do it because you don't have money. So this is a peak. And so we can use this kind of information in order to detect crisis also. I will mention this later. But let's focus in this part. So now we want to do a very small model of the World Trade Network. Usually the models of the World Trade Networks are defined in this way, are called by the gravity model of trade. These are defined like the money that one country sends to the other, buys to the products to the other, depends on the mass of each country, a constant and the geographical distance between these two countries. This model works well, but it's a symmetric model. From IEJ to GI, it is exactly the same. In these pictures, we have the plot of money matrix elements IEJ as a function of GI. So this is a symmetric thing, but you can see that if the matrix, the money matrix, the symmetric will be in a line here. This is from 1962, and this is from 2008, which looks more or less similar. But this is far away to be aligned. We have a very directed network, which is far to be symmetric. So this in log-log scale, you can see this state that this gravity model will show only one line here. The only change between 62 and 2008 is, OK, it's more dense. And also the scale has changed 100 times. The money matrix in dollars. And this is how it looks. The money matrix is supposed for a given product like petroleum, which is other kind of things, very different. So we can define a very simple model in order to analyze this, which was the simplest model of random model you can do. It's like taking two different random numbers between 0 and 1 and divided by a j, and you will end up with this kind of money matrix. OK, normalized in another way, but this is very simple and it works very well for our system. So this is the real data for money, this difference between position of chair and patient, and the sum over two of these two of the 50 years we have analyzed for the old countries. And this is density plot. And with the model, we use the same number of points. We have this kind of for one run of the random model when we obtain this. So it looks similar. OK, we were talking about crisis. We can define a balance, which is the difference between chair and patient values divided by this weight, which is the sum of the patient and chair values. So in 2008, suppose we can do a kind of table for top countries with negative balance, in this case. And taking into account only the most important countries with weight larger than 0.05, so almost 20% of countries. This is the rank, but this is a global rank without this. So Greece was in the first position, Spain, Romania, all countries that they were close to the crisis, they were in a bad position of balance. So if we do the change of balance in time, and we analyze the list for the whole period, we can see very specific crisis only ordering these values of the change of balance. Usually it works like I said. When you are in a bad position, you have to increase your balance, and then you can decrease it if you have a good balance. OK, let's move a little bit to this multi-product watery network. All of the first part was done with the first approach. With multi-product, we can define also balance, but balance of countries, for this is also 2008. But now analyzing a large matrix and tracing out over products in order to obtain the balance of countries, this is how it looks, the import-exports balance. With import-exports, everything is more uniform close to zero, but some African countries will have different values here for import and export of balance. But with patron and chair rank, we can see that, for example, the balance of China is very good, because the export position is much better than the imports, and we have other kind of information, and we think it's more visible with patron and chair rank. Other important thing we can do with multi-product network is we can also make a kind of ranking of products products. They don't have interactions in the data we are using, because it sells all kind of products, what are the sales from one country to the other, but doesn't use any information of what does the country inside. We don't have information that one country sells, buys one thing in order to produce the other, so we don't have interactions between products. But with this approach, we can do it in the same way. We have two parameters. We have countries and products. So if we trace out our countries, we obtain the ranking on products. So these are the ranking of products in the diagonal using the imports and export rank, which is diagonal. And using this patron and chair rank approach, we can see a very rich behavior that we have the same kind of information of countries now in products. Here we can see that the most imported products here are related with technology, machinery, cars, and these kind of things that are imported for a lot of countries, but exported only for a few. And on the other hand, these positions are occupied by the food. Suppose this red belongs to the first category, which are different kind of food, it could be not very visible, but different kind of food. And this is how evolves also in time the ranking of products. OK, let's move to what's happening with the spectrum of the system. For the multi-product world trade network, we have almost 14,000 nodes, so it can be diagonalized numerically in exact way. So these are the eigenvalues. And we have seen, Wikipedia, Klaus mentioned yesterday, that there are some in the eigenstates of given eigenvalue different from one. We have the information of a community. We can see the same kind of things here in the world trade network. We will see, for example, this is G, and this is G star spectrum. And we will analyze these four eigenvalues. What are the eigenstates related with these eigenvalues? Here, we take two real eigenvalues, and here two complex eigenvalues. But all of them with the large modulus, because in other case, the information is not very reliable. These are the participation ratio of this spectrum of all the eigenstates. So all these four cases are more or less localized, very large small values of participation ratio, which is this quantity, which measures the number of nodes that appears in a given eigenstate. So close to 0, we see that we have a delocalized, also because we have a degeneracy close to 0. But these four eigenstates in red, violet, green, and blue, and these are ordered by decreasing order of modulus of the values, normalized to 1 in this case. So we can see that they are localized in 10, more or less in 10 nodes. Each node here is a given product for a given country. So if we analyze what are the important nodes in these four eigenstates, we have that in the red, the top 10 nodes are related with the same product. In all cases, they are related always with the same product. And these are the communities. Suppose in red, we have sugar. And this community on blue, we have the community of the products of fertilizers, and also not only the community of fertilizers, but for a given region. This is a community. So we have South America here, Brazil, Bolivia. So of course, this is not a very good tool in order to obtain communities because you have to diagonalize a very big matrix. But on the other hand, you can analyze that the communities are present in the eigenstates of this matrix. We have done this also with larger matrices, but could be more expensive using different numerical methods, like Arnoldi methods, and could be very difficult. OK, now let's move to the reduced Google matrix that was used, explained yesterday by Klaus also, and analyzed by, in some cases, with Katia. The idea, I won't talk again, I want to explain everything, but the main idea is that we have the system, and we want to analyze what happened in the subset of the system. Now the subset will be our system, and the rest of the nodes of the network will be the environment. So the Google matrix can be divided in these four blocks. GRR, we have the direct links, and then we can write GR in this NR space. And GR has three different parts. The first one is the direct link. The second one is related with the projector, with the eigenstate, the page rank, and is more or less proportional to page rank. Could be maybe the largest part, but the most interesting one is the one, the part that is undirected links, but they have more information than page rank. So this is the approach that was introduced in this paper. I will mention here some plots that were done with Celestan, who is there, and in collaboration with Dima and Joseph. So for petroleum, we have chosen here our subset of interest, is the petroleum of the European countries. And we have also Russia here, because it's the first exporter in petroleum, the first in chain rank. So we want to see how, for example, how Russia affects on the other countries. But first of all, we will see these 28 countries, how it looks the matrix. And you can see here direct links of the matrix. And these are the 28 countries. And this is, sorry, this is GR, the reduced Google matrix. This is the direct links. These are the direct links. These are the part of the matrix, which is more or less proportional to page rank, the projector with the first eigenstate. And these are the second and third and all the other eigenstate projection. So the main thing we can see that here, the direct links between countries are very large. In this case, these are the trace over this matrix divided by n, not the trace, but the sum over all the elements of the matrix divided by n. And it's close to 0.3. It means that it has 30% of the weight here. And usually, for Wikipedia and some other networks, it is close to 1%, because here it's very well connected. The market of petroleum in this is a community in some sense that is well connected. The part proportional to page rank is, in this case, 0.65. But usually, it's very large. It's 95%. And here we have the undirected links without the diagonal term. And in this case, it's almost 4% of the weight. So we can see here some undirected links from, suppose here is from Malta to UK. And here we have different cases, like direct links here. We have, for example, Netherlands to Belgium. It has a very direct link. We can build from this g, a reduced network. We can build a kind of other kind of network, which is the four-friend networks. The idea is to take the four most important outgoing links of these Google metrics. And if we do this, we obtain this network. Here you can see that there are some countries like Netherlands or then France, maybe Spain or Germany that are in the central position. It means that all the flow of petroleum, the four most important elements, goes to Netherlands or these important countries. This is what we have. If we change the order of the links or the time, we reverse the time, it means the direction of the arrows. And we take also the four most more important. And we obtain this, that Russia is the most important. Here, this is the order of the money flow, if we want to see it like this. So it means that Russia exports to almost all countries of Europe and also Netherlands here. A lot of things can be done with this database. We can compute also the sensitivity of balance. This is the change of balance when we change a matrix element of the Google metrics. So what happens for it? And this will indicate how it depends on the page rank of these countries and chain rank of these countries, the difference by a change of one small change of the Google metrics element. This is for Russia. And for Russia, we see that here in blue, the change is very small. And in red, we have Italy and Netherlands, which are more affected. With Saudi Arabia, if we change Saudi Arabia, how it affects to European community countries and with USA also. And you can see that they are very different. So you can identify the sensitivity for a given country and given product, how it affected the change in price of only one element. The same thing can be done with gas. And with gas, it's more visible that if we change gas in Russia, it has very different behavior. Here was before 2008. So before the pipeline that has from Russia to Germany. So gas affects the neighbors of Russia because it has a very different market. OK. I will mention that we were also working in other kind of networks. We have analyzed Bitcoin network, which is open. The only thing you have to do is to build the metrics. And also, we would like to analyze Interbank Payment Network, but this is very difficult to do because information usually is hidden. We have tried to convince some people in order to give the information. But this was not possible. So this is an example of a paper of Soramaki in 2007, how it looks the Interbank Payment Network. And the thing we have done with Bitcoin is to obtain all the information now until the year 2013. The number of notes for Bitcoin is the order of 10 millions or something like this. So it's large. Now we cannot analyze 10 million metrics. But we can take some part of the spectrum. We did it with Klaus. And we have seen that the genetic efficient of this Bitcoin network is much worse for money, for volume than for patient and children, which is more democratic. But we have seen that Bitcoins are also the largest amount of Bitcoin belongs to only few users. And other things, Bitcoin network is not very good because they are changing users' names all the time and they move the money from one account to the other and we cannot identify people from users and there could be. And this is the good thing of Bitcoin for that it's very difficult to identify someone there. OK, I will move to a different thing only to mention that all these rankings we have done with Google Matrix analysis. But here we will use other kind of protocol of algorithm that comes from ecology in order to rank also countries and products. The idea is very simple. In biology you have a network, bipartite networks, which are networks that you have links from one set to the other and you don't have links between these inside each set. So usually in ecology you have, suppose this is a network of bacterias and phages that affects the bacterias and you have bacterias that are more susceptible here. That is more susceptible means that you have a lot of phages pointing to them that affects the bacteria. And very bacterias that are hard to infect like this one. So only a few phages, this kind of networks occurs in real network. And you have generally phages that points to almost all the bacterias and more especially that points only to this part of the bacterias. It's not, you don't have especially that affects too hard to infect the bacterias. The same thing can be done, for example, with animals, some species and some islands, some regions or some plants or insects and plants and these kind of systems. The idea is to order here suppose plants and animals in a way that given the dimensions of the bacterias and the dimension of phages here and the number of links you could make a curve here. And the idea is to do permutation of the matrix in order to have almost all ones here on all zeros here to decrease the distance of the ones, the zeros here with this curve and the ones with this curve also. So after all permutations you can do, you obtain a ranking of plants suppose here and of animals here. That animals that could live in some places and places that they live very much animals. OK, the same thing can be done for imports and export. Now we need a binary network. So we have a weighted network but we define a threshold in order to have this matrix. And after that the idea is to decrease zeros here and ones here and this is for 2008 for countries and for products. In this way, we have other ranking that we call ecological ranking. This is how the money matrix looks for products and countries when we analyze imports and here when we analyze exports. And but we define a threshold and we put one if it is larger than the threshold and zero if it is below. So the rankings we have found are different from patron and children rankings and they are different also with volume matrix we can say some examples but they are very different. And the information we have for example is something similar we have obtained for products, for multi-products that here for imports the most common products are related for imports with this technology and machinery blue and black and for exports more related with food. It's more easy to export food than machinery. But if we see only the money rank of products we will see that it's more related with imports because rich countries that takes imports are more expensive with these products so it governs between these two, these are the least. OK, I think I will stop here. The conclusions will be that we have analyzed the water network with the Google matrix approach and we have done in this two dimensional space we have done several things like modeling, visualizing and analyze also the spectral properties and obtain some communities there and the good thing of this approach is that once you have the information you can do a lot of things with Bitcoin we did it and we hope to have more information about some other economical systems. OK, thank you very much for that. Thank you very much. Questions? Were you able to identify the origin of the crisis of products in terms of imports? The origin in products? With crisis here the bad thing is you cannot predict nothing. The only thing you can do at least for us is to analyze the past and see, OK, we have a crisis because you need a difference between two years in order to identify. We have seen crisis in countries. We have analyzed crisis in products. You mean maybe I don't know what would be a crisis in a product? Yes, because we have seen a peak for Argentina. Yes. And what is this peak? The meaning of this peak is that this is a ranking. So it's a ranking of countries. So this peak is that it goes from, suppose, 45th position to, I don't know, 70th position in the whole world. These are crisis of a specific country or a specific region because they move a lot compared with other countries. Global crisis are not so easy to detect because the whole world is moving in a bad position. But so USA in a global crisis will remain to be the first. But what the meaning of this peak is that the position in exports or in J rank, which is related with export here, is more or less the same. But the import position, like a page rank position, goes from a good position to a bad position in some way. So it means that you cannot import more for this year because you don't have money. You cannot take debt. So the only thing you can do is to go to a worse position. Then due to the exports, maybe you can spend the next year because you're moving in balance a lot. The next year, maybe you can buy more things and recover your position, more or less. Here we can see that it never recovered perfectly, but this is the meaning of it. Yes? I have a question for the same slide. Yes. So to detect crisis is interesting, but maybe it's already too late. Yes. Can this technology be used to anticipate in crisis? It's not a big signal. The only thing, OK, I mentioned that these regions are dangerous in some way. So if you are all the time in this region or you move a lot in this in K position, you move to the left here and then you move to the top. So for sure you will have a crisis. For example, if you see evolution, you can see the movement. So Greece is very visible here. It's always in this position, but I don't know exactly when the crisis will arrive. But we know that if it spends a lot of time here, we will have a crisis. But to detect something, to say something about the future in economics is not so easy. We can say that this is a dangerous region and this will be a good region, but no more than that, at least for myself. More questions? Where is France on the graph? Sorry, can you show me where is France? France is here. France is here, but we have a lot of flags. And it's in a solid state. So it's not very visible, but it's here. Now it's with the United States, UK. Maybe it moves a little bit, but we have to do some zoom here in order to see France. Here you can see the position of France. It's in the fourth, goes to the third, in fourth, fifth, and it moves like this, the red one. I have one question. I think you had a slide that was showing that the page rank chain rank difference is better than the balance difference. Try to understand why page rank is a better indication of the crisis than the balance itself. So this slide you're showing so quickly. I'm curious of like this one. No, it has other information. We are not saying that we can detect the crisis better, but it has other kind of information. This is important and export. And you will see that the most largest and smallest values are most of them here in Africa that maybe small countries can affect a lot because this balance is normalized with the SAM over P. So small countries can have peaks. And here, using page rank and chain rank, you can see, for example, what would be the most important exporter in the world in those years was China. But China here, using import and export, because you divide by the total weight, it's not very visible here. It's in the Sian part. But with page rank and chain rank, it's very easy to see it. But it has other kind of information. Our information is global, and it's related with Google metrics. And the other is local information. And it can be done only with volume. OK, thanks.