Hello, this week, we've been looking at networks as dynamic structures that change over time. In the past two videos, we looked at different models of network evolution. That given the dynamics for how a network evolves from the beginning to the end. Today we're going to look at a related problem which is given a fixed network, can we predict how this network is going to look in the future. And more specifically, we're going to be looking at the link prediction problem. Which is given a network, can we predict which edges are going to be formed in the future? This problem has a lot of applications, for example if you're Facebook and you want to create a friend recommendation algorithm or a friend recommender to tell people who they should friend. Then, basically you're solving this problem. You're looking at the current Facebook network and you're trying to predict which new friendships are likely to form in the future. Let's take this small network as an example and let's try to ask the question, what new edges are likely to be formed? And more specifically, let's formulate the problem in this way. Let's say that we take a specific pair of nodes and we want to assess whether or not they're likely to connect. What we're going to do today is we're going to go over several measures, seven of them, to try to make this assessment. To try to decide whether a specific pair of nodes are likely to become friends or not. And so, we've actually thought about this kind of question before. If you remember, we talked about triadic closure which is the tendency for people who share connections in a social network to become connected themselves. So triadic closure actually gives us a hint for what the first measure that we're going to look at is. And that is very simple measure, just look at the number of common neighbors that the two nodes have. Now this is very simple, but let me define this measure to introduce some notation. So we're going to say that the number of common neighbors of nodes X and Y is going to be the size of a set, which is the intersection of the sets N(X) and N(Y), where N(X) defines the set of neighbors of the node X. And so for example, the common neighbor measure for nodes A, C in this network is going to be 2 because nodes A and C have two common neighbors. That is node B and node D. So A, C have two common neighbors. In network X, we can use the function common neighbors, which takes in as input the graph to nodes. And it outputs an iterator of all the common neighbors of the two nodes. And so here, what I'm doing is I'm creating a list of tuples which have the two nodes and the number of common neighbors. And I'm only including the nodes that are not connected with each other. The ones that don't have an edge between them by using the function non_edges. And if I sort this list, we can see the pairs of nodes that have the most common neighbors between them. And we see that the pair A, C has two neighbors in common, there are many others that have only one neighbor in common. And then many others that have zero neighbors in common. And so if we start to compare between different edges, for example we look at the pair A, G and the pair H, I, and ask, which one of these two is more likely to become connected? Then by looking at the number of common neighbors, we actually can't tell, because both of these have exactly one neighbor in common. And so let's look at other measures that may potentially give us different answers for these particular pairs of nodes. The next measure we're going to look at is called the Jaccard coefficient. And what it does is that it looks at the number of common neighbors but it normalizes it by the total number of neighbors of the two nodes. So the way that we write it down is we say the Jaccard coefficient of nodes X and Y is going to be the fraction of the number of common neighbors. That's in the numerator, so it's the intersection of the sets N(X) and N(Y), divided by the number of neighbors of X and Y which would be the union of N(X) and N(Y). And so here, the pair of nodes of A and C, have a Jaccard coefficient of one-half because they have two common neighbors, B and D. And they have four total neighbors, nodes B, D, E, and F. And so that's 2 over 4 which is one-half. In network X, we can use the function jaccard_coefficient which takes as input the graph and it outputs an iterator of tuples which have the two nodes and the Jaccard coefficient of the two nodes. But it only outputs the pairs of nodes that are not already connected, so the non edges. So if we sort this list, we now find that I, H have the highest Jaccard coefficient of 1. And that's because I and H are both connected to a single neighbor which is a common neighbor G. Whereas the nodes A and G, they have one common neighbor, but they have more neighbors that are in the union of the neighbors of A and G. And so, therefore, they have a lower Jaccard coefficient. The next measure is called the Research Allocation Index. And the intuition behind it is that it considers a fraction of a resource for example, information or something else that a node can send to another node through their common neighbors. And so first, let me tell you how it's defined mathematically and then I'll tell you a little bit more about what the intuition is behind the measure. So the Resource Allocation index of the nodes X, Y is going to be the sum over all the common neighbors of X and Y of one over the degree of the nodes. So in this case, if X and Y have a lot of common neighbors and they're going to have a large Resource Allocation index. But if they have a lot of neighbors that have low degree, then they're going to have an even larger Resource Allocation index. Now what's the intuition behind this? Let's consider two nodes X and Y, and let's say that we're measuring the Resource Allocation index between these two nodes. And let's say that they have exactly one common neighbor, Z. Now imagine that X is trying to send to Y a unit of something, let's say information or something else. And is going to do so by passing it for X to Z and then hoping that Z will pass this unit to Y. But actually what Z does is that when it receives this unit from X is going to distribute this unit evenly among all the neighbors of Z. Then in that case, well Y is only going to get a fraction of that unit. And which fraction depends on what the degree of Z is. So if Z has degree N, then Y is only going to get 1 over N of that unit. And so if Z is the only common neighbor of X and Y, and Z has a lot of neighbors, a very large degree. Then X is going to be able to send less to Y, than if Z had a very few neighbors. And so that's why this research allocation index penalizes pairs of nodes that have common neighbors that themselves have lots of other neighbors. So for example, when we measure the Resource Allocation index of nodes A and C, we would have one-third for node B. because node B is the common neighbor of A and C and has degree of 3, plus 1 over 3 which is for node D, which also has degree 3 and it's also a common neighbor of A and C. So the Resource Allocation index of A, C is going to be two-thirds. We can use the function of Resource Allocation index to compute the Resource Allocation index of all pairs of nodes that are not connected by an edge already in the graph. And if we sort it, we can see the Resource Allocation index of all pairs in this network. And if we look at the two edges that we're being kind of following, we see that in this case, A, G has a higher Resource Allocation index than I, H. And that is because I, H has a common neighbor which is G. But G has a high degree, has a degree of four. Whereas A, G, they're common neighbor is E, and E has a slightly smaller degree of three. So, that will give A, G a larger Resource Allocation index than I, H. The next measure is called the Adamic-Adar index. Which is very similar to the research allocation index. The only difference is that rather than dividing by the degree, it divides by the log of the degree. So in this case, the Adamic-Adar index for nodes A, C instead of being 2 over 3 is going to be 1 over the log of 3 plus 1 over the log of 3, which is 1.82. And there's a function for it too that you can use in network X is Adamic-Adar index. And again, we can compare A, G and H, I but in this case, it's going to be very similar to the Resource Allocation index. And so all we did was to put the log in the denominator. Next, measure 5 is going to be the preferential attachment score. So, if you recall, the preferential attachment model has the feature that nodes that have very high degree are more likely to get more neighbors. And so, the intuition behind this measure is that, well, if I'm looking at a pair of nodes and they both have a very high degree, then they're more likely to be connected to each other in the future. And so, what it does, is very simply, to take the product of the degree of the two nodes. And so, it's very simple. The preferential attachment score of X, Y is going to be the product of the number of neighbors of X, times the number of neighbors of Y. And A, C would have a preferential attachment score of 9. In network X, we can use the preferential_attachment function to compute the preferential attachment score over all the pairs of nodes that are not connected by an edge already. And here we can look at two edges that we've been following, A, G and H, I. And we see that A, G has a higher preferential attachment score than I, H, and that's because A has a degree of 3 and G has a degree of 4, which makes the preferential attachment score of 12. Whereas, I and H both only have one neighbor and so they have a score of 1. Okay, next, we're going to look at two other measures, that take in to account the community structure of the network. So in cases where you have a network and on top of the network, you have some knowledge about different communities. And here we'll think of communities as sets of nodes and we'll make the assumption that each node belongs only to one community. So one example is if this network was, for example, a network of communication among employees in a company. Then you may think that the department for which an employee works for would define a community. So for example, there's HR, and there's legal and so on, and so you could imagine thinking of those as communities. So if you had that type of structure, then you could use different measures for determining whether two nodes are likely to connect to each other or not. So in this case, let's assume that these network has two communities. So, these are the two communities. So there's A, B, D and C belong to Community 1, and the other nodes belong to Community 2. And what these two measures do, is they make the assumption that if two nodes belong to the same community, and they have many neighbors that also belong to the same community. Then they're more likely to form an edge, than if they had neighbors that belong to different communities, or if the two nodes themselves were in different communities. That's sort of the basic assumption that we make. And so this allows us to compute new measures, and so let's go over two of them. So the sixth measures that we are going to look at today is the number of common neighbors. But with a bonus for neighbors that belong to the same community as the two nodes we're looking at. So this is called the Common Neighbor Soundarajan-Hopcroft score. And if we're looking at nodes X and Y is simply going to be the number of common neighbors. So the size of intersection of N(X) and N(Y), plus some bonus, which is going to be the sum over all the common neighbors of this function, f(u). And f(u) is simply an indicator that tells us whether the u, which is a common neighbor of X and Y, belongs to the same community as X and Y, or not. And if it does, then it's a 1. If not, it's a 0. So it just simply gives a bonus for the common neighbors that belong to the same community as X and Y. And so for example, if we look at nodes A and C, their common neighbors are B and D, and they all belong to the same community. So A and C would get a score of 2 for having two common neighbors plus 2 bonus points, so we get 4. Nodes E and I would have a score of 2 because they both have a common neighbor, namely, G. And it belongs to the same community. So it would get a bonus point for a total of 2. But if we look at the edge A, G for example, they have one common neighbor, namely, E. But A and G are not in the same community. So E is not in the same community as A and G. Therefore A and G have a score of just 1. Now to use this on network X, we first have to tell network X which communities each node belongs to. So here, we're adding an attribute to each one of the nodes named, community, that tells which community the node belongs to. So for A through D, we'll say it belongs to community 0, and for the other ones it belongs to the community 1. And now we can use the function cn_soundarajan_hopcroft(G) which outputs an iterator of the tuples with the nodes and the score for each one, each pair, that is not already connected by an edge. And we can sort it and find which nodes have the highest score. And if we look at the two edges that we've been following, then we find that I, H has a score of 2, because their common neighbor G belongs to the same community as A do, whereas A, G has a score of 1. The last measure we're going to look at is a measure that is similar to the Resource Allocation index but it only takes into account nodes that are in the same community as the two nodes we're looking at. So if we're computing this measure which is called the Resource Allocation Soundarajan-Hopcroft score. And this is after the researchers that came up with this measure of X and Y. Then what we do is we sum over all the neighbors of X and Y. And rather than summing just one over the degree of the nodes of the common neighbors like we did in the standard Resource Allocation index. We now have this f(u) in the denominator of the fraction. And this function f(u) again is the same as before is 1, if u belongs to the same community as X and Y, and 0 otherwise. So in the case where you have a common neighbor that does not belong to the same community as X and Y, then that neighbor is not contributing anything to the sum because you have a 0 in the numerator. And so if we look at some examples for the nodes A and C, we would find that, well, because both of the neighbors B and D belong to the same community as A, C. Then we have the same score as we did before, and 1 over 3 plus 1 over 3, which is two-thirds for this Resource Allocation index. And if we look at for example, nodes E and I, we find that well, they have one common node which is node G. And node G has a degree of 4, so it has a score of 1 over 4 because G belongs to same community as E and I. However, if we look at two nodes that are not in the same community like A and G. Then they would have score 0 because their common neighbor, while they have one, namely E, belongs to a different community as at least one of the two nodes. And so, it doesn't contribute anything to the sum, so these two don't get anything in the sum. So we get a score of 0. We can use this function in network X, ra_index_soundarajan_hopcroft to compute the scores of all the non edges. And here, we find that I, H has a score of 0.25, whereas A, G has score of 0. Again because A, G, their common neighbor is not in the same community, at least one of A and G. And so in this case again, we find that I, H has a higher score. Okay, so these were the seven measures I wanted to talk about today and I want to point out two things. One, none of these measures actually tell you whether or not you should predict that a particular edge is going to come up in the future or not. It just gives a score that is supposed to give you a sense for whether or not these two nodes are likely to connect. The second thing is that different measures can give you different scores, right? So for example, we saw that some measures would give the edge H, I, a higher score that A,G and some measures would do the opposite. And so these measures aren't necessarily consistent with each other. So if you're actually trying to solve the link-prediction problem, typically what would happen is that you would use these measures as features. And then you would use a classifier, if you have some label data, you would train a classifier and use these measures as features in order to make the prediction. And so in summary, we talked about the link prediction problem which says that given a network, we´re tying to predict which edges are going to be formed in the future. And to do so, we've talked about the five different measures, the number of common neighbors, the Jaccard coefficient, which takes the number of common neighbors, but normalizes by the total number of neighbors that the two nodes have. The Resource Allocation index which is thinking along the lines of, if I know I needed to pass information or pass a resource to another node through their common neighbors, how much of that would actually get there? The Adamic-Adar index which is similar to the Resource Allocation index, but it takes the log in the denominator. And the preferential attachment score which bases the score on the idea that in preferential attachment, nodes with high degree tend to accumulate more edges. As well as two measures that require community information. So if you do have community information on your network, then you can use these two, and they're very similar to other ones we saw above. We have one that takes the common neighbors and it gives a bonus for the common neighbors that are in the same community as the two nodes. And then the Resource Allocation one which only considers common neighbors that are in the same community as the two nodes. And this is all for this video, I hope to you see next time, thanks.