So, consider the social graph on the right, which consists of 8 people, and their social relationships to one another. We've initialized Charlie to have flipped, and what we want to do is trace out the flipping behavior over time. This process is known as contagion. And what contagion does in each iteration, we go through person by person to see if their thresholds have been met. And if so, they'll become adopters. And then we use, well on the next one iteration to see the next, and then that iteration to see the next, and so forth. And one thing we're going to have to remember the key point here of the, the key consideration. When you're walking through the flipping examples is that when a node flips in a given iteration, you can't use that updated value until the next iteration. And we'll see what that means now, as we start. So, we're going to assume the threshold is 50%, just to keep things simple. So if half or more of a person's neighbors have flipped, then the node will flip. So let's start at time one, which is just from this graph. So, start by looking at Anna. So, we see Anna has two neighbors, right? So, social influence can come from two neighbors from Charlie and Dana. At time one, Charlie has flipped but Dana has not flipped, right? So, that means that one of her two neighbors or one out of two have flipped, which is 50%. And that's right at the threshold, which means that Anna will flip. Now for Ben, you see Ben has two neighbors Charlie and Dana again. And again, Charlie's flipped and Dana has not. Which means that half of Ben's neighbors have flipped, so Ben will also flip. Right to this threshold, just enough social influence. And for Dana, Dana has four neighbors, and at this time, none of them have flipped. Because remember, even though Anna and Ben have flipped, that's not going to come into effect until time two. Ben and Anna, and even Frank are her neighbors, and none of them have flipped. So, none of her four neighbors have flipped, so 0%. So she's not going to flip. For Eve, one of her neighbors has flipped. She has five of them, one, two, three, four, five. And Charlie has flipped, who's one of them, which is 1/5, that's only 20%, and so she will not flip. This 20% is less than 50%. So, we can also write that in here, 50% is equal to the flipping threshold of 50%. It means he flips is equal to 50%, this is less than 50% and this is less than 50%, which is their threshold. And so, this is the updated graph now at time one. So this is, this is time zero, if you will. Initially, this is at time one. What's just happened? As we see here now three of the nodes have flipped Anna, Charlie and Ben. And now, we move to time two. We're going to walk through the remaining nodes that have not flipped and see which of them will flip. Now we proceed to time two using this updated graph, and again, we'll consider all the unflipped notes. So first, let's look at Dana. Now, Dana has four neighbors, right? She has Ben, Ana, Eve and Frank. And, two of them, Ben and Ana have flipped, where as Eve and Frank have not. So that's two out of four, which is 50%, which is equal to the flipping threshold of 50%. Therefore, Dana will now flip. If we look at Eve, the situation hasn't changed at all from last time. She has five neighbors, only one of them have flipped. So that's 1/5, just 20%. It's just less than the flipping threshold of 50%, so she's not going to flip. And for Frank, George and Hannah, again none of their neighbors have flipped again. So, they're not going to flip. Frank, George and Hannah, this is no flip. And so this diagram now shows where we are at the end of time two. So this is time one, as we showed the last time that this is now time two. So now we have four flipped nodes, we have Anna, Charlie, Ben and Dana haven't flipped. So now let's look at what happens at time three. So we start now with all the unflipped nodes again. So, we start with Eve. Eve has two nodes flipped now, Charlie and Dana, out of five total. Two over five which is 40%, still less than 50%. So, no flip. For Frank, Frank has one of his four neighbors flipped, 1/4, 25%. Again, that's less than 50%, not enough, so no flip. And for George and Hannah, we have no nodes flipped. We have 0 out of, they have the same number of neighbors, which is 3. So 0%, no flip. So, none of the nodes are switching at time three. Because neither Eve, nor Frank, nor George, nor Hannah, none of these guys, are flipping at all, at time three. Now, why is this? Well, each person has too large a fraction of their neighbors, to friends that have not flipped, right? So, they have sort of density concentrated within here. And each of them have enough influence on each other that they're not going to flip, because there's not enough outside influence force. Out of all of them, Eve has the most outside influence at 40%, but that's still not quite high enough. And since none of the, none of the flipping states have changed for all the next iterations, if we try to go through uneven nodes again, we are going to get exactly the same results. So, there is no point in continuing further computation. And at time three we've reached what's what we could call the equilibrium of the contagion.
