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Suppose that G is a (two-way) in nite path, with nodes labeled {. . .

, 2, 1, 0, 1, 2, . . .

}, and there is a new behavior B with threshold q = 1/2. Now, rst, suppose that the single node 0 initially adopts B. Then in time step t = 1, nodes 1 and 1 will adopt B, but 0 (observing both 1 and 1 in their initial behaviors A) will switch to A.

As a result, in time step t = 2, nodes 1 and 1 will switch back to behavior A, and the new behavior will have died out completely. On the other hand, suppose that the set S = { 1, 0, 1} initially adopts B. Then in time step t = 1, these three nodes will stay with B, and nodes 2 and 2 will switch to B.

More generally, in time step t = k, nodes { k, (k 1), . . .

, k 1, k} will already be following behavior B, and nodes (k + 1) and k + 1 will switch to B. Thus, every node is converted by S = { 1, 0, 1}, the set S is contagious, and hence the contagion threshold of G is at least q = 1/2. (Note that it would in fact have been suf cient to start with the smaller set S = {0, 1}.

) In fact, 1/2 is the contagion threshold of G: given any nite set S adopting a new behavior B with threshold q > 1/2, it is easy to see that B will never spread past the rightmost member of S. It is instructive to try this oneself on other graphs; if one does, it quickly becomes clear that while a number of simple graphs have contagion threshold 1/2, it is hard to nd one with a contagion threshold strictly above 1/2. This suggests the following question: Does there exist a graph G with contagion threshold q > 1/2 We will shortly answer this question, after rst resolving a useful technical issue in the model.

Progressive vs. nonprogressive processes. Our model thus far has the property that as time progresses, nodes can switch from A to B or from B to A, depending on the states of their neighbors.

Many behaviors that one may want to model, however, are progressive, in the sense that once a node switches from A to B, it remains with B in all subsequent time steps. (Consider, for example, a professional community in which the behavior is that of returning to graduate school to receive an advanced degree. For all intents and purposes, this is a progressive process.

) It is worth considering a variation on our model that incorporates this notion of monotonicity for two reasons. First, it is useful to be able to capture these types of settings; and second, it will turn out to yield useful ways of thinking about the nonprogressive case as well. We model the progressive contagion process as follows.

As before, time moves in discrete steps t = 1, 2, 3, . . .

. In step t, each node v currently following behavior A switches to B if at least a q fraction of its neighbors is currently following B. Any node following behavior B continues to follow it in all subsequent time steps.

Now, if S is the set of nodes initially adopting B, we let hq (S) denote the set of nodes adopting k B after one round of updating in this progressive process, and we let hq (S) denote the result of applying hq to S a total of i times in succession. We can then de ne the notion of converted and contagious with respect to hq exactly as we did for hq . With a progressive process, it seems intuitively that it should be easier to nd nite contagious sets after all, in the progressive process, one does not have to worry about early adopters switching back to the old behavior A and thereby killing the spread of B.

In view of this intuition, it is perhaps a bit surprising that for any graph G, the.
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