X How to Schedule a Cascade in an Arbitrary Graph

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1 X How to Schedule a Cascade in an Arbitrary Grah Flavio Chierichetti, Cornell University Jon Kleinberg, Cornell University Alessandro Panconesi, Saienza University When individuals in a social network make decisions that deend on what others have done earlier, there is the otential for a cascade to form a run of behaviors that are highly correlated. In an arbitrary network, the outcome of such a cascade can deend sensitively on the order in which nodes make their decisions, but to do date there has been very little investigation of how this deendence works, or how to choose an order to otimize various arameters of the cascade. Here we formulate the roblem of ordering the nodes in a cascade to maximize the exected number of favorable decisions those that suort a given otion. We rovide an algorithm that ensures an exected linear number of favorable decisions in any grah, and we show that the erformance bounds for our algorithm are essentially the best achievable assuming P NP. 1. INTRODUCTION When eole in a social network are influenced by each other s decisions, one can see long runs of behavior in which eole s choices become highly correlated, resulting in a cascade of decisions. A striking asect of such henomena is the fact that the oulation s references might be quite heterogeneous at an individual level, but this heterogeneity is washed away by a global attern of behavior in which most eole arrive at the same observable decision. For examle, it often haens that two cometing roducts A and B each erform well in reliminary market research, but one quickly catures most of the market once they are both introduced. Cascades of this tye are sensitive to the order in which eole make decisions; when a grou of eole are choosing sequentially from among a set of otions, the consequences of certain early decisions can be magnified through the effects they roduce on the rest of the oulation [Arthur 1989; Banerjee 1992; Bikhchandani et al. 1992; Salganik et al. 2006]. Thus, if we imagine the roblem faced by a lanner who is trying to romote a certain otion within a oulation for examle, to manage the marketing of a new roduct there is a comlex scheduling roblem inherent in the question of how to bring this roduct to eole s attention over time. However, desite considerable research on the roerties of cascades, as well as aroaches to seeding cascades with initial adoters [Domingos and Richardson 2001; Keme et al. Flavio Chierichetti was suorted in art by the NSF grant CCF Jon Kleinberg was suorted in art by a John D. and Catherine T. MacArthur Foundation Fellowshi, a Google Research Grant, a Yahoo! Research Alliance Grant, and NSF grants IIS , IIS , and CCF Alessandro Panconesi was suorted in art by a Google Research Grant. Author s addresses: F. Chierichetti, Deartment of Comuter Science, Cornell University, Ithaca, NY 14853; J. Kleinberg, Deartment of Comuter Science, Cornell University, Ithaca, NY 14853; A. Panconesi, Diartimento di Informatica, Saienza University, Rome, Italy. Permission to make digital or hard coies of art or all of this work for ersonal or classroom use is granted without fee rovided that coies are not made or distributed for rofit or commercial advantage and that coies show this notice on the first age or initial screen of a dislay along with the full citation. Coyrights for comonents of this work owned by others than ACM must be honored. Abstracting with credit is ermitted. To coy otherwise, to reublish, to ost on servers, to redistribute to lists, or to use any comonent of this work in other works requires rior secific ermission and/or a fee. Permissions may be requested from Publications Det., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY USA, fax +1 (212) , or ermissions@acm.org. c 2012 ACM /2012/02-ARTX $10.00 DOI / htt://doi.acm.org/ /

2 X:2 2003; Richardson and Domingos 2002] and offering time-varying incentives [Hartline et al. 2008], very little is known about this scheduling asect of the roblem. In this aer, we rovide an algorithm for this tye of scheduling roblem in an arbitrary grah reresenting an underlying social network. We work within a model adated from a widely-used framework in the economic theory literature; in what follows, we first describe the model, and then our results. A model: Sequential decisions with ositive externalities. In an influential aer in the economic theory literature, Brian Arthur roosed a simle model for how cascades form [Arthur 1989]. He based his model on a scenario with two cometing roducts Y and N (mnemonic shorthand for yes and no ) that each exhibit ositive externalities they become more valuable as more eole use them. 1 In the model, there are two tyes of consumers those with a reference for Y, and those with a reference for N. (We will refer to these two tyes of consumers as Y - tyes and N-tyes, to avoid confusion between decisions and references.) We assume that when there are no other users of a roduct, each tye gets a ayoff of π 1 from using her referred roduct, and a ayoff of π 0 from using her non-referred roduct, with π 1 > π 0. The ositive externalities are manifested by a term that adds δ > 0 to the ayoff of a roduct for each user it has; thus, if there are currently m Y users of Y and m N users of N, then the ayoff of a Y -tye for roduct Y is π 1 + δm Y, and the ayoff of a Y -tye for roduct N is π 0 + δm N. Analogous ayoffs hold for an N-tye: π 1 + δm N for roduct N and π 0 + δm Y for roduct Y. Given this, if a consumer must decide which roduct to urchase based on the current number of users of each roduct, then the decision rule is very simle: a Y -tye comares π 1 + δm Y to π 0 + δm N, while an N-tye comares π 1 + δm N to π 0 + δm Y. 2 In both cases, if we define c = δ 1 π 1 π 0, the rule is equivalent to the following: ( ) When m Y m N c the consumer should choose the roduct with more users; when m Y m N < c, the consumer should choose based on her own reference. We will call c the decision arameter in this rule. Now, suose that consumers make urchasing decisions one at a time, starting from m Y = m N = 0, and each new consumer who arrives is a Y -tye with robability > 0 and an N-tye with robability 1 > 0. The key oint of the model is that if consumers can observe the current values of m Y and m N as they make their decisions, then as soon as one roduct has c users more than the other, it becomes locked in all future decisions will be in favor of the majority roduct. And since the decisions before this moment will favor each consumer s referred roduct, the quantity m Y m N over time can be analyzed quite simly: it is erforming a random walk in which it changes by +1 each ste with robability and by 1 with robability 1. Once this walk reaches the value c or c, one of the roducts has become locked in. If (by symmetry) we assume that < 1/2, it is not hard to show that the robability roduct Y is the one that becomes locked in is Θ( c n). Arthur s model thus catures, in a very simle fashion, the way in which cascades can lead to highly correlated behavior, and can be sensitive to the outcomes of a few early decisions. Subsequent models by Banerjee [Banerjee 1992] and Bikhchandani, Hirshleifer, and Welch [Bikhchandani et al. 1992], also very influential in economic 1 Examles of such roducts include cell hones, social media sites, and comuter oerating systems, since in all these cases they tend to acquire better service and more third-arty alications as their audiences grow. 2 We can assume that in the case of an exact tie in ayoffs, a consumer uses a canonical tie-breaking rule such as following the majority.

3 theory, have shown how similar henomena can arise when eole are influenced by earlier decisions not because of ositive externalities, but because these earlier decisions convey information that they can use in their resent decision. Given the generality of the henomenon, in what follows we abstract away from the language of roducts and urchases, and instead cast the discussion more broadly simly in terms of decisions and tyes (or signals): each erson chooses Y (which we also refer to as yes ) or N (also referred to as no ), and is of tye Y or N (or, equivalently, gets a ositive or negative signal). The roblem: Scheduling a cascade in a grah. This model can be directly generalized to oerate on an arbitrary grah G = (V, E), reresenting a social network on the individuals. As before, each individual i is a Y -tye (that is, individual i gets a ositive signal) with robability > 0 and an N-tye (the individual gets a negative signal) with robability 1 > 0, and i has the same ayoffs as before, excet that m Y and m N now denote the number of neighbors of i in G who have already chosen Y and N resectively. This reflects the fact that i only derives benefit when a neighbor in G makes the same decision that i does. With this new definition of m Y and m N, the decision rule governing i s decision is the same as before. Note that Arthur s original model corresonds recisely to this generalized model in the case when G is a comlete grah. We now return to the scheduling question raised at the outset. Suose that eole choose between Y and N using rule ( ) with some decision arameter c. Suose further that we are interested in heling to maximize the number of eole who choose Y. While the relative attractiveness of Y and N to any one individual is fixed by rule ( ), we have the ability to affect the order in which the cascade rogresses that is, the order in which individuals will make their decisions. Can we do this in a way that guarantees a reasonably large exected number Y decisions? There are several oints to note about this tye of question. First, this scheduling roblem will turn out to be trivial in comlete grahs, but as we will see, in general grahs it becomes much more comlex. Second, the question is interesting both when 1/2 and when < 1/2 that is, both when Y is the majority reference and when Y is the minority reference and we will consider both of these; but our main focus will be on the case when < 1/2, since this is the more challenging situation when we are trying to romote an otion that is inherently favored by a smaller fraction of eole. Finally, it is also imortant to note that Arthur s model of cascades, which we are following here, is distinct in some fundamental and imortant ways from the notion of threshold contagion that has been studied in a number of recent aers (e.g. [Blume et al. 2011; Centola and Macy 2007; Dodds and Watts 2004; Keme et al. 2003; Mossel and Roch 2007; Watts 2002]). In threshold contagion, an individual decides whether to change to a state (say, Y ) based only on the number of neighbors who have reviously chosen Y. This makes the outcome order-indeendent, and hence there is no scheduling issue. In Arthur s cascade model, on the other hand, nodes base their decision on the number of neighbors who have reviously chosen Y and the number of neighbors who have reviously chosen N. As we will see, in general grahs this makes the outcome sensitively deendent on the order in which the decisions are made, and hence leads naturally to questions of scheduling. Thus, concretely, the roblem we consider is as follows. We are given a grah G and the (constant) value of c used as the decision arameter in rule ( ) above; at the outset, we do not know the tyes of the nodes. One at a time, we select a node from G, reveal its tye to be a Y -tye or an N-tye (with robability and 1 resectively), and then have it make its decision according to rule ( ). The selection of the next node can X:3

4 X:4 deend on the outcome of the decisions made by earlier nodes. Our overall goal is to have a large exected number of nodes choose Y ; note here that (even for deterministic selection rules) the exectation is comuted over the random exosure of a tye (Y or N) at each node as the selection roceeds. Here is a basic question about the attainability of this goal. Let G = {G 1, G 2, G 3,...} be a family of grahs, with n i = V (G i ) denoting the number of nodes in G i. We say that the set G achieves constant adotion with decision arameter c if it is ossible to achieve a linear exected number of nodes choosing Y for all grahs in the family that is, if there exists a function f c () such that for all grahs G i in the family and all > 0, there is a way of selecting the nodes of G i for which the exected number of decisions equal to Y is at least f c () n i. A natural question is then the following: which families of grahs achieve constant adotion? We note first that the discussion of Arthur s model above shows that the family of comlete grahs achieves constant adotion with f c () = c. In a different direction, it is not hard to see that the family of emty grahs (those with no edges) achieves constant adotion with f c () =, since in such a grah each node is simly following its own reference. Of course, the scheduling roblem in both of these tyes of grahs is trivial, since all orderings yield the same exected value. But one intriguing oint about even this very simle air of examles cliques and emty grahs is that it shows how different families of grahs can achieve constant adotion for very different reasons. In the former case, the tight deendence among all nodes in the comlete grah results in a constant robability that all but a constant number of nodes will choose Y ; in the latter case, the comlete indeendence among all nodes in the emty grah ensures that a fraction of them will choose Y in exectation. For more comlex grahs, it is not a riori clear whether the cascade dynamics can exhibit roerties qualitatively different from either of these. Our Results. Our first main result is that the family of all grahs achieves constant adotion. Secifically, we show that for any n-node grah G, there is a way of ordering nodes so that the exected number of nodes choosing Y is at least Θ( c n). Thus, the family of cliques is in fact the worst case for ensuring many decisions equal to Y. We rove this result through an exlicit method for ordering the nodes, which roughly seaking combines ideas from the analysis of the comlete grah and the emty grah. In articular, we first identify a maximal set S of nodes that is sufficiently sarse that decisions within S can be made indeendently; we then try to extend S by adding nodes whose decisions have been forced by the decisions made in S. We also show that there are some grahs in which it is ossible to do much better than is ossible in either the comlete grah or the emty grah: there exist n-node grahs for which one can order the nodes to achieve an exected number of Y decisions that is n O(1/), and this is the best ossible for any grah. Moreover, it is comutationally hard to distinguish among these extremes: for any integer c 1, we show that it is NP-hard to distinguish between grahs for which the maximum exected number of nodes choosing Y is at least n o(n) and those for which it is at most c ɛ n. Thus far our model for ordering has been adative when we go to choose the next node in the ordering, we are able to see the tyes and decisions of all earlier nodes. It is also interesting to consider a non-adative version of the roblem; here the requirement is that we choose an ordering of the full node set in advance, before seeing the tyes or decisions of any nodes. For this non-adative version of the roblem, we show that when 1 2, it is not ossible to get n o(n) nodes in exectation to choose Y. We also show that if 1 2, then any ordering guarantees that the exected number of nodes choosing Y is at least

5 X:5 n 2 so any ordering is a 2-aroximation if 1 2. Finally, we rove that for small, it is NP-hard to distinguish between the case where the maximum exected number of nodes choosing Y is at least 1+ɛ n and the case where it is at most c ɛ n. 2. AN ALGORITHM In this section we develo and analyze an algorithm that allows us to obtain an exected number of Y s that is at least c n in any grah. The algorithm makes use of the notions of t-degenerate induced subgrahs and Erdös-Hajnal sequences, so we begin by defining these here. Given a grah G, the set of nodes S V (G) induces a t-degenerate grah G[S] if there exists an ordering v 1,..., v S of the nodes in S such that, for each i = 1,..., S, the degree of v i in G [{v 1,..., v i 1 }] is at most t. Such an ordering of the nodes in S is called a Erdös-Hajnal sequence for the t-degenerate grah G[S], and can be easily comuted in olynomial time. Finally, an induced subgrah G[S] is maximal t-degenerate if it is t-degenerate, and for every node v V (G) S it holds that G[S {v}] is not t-degenerate. We are now ready to introduce Algorithm 1. If a node v has already made its choice at some oint in the rocess, we say that it has been activated ; otherwise that it is unactivated. Algorithm 1 starts by selecting a maximal induced subgrah S with the informal roerties that (i) any node in the subgrah will make its choice somewhat indeendently from other nodes in the subgrah, (ii) every node outside the subgrah will get a chance of being forced to choose Y with robability at least c. Then, it schedules the maximal induced subgrah S interleaved with the nodes in V (G) S, whenever they get a chance of being forced to choose Y. ALGORITHM 1: A scheduling algorithm. 1: Let W V (G) be any set inducing a maximal (c 1)-degenerate grah in G 2: Let v 1, v 2,..., v W be a Erdös-Hajnal sequence of the nodes in W 3: for i = 1,..., W do 4: Schedule v i 5: while there exists an unactivated node v V (G) W with exactly c activated neighbors in W, all of which have chosen Y do 6: Schedule v 7: Schedule the remaining nodes arbitrarily The following theorem characterizes the erformance of Algorithm 1. THEOREM 2.1. Let G be any grah on n nodes, let be the robability that a node receives a ositive signal, and let c 1 be the threshold. Then, the random variable X reresenting the number of nodes having chosen Y with the scheduling roduced by Algorithm 1, is such that E[X] c n. We will rove Theorem 2.1, after having established a series of lemmas. The first lemma claims that every node in V (G) W will have exactly c activated neighbors in W at some oint in the execution. LEMMA 2.2. For every node v V (G) W, there will be (at least) one iteration k(v) in which v will have exactly c neighbors in W that have been scheduled. PROOF. By fixing a maximal (c 1)-degenerate set W, we have that every node v V (G) W will be connected to at least c nodes in W (for otherwise, we could add

6 X:6 v to W while still keeing W {v} has a (c 1)-degenerate grah, and therefore W would not be maximal.) Given W, v V (G) W, and the ordering v 1,..., v W of the nodes in W, let k = k(v) be the smallest integer such that v has at least c neighbors in the refix v 1, v 2,..., v k. After having scheduled v k, node v will have exactly c activated neighbors in W. The next lemma shows how nodes in V (G) W will not choose N until the end of the algorithm. LEMMA 2.3. If v V (G) W, either node v will choose Y at iteration k(v), or it will remain unactivated until line 7. PROOF. By induction, assume that l 0 nodes in V (G) W have been scheduled. By induction, each of them chose Y. By the test at line 5, if a new node v is scheduled at line 6, its only c activated neighbors w 1, w 2,..., w c in W have chosen Y and by the induction hyothesis all its actived neighbors (if any) in V (G) W have also chosen Y. Therefore v will choose Y. If, on the other hand, one of w 1, w 2,..., w c chose N, then v will not be scheduled until line 7 is reached. We then rove that each node in W will choose Y if its random signal is ositive. LEMMA 2.4. When a node v W is activated, it will choose Y if its random signal is ositive; therefore it will choose Y with robability at least. PROOF. Indeed, before reaching line 7, every activated node in V (G) W will choose Y. Thanks to our choice of the ordering of W s nodes, when we activate a node v W, there will be at most c 1 of its neighbors in W that have already been activated. Therefore, either v will be forced to choose Y, or its choice will be equal to its random signal. We now rove a lower bound on the robability that a node in V (G) W will choose Y. LEMMA 2.5. Every node v V (G) W will choose Y with robability at least c. PROOF. At iteration k(v), when v has exactly c active neighbors w 1, w 2,..., w c in W, we execute v iff each of the w i s chose Y. Since, for i = 1, 2,..., c, w i will choose Y if its signal is ositive, and since w i s signal is indeendent of other signals (and ositive with robability ), we have that w 1, w 2,..., w c will all choose Y, and therefore v will choose Y, with robability at least c. We can finally rove Theorem 2.1. PROOF PROOF OF THM Since every node v of G is either art of W or V (G) W, we have that the exected value of the random variable indicating a yes-choice of v is at least c. Linearity of exectation then gives the claim. We observe how the revious roof actually entails the following stronger corollary: COROLLARY 2.6. Let W be a maximum (c 1)-degenerate induced subgrah of G. Then, there exists a scheduling that guarantees an exected number of Y s of value at least W + c V (G) W. It is now useful to recall that the size of the (c 1)-degenerate subgrah is lower bounded by the size of the maximum indeendent set, and the size of the maximum indeendent set is lower bounded by the size of the maximum (c 1)-degenerate subgrah divided by c. Therefore, since c is a constant, Corollary 2.6 claims that it is

7 always ossible to get a scheduling of value Θ ( α(g) + c (n α(g))), where α(g) is the size of the maximum indeendent set of G. We now give an easy examle showing that Corollary 2.6 is tight: OBSERVATION 2.1. For any n c, and any c w n, there exists a grah of n nodes whose maximum (c 1)-degenerate induced subgrah has size w, whose maximum exected number of Y s has value w + Θ ( c (n w)). PROOF. Take an indeendent set on w c nodes and a disjoint clique on n w + c nodes. Clearly the maximum exected number of Y s is the sum of the maximum exected number of Y s for a scheduling of the clique lus the maximum exected number of Y s for a scheduling of the indeendent set. Any scheduling for the indeendent set, and for the clique, has the same exectation. For the indeendent set it is times its size. For the clique it is Θ( c ) times its size this is a consequence of the unfair gambler s ruin roblem. Suose that two gamblers, starting with c coins each, lay a sequence of games in such a way that layer 1 will win any fixed game indeendently with robability < 1 2, and layer 2 will win that same game with robability 1. After a game is layed, the loser gives one coin to the winner. The rocess ends when one layer goes bankrut. It is known (see, for instance, exercise 7.20 of [Mitzenmacher and Ufal 2005]) that the robability that layer 1 will end u with 2c coins (and therefore, the robability that layer 2 will go bankrut) is ( ) c 1 1 ( 1 ) 2c 1 = Θ ( c ). Since the exected number of stes needed to end the rocess is a constant, if c is a constant, the claim that the exected number of Y s in any scheduling of a clique of size n w is Θ ( c (n w)) is roved therefore the main claim is also roved. 3. MAXIMUM NUMBER OF Y S We just saw that, on every grah on n nodes, one can find a scheduling guaranteeing at least c n Y s on exectation, and that this lower bound is tight. The next question we answer is: what is the largest ossible number of Y s? ( ) LEMMA 3.1. There exists a grah G on n nodes that, for any Ω log n c n guarantees an exected number of Y s, for the adative scheduling with threshold c, of at least n O(1/). PROOF. Let n be n = c ( ( t c) + t + c, for a ositive integer t. Create a clique A on c t c) + c nodes: c nodes x 1, x 2,..., x c and one node v j {i 1,...,i c} for each j = 1,..., c, and for each {i 1,..., i c } ( ) [t] c. Also, create a disjoint indeendent set B on t nodes w 1,..., w t. Connect each node w i to each node v j A such that i A. We choose Ω ( ) ln n t. Observe that t = Θ ( c n). Now we describe the scheduling. Until we get c distinct Y choices, activate in order the nodes w 1, w 2,.... Suose we get c choices of tye Y at nodes w i1, w i2,..., w ic. Observe that those c nodes form{ a side of a} comlete biartite grah whose other side is c comosed by the nodes X = w j i 1,i 2,...,i c We execute the nodes in X in any order. j=1. Since each of them has at least c neighbors that have chosen Y (the w i1,..., w ic nodes) X:7

8 X:8 and zero N neighbors, they will all choose Y. We then schedule the nodes x 1, x 2,..., x c, that are not connected to any node w i. They will all choose Y. We then schedule the remainder of the clique since every remaining node in the clique has at least 2c neighbors that have chosen Y, and at most c neighbors that have chosen N, they will all choose Y. Finally, we schedule every remaining w i node each of them is only connected to c different Y nodes in the clique. Therefore they will all choose Y. The above scheduling has the roerty that after we obtain at least c different Y s every remaining node will choose Y. Furthermore each w i node activated before we get the (c+1)st choice of tye Y, is activated indeendently from the others. Therefore the exected number of N s is at most c 1 c 1 + n i=0 (( t i ) ) i (1 ) t i c + n (1 c 1 )t c (t) i = O ( Therefore the exected number of Y s is at least n O Again, we rove tightness of the above bound. OBSERVATION 3.1. For every grah G on n nodes, and for every Ω ( ) smallest exected number of N s is at least Ω. 1 1 ). i=0 ( ) 1 ( ) log n n, the PROOF. The exected number of N s, with any scheduling, and on any grah is at least the exected number of N s until a ositive signal is received, that is: n ( i (1 )i ) = 1 ( (1 ) n+1 n + 1 ). i=0 4. APPROXIMATING THE MAXIMUM Since the maximum number of Y s is at most n, Algorithm 1 gives a c aroximation to the roblem of finding the scheduling that maximizes the (exected) number of Y s. In this section we show that a better aroximation is not achievable if P NP. We start with a lemma that will turn out to be a crucial ingredient in the hardness of aroximation roof. LEMMA 4.1. In the adative case, ( if G is a grah of order n, with maximum indeendent set of size α, and if α Ω c n 1 ), then the maximum exected number of Y s is at most O(n α c c ). 1 PROOF. If > 2(3c 2)α, there is nothing to rove. We assume the contrary. We will base our analysis on the following consideration: if at most c 1 nodes chose Y during the rocess, then if we activate a node v connected to 3c 2 or more already activated nodes, it will necessarily choose N since at least 2c 1 of the 3c 2 activated neighbors will have chosen N themselves, and at most c 1 of them will have chosen Y. Therefore in the neighborhood of v there will be at least c more N s than Y s. Take any scheduling, and let S be the set of nodes that, at the moment they are activated, are connected to at most 3c 3 already-activated nodes. If we induce G on S, we obtain a grah G[S] that is (3c 2)-colorable: indeed, a (3c 2)- coloring can be obtained by coloring the nodes of S greedily in the order dictated by the scheduling. Since any indeendent set in G[S] is also an indeendent set in G, we have

9 that the indeendence number of G[S] is at most α since G[S] is (3c 2)-colorable, this imlies that the cardinality of S is at most S (3c 2) α. Suose that at most c 1 nodes in S get a ositive signal. Then, the number of nodes in S that choose Y is at most c 1. Furthermore, every node outside S will choose deterministically N. Therefore, the total number of Y s will be at most c 1. We comute the robability that at least c nodes in S will have a ositive signal: S (( ) ) S S i (1 ) S i ( S ) i ((3c 2) α ) i = i i=c ((3c 2)α) c 1 (3c 2)α 2 ((3c 2)α)c = 2 (3c 2) c α c c. i=c In general, the number of Y s we can get is at most n. It follows that the exected number of Y s is at most n O ( (c c ) α c c )+c 1 = O (n α c c ), where the last ste follows from c = O(1) and α Ω n 1 c. We are now ready to rove our hardness result. As already mentioned it makes use of Lemma 4.1, and of ideas develoed in the roof of Lemma 3.1. THEOREM 4.2. In the adative case, it is NP-hard to distinguish between grahs G for which the maximum number of Y s is at least n o(n) and grahs for which the maximum number of Y s is at most n c ɛ, for equal to an inverse olynomial in n. PROOF. We reduce from an indeendent set instance H on k = V (H) nodes. Zuckerman [Zuckerman 2007] roved that it is NP-hard to distinguish whether H has an indeendent set of size Ω ( k 1 ɛ), or if the maximum indeendent set of H has size O (k ɛ ), for an arbitrary small constant ɛ > 0. We choose = k 1+2ɛ ; our grah G will contain a coy of H and a disjoint clique C of order c ( k c) + k d, where d is an arbitrary constant larger than c. Therefore, the number of nodes will be n = k + c ( k c) + k d. (Observe that it holds = n 1 +Θ( ɛ d d).) For each set S of k nodes in H, we choose a set S of k unique nodes in C, and we add each edge from S to S. Observe that if the maximum indeendent set of H has size O (k ɛ ), then the maximum indeendent set of G has also size O (k ɛ ), and by Lemma 4.1 the maximum exected number of Y s in G will be at most O (n k cɛ c ) = O ( k d k cɛ k c+2cɛ) = O ( k d c+6cɛ) = n c O(ɛ). Lemma 4.1 can be alied since k = k k 1+2ɛ = k 2ɛ 1. On the other hand, if the maximum indeendent set of H has size Ω ( k 1 ɛ), by scheduling all the nodes in that indeendent set we will have that with robability 1 o(1) at least c nodes in the indeendent set will choose Y. Let T be the set of c nodes in the indeendent set that chose Y, and let T be the set of c nodes in the clique that they were maed to by the reduction. We schedule all the nodes in T they will all choose Y deterministically; then we can schedule the subset (of size k d ) of the nodes in C that are not connected to any node in H. Every such node will choose Y deterministically. Since k d = n o(n) (for any d > c), the claim is roved. i=c X:9

10 X:10 5. THE NON-ADAPTIVE PROBLEM Here, we consider the non-adative version of the scheduling roblem: first we fix a scheduling, and then we activate the nodes in the ordering it dictates. As the next theorem shows, in the non-adative case there is an interesting threshold henomenon at = 1 2 : THEOREM 5.1. If 1 2 (res., 1 2 ), then for each non-adative scheduling, the exected number of Y s is greater than or equal (res., less than or equal) 1 2 n. PROOF. Let us label the nodes in terms of an arbitrary scheduling, v 1, v 2,..., v n, so that node v i will be activated before node v j iff i < j. Our random rocess assigns a signal X i {Y, N} to each node v i indeendently in such a way that Pr[X i = Y ] =. Furthermore, after having fixed a schedule, and having samled the variables X 1,..., X n, the rocess becomes deterministic. Take the ith node in the scheduling v i, for any 1 i n. Let X i be the random vector X i = (X 1, X 2,..., X i ). For now, let us assume = 1 2. Then, clearly, every realization of X i is equirobable. Given a realization x = (x 1, x 2,..., x i ) of X i (that is, given a boolean vector on i coordinates), we denote its coordinate-wise comlement by x = ( x 1, x 2,..., x i ). Observe that the ith node chooses Y with realization x iff the ith node chooses N with realization x. This is trivially true if i = 1 (since in that case, v i = v 1 will make the choice based only on its signal). Now, if it is true u until i, then it is also true at i + 1: indeed, each neighbor of v i+1 has an index less than or equal i, and therefore will change its choice. If, given x, the neighbors of v i+1 forced it to make a choice, then with x they will force it to make the oosite choice: that is, if, with x, there were at least c more Y s than N s (res., N s than Y s ), then v i+1 was forced to choose Y (res., N ) but, the oosite will haen on x and the ith node will be forced to choose N (res., Y ). If on the other hand, given x, v i+1 made its choice according to its signal, it will make the choice dictated by the oosite signal with x. Since all realizations of X i are equally likely, and since we gave a erfect matching between that guarantees that only contains edges connecting a realization where v i chose Y with a realization where v i chose N, it holds that the robability that node i chooses Y is exactly 1 2. Therefore the claim for = 1 2, follows from linearity of exectation. Let us now consider > 1 2 (res., < 1 2 ). We define a couling between the user signal rocess P and a new rocess P : namely, each node first get a ositive signal with robability 1 2 and a negative signal with robability 1 2 if the first signal is negative (res., ositive), then the node gets a second signal that is ositive (res., negative) with robability 2 1 (res., 1 2). The actual user signal in P will be the last signal that the user receives. A simle calculation shows that the robability of a ositive signal in P is exactly. Since the function that mas a node s neighbors choices and the node signal to the node choice is monotonically non-decreasing in each coordinate (i.e., in the neighbors choices and the node signal), if > 1 2 (res., < 1 2 ), then the first signal that a node receives is sufficient to guarantee that the robability that the node chooses Y is at least (res., at most) 1 2. The second signal can only increase (res., decrease) that robability. Theorem 5.1 directly translates to a 2-aroximation algorithm for finding the nonadative schedule that maximizes the number of Y s, when 1 2 : any scheduling will

11 X:11 guarantee at least n 2 Y s on exectation, and the maximum number of Y s is n. We will show later that, for small, a constant aroximation is not achievable. Clearly, the bound of n 2 is not always tight: if < 1 2, and the grah is a clique, the exected number of Y s is at most O( c ) n; on the other hand, if > 1 2, then the exected number of Y s is at least (1 O(1 ) c ) n. Also, if the grah is an indeendent set, then the exected number of Y s is exactly n. More generally, we have the following easy lemma: LEMMA 5.2. If the grah G contains an induced (c 1)-subgrah of size k, then the maximum exected number of Y s, with a non-adative scheduling, is at least k. PROOF. The scheduling is similar to the one of Algorithm 1. Secifically, given a set W of nodes that induces a (c 1)-degenerate subgrah of G, we schedule the nodes in W according to some Erdös-Hajnal sequence (that is, we schedule them in such a way that whenever a node gets activated, that node has at most c 1 activated neighbors). The choice of v W, then, deends only on its rivate signal; that is, v will choose Y indeendently with robability. The claim follows from linearity of exectation. We observe that the uer bound of Lemma 4.1, that held in the adative setting, holds also in the non-adative case. We now show that the non-adative scheduling roblem cannot be aroximated to better than Ω ( c 1 ɛ) for sufficiently small. THEOREM 5.3. In the non-adative case, it is NP-hard to distinguish between grahs G for which the maximum number of Y s is at least n 1+ɛ and grahs for which the maximum number of Y s is at most n c ɛ, for equal to an inverse olynomial in n. PROOF. As in our other reduction, we start from an indeendent set instance G on n = V (G) nodes. It is NP-hard to distinguish whether G has an indeendent set of size Ω ( n 1 ɛ), or if the maximum indeendent set of G has size O (n ɛ ), for an arbitrary small constant ɛ > 0 (see [Zuckerman 2007]). We choose = n 1 c +ɛ. Now, if the indeendent set of G had size at most O (n ɛ ), then Lemma 4.1 guarantees that in the adative-case, and therefore in the non-adative case, the maximum number of Y s is at most O (n c n cɛ ) = n c O(ɛ). Suose instead that the indeendent set of G had size at least Ω ( n 1 ɛ), then Lemma 5.2 guarantees that in the non-adative case the maximum number of Y s is at least O ( n 1 ɛ ) = n 1+O(ɛ). REFERENCES ACEMOGLU, D., DAHLEH, M. A., LOBEL, I., AND OZDAGLAR, A. to aear. Bayesian learning in social networks. Review of Economic Studies. ARTHUR, W. B Cometing technologies, increasing returns, and lock-in by historical events. The Economic Journal 99, 394,

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