arXiv:1405.2486v1 [math.PR] 11 May 2014 CONVERGENCE, UNANIMITY AND DISAGREEMENT IN MAJORITY DYNAMICS ON UNIMODULAR GRAPHS AND RANDOM GRAPHS ITAI BENJAMINI, SIU-ON CHAN, RYAN O’DONNELL, OMER TAMUZ, AND LI-YANG TAN Abstract. In majority dynamics, agents located at the vertices of an undirected simple graph update their binary opinions synchronously by adopting those of the majority of their neighbors. On infinite unimodular transitive graphs (e.g., Cayley graphs), when initial opinions are chosen from a distribution that is invariant with respect to the graph automorphism group, we show that the opinion of each agent almost surely either converges, or else eventually oscillates with period two; this is known to hold for finite graphs, but not for all infinite graphs. √ On Erdős-Rényi random graphs with degrees Ω( n), we show that when initial opinions are chosen i.i.d. then agents all converge to the initial majority opinion, with constant probability. Conversely, on random 4-regular finite graphs, we show that with high probability different agents converge to different opinions. Contents 1. Introduction Acknowledgments 2. The almost sure period two property 2.1. Proof of Theorem 4 2.2. Example 1.2: an infinite graph without the almost sure period two property 3. Majority dynamics on G(n, p) 3.1. Heuristic analysis for the high degree case 3.2. Constant time to unanimity for the very high degree case 4. Majority dynamics on R(n, d) References Date: May 13, 2014. R. O’Donnell is supported by NSF grants CCF-1319743 and CCF-1116594. 1 2 6 6 8 10 11 11 12 18 18 2 I. BENJAMINI, S. CHAN, R. O’DONNELL, O. TAMUZ, AND L. TAN 1. Introduction Let G = (V, E) be a finite or countably infinite, locally finite, undirected simple graph. Consider time periods t ∈ {0, 1, 2, . . .} and, for each time t and i ∈ V , let Xt (i) ∈ {−1, +1} be the opinion of vertex i at time t. We define majority dynamics by X (1.1) Xt+1 (i) = sgn Xt (j), j∈∂(i) where ∂(i) is the set of neighbors of i in G. To resolve (or avoid) ties, we either add or remove i from ∂(i) so that |∂(i)| is odd. This ensures that the sum in the r.h.s. of (1.1) is never zero. Equivalently, we let ties be broken by reverting to the agent’s existing opinion. A well known result is the period two property of finite graphs, due to Goles and Olivos [8]. Theorem 1.1 (Goles and Olivos). For every finite graph G = (V, E), initial opinions {X0 (i)}i∈V and vertex i it holds that Xt+2 (i) = Xt (i) for all sufficiently large t. That is, every agent’s opinion eventually converges, or else enters a cycle of length two. This theorem also holds for some infinite graphs [7, 11]; in particular for those of bounded degree and subexponential growth, or slow enough exponential growth. In [15] it is furthermore shown that on graphs of maximum degree d the number of times t for which Xt+2 (i) 6= Xt (i) is at most −r ∞ X d+1 d+1 ·d· nr (G, i), d−1 d−1 r=0 where nr (G, i) is the number of vertices at graph distance r from i in G. However, on some infinite graphs there exist initial configurations of the opinions such that no agent’s opinion converges to any period; this is easy to construct on regular trees. A natural question is whether such configurations are “rare”, in the sense that they appear with probability zero for some natural probability distribution on the initial configurations. In [9] it was shown that on a regular trees, when initial opinions are chosen i.i.d. with sufficient bias towards +1, then all opinions converge to +1 with probability one. It was shown also that this is not the case in some odd degree regular trees, when the bias is sufficiently small. However, the question of whether opinions converge at all when the bias is small was not addressed. CONVERGENCE, UNANIMITY AND DISAGREEMENT 3 We show that indeed opinions almost surely converge (or enter a cycle with period two) on regular trees, whenever the initial configuration is chosen i.i.d. In fact, we prove a much more general result. A graph isomorphism between graphs G = (V, E) and G′ = (V ′ , E ′ ) is a bijection h : V → V ′ such that (i, j) ∈ E iff (h(i), h(j)) ∈ E ′ . Intuitively, two graphs are isomorphic if they are equal, up to a renaming of the vertices. The automorphism group Aut(G) is the set of isomorphisms from G to G, equipped with the operation of composition. G is said to be transitive if Aut(G) acts transitively on V . That is, if there is a single orbit V /G, or, equivalently, if for every i, j ∈ V there exists an h ∈ Aut(G) such that h(i) = j. G is said to be unimodular if Aut(G) is unimodular (see, e.g., Aldous and Lyons [1])1. G is unimodular if and only i the following “mass transport principle” holds: informally, in every flow on the graph that is invariant to Aut(G), the sum of what flows into a node is equal to the sum of what flows out. Formally, for every F : V × V → R+ that is invariant with respect to the diagonal action of Aut(G) it holds that X X f (i, j) = f (j, i), j∈∂(i) j∈∂(i) where i ∈ V is arbitrary. Many natural infinite trasitive graphs are unimodular. These include all Cayley graphs, all transitive amenable graphs, and, for example, transitive planar graphs with one end [10]. Our first result is the following. Theorem 1 (The almost sure period two property for unimodular transitive graphs). Let G be a unimodular transitive graph, and let the agents’ initial opinions {X0 (i)}i∈V be chosen from a distribution that is Aut(G)-invariant. Then, under majority dynamics, h i (1.2) P lim Xt+2 (i) − Xt (i) = 0 = 1, t and furthermore E [#{t : Xt+2 (i) 6= Xt (i)}] ≤ 2d, where d is the degree of G. That is, each node’s opinion almost surely converges to a cycle of period at most two. 1See [1] for an example (the “grandfather graph”) of a transitive graph that is not unimodular. 4 I. BENJAMINI, S. CHAN, R. O’DONNELL, O. TAMUZ, AND L. TAN In fact, this result is a special case of our Theorem 4 below, which applies to unimodular random networks. These include many natural random graphs such as invariant percolation clusters, uniform infinite planar triangulations [4] and any limit of finite graphs, in the sense of [5]; see Section 2 for a formal definition. In fact, this is such a large family that one may guess that any graph has what we call the almost sure period two property: if initial opinions are chosen i.i.d. from the uniform distribution over {−1, −1}, then each node’s opinion almost surely converges to a cycle of period at most two. This, however, is not true, as we show in the next example. Example 1.2. There exists an infinite graph G that does not have the almost sure period two property. As a reading of the details of this example will reveal, this graph is not of bounded degree. We conjecture that Conjecture 1.3. Every bounded degree graph has the almost sure period two property. We next consider the process of majority dynamics on a random finite graph, where initial opinions {X0 (i)}i∈V are chosen i.i.d. from the uniform distribution over {−1, +1}. Here convergence to period two is guaranteed by the Goles-Olivos Theorem. The question we tackle is whether agents all converge to the same opinion. The Erdős-Rényi graph G(n, p) is the distribution over graphs with n vertices in which each edge exists independently with probability p. A random regular graph R(n, d) is the uniform distribution over all d-regular connected graphs with n vertices. √ We first study G(n, pn ), where pn = Ω( n). Following the usual convention, we say that an event happens with high probability when it happens with probability that tends to one as n tends to infinity. Let µ0 = avgi∈V {X0 (i)}. Theorem 2 (Unanimity on high degree Erdős-Rényi graphs). Assume n ≥ n0 and p ≥ cn−1/2 , where n0 , c > 0 are sufficiently large universal constants. Then with probability at least .4 over the choice of G ∼ G(n, p) and the initial opinions, the vertices unanimously hold opinion sgn(µ0 ) at time 4. Next, we consider R(n, d), with d = 4. In this setting we prove the following result. We say that unanimity is reached at time t when Xt (i) = Xt (j) for all i, j ∈ V . Theorem 3 (Disagreement on random regular low degree graphs). Let Gn be drawn from R(n, 4), or be any sequence of 4-regular expanders CONVERGENCE, UNANIMITY AND DISAGREEMENT 5 with growing girth. Choose the initial opinions independently with probability 1/3 < p < 2/3. Then, with high probability, unanimity is not reached at any time. The following result on finite graphs is an immediate corollary of Theorem 4, which is a statement on infinite graphs. Corollary 1.4. Let G be drawn from R(n, d) with d ≥ 3, or from G(n, d/n) with d > 1. Then for every ε > 0 there exists a time t such that, with high probability, Xt+2 (i) = Xt (i) for all i ∈ V except a set of size ε · |V |. Furthermore, at this time t, the fraction of nodes for which Xt (i) = 1 is, with high probability, in [1/2 − ε, 1/2 + ε]. Hence at some time t almost all nodes will have already reached period at most two (at least temporarily), and without having reached agreement. This, together with the results above, motivates the following conjecture. Conjecture 1.5. Let G be drawn from G(n, dn /n). • When dn is a bounded, then for every ε > 0, with high probability, the fraction of nodes for which limt X2t (i) = +1 will be in [1/2 − ε, 1/2 + ε]. • When dn → ∞, then for every ε > 0, with high probability, the fraction of nodes for which limt X2t (i) = +1 will be in [0, ε] ∪ [1 − ε, 1]. That is, stark disagreement is reached for constant degrees, and unanimity is reached for super-constant degrees. An alternative, equally reasonable conjecture stipulates that this phase transition occurs, in fact, when degrees become high enough so that locally the graph ceases to resemble a tree. Given a vertex i in a large finite transitive graph and random uniform initial opinions, consider the Boolean function which is the eventual opinion of the majority dynamics at i, say at even times. An interesting question is whether this function is local; that is, is it determined with high probability by the initial opinions in a bounded neighbourhood of i? If it is non-local, can it be noise-sensitive [14] or it is correlated with the majority of the initial opinions? Our results so far heuristically suggest that in the bounded degree regime, majority dynamics is local, while when the degrees are growing fast enough the majority of the initial opinions determines the final outcome. In this respect we still did not find (or even conjecturally suggest) a family of graphs in which more interesting global behaviour occurs, such as in noise-sensitive Boolean functions. Indeed, we are curious to know if such a family exists. 6 I. BENJAMINI, S. CHAN, R. O’DONNELL, O. TAMUZ, AND L. TAN Acknowledgments The authors would like to thank Microsoft Research New England, where this research work was substantially performed. 2. The almost sure period two property In this section we shall consider generalized majority dynamics, or weighted majority dynamics. In this case we fix a function w : E → R+ and let X (2.1) Xt+1 (i) = sgn Xt (j) · w(i, j). j∈∂(i) Note that w(i, j) = w(j, i), since w is a function of the (undirected) edges. Note also that w(i, i) is possibly positive. We here too assume that w is chosen so that the sum in the r.h.s. can never be zero. A network is a triplet N = (G, w, X), where G = (V, E) is a graph as above, X : V → {−1, +1} is a labeling of the nodes, and w : E → R+ is a weighting of the edges. In the context of networks, we think of the process of generalized majority dynamics as a sequence of networks {Nt }, which all share the same graph Gt = G = (V, E) and edge weights wt = w, and where the node labels Xt are updated by (2.1). A rooted network is a pair (N, i) with N a network and i ∈ V . An isomorphism between two rooted networks (N, i) and (N ′ , i′ ) is a graph isomorphism h between G and G′ such that h(i) = i′ , X = X ′ ◦ h and w = w ′ ◦ h, where we here extend h to a bijection from E to E ′ . A directed edge rooted network is a triplet (N, i, j) with (i, j) ∈ E. Isomorphisms of directed edge rooted networks are defined similarly to those of rooted networks. A rooted network isomorphism class [N, i] is the set of rooted graphs isomorphic to (N, i). The set of connected, rooted network isomorphism classes, which we shall denote by G• , is equipped with the natural topology of convergence of finite balls around the root (see [2, 5]). This topology provides a Borel structure for probability measures on this space. A random network, or, more precisely, a random rooted network isomorphism class (we shall use the former term), is a rooted-networkisomorphism-class-valued random variable [N, I]; its distribution is a measure on G• . Denote by G•• the space of isomorphism classes of directed edge rooted networks [N, i, j]. [N, I] is a unimodular random CONVERGENCE, UNANIMITY AND DISAGREEMENT 7 network if, for every Borel f : G•• → [0, ∞], it holds that X X E f (N, I, j) = E f (N, j, I) . (2.2) j∈∂(I) j∈∂(I) We direct the reader to Aldous and Lyons [1] for an excellent discussion of this definition. Let {[Nt , I]}t∈N be a sequence of random networks defined as follows. Fix some random network [N0 , I] = [G, w, X0 , I]. For t > 0, let [Nt , I] = [G, w, Xt , I], where X (2.3) Xt (i) = sgn Xt−1 (j) · w(i, j). j∈∂(i) This sequence of random networks is coupled to share the same (random) graph, weights and root; only the labeling of the nodes Xt changes with time. We say that such a sequence is related by generalized majority dynamics. We impose the condition that w is such that almost surely no ties occur (i.e., the sum in (2.3) is nonzero). Claim 2.1. If [N0 , I] is a unimodular random network then so is [Nt , I], for all t ∈ N. This follows immediately from the fact that the majority dynamics map (G, w, Xt−1) 7→ Xt (i) given by (2.3) is indeed a function of the rooted network isomorphism class [Nt−1 , i] ∈ G• . For W, ε > 0 we say that (the weights w of) a random network [N, I] is (ε, W )-regular if the following two conditions hold. First, we require that X E w(I, j) ≤ W. j∈∂(I) Note that in the case that w is the constant function one, this is equivalent to having finite expected degree. Next, we require that X X (j)w(i, j) min ≥ε t ∂(I) x∈{−1,+1} j∈∂(I) almost surely. This is an “ellipticity” condition that translates to requiring that one is always ε-far from a tie. In the case that w is the constant function one and degrees are odd, this holds with ε = 1. We are now ready to state our main result of this section, which is a generalization of Theorem 1 from a fixed unimodular graph setting to a unimodular random network setting. 8 I. BENJAMINI, S. CHAN, R. O’DONNELL, O. TAMUZ, AND L. TAN Theorem 4. Let {[Nt , I]} be a sequence of (ε, W )-regular, unimodular random networks related by majority dynamics. Then h i (2.4) P lim Xt+2 (I) − Xt (I) = 0 = 1, t and furthermore E [#{t : Xt+2 (I) 6= Xt (I)}] ≤ 2W . ε Before proving this theorem, we show that it implies Theorem 1. When the underlying graph of a random network is a fixed transitive unimodular graph, and when the distribution of the labels Xt (i) is invariant to the automorphism group of this graph, then this random network is a unimodular random network [1]. Furthermore, since majority dynamics is generalized majority dynamics with weights 1, this random network is (1, d)-regular, where d is the degree. Hence Theorem 1 follows. 2.1. Proof of Theorem 4. In this section we prove Theorem 4. Our proof follows the idea of the proof of the period two property for finite graphs by Goles and Olivos [8]. Let {[Nt , I]} be a sequence of finite expected weighted degree, unimodular random networks related by majority dynamics. Define the function f : G•• → [0, ∞] by X w(i, k)Xk , f (N, i, j) = w(i, j) 1 + Xj sgn k∈∂(i) where N = (G, w, X) is a network and (i, j) is an edge in G. If [N, I] is unimodular then X X (2.5) f (N, j, I) . f (N, I, j) = E E j∈∂(I) j∈∂(I) Note that Xt+1 (i) = sgn P k∈∂(i) w(i, k)Xk , and so f (Nt , I, j) = w(I, j) (1 + Xt+1 (I)Xt (j)) and f (Nt , j, I) = w(I, j) (1 + Xt+1 (j)Xt (I)) . CONVERGENCE, UNANIMITY AND DISAGREEMENT 9 Hence we can write 2.5 for Nt as (2.6) E X j∈∂(I) w(I, j)Xt+1 (I)Xt (j) = E X j∈∂(I) w(I, j)Xt+1 (j)Xt (I) . Next, we define a “potential” X 1 w(I, j) (Xt+1 (I) − Xt (j))2 . ℓt = E 4 j∈∂(I) Note that ℓt is positive for all t, and also that it is finite for all t, since it is bounded from above by W , as a consequence of the (ε, W )-regularity of w. We would like to show that ℓ is non-increasing. By definition, X X 1 1 ℓt − ℓt−1 = − E w(I, j)Xt+1(I)Xt (j) + E w(I, j)Xt(I)Xt−1 (j) . 2 2 j∈∂(I) j∈∂(I) By (2.6) we can, in the expectation on the right, switch the roles of I and j. Rearranging, we get X 1 w(I, j)Xt(j) . ℓt − ℓt−1 = − E Xt+1 (I) − Xt−1 (I) 2 j∈∂(I) Now, Xt+1 (I) = sgn P w(I, j)Xt(j), and so X Xt+1 (I) − Xt−1 (I) w(I, j)Xt(j) j∈∂(I) j∈∂(I) X w(I, j)Xt (j) = |Xt+1 (I) − Xt−1 (I)| · j∈∂(I) X = 1{Xt+1 (I)6=Xt−1 (I)} · w(I, j)Xt(j). j∈∂(I) Hence ℓt − ℓt−1 X 1 = − E 1{Xt+1 (I)6=Xt−1 (I)} · w(I, j)Xt(j) , 2 j∈∂(I) and we have shown that ℓt is non-increasing. 10 I. BENJAMINI, S. CHAN, R. O’DONNELL, O. TAMUZ, AND L. TAN Now, by the (ε, W )-regularity of w we have that X w(I, j)Xt(j) ≥ ε, j∈∂(I) and so 1 P [Xt+1 (I) 6= Xt−1 (I)] · ε. 2 P Since ℓ1 ≤ W , and since ℓ1 ≥ ∞ t=2 ℓt−1 − ℓt = ℓ1 − limt ℓt , we can conclude that ℓt − ℓt−1 ≤ − ∞ X t=2 P [Xt+1 (I) 6= Xt−1 (I)] ≤ 2W . ε Hence E [#{t : Xt+2 (I) 6= Xt (I)}] < 2W , ε and by the Borel-Cantelli lemma h i P lim Xt+2 (I) − Xt (I) = 0 = 1. t This completes the proof of Theorem 4. 2.2. Example 1.2: an infinite graph without the almost sure period two property. Consider an infinite, locally finite graph defined as follows. Divide the set of nodes into “levels” L1 , L2 , . . ., where level Ln has 2n − 1 vertices. Connect each node in Ln with each of the nodes in Ln−1 , Ln and Ln−1 , except for the nodes in L0 , which are connected only to L1 . It follows that • Every pair of nodes in the same level have the same set of neighbours. • The majority of the neighbors of i ∈ Ln are in Ln+1 . Therefore, for all n and for all i, j ∈ Ln , it holds that X1 (i) = X1 (j). By induction, it follows that Xt (i) = Xt (j) for all t ≥ 1, and we accordingly denote Xt (Ln ) = Xt (i) for some i ∈ Ln . Furthermore, Xt (Ln ) = Xt−1 (Ln+1 ) for t ≥ 2, and so Xt (L0 ) = X1 (Lt+1 ) for t ≥ 2. Finally, {X1 (L3n )}n∈N are independent random variables, each uniformly distributed over {−1, +1}. Hence so are the random variables {X3t−1 (L0 )}t≥1 , and the single node in L0 (and in fact all the other nodes too) does not converge to period two. CONVERGENCE, UNANIMITY AND DISAGREEMENT 11 3. Majority dynamics on G(n, p) 3.1. Heuristic analysis for the high degree case. Herein we describe a “heuristic” analysis suggesting what should happen for majority dynamics in G(n, dn /n) when dn = ω(1) is sufficiently large. We suggest the reader keep in mind the parameter range dn = nδ where 0 < δ < 1 is an absolute constant. Our heuristic reasoning will suggest that unanimity is reached at time roughly 1/δ + O(1). Unfortunately, we will only be able to make some of this reasoning precise in the case that δ ≥ 1/2. That case is handled formally in Section 3.2. The global mean at time t is defined to be µt = avgi∈V {Xt (i)}. To analyze convergence to unanimity we will track the progression of µ2t over time. The quantity is nonnegative and it is easy to estimate it initially: Proposition 3.1. E [µ20 ] = n1 . On the other hand, we also have µ2t ≤ 1 with equality if and only if there is unanimity at time t. We suggest the following heuristic:2 Heuristic 3.2. In G(n, dn /n), assuming d = dn = ω(1) is sufficiently large, we expect µ2t+1 & dµ2t , provided dµ2t ≤ 1. Granting this heuristic, we expect the sequence µ20 , µ21 , µ22, . . . , µ2t to 2 t behave (up to constant factors) as n1 , nd , dn , . . . , dn until dt ≈ n. Once dt is within a constant factor of n we expect to reach near-unanimity in one more step, and to reach perfect unanimity after an additional step. For these reasons, we suggest that for dn = nδ , one may expect convergence to unanimity after 1δ + O(1) steps. Our intuition for how well this heuristic should hold when δ is “subconstant” is not very strong, but perhaps it indeed holds so long as dn = ω(1). The remainder of this section is devoted to giving some justification for Heuristic 3.2. Let us suppose that we have reached time t and that dµ2t ≪ 1. Computing just the expectation we have E µ2t+1 = avgi,j∈V E [Xt+1 (i)Xt+1 (j)] ≈ avgi6=j E [Xt+1 (i)Xt+1 (j)] . Here the approximation neglects the case i = j; this only affects the average by an additive quantity on the order of n1 , which is negligible even compared to dµ21 . In a random graph drawn from G(n, d/n) we 2We here use the notation A & B for A = Ω(B). 12 I. BENJAMINI, S. CHAN, R. O’DONNELL, O. TAMUZ, AND L. TAN expect all pairs of distinct vertices i, j to behave similarly, so we simply consider E [Xt+1 (i)Xt+1 (j)] for some fixed distinct i, j ∈ V . Here we come to the weakest point in our heuristic justification; we imagine that the neighbors of i and j are “refreshed” — i.e., that we can view them as chosen anew from the G(n, d/n) model. For simplicity, we also assume that i and j both have exactly d neighbors (an odd 2 number). We might also imagine that they have roughly dn neighbors in common, though we won’t use this. Under these assumptions we have E [Xt+1 (i)Xt+1 (j)] = E [sgn(R1 + · · · + Rd ) sgn(S1 + · · · + Sd )] where R1 , . . . , Rd are independent {−1, +1}-valued random variables with E [Ri ] = µt , the same is true of S1 , . . . , Sd , and we might assume 2 that some dn of the Ri ’s and Si ’s are identical. In any case, by the FKG Inequality (say), we have E [sgn(R1 + · · · + Rd ) sgn(S1 + · · · + Sd )] ≥ E [sgn(R1 + · · · + Rd )] E [sgn(S1 + · · · + Sd )] . Thus to finish our heuristic justification of µ2t+1 & dµ2t it suffices to argue that √ (3.1) |E [sgn(R1 + · · · + Rd )]| & d|µt |. Without loss of generality we assume µt ≥ 0. By the Central Limit Theorem, R1 +· · ·+Rd is distributed essentially as Z ∼ N(dµt , d(1 − µ2t )) ≈ N(dµt , d). (We are already assuming dµ2t ≪ 1, so µ2t ≪ 1 as well.) By the symmetry of normal random variables around their mean we have h √ √ i ′ E [sgn(Z)] = P [0 ≤ Z ≤ 2E [Z]] = P − dµt ≤ Z ≤ dµt , where Z ′ is a standard normal random variable. This last quantity q √ √ 2 · is asymptotic to dµ assuming dµt ≪ 1 ⇐⇒ dµ2t ≪ 1, t π “confirming” (3.1) 3.2. Constant time to unanimity for the very high degree case. In this section we give a precise argument supporting the heuristic √ analysis from Section 3.1 in the case of G(n, pn ) when p = pn ≫ 1/ n. The main task is to analyze what happens at time 1; after that we can apply a result from [12], relying on the fact that a random graph is a good expander. For simplicity we will assume n is odd, so that µt is never 0. √ 1 . Proposition 3.3. (Assuming n is odd,) E [sgn(µ0 )µ1 ] ≥ π2 p − n√ p CONVERGENCE, UNANIMITY AND DISAGREEMENT 13 Proof. We have E [sgn(µ0 )µ1 ] = avgi∈V [sgn(µ0 )X1 (i)] and by symmetry the expectation is the same for all i. Let’s therefore compute it for a fixed i ∈ V ; say, i = n. Now suppose we condition on vertex n having exactly d neighbors when the graph is chosen from G(n, p). The conditional expectation does not depend on the identities of these neighbors; thus we may as well assume they are vertices 1, . . . , d. Writing X(j) = X0 (j) for brevity, we therefore obtain E [sgn(µ0 )µ1 ] = n−1 X d=0 (3.2) Pr[Bin(n − 1, p) = d]× E [Majn (X(1), . . . , X(n))Majd′ (X(1), . . . , X(d), X(n))] . Here d′ denotes d when d is odd and d + 1 when d is even, and Majk (x1 , . . . , xℓ ) denotes sgn(x1 + · · · + xk ). We can lower-bound the expectation in line (3.2) using Fourier analysis; by Parseval’s identity, X [n (S)Maj \ Maj (3.2) = d′ (S). S⊆[n] [k (S) only depends on |S|; furthermore, By symmetry, the value of Maj [k (S) depends only on |S| and not it’s well known that the sign of Maj on k [14]. Thus all summands above are nonnegative so we obtain X [n (S)Maj \ Maj (3.2) ≥ d′ (S). |S|=1 [k (S) = 2k k−1 Finally, for odd k we have the explicit formula Maj ≥ k−1 2 2 √ 2/π √ for any |S| = 1. Since the two majorities have exactly d′ coordik nates in common, we conclude r r p p ′ 2/π 2/π d d 2 2 √ ≥ . (3.2) ≥ d′ √ = π n π n n d′ Putting this into the original identity we deduce i 2 1 hp E [sgn(µ0 )µ1 ] ≥ √ E Bin(n − 1, p) . π n We have the standard estimates3 hp i p √ 1 √ ≥ np − 1.5/ np. E Bin(n − 1, p) ≥ (n − 1)p − p 2 (n − 1)p 3For the first inequality see, e.g., http://mathoverflow.net/questions/121411/expectation-of-square-r 14 I. BENJAMINI, S. CHAN, R. O’DONNELL, O. TAMUZ, AND L. TAN Thus we finally obtain E [sgn(µ0 )µ1 ] ≥ 2√ 1 p− √ , π n p as claimed. Proposition 3.4. We have E (sgn(µ0 )µ1 )2 = E µ21 ≤ p + 3 . pn Proof. We have E [µ21 ] = n1 + avgi6=j {E [X1 (i)X1 (j)]}; by symmetry it therefore certainly suffices to show E [X1 (i)X1 (j)] ≤ p + (3.3) 2 pn for some fixed pair of vertices i 6= j. Let us condition on the neighborhood structure of vertices i and j. Write X(j) = X0 (j) as in the proof of Proposition 3.3, and write N1′ = ∂(i) \ {j}, N2′ = ∂(j) \ {i}. Then E [X1 (i)X1 (j)] = E [Maj((X(k))k∈N1 ) · Maj((X(k))k∈N2 )] for some sets N1′ ⊆ N1 ⊆ N1 ∪ {i, j} and similarly N2 . Writing M = N1 ∩ N2 and also MajN = Maj(X(r) : k ∈ N) for brevity, the above is equal to EN1 ,N2 ,(X(k))k∈M E MajN1 (X(k))k∈M · E MajN2 (X(k))k∈M r h 2 i ≤ EN1 ,N2 ,(X(k))k∈M E MajN1 (X(r))k∈M r h 2 i E MajN (X(r))k∈M × EN ,N ,(X(k)) 1 2 k∈M 2 (3.4) h 2 i , = EN1 ,N2 ,(X(k))k∈M E MajN2 (X(r))k∈M where the inequality is Cauchy–Schwartz and the final equality is by symmetry of i with j. To analyze (3.4), suppose we condition on N1 and N2 (hence also M). By symmetry, the conditional expectation depends only on |N1 | = n1 and |M| = m; by elementary Fourier analysis [14] it equals m X |S| X 2 2 \ \ Maj Maj n (S) = n (S) 1 S⊆[m] S⊆[n1 ] n1 |S| 1 X m |S| 2 \ m Maj ≤ n1 (S) = Stab n1 [Majn1 ]. n1 S⊆[n1 ] CONVERGENCE, UNANIMITY AND DISAGREEMENT 15 Finally, we have the bounds [13] m Stab nm [Majn1 ] ≤ , 1 n1 (3.5) 2 m Stab nm [Majn1 ] ≤ arcsin +O 1 π n1 1 p √ 1 − (m/n1 )2 n ! . Although the second bound here would save us a factor of roughly π2 , for simplicity we’ll only use the first bound. It yields |M| E [X1 (i)X1 (j)] ≤ E . |N1 | Each vertex in N1 \ {j} has an (independent) probability p of being in M; as for j, we’ll overestimate by assuming that if j ∈ N1 then it is always in M as well. This leads to |M| 1 E ≤p+E . |N1 | |N1 | Finally, recall that |N1 | is distributed as Bin(n − 1, p) rounded up to the nearest even integer. Thus (see, e.g., [6]) 1 2 1 2 E ≤E = (1 − (1 − p)n ) ≤ . |N1 | (Bin(n − 1, p) + 1)/2 pn pn This completes the proof. Proposition 3.5. Assume n ≥ n0 and p ≥ √cn , where n0 , c > 0 are suf √ ficiently large universal constants. Then P sgn(µ0 )µ1 ≥ .006 p ≥ .4004. Proof. Write W = sgn(µ0 )µ1 . The “one-sided Chebyshev inequality” implies that P [W ≥ .01E [W ]] ≥ .992 E[W 2 ] E[W ]2 + .992 − 1 . Combining Propositions 3.3, 3.4, we have As .02 π E [W 2 ] p + 3/(pn) π2 + O p21n . √ 2 ≤ 2 ≤ 2√ 4 ( π p − 1/(n p)) E [W ] > .006 and .992 π 2 /4+.992 −1 > .4004, the claim follows. For good expander graphs of degree d, the results of [12] show that unanimity √ will be reached quickly if the global mean ever significantly exceeds 1/ d (in magnitude). In our situation, we essentially have √ degree-pn graphs with a constant chance of global mean exceeding Ω( p) 16 I. BENJAMINI, S. CHAN, R. O’DONNELL, O. TAMUZ, AND L. TAN at time 1. Consequently we are able to show convergence to unanimity √ provided p ≫ 1/ n. We’ll need the following result, which is essentially Proposition 6.2 from [12] (but slightly modified since we need not have perfectly regular graphs): Lemma 3.6. Assume G = (V, E) satisfies the following form of the “Expander Mixing Lemma”: for all A, B ⊆ V , p E(A, B) − p|A||B| ≤ λ |A||B|, where E(A, B) denotes #{(u, v) ∈ E : u ∈ A, v ∈ B}. Then if majority dynamics on G ever has µt ≥ |α| then #{i ∈ V : Xt+1 (i) = − sgn(µt )} ≤ 2λ2 . α 2 p2 n To use this, we’ll also need the following claim that follows easily from a result of Vu on the spectra of G(n, p) [16]. Lemma 3.7. For G ∼ G(n, p) with p ≫ (log n)4 /n and n high enough, √ the Expander Mixing Lemma holds for G with λ = 4 np, except with probability at most o(1). Proof. Let P0 be the (random) adjacency matrix of G, and let P be given by P = P0 + D, where D is a random diagonal matrix whose each diagonal entry is one with probability p and zero otherwise. P can be thought of as the adjacency matrix of a graph G′ which is obtained from G by adding each self-loop with probability p. Let Q be the n × n matrix whose entries are all equal to p. Then, since p ≫ (log n)4 /n, by [16] it holds with high probability that √ |P − Q| ≤ 3 np, where | · | is here the L2 operator (spectral) norm. Equivalently, for any two vectors v, w ∈ Rn , √ (3.6) |v ⊤ (P − Q)w| ≤ 3 np · |v||w|. Let A, B ⊂ V be any two subsets of vertices. Then the number of edges between A and B is given by ⊤ ⊤ ⊤ E(A, B) = 1⊤ A P0 1B = 1A (P − D)1B = 1A P 1B − 1A D1B . Now, 1⊤ A D1B is at most p |A ∩ B| and therefore, for n high enough it ⊤ holds that 1A D1B ≤ np|A||B|. By (3.6), p √ ⊤ np|1 ||1 | = 3 np|A||B|. |1⊤ P 1 − 1 Q1 | ≤ 3 A B B B A A CONVERGENCE, UNANIMITY AND DISAGREEMENT 17 Since 1⊤ A Q1B = p|A||B|, with high probability p ⊤ ⊤ P 1 − 1 D1 − 1 Q1 | ≤ 4 np|A||B|. E(A, B) − p|A||B| = |1⊤ B B B A A A Combining the previous two lemmas with Proposition 3.5 we obtain: Proposition 3.8. Assume n ≥ n0 and p ≥ √cn , where n0 , c > 0 are sufficiently large universal constants. Then with probability at least .4003 we have c #{i ∈ V : X2 (i) 6= sgn(µ0 )} ≤ 2 . p √ In G(n, p) (with p ≫ 1/ n, say), almost surely each vertex has degree at least (p/2)n, which in turn exceeds 2c/p2 if p > (4c/n)1/3 . Thus we may conclude: Theorem 3.9. Assume n ≥ n0 and p ≥ cn−1/3 , where n0 , c > 0 are sufficiently large universal constants. Then with probability at least .4 over the choice of G ∼ G(n, p) and the initial opinions, the vertices unanimously hold opinion sgn(µ0 ) at time 3. In case n−1/2 ≪ p . n−1/3 we need an extra time period. By assuming p ≫ n−1/2 , the right-hand side in Proposition 3.8 can be made smaller than any desired positive constant. Then applying the two lemmas again we obtain: Proposition 3.10. Assume n ≥ n0 and p ≥ √cn , where n0 , c > 0 are sufficiently large universal constants. Then with probability at least .4002 we have c #{i ∈ V : X3 (i) 6= sgn(µ0 )} ≤ . p Now we can finish as in the case of p ≫ n−1/3 ; we get: Theorem (2). Assume n ≥ n0 and p ≥ cn−1/2 , where n0 , c > 0 are sufficiently large universal constants. Then with probability at least .4 over the choice of G ∼ G(n, p) and the initial opinions, the vertices unanimously hold opinion sgn(µ0 ) at time 4. As a final remark, when p = o(1) we can (with slightly more effort) improve the probability bound of .4 to any constant smaller than 2/π ≈ .6366 by using (3.5). 18 I. BENJAMINI, S. CHAN, R. O’DONNELL, O. TAMUZ, AND L. TAN 4. Majority dynamics on R(n, d) Proposition 4.1. Let Gn be drawn from R(n, 4), or be a sequence of 4-regular expanders with growing girth. Choose the initial opinions independently with probability 1/3 < p < 2/3. Then, with high probability, unanimity is not reached at any time. Note that Theorem 3 is rephrasing of this proposition. Proof. Consider a growing sequence of d-regular expanders with girth growing to infinity, denoted by Gn . By [3], p-Bernoulli percolation will contains a unique giant component of size proportional to Gn , provided p > 1/(d − 1). The same holds for random d-regular graphs. In particular if d = 4 and 1/3 < p < 2/3 an open giant component and a closed giant component will coexist. Since d = 4 is even, we take majorities over the four neighbors and the vertex itself, as we explain above. To show that with high probability unanimity is not reached on these graphs, it is enough then to show that the giant component of 1/2-Bernoulli percolation contains cycles. The opinions on the cycles will not change in the process of majority dynamics, since each node will have three (including itself) neighbors on the cycle with which it agrees. To see this, perform the percolation in two stages: first carry out (p− ǫ)-Bernoulli percolation (such that 1/3 < p − ǫ), and then sprinkle on top of it an independent ǫ-Bernoulli percolation. If the first percolation already contains a cycle we are done. Otherwise the giant component is a tree. Pick an edge of the random giant tree that splits the tree to two parts, so that each part has size at least 1/4 of the tree. Denote these two parts by A and B. As in the uniqueness proof of [3], since Gn is an expander and A and B has size proportional to the size of Gn . there are order Θ(n) disjoint paths of length bounded by a function depending only on the expansion. Thus the ǫ-sprinkling connects A and B with order Θ(n) disjoint open paths, creating many cycles with probability tending to 1 with n, and we are done. References [1] David Aldous and Russell Lyons, Processes on unimodular random networks, Electronic Journal of Probability 12 (2007), no. 54, 1454–1508. MR2354165 (2008m:60012) [2] David Aldous and J. Michael Steele, The objective method: probabilistic combinatorial optimization and local weak convergence, Probability on discrete structures, 2004, pp. 1–72. MR2023650 (2005e:60018) CONVERGENCE, UNANIMITY AND DISAGREEMENT 19 [3] Noga Alon, Itai Benjamini, and Alan Stacey, Percolation on finite graphs and isoperimetric inequalities, The Annals of Probability 32 (2004), no. 3A, 1727– 1745. MR2073175 (2005f:05149) [4] Omer Angel and Oded Schramm, Uniform infinite planar triangulations, Communications in Mathematical Physics 241 (2003), no. 2-3, 191–213 (English). [5] Itai Benjamini and Oded Schramm, Recurrence of distributional limits of finite planar graphs, Electronic Journal of Probability 6 (2001), no. 23, 13 pp. (electronic). MR1873300 (2002m:82025) [6] Min-Te Chao and William Strawderman, Negative moments of positive random variables, Journal of the American Statistical Association 67 (1972), no. 338, 429–431. [7] Yuval Ginosar and Ron Holzman, The majority action on infinite graphs: strings and puppets, Discrete Mathematics 215 (2000), no. 1-3, 59–72. [8] Eric Goles and Jorge Olivos, Periodic behaviour of generalized threshold functions, Discrete Mathematics 30 (1980), no. 2, 187–189. [9] Yashodhan Kanoria and Andrea Montanari, Majority dynamics on trees and the dynamic cavity method, The Annals of Applied Probability 21 (2011), no. 5, 1694–1748. [10] Russell Lyons and Yuval Peres, Probability on trees and networks, 2013. [11] Gadi Moran, On the period-two-property of the majority operator in infinite graphs, Transactions of the American Mathematical Society 347 (1995), no. 5, 1649–1667. [12] Elchanan Mossel, Joe Neeman, and Omer Tamuz, Majority dynamics and aggregation of information in social networks, Autonomous Agents and MultiAgent Systems (June 2013). [13] Ryan O’Donnell, Computational applications of noise sensitivity, Ph.D. Thesis, 2003. , Analysis of Boolean functions, Cambridge Univesity Press, 2014. [14] [15] Omer Tamuz and Ran J Tessler, Majority dynamics and the retention of information, arXiv preprint arXiv:1307.4035 (2013). [16] Van Vu, Spectral norm of random matrices, Combinatorica 27 (2007), no. 6, 721–736. (I. Benjamini) Faculty of Mathematics and Computer Science, Weizmann Institute of Science. (R. O’Donnell) Department of Computer Science, Carnegie Mellon University. (S. Chan, O. Tamuz) Microsoft Research New England. (L. Tan) Department of Computer Science, Columbia University.

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