CONVERGENCE, UNANIMITY AND DISAGREEMENT IN MAJORITY DYNAMICS ON UNIMODULAR

CONVERGENCE, UNANIMITY AND DISAGREEMENT IN MAJORITY DYNAMICS ON UNIMODULAR
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.
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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.
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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.
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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
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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.
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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.
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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).
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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.
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(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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