arXiv:1411.4722v1 [math.PR] 18 Nov 2014
ASYMPTOTICS FOR SPARSE EXPONENTIAL RANDOM
GRAPH MODELS
MEI YIN AND LINGJIONG ZHU
Abstract. We study the asymptotics for sparse exponential random graph
models where the parameters may depend on the number of vertices of the
graph. We obtain a variational principle for the limiting free energy, an associated concentration of measure, the asymptotics for the mean and variance of
the limiting probability distribution, and phase transitions in the edge-triangle
model. Similar analysis is done for directed sparse exponential random graph
models parametrized by edges and outward stars.
1. Introduction
Exponential random graphs are a class of graph ensembles of fixed vertex number n defined by analogy with the Boltzmann ensemble of statistical mechanics.
Let {p } be a set of local features of a single graph, for example the number of
edges or copies of any finite subgraph, as well as more complicated characteristics
including the degree sequence or degree distribution, and combinations thereof.
These quantities play a role similar to energy in statistical mechanics. Let {βp }
be a set of inverse temperature parameters whose values we are free to choose. By
varying these parameters, one could analyze the influence of different local features
on the global structure of the graph. Let Gn be the set of all possible graphs (undirected and with no self-loops or multiple edges in the simplest case) on n vertices.
The k-parameter family of exponential random graphs is defined by assigning a
probability P(n) (Gn ) to every graph Gn in Gn :
" k
#
X
(n)
(n)
(n) −1
(n)
(n)
P (Gn ) = Zn (β1 , β2 , . . . , βk ) exp
βp p (Gn ) ,
(1.1)
p=1
where Zn is the partition function,
"
(n)
(n)
(n)
Zn (β1 , β2 , . . . , βk )
=
X
Gn ∈Gn
exp
k
X
#
βp(n) p (Gn )
.
(1.2)
p=1
These rather general models are widely used to model real-world networks, such as
the Internet, the World Wide Web, social networks, and biological networks, as they
are able to capture a wide variety of common network tendencies by representing a
complex global structure through a set of tractable local features, see e.g. Newman
[23] and Wasserman and Faust [32]. They are particularly useful when one wants
to construct models that resemble observed networks as closely as possible but
without going into details of the specific process underlying network formation.
Date: 17 November 2014. Revised: 17 November 2014.
2000 Mathematics Subject Classification. 05C80,82B26,05C35.
Key words and phrases. sparse random graphs, exponential random graphs, phase transitions.
1
2
MEI YIN AND LINGJIONG ZHU
Since real-world networks are often very large in size, a pressing objective is to
understand the asymptotics of the limiting partition function Zn , the limiting probability distribution P(n) (Gn ), and the limiting free energy ψ := limn→∞ γ1n log Zn ,
for some proper scaling γn → ∞ as n → ∞, which encodes essential information
about the structure of the limiting probability measure. By differentiating ψ with
respect to appropriate parameters βp , averages of various quantities of interest may
be derived. In particular, a phase transition occurs when ψ is non-analytic, since it
is the generating function for the limiting expectations of other random variables.
Computation of ψ is also important in statistics because it is crucial for carrying
out maximum likelihood estimates and Bayesian inference of unknown parameters.
Exponential models have been extensively studied over the last decades. We
refer to Besag [4], Snijders et al. [31], Rinaldo et al. [30], and Fienberg [11, 12]
for history and a review of developments. In recent years, exponential random
graph models and their variations have received (exponentially) growing attention,
where the emphasis has been made on the variational principle of the limiting
free energy, concentration of the limiting probability distribution, phase transitions
and asymptotic structures, see e.g. Chatterjee and Varadhan [10], Chatterjee and
Diaconis [9], Radin and Yin [25], Lubetzky and Zhao [22], Radin and Sadun [26, 27],
Radin et al. [28], Kenyon et al. [17], Yin [34], Yin et al. [35], Aristoff and Zhu
[2, 3], and Zhu [36].
However, in the real world, most networks data are sparse, see e.g. Golub et
al. [13], Guyon et al. [14], Hromádka et al. [16] etc. For example, a gene network
is sparse since a regulatory pathway involves only a small number of genes; the
neural representation of sounds in the auditory cortex of unanesthetized animals
is sparse, since the fraction of neurons active at a given instant is small; many
biomedical signals have sparse depictions when expressed in a proper basis, see Ye
and Liu [33]. Therefore it is important to understand sparse exponential random
graph models. Nevertheless, all previous investigations have been centered on dense
graphs (number of edges comparable to the square of number of vertices) except
some partial results in a very recent paper by Chatterjee and Dembo [8] where
very strong assumptions are imposed. A systematic study in the sparse regime is
currently lacking, and this will be the main focus of the present paper.
The rest of this paper is organized as follows. In Section 2 we introduce notation and results concerning the theory of graph limits and its use in (undirected)
exponential random graph models. In Section 3 we analyze the asymptotic features
of the undirected exponential model parametrized by various subgraph densities.
Our main results are: a variational principle for the limiting free energy (Theorem
1), an associated concentration of measure (Theorem 2) indicating that almost all
large graphs lie near the maximizing set, and the mean and variance of the limiting
probability distribution under a scaling assumption about the parameters (Propositions 5-6). We then resort to the large deviations result of Chatterjee and Dembo
[8] and obtain exact asymptotics for the limiting partition function of edge-(single)star model (and beyond) in the sparse regime (Theorem 7). Lastly, we specialize to
the edge-triangle model and show the existene of countably many first-order phase
transitions (Proposition 10). In Section 4 we analyze the asymptotic features of the
directed exponential model parametrized by edges and multiple outward stars. Our
main results are: a variational principle for the limiting free energy (Theorem 11),
and the mean and variance of the limiting probability distribution under different
ASYMPTOTICS FOR SPARSE EXPONENTIAL RANDOM GRAPH MODELS
3
scaling assumptions about the parameters (Propositions 13-16). We then specialize
to the edge-(single)-star model and show the existence of first- and second-order
phase transitions.
2. Background
We present some background on the theory of graph limits and its use in (undirected) exponential random graph models. Following the earlier work of Aldous
[1] and Hoover [15], Lovász and coauthors (V. T. Sós, B. Szegedy, C. Borgs, J.
Chayes, K. Vesztergombi, etc.) have constructed a unified and elegant theory of
graph limits in a sequence of papers [5, 6, 7, 19, 21]. See also the recent book of
Lovász [20] for a comprehensive account and references. This emerging theory has
provided a new set of tools for representing and studying the asymptotic behavior
of graphs, and has become the object of intense research in many fields, such as
discrete mathematics, statistical mechanics, and probability.
Here are the basics of this beautiful theory. Any undirected graph Gn that
has no self-loops or multiple edges, irrespective of the number of vertices, may be
represented as an element hGn of a single abstract space W that consists of all
symmetric measurable functions from [0, 1]2 into [0, 1], by defining
1, if (dnxe, dnye) is an edge in Gn ;
hGn (x, y) =
(2.1)
0, otherwise.
A sequence of graphs {Gn }n≥1 is said to converge to a function h ∈ W (referred to
as a “graph limit” or “graphon”) if for every finite simple graph H with vertex set
V (H) = [k] = {1, . . . , k} and edge set E(H),
lim t(H, hGn ) = t(H, h),
n→∞
where
Z
t(H, h) =
Y
(2.2)
h(xi , xj )dx1 · · · dxk ,
(2.3)
|hom(H, Gn )|
,
|V (Gn )||V (H)|
(2.4)
[0,1]k {i,j}∈E(H)
and so by construction,
t(H, hGn ) = t(H, Gn ) :=
the homomorphism density of H in Gn . It was shown in Lovász and Szegedy [21]
that every function in W arises as the limit of a certain graph sequence. Intuitively,
the interval [0, 1] represents a “continuum” of vertices, and h(x, y) denotes the
probability of putting an edge between x and y. For example, for the ErdősRényi random graph G(n, ρ), the “graphon” is represented by the function that is
identically equal to ρ on [0, 1]2 .
This “graphon” interpretation enables us to capture the notion of convergence in
terms of subgraph densities by an explicit metric on W, the so-called “cut distance”:
Z
d (f, h) = sup (f (x, y) − h(x, y)) dx dy (2.5)
S,T ⊆[0,1]
S×T
for f, h ∈ W. A non-trivial complication is that the topology induced by the cut
metric is well defined only up to measure preserving transformations of [0, 1] (and
up to sets of Lebesgue measure zero), which in the context of finite graphs may be
thought of as vertex relabeling. The solution is to work instead on an appropriate
quotient space. To that end, an equivalence relation ∼ is introduced in W. We say
4
MEI YIN AND LINGJIONG ZHU
that f ∼ h if f (x, y) = hσ (x, y) := h(σx, σy) for some measure preserving bijection
σ of [0, 1]. Let h̃ (referred to as a “reduced graphon”) denote the equivalence class
of h in (W, d ). Since d is invariant under σ, one can then define on the resulting quotient space W̃ the natural distance δ by δ (f˜, h̃) = inf σ1 ,σ2 d (fσ1 , hσ2 ),
where the infimum ranges over all measure preserving bijections σ1 and σ2 , making
(W̃, δ ) into a metric space. With some abuse of notation we also refer to δ as
the “cut distance”. The space (W̃, δ ) enjoys many important properties that are
essential for the study of exponential random graph models. For example, it is a
compact space and homomorphism densities t(H, ·) are continuous functions on it.
3. Undirected Graphs
Consider undirected graphs Gn on n vertices, where a graph is represented by
a matrix X = (Xij )1≤i<j≤n with each Xij ∈ {0, 1}. Here, Xij = 1 means there is
an edge between vertex i and vertex j; otherwise, Xij = 0. Give the set of such
graphs the probability
!#
"
k
X
(n)
(n)
(n)
βp(n) t(Hp , Gn )
,
(3.1)
P(n) (Gn ) = Zn (β1 , β2 , . . . , βk )−1 exp n2
p=1
(n)
(n)
(n)
where Zn (β1 , β2 , . . . , βk ) is the appropriate normalization.
We are interested in the sparse graph, i.e., the probability that there is an edge
(n)
between vertex i and vertex j goes to 0 as n → ∞. This requires that βp → −∞,
for some 1 ≤ p ≤ k, as n → ∞. Let us assume that
βp(n) = βp αn ,
p = 1, 2, . . . , k,
(3.2)
where αn → ∞ as n → ∞.
Theorem 1. Assume that H1 denotes a single edge.
ψ(β1 , . . . , βk ) := lim
1
n→∞ n2 αn
log Zn =
sup
{β1 t(H1 , h) + · · · + βk t(Hk , h)} .
h:[0,1]2 →[0,1]
h(x,y)=h(y,x)
(3.3)
Proof. We can compute that
h 2 (n)
i
(n)
n
(n)
(n)
Zn (β1 , . . . , βk ) = 2( 2 ) E en (β1 t(H1 ,Gn )+···+βk t(Hk ,Gn ))
(3.4)
X
(n)
(n)
2
=
en (β1 t(H1 ,x)+···+βk t(Hk ,x)) .
(xij )1≤i<j≤n ∈{0,1}n2
On the one hand, we have
n
Zn (β1 , . . . , βk ) ≤ 2( 2 ) e
(n)
(n)
n2 αn max
2
(xij )1≤i<j≤n ∈{0,1}n
{β1 t(H1 ,x)+···+βk t(Hk ,x)}
, (3.5)
and on the other hand,
(n)
(n)
Zn (β1 , . . . , βk ) ≥ e
αn n2 max
2
(xij )1≤i<j≤n ∈{0,1}n
{β1 t(H1 ,x)+···+βk t(Hk ,x)}
.
(3.6)
ASYMPTOTICS FOR SPARSE EXPONENTIAL RANDOM GRAPH MODELS
5
The proof is complete by noticing that
lim
max
n→∞ (xij )
1≤i<j≤n ∈{0,1}
=
n2
{β1 t(H1 , x) + · · · + βk t(Hk , x)}
(3.7)
{β1 t(H1 , h) + · · · + βk t(Hk , h)} .
sup
h:[0,1]2 →[0,1]
h(x,y)=h(y,x)
First, it is clear that the LHS is less than or equal to the RHS in (3.7). Second,
there exists a graphon h∗ , so that
( k
)
k
X
X
βp t(Hp , h∗ ) =
sup
βp t(Hp , h) .
(3.8)
h:[0,1]2 →[0,1],h(x,y)=h(y,x)
p=1
p=1
For a fixed graphon h∗ , we can find a sequence of graphs that converge to h∗ in the
cut norm as n → ∞. Hence we proved (3.7).
Pk
Let H̃ be the subset of W̃ where p=1 βp t(Hp , h̃) is maximized. By the comPk
pactness of W̃ and the continuity of p=1 βp t(Hp , ·), H̃ is a non-empty compact
set. Let Gn be a random graph on n vertices drawn from the sparse exponential
random graph model P(n) (3.1). The following theorem shows that for n large, G̃n
must lie close to H̃ with high probability. In particular, if H̃ is a singleton set, then
the theorem gives a weak law of large numbers for Gn .
Theorem 2. Let H̃ be defined as above. Then for any η > 0 there exist C, γ > 0
such that for all n large enough,
P(n) (δ (G̃n , H̃) > η) ≤ Ce−n
2
αn γ
.
(3.9)
Proof. Take any η > 0. Let à be the subset of H̃ consisting of reduced graphons
that are at least η-distance away from H̃,
à = {h̃ : δ (h̃, H̃) > η}.
(3.10)
Pk
By the compactness of W̃ and H̃ and the continuity of p=1 βp t(Hp , ·), it follows
that
k
k
X
X
βp t(Hp , h̃) > 0.
(3.11)
βp t(Hp , h̃) − sup
γ 0 := sup
h̃∈Ã p=1
h̃∈W̃ p=1
Then
P(n) (G̃n ∈ Ã)
2
= e−n
X
αn ψn
en
2
αn
Pk
p=1
βp t(Hp ,Gn )
(3.12)
δ (G̃n ,H̃)>η
2
≤ e−n
αn ψn
n
2( 2 ) e
n2 αn supδ
(G̃n ,H̃)>η
Pk
p=1
βp t(Hp ,Gn )
.
Notice that
lim
n→∞
sup
k
X
βp t(Hp , Gn ) = sup
δ (G̃n ,H̃)>η p=1
k
X
βp t(Hp , h̃).
(3.13)
h̃∈Ã p=1
First, it is clear that the LHS is less than or equal to the RHS. Second, for any
> 0, there exists a reduced graphon h̃∗ ∈ Ã so that
k
X
p=1
βp t(Hp , h̃∗ ) ≥ sup
k
X
h̃∈Ã p=1
βp t(Hp , h̃) − .
(3.14)
6
MEI YIN AND LINGJIONG ZHU
For a fixed reduced graphon h̃∗ , we can find a sequence of graphs Gn that converge
to h̃∗ in the cut norm as n → ∞. This implies that
lim sup
n→∞
k
k
X
X
log P(n) (G̃n ∈ Ã)
≤
−
sup
β
t(H
,
h̃)
+
sup
βp t(Hp , h̃) = −γ 0 .
p
p
n2 αn
h̃∈W̃ p=1
h̃∈Ã p=1
(3.15)
Hence (3.9) holds for any γ < γ 0 .
When ψ = 0, h ≡ 0 is an optimal graphon and in the limit n → ∞, we have an
empty graph. That translates to sparse graphs before we take the limit n → ∞.
One natural question to ask is for what set of parameters (β1 , . . . , βk ) we will get
ψ(β1 , . . . , βk ) = 0. Note that if ψ = 0 then h ≡ 0 is an optimal graphon and
therefore
0 = ψ(β1 , . . . , βk ) = sup {β1 x + β2 x2 + · · · + βk xk }.
(3.16)
0≤x≤1
So it is interesting to understand when we have
sup {β1 x + β2 x2 + · · · + βk xk } = 0.
(3.17)
0≤x≤1
Remark 3. Three trivial observations.
(i) If (3.17) holds, then we must have β1 + β2 + · · · + βk < 0.
(ii) If (3.17) holds and βP
1 = · · · = β`−1 = 0, then we must have β` < 0.
(iii) If β1 < 0 and β1 + j:βj >0 j < 0, then (3.17) holds.
(i) is true because otherwise x = 1 is more optimal than x = 0. (ii) is true since
when x > 0 is very small, β` x is the dominating term if β1 = · · · = β`−1 = 0. (iii)
is true since
X
X
X
`0 (x) = β1 +
jβj xj−1 +
jβj xj−1 ≤ β1 +
j,
0 ≤ x ≤ 1. (3.18)
j>1:βj <0
j:βj >0
j:βj >0
Remark 4. Let us derive the sufficient and necessary conditions for
sup {β1 x + β2 x2 + β3 x3 } = 0,
β1 , β2 , β3 6= 0.
(3.19)
0≤x≤1
First, we must have β1 < 0. We can compute that
`0 (x) = β1 + 2β2 x + 3β3 x2 .
(3.20)
Thus `0 (0) = β1 < 0. If β3 > 0, then as x increases from 0 to ∞, `0 (x) changes from
being negative to positive. Thus as x increases from 0 to ∞, `(x) first decreases and
then increases. Hence when β3 > 0, sup0≤x≤1 `(x) = 0 if and only if β1 +β2 +β3 ≤ 0.
If β3 < 0 and β2 ≤ 0, then `0 (x) ≤ 0 for any x ≥ 0 and sup0≤x≤1 `(x) = 0.
0
If β3 < 0√and β2 > 0, then there are two positive
√ roots of ` (x) = 0, given by
−β ±
β 2 −3β β
−β +
β 2 −3β β
1 3
2
1 3
2
2
2
if β22 > 3β1 β3 . If 1 ≤
, then on the interval
x=
3β3
3β3
[0, 1], `(x) first decreases and then increases. Hence, sup0≤x≤1 `(x) = 0 if and only
√
−β2 + β22 −3β1 β3
if β1 + β2 + β3 ≤ 0. If 1 >
, then on the interval [0, 1], `(x) first
3β3
decreases and then increases and finally decreases. Therefore, sup0≤x≤1 `(x) = 0
√
−β + β 2 −3β β
if and only if `( 2 3β23 1 3 ) ≤ 0. Finally if β3 < 0, β2 > 0 and β22 ≤ 3β1 β3
ASYMPTOTICS FOR SPARSE EXPONENTIAL RANDOM GRAPH MODELS
7
then `0 (x) ≤ 0 and sup0≤x≤1 `(x) = 0. Hence, to summarize, the sufficient and
necessary condition for (3.19) is that (β1 , β2 , β3 ) belongs to the set
{β1 < 0, β3 > 0, β1 + β2 + β3 ≤ 0} ∪ {β1 < 0, β3 < 0, β2 ≤ 0}
∪ {β1 < 0, β3 < 0, β2 > 0, β22 ≤ 3β1 β3 }
∪ β1 < 0, β3 < 0, β2 > 0, β22 > 3β1 β3 ,
1>
−β2 +
p
β22 − 3β1 β3
,`
3β3
−β2 +
!
p
β22 − 3β1 β3
≤0
3β3
∪ β1 < 0, β3 < 0, β2 > 0, β22 > 3β1 β3 ,
p
−β2 + β22 − 3β1 β3
, β1 + β2 + β3 ≤ 0 .
1≤
3β3
For a sparse random graph, P (n) (Xij = 1) → 0 as n → ∞. It will be interesting
to know how sparse the random graph is and how fast P (n) (Xij = 1) converges to
zero as n → ∞.
Proposition 5. Assume that β1 , . . . , βk are all negative and H1 denotes a single
edge. Let us further assume that limn→∞ n2 e2αn β1 = 0 and limn→∞ αnn = 0.
P(n) (X1i = 1)
= 1.
n→∞
e2β1 αn
lim
(3.21)
Proof. By symmetry,
P(n) (X1i = 1) = E(n) [X1i ]

1 (n)  X
= n E
2
(3.22)

Xij 
1≤i<j≤n
1 E[
= n
Pk
(n) 2
p=1 βp n t(Hp ,X) ]
1≤i<j≤n Xij e
Pk
(n) 2
p=1 βp n t(Hp ,X)
P
E[e
]
P
2 Pk
α
n
( )
p=1 βp t(Hp ,X) ]
1 2 E[ 1≤i<j≤n Xij e n
.
= n
Pk
n
2
2( 2 ) E[eαn n p=1 βp t(Hp ,X) ]
2
2
n
2
First, let us analyze the denominator. It is clear that
n
2( 2 ) E[eαn n
2
Pk
p=1
βp t(Hp ,X)
n
1
] ≥ 2( 2 ) n = 1.
(
2 2)
On the other hand, since βi ’s are negative,
n
n
2 Pk
2
2( 2 ) E[eαn n p=1 βp t(Hp ,X) ] ≤ 2( 2 ) E[eαn n β1 t(H1 ,X) ]
n
2
= 2( ) E[e2αn β1 i<j Xij ]
n
2αn β1 ( 2 )
n
1
+
e
= 2( 2 )
2
→ 1,
as n → ∞ since we assumed that limn→∞ n2 e2αn β1 = 0.
P
(3.23)
(3.24)
8
MEI YIN AND LINGJIONG ZHU
Next, let us analyze the numerator. On the one hand,


n
2
2( ) E 
X
Xij eαn n
2
Pk
p=1
βp t(Hp ,X) 
(3.25)
1≤i<j≤n
n
≥ 2( 2 )
X
1≤i<j≤n
= e2αn β1
2 Pk
E eαn n p=1 βp t(Hp ,X) Xij = 1, Xi0 j 0 = 0, (i0 , j 0 ) 6= (i, j)
· P (Xij = 1, Xi0 j 0 = 0, (i0 , j 0 ) 6= (i, j))
X
2 Pk
E eαn n p=2 βp t(Hp ,X) Xij = 1, Xi0 j 0 = 0, (i0 , j 0 ) 6= (i, j)
1≤i<j≤n
=
X
e2αn β1 +αn n
2
Pk
p=2
βp cp n−v(Hp )
1≤i<j≤n
n 2αn β1 +αn n2 Pkp=2 βp cp n−v(Hp )
=
e
,
2
where v(Hp ) ≥ 3 denotes the number of vertices of Hp and cp ≥ 0 is a constant
that only depends on Hp .
On the other hand,

n
2
2( ) E 

X
Xij eαn n
2
Pk
p=1
βp t(Hp ,X) 

n
2
≤ 2( ) E 
1≤i<j≤n

X
Xij eαn n
2
β1 t(H1 ,X) 
1≤i<j≤n
(3.26)


n
= 2( 2 ) E 
X
Xij e2αn β1
P
i<j
Xij 
1≤i<j≤n
i
h
P
1 ∂
E e2αn β1 i<j Xij
= 2( )
2αn ∂β1
n
2αn β1 ( 2 )
n
1
∂
1
+
e
(
)
=2 2
2αn ∂β1
2
(n)−1
n
= e2αn β1
1 + e2αn β1 2
.
2
n
2
Putting everything together, we proved the desired result.
Proposition 6. Assume that β1 , . . . , βk are all negative and H1 denotes a single
edge. Let us further assume that limn→∞ n2 e2αn β1 = 0 and limn→∞ αnn = 0.
P(n) (X1i = 1, X1j = 1)
= 1, i 6= j.
n→∞
e4β1 αn
lim
(3.27)
ASYMPTOTICS FOR SPARSE EXPONENTIAL RANDOM GRAPH MODELS
9
Proof. By symmetry,
P(n) (X1i = 1, X1j = 1) = E(n) [X1i X1j ]


=

2 
X

Xij   − E(n) 
4
 (n)  X
E 
(n + 1)n(n − 1)(n − 2)
1≤i<j≤n
1≤i<j≤n
4E
=
P
1≤i<j≤n Xij
2
−
P
1≤i<j≤n Xij e
Pk
(n)
Pk
(3.28)


Xij 
p=1
βp(n) n2 t(Hp ,X)
2
(n + 1)n(n − 1)(n − 2)E[e p=1 βp n t(Hp ,X) ]
2 P
P
n
P
αn n2 k
p=1 βp t(Hp ,X)
X
−
4 · 2( 2 ) E
X
e
1≤i<j≤n ij
1≤i<j≤n ij
=
n
2
(n + 1)n(n − 1)(n − 2)2( 2 ) E[eαn n
Pk
p=1
βp t(Hp ,X)
n
Under the assumption limn→∞ n2 e2β1 αn = 0, we have 2( 2 ) E[eαn n
1 as n → ∞. Moreover, it is clear that

X
n

2( 2 ) E 
X
Xij  −
1≤i<j≤n
Pk
p=1
βp t(Hp ,X)
]→


2
2
.
]
2 Pk


Xij  eαn n p=1 βp t(Hp ,X) 
(3.29)
1≤i<j≤n
X
X
n 1
≥ 2( 2 )
2
1≤i<j≤n 1≤i0 <j 0 ≤n:(i0 ,j 0 )6=(i,j)
2 Pk
E (22 − 2)eαn n p=1 βp t(Hp ,X) Xij = Xi0 j 0 = 1, Xi00 j 00 = 0, (i00 , j 00 ) 6= (i, j) and (i00 , j 00 ) 6= (i0 , j 0 )
= e4αn β1
· P (Xij = Xi0 j 0 = 1, Xi00 j 00 = 0, (i00 , j 00 ) 6= (i, j) and (i00 , j 00 ) 6= (i0 , j 0 ))
X
X
1≤i<j≤n 1≤i0 <j 0 ≤n:(i0 ,j 0 )6=(i,j)
P
αn n2 k
βp t(Hp ,X) 00 00
00 00
0 0
p=2
E e
Xij = Xi0 j 0 = 1, Xi00 j 00 = 0, (i , j ) 6= (i, j) and (i , j ) 6= (i , j )
=
X
X
e4αn β1 +αn n
2
Pk
p=2
βp cp n−v(Hp )
1≤i<j≤n 1≤i0 <j 0 ≤n:(i0 ,j 0 )6=(i,j)
=
(n + 1)n(n − 1)(n − 2) 4αn β1 +αn n2 Pkp=2 βp cp n−v(Hp )
e
,
4
10
MEI YIN AND LINGJIONG ZHU
where v(Hp ) ≥ 3 denotes the number of vertices of Hp and cp ≥ 0 is a constant
that only depends on Hp . And

X
n

2( 2 ) E 

2
X
Xij  −
1≤i<j≤n


Xij  eαn n
2 Pk
p=1
βp t(Hp ,X) 

(3.30)
1≤i<j≤n


2
X
n

≤ 2( 2 ) E 
Xij  −
1≤i<j≤n

X
n

= 2( 2 ) E 
X

2


Xij  eαn n β1 t(H1 ,X) 
1≤i<j≤n
Xij  −
1≤i<j≤n


2
X

Xij  e2αn β1
P
i<j
Xij 

1≤i<j≤n
i
h
P
1 ∂ 2 h 2αn β1 Pi<j Xij i
1 ∂
2αn β1 i<j Xij
−
E
e
E
e
4α2 ∂β 2
2αn ∂β1
 n 1

n
(n2 )
(
2
2α
β
2α
β
2)
n
1
n
1
n
1 ∂
1 ∂
1+e
1+e

= 2( 2 )  2
−
4αn ∂β12
2
2αn ∂β1
2
n
= 2( 2 )
= e4αn β1
(n)−2
(n + 1)n(n − 1)(n − 2)
.
1 + e2αn β1 2
4
From the assumptions limn→∞ n2 e2αn β1 = 0 and limn→∞
(n)
αn
n
= 0, our claim follows.
(n)
In Chatterjee and Dembo [8], when |β1 | + · · · + |βk | does not grow too fast,
then logn2Zn can be approximated by
Ln := sup
x∈Pn
(n)
(n)
β1 t(H1 , x) + · · · + βk t(Hk , x) −
I(x)
n2
,
(3.31)
where Pn := {(xij )1≤i<j≤n : xij ∈ [0, 1], 1 ≤ i < j ≤ n}.
Chatterjee and Dembo [8] showed that
−
cB
log Zn
≤
− Ln
n
n2
(3.32)
log B
≤ CB 8/5 n−1/5 (log n)1/5 1 +
+ CB 2 n−1/2 ,
log n
(n)
(n)
where B := 1 + |β1 | + · · · + |βk | and c and C are constants only depending on
H1 , . . . , H k .
j−1 j
i
By considering h(x, y) = xij for any [ i−1
n , n ] × [ n , n ], we have
Ln ≤
sup
h:[0,1]2 →[0,1],h(x,y)=h(y,x)
( k
X
p=1
βp(n) t(Hp , h)
1
−
2
)
ZZ
I(h(x, y))dxdy
.
[0,1]2
(3.33)
ASYMPTOTICS FOR SPARSE EXPONENTIAL RANDOM GRAPH MODELS
11
It was proved in Chatterjee and Diaconis [9] that when Hp , p ≥ 2 are stars or when
βp ’s are non-negative for any p ≥ 2,
( k
)
ZZ
X
1
(n)
sup
βp t(Hp , h) −
I(h(x, y))dxdy
(3.34)
2
h:[0,1]2 →[0,1],h(x,y)=h(y,x) p=1
[0,1]2
1
= sup αn β1 x + αn β2 x2 + · · · + αn βk xk − I(x) .
2
0≤x≤1
On the other hand, by considering xij ≡ x,
1
2
k
Ln ≥ sup αn β1 x + αn β2 x + · · · + αn βk x − I(x) .
2
0≤x≤1
Therefore, for edge-star model or when βi ’s are non-negative, 2 ≤ i ≤ k,
1
Ln = sup αn β1 x + αn β2 x2 + · · · + αn βk xk − I(x) .
2
0≤x≤1
(3.35)
(3.36)
Let us analyze the edge and p-star model in more detail. The analysis also works
for edge and H2 model where the number of edges in H2 is p and β2 ≥ 0.
1
Ln = sup αn β1 x + αn β2 xp − I(x) .
(3.37)
2
0≤x≤1
We know that
ψ(β1 , β2 ) = 0,
when (β1 , β2 ) ∈ {β1 + β2 ≤ 0} ∩ {β1 ≤ 0}.
(3.38)
This represents the regime in which we expect sparse random graphs.
Theorem 7. Take p ≥ 3. When β1 < 0 and β1 + β2 ≤ 0,
lim
n→∞
1
Ln
= .
e2αn β1
2
Moreover, if we further assume that limn→∞
8/5
αn
(log n)1/5 e2αn |β1 |
n1/5
(3.39)
= 0, then
lim
log Zn
1
= .
n2 e2αn β1
2
(3.40)
lim
Ln
1−p
,
=
γ n e γn
2p
(3.41)
n→∞
When β1 = 0 and β2 < 0,
n→∞
where γn is uniquely defined via the equation 2αn β2 e(p−1)γn p = γn and γn → −∞
α8/5 (log n)1/5 e|γn |
as n → ∞. Moreover, if we further assume that limn→∞ n |γn |n1/5
= 0, then
lim
n→∞
log Zn
1−p
=
.
2
γ
n
n γn e
2p
(3.42)
Proof. The optimization problem (3.37) has been well studied in Radin and Yin
[25] and Aristoff and Zhu [2]. In Radin and Yin [25], it was proved that there
(n)
(n)
exists a phase transition curve β2 = q(β1 ) below which the maximizer of (3.37)
(n)
(n)
is either unique or the smaller one. They also proved that β1 + q(β1 ) → 0 as
n → ∞. Moreover, in Aristoff and Zhu [2], it was shown that the phase transition
12
MEI YIN AND LINGJIONG ZHU
(n)
(n)
curve always lies above the curve β1 + β2 = 0 when p ≥ 3. Therefore, the
optimizer in (3.37) is either the unique or the smaller solution to the equation
1
x∗
αn β1 + αn β2 p(x∗ )p−1 = log
.
(3.43)
2
1 − x∗
In the sparse regime we consider, (β1 , β2 ) ∈ {β1 + β2 ≤ 0} ∩ {β1 ≤ 0}, at least one
of |αn β1 | and |αn β2 | go to ∞ as n → ∞, so we must have x∗ → 0 as n → ∞, see
either [25] or [2]. We can rewrite (3.43) as
(1 − x∗ )e2αn β2 p(x
∗ p−1
)
x∗
=
e2αn β1
For the edge and p-star model, if β2 ≤ 0, then we have
∗ p−1
2αn β2 p(x )
= 2αn β2 pe
(3.44)
x∗
e2αn β1
≤ 1 and thus
p−1
x∗
2(p−1)αn β1
.
e2αn β1
→ 0,
(3.45)
as n → ∞. Hence, we conclude that
lim
x∗
= 1.
n→∞ e2αn β1
(3.46)
Now, let us consider the more general case β1 < 0 and β1 + β2 ≤ 0. By letting
∗
y = e2αxn β1 , we can rewrite (3.44) as
∗
(1 − y ∗ e2αn β1 )e2αn β2 e
2αn β1 (p−1)
Let us define F (y) = (1 − ye2αn β1 )e2αn β2 e
F (0) = 1 and
p(y ∗ )p−1
2αn β1 (p−1)
py p−1
2αn β1 (p−1)
p
F (1) = (1 − e2αn β1 )e2αn β2 e
= y∗ .
(3.47)
− y. We can compute that
− 1 < 0,
(3.48)
for any sufficiently large n. This can be seen through the following argument. First
2αn β1 (p−1)
p
we notice that when β1 + β2 ≤ 0, F (1) ≤ (1 − e2αn β1 )e−2αn β1 e
− 1. Take
2αn β1
−pz p−1
z=e
, it suffices to show that for z sufficiently close to 0, (1 − z)z
< 1,
which is equivalent to pz p−1 log z > log(1 − z). But this is clear when p ≥ 3 since
the derivative on the left tends to 0 whereas the derivative on the right tends to
−1 as z approaches 0+. Moreover, we can compute that
F 0 (y) = −e2αn β1 e2αn β2 e
2αn β1 (p−1)
2αn β1
+ (1 − ye
py p−1
)2αn β2 e
2αn β1 (p−1)
(3.49)
p(p − 1)y
p−2 2αn β2 e2αn β1 (p−1) py p−1
e
− 1,
and for sufficiently large n we have F 0 (y) < 0 for any 0 ≤ y ≤ 1. Since we know
that y ∗ is the unique or the smaller solution of F (y) = 0, we conclude that y ∗
is the unique solution on the interval (0, 1). Furthermore, we can also check that
F (1) → 0 and F 0 (1) → −1 as n → ∞. Hence, we conclude that y ∗ → 1 as n → ∞
and therefore
x∗
(3.50)
lim 2αn β1 = 1.
n→∞ e
ASYMPTOTICS FOR SPARSE EXPONENTIAL RANDOM GRAPH MODELS
13
Hence, by using (3.43), we get
1
1
Ln = αn β1 x∗ + αn β2 (x∗ )p − x∗ log x∗ − (1 − x∗ ) log(1 − x∗ )
2
2 ∗
1
1
x
− log(1 − x∗ )
= αn β1 x∗ + αn β2 (x∗ )p − x∗ log
∗
2
1−x
2
1
= αn β2 (x∗ )p (1 − p) − log(1 − x∗ )
2
1
= αn β2 (x∗ )p (1 − p) + x∗ + O((x∗ )2 ).
2
Therefore, limn→∞
Ln
x∗
=
1
2
(3.51)
and
1
Ln
= .
e2αn β1
2
Chatterjee and Dembo [8] showed that
lim
(3.52)
n→∞
cαn (|β1 | + |β2 |)
n
log Zn
≤
− Ln
n2
−
(3.53)
log αn + log(|β1 | + |β2 |)
≤ Cαn8/5 (|β1 | + |β2 |)8/5 n−1/5 (log n)1/5 1 +
log n
+ C(|β1 | + |β2 |)2 αn2 n−1/2 .
α8/5 (log n)1/5 e2αn |β1 |
= 0, we
Therefore, under the further assumption that limn→∞ n
n1/5
have
log Zn
Ln as n → ∞,
(3.54)
n2 e2αn β1 − e2αn β1 → 0,
and hence
log Zn
1
lim
(3.55)
= .
n→∞ n2 e2αn β1
2
The boundary of phases are {β1 = 0, β2 < 0} and {β1 + β2 = 0, β1 < 0}. Along
the curve {β1 = 0, β2 < 0}, we have
1
x∗
αn β2 p(x∗ )p−1 = log
.
(3.56)
2
1 − x∗
Let γn < 0 be defined via the equation 2αn β2 e(p−1)γn p = γn . First, let us check
that γn is well defined. Let us consider the function F (x) = 2αn β2 e(p−1)x p − x.
Then F (0) < 0 and F (−∞) = ∞. Moreover, F 0 (x) = 2αn β2 e(p−1)x (p − 1)p − 1 < 0.
Thus F (x) = 0 has a unique negative solution, which is denoted by γn . For any
fixed x < 0, F (x) = 2αn β2 e(p−1)x p − x < 0 for sufficiently large n. Therefore,
γn → −∞ as n → ∞. We can rewrite (3.56) as
x∗
x∗ γn 2αn β2 p( xγ∗n )p−1 e(p−1)γn −γn
e
= γn ,
1 − γn e
e
(3.57)
e
e
which is equivalent to
x∗ γn γn (( xγ∗n )p−1 −1)
x∗
e
1 − γn e
e
= γn .
e
e
(3.58)
14
MEI YIN AND LINGJIONG ZHU
Let y ∗ =
x∗
eγ n
. The equation reduces to
(1 − y ∗ eγn ) eγn ((y
∗ p−1
)
−1)
= y∗ .
(3.59)
p−1
Let us define the function G(y) = (1 − yeγn ) eγn (y −1) − y. We can check that
p−1
G(1) = −eγn and G(0) = e−γn . Moreover, G0 (y) = −eγn eγn (y −1) +(1 − yeγn ) γn (p−
p−1
1)y p−2 eγn (y −1) − 1 < 0 on (0, 1). Therefore y ∗ is the unique solution of G(y) = 0
on (0, 1). Since γn → −∞ as n → ∞, we have G(1) = −eγn → 0 and G0 (1) =
−eγn + (1 − eγn )γn (p − 1) − 1 → −∞ as n → ∞. Thus y ∗ → 1 as n → ∞. Hence,
we conclude that
x∗
lim γn = 1,
(3.60)
n→∞ e
where γn is uniquely defined via the equation 2αn β2 e(p−1)γn p = γn and γn → −∞
as n → ∞. Moreover, we have
lim
n→∞
Ln
1−p
1
1−p
=
+ lim
=
.
n→∞ 2γn
γn eγn
2p
2p
If we further assume that limn→∞
8/5
αn
(log n)1/5 e|γn |
|γn |n1/5
(3.61)
= 0, then
log Zn
Ln n2 γn eγn − γn eγn → 0,
(3.62)
log Zn
1−p
=
.
2
γ
n
n→∞ n γn e
2p
(3.63)
as n → ∞ and
lim
Along the curve {β1 + β2 = 0, β1 < 0}, we have
∗ p−1
αn β1 − αn β1 p(x )
1
= log
2
x∗
1 − x∗
,
(3.64)
and the asymptotic estimates follow exactly as in the case β1 + β2 ≤ 0 and β1 < 0
discussed earlier.
Remark 8. Intuitively, the edge density is given by
P(n) (Xij = 1) = E(n) [Xij ] =
∂ log Zn
n2
“'”x∗ ,
2 n2 ∂β1(n) n2
(3.65)
where x∗ is the maximizer in (3.36) and x∗ ' e2αn β1 . This is consistent with the
results in Proposition 5.
Let us recall that in Theorem 1, we proved that
ψ(β1 , . . . , βk ) =
sup
{β1 t(H1 , h) + · · · + βk t(Hk , h)} ,
(3.66)
h:[0,1]2 →[0,1]
h(x,y)=h(y,x)
where H1 denotes a single edge. For the edge-(single)-star model, i.e., H1 is an
edge and H2 is a p-star, it is easy to see that the optimal graphon is uniform.
ASYMPTOTICS FOR SPARSE EXPONENTIAL RANDOM GRAPH MODELS
Proposition 9. For the edge-(single)-star model,
( Z Z
Z
1
1
h(x, y)dxdy + β2
ψ(β1 , β2 ) =
sup
β1
h:[0,1]2 →[0,1]
h(x,y)=h(y,x)
0
0
1
Z
p
1
h(x, y)dy
0
15
)
dx
0
(3.67)
p
= max {β1 x + β2 x }.
0≤x≤1
Proof. By optimizing over constant h, it is clear that ψ(β1 , β2 ) ≥ max0≤x≤1 {β1 x +
β2 xp }. On the other hand,
ψ(β1 , β2 )
(3.68)
(Z
=
1
"
sup
Z
1
0
1
Z
)
dx
0
0
0
p #
1
h(x, y)dy
h(x, y)dy + β2
β1
h:[0,1]2 →[0,1]
h(x,y)=h(y,x)
≤
Z
max {β1 x + β2 xp } dx
0≤x≤1
= max {β1 x + β2 xp }.
0≤x≤1
Next, let us consider the edge-triangle model, i.e., H1 is an edge and H2 is a
triangle.
Proposition 10. For the edge-triangle model,
ψ(β1 , β2 )
(
=
sup
ZZ
β1
h(x, y)dxdy + β2
h:[0,1]2 →[0,1]
h(x,y)=h(y,x)


β1 + β2



0
= l+1

 l+2 β1 +



(3.69)
)
ZZZ
[0,1]2
l(l+1)
(l+2)2 β2
h(x, y)h(y, z)h(z, x)dxdydz
[0,1]3
if β2 ≥ 0 and β1 + β2 ≥ 0 or if β2 < 0 and β1 + 3β2 ≥ 0
if β2 ≥ 0 and β1 + β2 < 0 or if β2 < 0 and β1 ≤ 0
.
if β2 < 0 and al β2 < β1 ≤ al+1 β2
l(3l+5)
where al = − (l+1)(l+2)
for l ≥ 0
Proof. When β2 ≥ 0, by generalized Hölder’s inequality,
( ZZ
)
ZZZ
sup
β1
h(x, y)dxdy + β2
h(x, y)h(y, z)h(z, x)dxdydz
h:[0,1]2 →[0,1]
h(x,y)=h(y,x)
[0,1]2
[0,1]3
(3.70)
(
≤
sup
2
h:[0,1] →[0,1]
h(x,y)=h(y,x)
ZZ
)
ZZZ
β1
h(x, y)3 dxdy
h(x, y)dxdy + β2
[0,1]2
[0,1]2
≤ sup {β1 x + β2 x3 }.
0≤x≤1
The other direction ψ(β1 , β2 ) ≥ sup0≤x≤1 {β1 x + β2 x3 } is trivial.
16
MEI YIN AND LINGJIONG ZHU
When β2 < 0 and β1 ≤ 0, it is clear that ψ(β1 , β2 ) = 0. When β2 < 0 and
β1 > 0, we need to do a more careful analysis. We have
ψ(β1 , β2 ) = sup {β1 + β2 τ ()} ,
(3.71)
0≤≤1
where
ZZZ
τ () =
h(x, y)h(y, z)h(z, x)dxdydz (3.72)
inf
2
h:[0,1]
RR →[0,1]
h(x,y)=h(y,x), [0,1]2 h(x,y)dxdy=
[0,1]3
is the smallest possible triangle density given the edge density . It is well known
that τ () = 0 for 0 ≤ ≤ 21 and τ () is a nontrivial scallop curve defined as
p
p
(` − 1)(` − 2 `(` − (` + 1)))(` + `(` − (` + 1)))2
τ () =
,
(3.73)
`2 (` + 1)2
1
where ` = b1/(1 − )c is the integer so that ∈ [1 − 1` , 1 − `+1
] for 21 ≤ ≤ 1, see
e.g. Pikhurko and Raborov [24] and Razborov [29]. Therefore,
(
)
β1
ψ(β1 , β2 ) = max
, sup {β1 + β2 τ ()} = sup {β1 + β2 τ ()} . (3.74)
1
2 1 ≤≤1
≤≤1
2
2
Assume β1 = aβ2 for some a < 0. Then
ψ(β1 , β2 ) = β2 1min {a + τ ()}.
(3.75)
2 ≤≤1
The rest of the proof follows a similar line of reasoning as in the proof of Theorem
3.3 of [35]. The derivative of a + τ () with respect to is given by
p
3(l − 1) a+
l + l(l − (l + 1)) .
(3.76)
l(l + 1)
1
It is a decreasing function of on each subinterval [1 − 1l , 1 − l+1
], hence we further
l
conclude that the minimizer can only be obtained at the connection points l = l+1
.
Consider two adjacent connection points (l , τ (l )) and (l+1 , τ (l+1 )), where
l
l(l − 1)
l + 1 l(l + 1)
(l , τ (l )) =
,
and
(
,
τ
(
))
=
,
. (3.77)
l+1
l+1
l + 1 (l + 1)2
l + 2 (l + 2)2
Let Ll be the line segment joining these two points. The slope of the line passing
through Ll is
l(3l + 5)
= −al .
(3.78)
(l + 1)(l + 2)
It is clear that al is a decreasing function of l and al → −3 as l → ∞. More
importantly, if a > al , then al +τ (l ) < al+1 +τ (l+1 ); if a = al , then al +τ (l ) =
al+1 + τ (l+1 ); and if a < al , then al + τ (l ) > al+1 + τ (l+1 ). Decreasing a thus
moves the location of the minimizer upward along the scallop curve, with sudden
jumps happening at special angles a = al , which correspond to first-order phase
transitions. To illustrate the phase diagram, we refer to Figure 1.
17
1.0
ASYMPTOTICS FOR SPARSE EXPONENTIAL RANDOM GRAPH MODELS
0.0
0.5
Phase I
-0.5
Phase II
Phase
IV
-1.0
Phase III
-1.0
-0.5
0.0
0.5
1.0
Figure 1. This is a plot of the phase diagram for the edge and
triangle model. The horizontal axis denotes β1 and the vertical
axis denotes β2 . There are countably many phases with boundaries
given by {β1 + β2 = 0, β1 < 0}, {β1 = 0, β2 < 0}, {β1 = al β2 , β1 >
0}, ` = 1, 2, . . ., and {β1 = −3β2 , β1 > 0}.
4. Directed Graphs
Consider directed graphs on n vertices, where a graph is represented by a matrix
X = (Xij )1≤i,j≤n with each Xij ∈ {0, 1}. Here, Xij = 1 means there is a directed
edge from vertex i to vertex j; otherwise, Xij = 0. Give the set of such graphs the
probability
"
!#
k
X
(n)
(n) −1
(n)
(n)
2
(n)
P (X) = Zn (β1 , β2 , . . . , βk ) exp n
βp sp (X)
,
(4.1)
p=1
where
sp (X) := n−p−1
X
Xij1 Xij2 · · · Xijp .
(4.2)
1≤i,j1 ,j2 ,...,jp ≤n
(n)
(n)
(n)
Here, Zn (β1 , β2 , . . . , βk ) is the appropriate normalization. Note sp (X), defined
in (4.2), represents outward directed p-star homomorphism densities of X. When
p = 1, it represents the directed edge homomorphism density of X. It is easy to
see that sp (X) has the alternative expression

p
n
n
X
X

Xij  .
(4.3)
sp (X) = n−p−1
i=1
j=1
18
MEI YIN AND LINGJIONG ZHU
We allow Xii to equal 1 for ease of notation. It is not hard to see that without this
simplification, our main results still hold.
We are interested in the sparse graph, i.e., the probability that there is a directed
(n)
edge from vertex i to vertex j goes to 0 as n → ∞. This requires that βp → −∞,
for some 1 ≤ p ≤ k, as n → ∞. Let us assume that
βp(n) = βp αn ,
p = 1, 2, . . . , k,
(4.4)
where αn → ∞ as n → ∞.
Theorem 11.
ψ(β1 , . . . , βk ) := lim
n→∞
1
log Zn = sup {β1 x + β2 x2 + · · · + βk xk },
n2 αn
0≤x≤1
Proof. It is therefore easy to compute that
h 2 Pk
i
(n)
2
(n)
(n)
Zn (β1 , . . . , βk ) = 2n E en ( p=1 βp sp (X))
h Pk
i
Pn
(n) −p+1 Pn
p
2
i=1 (
j=1 Xij )
= 2n E e p=1 βp n
h Pk
in
(n) −p+1 Pn
2
( j=1 Xij )p
= 2n E e p=1 βp n
n

n Pk
X
(n) j p
n
=
en p=1 βp ( n )  ,
j
j=0
(4.5)
(4.6)
where E denotes the expectation under which Xij are i.i.d. P(Xij = 0) = P(Xij =
1) = 21 .
On the one hand, we have
in
h
Pk
(n) j p
(n)
(n)
(4.7)
Zn (β1 , . . . , βk ) ≤ 2n emax0≤j≤n {n p=1 βp ( n ) }
2
= 2n en
2
2
2
≤ 2n en
Pk
j p
βp ( n
) }
Pk
βp xp }
αn max0≤j≤n {
αn max0≤x≤1 {
p=1
.
P
k
Therefore, lim supn→∞ n21αn log Zn ≤ max0≤x≤1 { p=1 βp xp }.
Pk
Pk
On the other hand, suppose that max0≤x≤1 { p=1 βp xp } = p=1 βp xp∗ for some
0 ≤ x∗ ≤ 1. There exists such an x∗ since the maximum of a continuous function
on a compact set is achieved though it may not be unique. Then, for any > 0, for
sufficiently large n, there exists some j∗ ∈ {0, 1, . . . , n} so that
( k
)
p
k
X
X
j∗
p
≥ max
βp x − .
(4.8)
βp
0≤x≤1
n
p=1
p=1
p=1
Therefore, we have
h Pk
i
(n) j∗ p n
(n)
(n)
Zn (β1 , . . . , βk ) ≥ en p=1 βp ( n )
2
≥ en
αn (max0≤x≤1 {
Pk
Pk
p=1
(4.9)
p
βp x }−)
.
Therefore, lim inf n→∞ n21αn log Zn ≥ max0≤x≤1 { p=1 βp xp } − . Since it holds for
any > 0, together with the upper bound, we proved (4.5).
ASYMPTOTICS FOR SPARSE EXPONENTIAL RANDOM GRAPH MODELS
19
Remark 12. When the parameters β1 , β2 , . . . , βk are negative, it gives the sparse
random graphs. We have already computed that
n X
1
n nαn Pkp=1 βp ( nj )p
(Zn ) n =
e
,
(4.10)
j
j=0
and limn→∞ n21αn log Zn = 0. That indicates that when the parameters β1 , β2 , . . . , βk
are negative, n21αn is not the optimal scaling for log Zn as n → ∞.
(i) When β1 , β2 , . . . , βk are negative,
n X
1
n nαn β1 ( j )
n
n
(Zn ) ≤
(4.11)
e
= (1 + eβ1 αn )n .
j
j=0
1
1
Therefore,
we always have lim supn→∞ (Zn ) n2 ≤ 1. On the other hand, Znn ≥
P
0 p
n nαn k
p=1 βp ( n ) = 1. Thus,
0 e
1
lim (Zn ) n2 = 1.
(4.12)
n→∞
(ii) Furthermore, if we assume that limn→∞ neβ1 αn = 0, then
1
lim (Zn ) n = 1.
(4.13)
n→∞
(iii) If instead we assume that limn→∞ neβ1 αn = λ ∈ (0, ∞), then as will be
shown in Proposition 15,
1
lim (Zn ) n = eλ .
(4.14)
n→∞
(iv) We can get more precise asymptotics. Let us assume that limn→∞ neβ1 αn =
0 and limn→∞ αnn = 0. On the one hand,
1
log Zn ≤ log(1 + eβ1 αn ).
n2
(4.15)
Therefore,
lim sup
n→∞
log Zn
≤ 1.
n2 eβ1 αn
(4.16)
On the other hand,
1
(Zn ) n ≥
1 X
n
j=0
j
enαn
Pk
p=1
j p
βp ( n
)
= 1 + nenαn
Pk
p=1
1 p
βp ( n
)
(4.17)
Therefore,
Pk
1 p
log Zn
log(1 + nenαn p=1 βp ( n ) )
lim inf 2 β1 αn ≥ lim inf
(4.18)
n→∞ n e
n→∞
neβ1 αn
Pk
P
p
k
p
1
1
log(1 + nenαn p=1 βp ( n ) ) nenαn p=1 βp ( n )
Pk
= lim inf
1 p
n→∞
neβ1 αn
nenαn p=1 βp ( n )
= 1,
where we used the assumptions that limn→∞ neβ1 αn = 0 and limn→∞
Hence, we conclude that
log Zn
lim
= 1.
n→∞ n2 eβ1 αn
αn
n
= 0.
(4.19)
20
MEI YIN AND LINGJIONG ZHU
For a sparse random graph, P(n) (Xij = 1) → 0 as n → ∞. It will be interesting
to know how sparse the random graph is and how fast P(n) (Xij = 1) converges to
zero as n → ∞.
Proposition 13. Assume that β1 , . . . , βk are all negative. Let us further assume
that limn→∞ neβ1 αn = 0 and limn→∞ αnn = 0.
P(n) (X1i = 1)
= 1.
n→∞
eβ1 αn
lim
(4.20)
Proof. By symmetry,
P(n) (X1i = 1) = E(n) [X1i ]
" n
#
1 (n) X
= E
X1i
n
i=1
Pk
Pn
(n) −p+1 Pn
( i=1 X1i )p
]
1 E[ i=1 X1i e p=1 βp n
=
Pk
(n) −p+1 Pn
p
β
n
(
X
)
n
1i
i=1
E[e p=1 p
]
Pk
j p
Pn
n
nαn p=1 βp ( n )
1 j=0 j je
= Pn
P
j p .
n nαn k
n
p=1 βp ( n )
j=0 j e
(4.21)
The denominator converges to 1 as n → ∞ from the assumption limn→∞ neβ1 αn =
0. For the numerator, it is clear that
n Pk
Pk
X
j p
1 p
n
jenαn p=1 βp ( n ) ≥ nenαn p=1 βp ( n ) .
(4.22)
j
j=0
On the other hand,
n n P
X
X
j p
n
n
nαn k
βp ( n
)
p=1
je
jejαn β1
≤
j
j
j=0
j=0
n 1 ∂ X n jαn β1
e
=
αn ∂β1 j=0 j
(4.23)
1 ∂
(1 + eαn β1 )n
αn ∂β1
= eαn β1 n(1 + eαn β1 )n−1 .
=
Finally, from the assumptions limn→∞ neβ1 αn = 0 and limn→∞
that
nαn Pk βp ( j )p
Pn
n
p=1
n
j=0 j je
= 1.
lim
β
α
n→∞
ne 1 n
αn
n
= 0, we conclude
(4.24)
Proposition 14. Assume that β1 , . . . , βk are all negative. Let us further assume
that limn→∞ neβ1 αn = 0 and limn→∞ αnn = 0.
P(n) (X1i = 1, X1j = 1)
= 1, i 6= j.
n→∞
e2β1 αn
lim
(4.25)
ASYMPTOTICS FOR SPARSE EXPONENTIAL RANDOM GRAPH MODELS
21
Proof. By symmetry,
P(n) (X1i = 1, X1j = 1)
(4.26)
(n)
=E
[X1i X1j ]


" n
!2 
#
n
X
X
1
E(n) 
=
X1i  − E(n)
X1i 
n(n − 1)
i=1
i=1
h P
Pk β (n) n−p+1 (Pn X1i )p i
Pn
n
2
i=1
E
(
X
)
−
X
e p=1 p
1i
1i
i=1
i=1
1
=
Pk
(n) −p+1 Pn
( i=1 X1i )p
n(n − 1)
E[e p=1 βp n
]
P
2
j p
Pn
n
nαn k
)
β
(
p=1 p n
1
j=0 j (j − j)e
=
.
Pk
j p
P
n
n nαn p=1 βp ( n
)
n(n − 1)
e
j=0
j
Pk
j p
Pn
Under the assumption limn→∞ neβ1 αn = 0, we have j=0 nj enαn p=1 βp ( n ) → 1
as n → ∞. Moreover, it is clear that
n Pk
Pk
X
j p
2 p
n
n
(j 2 − j)enαn p=1 βp ( n ) ≥
(22 − 2)enαn p=1 βp ( n )
(4.27)
j
2
j=0
= n(n − 1)e2αn β1 +nαn
Pk
p=2
2 p
βp ( n
)
,
and
n Pk
X
j p
n
(j 2 − j)enαn p=1 βp ( n )
j
j=0
n
X n
≤
(j 2 − j)ejαn β1
j
j=0
n n 1 ∂ 2 X n jαn β1
1 ∂ X n jαn β1
= 2
e
−
e
αn ∂β12 j=0 j
αn ∂β1 j=0 j
=
(4.28)
1 ∂2
1 ∂
(1 + eαn β1 )n −
(1 + eαn β1 )n
αn2 ∂β12
αn ∂β1
= n(n − 1)(1 + eαn β1 )n−2 e2αn β1 .
From the assumptions limn→∞ neβ1 αn = 0 and limn→∞
αn
n
= 0, our claim follows.
For the directed exponential random graph model, under the assumptions βi ,
1 ≤ i ≤ k are all negative and limn→∞ neβ1 αn = 0 and limn→∞ αnn = 0, we showed
(n)
1i =1)
that limn→∞ P e(X
= 1. What if we have limn→∞ neβ1 αn = λ ∈ (0, ∞)?
β1 αn
If that is the case, then limn→∞ αnn = 0 is automatically satisfied. We have the
following result.
Proposition 15. Assume that βi , 1 ≤ i ≤ k are all negative and limn→∞ neβ1 αn =
λ ∈ (0, ∞). Then, we have
P(n) (X1i = 1)
= 1.
n→∞
λn−1
lim
(4.29)
22
MEI YIN AND LINGJIONG ZHU
P(n) (X1i = 1, X1j = 1)
= 1, i 6= j.
n→∞
λ2 n−2
lim
(4.30)
Moreover, the degree of any vertex is asymptotically Poisson with parameter λ, i.e.,
n
X
X1i → Poisson(λ),
(4.31)
i=1
in distribution as n → ∞.
Proof. By symmetry,
"
(n)
nP
(n)
(X1i = 1) = E
n
X
#
X1i
(4.32)
i=1
Pn
j=0
= P
n
j
n
j
n
j=0 j
Pk
eαn β1 j+
eαn β1 j+αn
p=2
jp
np−1
αn βp
Pk
p=2
βp
jp
np−1
.
First, let us analyze the denominator. Since we assumed that limn→∞ neαn β1 =
λ, for any fixed M ,
n X
n
j=0
j
e
αn β1 j+αn
Pk
p=2
βp
jp
np−1
≥
M X
n
j
j=0
→
M
X
λj
j=0
j!
eαn β1 j+αn
Pk
p=2
βp
jp
np−1
(4.33)
,
as n → ∞. Since it’s true for any M , let M → ∞, and we obtain an asymptotic
P∞ j
lower bound j=0 λj! = eλ .
Moreover, since βp ’s are negative,
n n X
X
jp
n αn β1 j+αn Pkp=2 βp p−1
n αn β1 j
n
e
≤
e
j
j
j=0
j=0
(4.34)
= (1 + eαn β1 )n
→ eλ ,
as n → ∞.
Next, let us analyze the numerator. On the one hand, for any fixed M ,
n
M X
X
jp
jp
n αn β1 j+Pkp=2 αn βp p−1
n αn β1 j+Pkp=2 αn βp p−1
n
n
j
e
≥
j
e
j
j
j=0
j=0
→
M
−1
X
j=0
(4.35)
λj+1
,
j!
as n → ∞. Since it’s true for any M , let M → ∞, and we obtain an asymptotic
P∞ j+1
lower bound j=0 λ j! = λeλ .
ASYMPTOTICS FOR SPARSE EXPONENTIAL RANDOM GRAPH MODELS
On the other hand,
n
n
X
X
jp
n αn β1 j+Pkp=2 αn βp p−1
n αn β1 j
n
j
e
≤
j
e
j
j
j=0
j=0
23
(4.36)
1 ∂
(1 + eαn β1 )n
αn ∂β1
= neαn β1 (1 + eαn β1 )n−1
=
→ λeλ .
Again by symmetry,

(n)
n(n − 1)P
n
X
(n)
(X1i = 1, X1j = 1) = E

!2 
X1i
"
(n)
−E
i=1
Pn
=
n
X
#
X1i
(4.37)
i=1
Pk
jp
− j) nj eαn β1 j+ p=2 αn βp np−1
.
P
jp
Pn
n αn β1 j+ k
p=2 αn βp np−1
e
j=0 j
j=0 (j
2
The denominator converges to eλ as n → ∞ from the assumption limn→∞ neβ1 αn =
λ. For the numerator, it is clear that for any fixed M ,
n
M
X
X
jp
n αn β1 j+Pkp=2 αn βp p−1
n nαn Pkp=1 βp ( nj )p
n
(j 2 − j)
e
(j 2 − j)
e
≥
(4.38)
j
j
j=0
j=0
→
M
−2
X
j=0
λj+2
,
j!
as n → ∞. Since it’s true for any M , let M → ∞, and we obtain an asymptotic
P∞ j+2
lower bound j=0 λ j! = λ2 eλ .
On the other hand,
n
n
X
X
jp
n αn β1 j
n αn β1 j+Pkp=2 αn βp p−1
2
2
n
(j − j)
e
(4.39)
(j − j)
e
≤
j
j
j=0
j=0
=
1 ∂2
1 ∂
(1 + eαn β1 )n −
(1 + eαn β1 )n
2
2
αn ∂β1
αn ∂β1
= n(n − 1)(1 + eαn β1 )n−2 e2αn β1
→ λ2 eλ .
Lastly, for any fixed j ∈ N ∪ {0},
!
n
X
P(n)
X1i = j = E(n) 1Pni=1 X1i =j
(4.40)
i=1
=P
n
n
j
j=0
Pk
jp
eαn β1 j+ p=2 αn βp np−1
.
P
jp
n αn β1 j+αn k
p=2 βp np−1
e
j
Under the assumption limn→∞ neβ1 αn = λ, the denominator converges to eλ and
j
the numerator converges to λj! as n → ∞. Hence, we proved the desired result.
24
MEI YIN AND LINGJIONG ZHU
Another natural question to ask is what if lim inf n→∞
case, then limn→∞ neβ1 αn = 0 is automatically satisfied.
αn
n
> 0? If that is the
Proposition 16. Assume that β1 , . . . , βk are all negative and lim inf n→∞
log 2
|β1 | , then
lim
n→∞
P(n) (X1i = 1)
enαn
Pk
p=1
1 p
βp ( n
)
= 1.
αn
n
>
(4.41)
Proof. Let us recall that
(n)
P
(X1i
Pn
1 j=0
= 1) = Pn
n
j=0
n
j
j
Pk
p
jenαn p=1 βp ( n )
P
j p .
n nαn k
p=1 βp ( n )
j e
(4.42)
Under the implied assumption limn→∞ neβ1 αn = 0, the denominator
n X
n nαn Pkp=1 βp ( nj )p
= 1.
e
lim
n→∞
j
j=0
(4.43)
For the numerator, on the one hand,
n Pk
Pk
X
j p
1 p
n
n nαn Pkp=1 βp ( n1 )p
jenαn p=1 βp ( n ) ≥
e
= nenαn p=1 βp ( n ) .
j
1
j=0
On the other hand, since βp ’s are negative,
n n P
X
X
j p
2
n
n
nαn k
βp ( n
)
p=1
je
≤
jenαn β1 n ≤ 2n ne2αn β1 .
j
j
j=2
j=2
(4.44)
(4.45)
Moreover,
2n ne2αn β1
ne
nαn
Pk
1 p
p=1 βp ( n )
2n e2αn β1
=
e
P
αn β1 + k
p=2 αn
βp
np−1
= en[log 2+
as n → ∞ under the assumption lim inf n→∞
desired result.
αn
n
>
αn
n
β1 − αnn
log 2
|β1 | .
βp
p=2 np−1
Pk
]
→ 0,
(4.46)
Hence, we proved the
Let us recall that in Theorem 11, we proved that
1
ψ(β1 , . . . , βk ) := lim 2
log Zn = sup {β1 x + β2 x2 + · · · + βk xk }.
n→∞ n αn
0≤x≤1
(4.47)
Now, let us consider the edge-(single)-star model, i.e. H1 is an edge and H2 is an
outward p-star. Thus, by Theorem 11,
ψ(β1 , β2 ) = sup {β1 x + β2 xp }.
0≤x≤1
Proposition 17.
h
p
i
−p
−1

β1p−1

p−1 − p p−1

p
1

(−β2 ) p−1
ψ(β1 , β2 ) =
0



β + β
1
2
if β1 + β2 p < 0 and β1 > 0
if β1 + β2 ≤ 0 and β1 ≤ 0
otherwise
.
(4.48)
There are second-order phase transitions across the phase transition curves {β1 =
0, β2 < 0} and {β1 +pβ2 = 0, β2 < 0}. There is a first-order phase transition across
the phase transition curve {β1 + β2 = 0, β2 > 0}.
ASYMPTOTICS FOR SPARSE EXPONENTIAL RANDOM GRAPH MODELS
25
Proof. Let us define
`(x) = β1 x + β2 xp ,
0 ≤ x ≤ 1.
(4.49)
Then, `00 (x) = β2 p(p − 1)xp−2 is always non-negative if β2 is and is always nonpositive if β2 is. Therefore, the local maximizer on the interval [0, 1] is unique.
1
p−1
1
Note that if β1 β2 < 0, then, `0 (x) = β1 + β2 pxp−1 = 0 gives x = −β
.
β2 p
1 p−1
1
. The maximizer is
The maximizer should be achieved in the set 0, 1, −β
β2 p
1
1
p−1
p−1
1
1
only if `00 (x) < 0 and −β
< 1 which is equivalent to
achieved at −β
β2 p
β2 p
β1 > 0 and β1 + β2 p < 0. One can compute that
p
1 !
1
p
h −1
i β p−1
−p
−β1 p−1
−β1 p−1
−β1 p−1
1
p−1
p−1
`
+ β2
= p
−p
= β1
1
β2 p
β2 p
β2 p
(−β2 ) p−1
(4.50)
Otherwise, the maximizer is achieved at either 0 or 1. To summarize, we have
h
p
i
−p
−1
 p−1
β1p−1

p−1

p
−
p
if β1 + β2 p < 0 and β1 > 0
1

(−β2 ) p−1
.
(4.51)
ψ(β1 , β2 ) =
0
if β1 + β2 ≤ 0 and β1 ≤ 0



β + β
otherwise
1
2
There are three phases, with boundaries given by
(1) {β1 = 0, β2 < 0}.
(2) {β1 + β2 = 0, β2 > 0}.
(3) {β1 + pβ2 = 0, β2 < 0}.
The three phases are illustrated in Figure 2.
As β1 → 0+ ,
p
i β p−1
h −1
−p
1
→ 0.
p p−1 − p p−1
1
(−β2 ) p−1
(4.52)
Therefore ψ(β1 , β2 ) is continuous across the phase transition curve
{β1 = 0, β2 < 0}.
(4.53)
Moroever,
p
1
i β p−1
h −1
i β p−1
−p
−p
−1
∂ h p−1
p
1
1
p
− p p−1
= p p−1 − p p−1
→ 0,
1
1
∂β1
p
−
1
p−1
p−1
(−β2 )
(−β2 )
(4.54)
as β1 → 0+ and
p
1
i β p−1
h −1
i β p−1 −1
−p
−p
−1
1
∂ 2 h p−1
p
1
1
p−1
p−1
p−1
p
−p
= p
−p
→ +∞,
1
1
2
∂β1
(−β2 ) p−1
(−β2 ) p−1 p − 1 p − 1
(4.55)
as β1 → 0+ . Hence, there is a second-order phase transition across the phase
∂k
transition curve {β1 = 0, β2 < 0}. On the other hand, we can check that ∂β
kψ = 0
as β1 → 0+ for any fixed β2 < 0 for any k = 0, 1, 2, . . ..
2
26
MEI YIN AND LINGJIONG ZHU
∂
There is a first-order phase transition, i.e. ∂β
ψ(β1 , β2 ) is not continuous when
2
the parameter β2 crosses the phase transition curve
{β1 + β2 = 0, β2 > 0}.
(4.56)
First, ψ(β1 , β2 ) is continuous across this curve so there is no zeroth-order phase
∂
ψ changes from 1 to 0 when β2 decreases across the curve.
transition. Second, ∂β
2
Hence, there is a first-order phase transition across the curve.
Finally, when β2 = − βp1 , we have
β1 + β2 = (1 − p)β2 ,
and
"
h
p
p
−1
p−1
−p
−p
p−1
i
(4.57)
#
β1p−1
1
(−β2 ) p−1
= (1 − p)β2 .
Therefore ψ(β1 , β2 ) is continuous across the curve {β1 + pβ2 = 0, β2 < 0}.
Moreover, one can compute that,
"
#
p
i β p−1
−p
−1
∂ h p−1
1
− p p−1
p
1
β
∂β2
p−1
(−β2 )
β2 =− p1
p
i β p−1 1 h −1
−p
1
p−1
p−1
−p
= p
p
(−β2 ) p−1 p − 1 β2 =− βp1
(4.58)
(4.59)
= 1.
∂
Hence, ∂β
ψ is continuous across the curve {β1 + pβ2 = 0, β2 < 0}. Similarly, one
2
can check that
"
#
p
i β p−1
−p
−1
∂ h p−1
1
= 1.
(4.60)
p
− p p−1
1
∂β1
(−β2 ) p−1 β2 =− β1
p
∂
∂β1 ψ
Hence,
is continuous across the curve {β1 + pβ2 = 0, β2 < 0}.
On the other hand,
"
#
p
i β p−1
−p
−1
∂ 2 h p−1
1
p−1
p
−p
1
2
∂β2
(−β2 ) p−1 β2 =− βp1
p
h −1
i β p−1
−p
p
1
1
= p p−1 − p p−1
2p−1
β
p
−
1
p
−
1
p−1
(−β2 )
β2 =− p1
=
p
p−1
−β2
(4.61)
.
Similarly,
"
#
p
i β p−1
−p
−1
−1
∂ 2 h p−1
1
p
− p p−1
=
.
1
2
∂β1
p(p − 1)β2
(−β2 ) p−1 β2 =− β1
(4.62)
p
Hence there is second-order phase transition across the curve {β1 + pβ2 = 0, β2 <
0}.
ASYMPTOTICS FOR SPARSE EXPONENTIAL RANDOM GRAPH MODELS
27
1.0
Phase I
0.5
0.0
Phase II
-0.5
Phase III
-1.0
-1.0
-0.5
0.0
0.5
1.0
Figure 2. This is a plot of the phase diagram for the edge and
outward p-star model, where p = 2. The horizontal axis denotes
β1 and the vertical axis denotes β2 . In Phase I, ψ(β1 , β2 ) =
β1 + β2 ; in Phase II, ψ(β1 , β2 ) = 0; in Phase III, ψ(β1 , β2 ) =
p
h −1
i
−p
β1p−1
p−1
p−1
p
−p
1 . The boundaries between the phases are
(−β2 ) p−1
given by {β1 = 0, β2 < 0}, {β1 + β2 = 0, β2 > 0}, and
{β1 + pβ2 = 0, β2 < 0}. There are second-order phase transitions between Phase II and Phase III, and between Phase III and
Phase I. There is a first-order phase transition between Phase I
and Phase II.
Remark 18. More generally, we can consider sup0≤x≤1 {β1 xq + β2 xp }, q < p. If
β1 ≥ 0 and β2 ≥ 0, ψ(β1 , β2 ) = β1 + β2 . If β1 ≤ 0 and β2 ≤ 0, ψ(β1 , β2 ) = 0. If
1
p−q
β1 q
β1 β2 < 0, `0 (x) = 0 is achieved at x = −β
. If β1 < 0, β2 > 0, then `(x)
2p
1
p−q
β1 q
decreases for small positive x and x = −β
cannot be a maximizer. Thus
2p
when β1 < 0 and β2 > 0, ψ(β1 , β2 ) = 0 if β1 + β2 ≤ 0 and ψ(β1 , β2 ) = β1 + β2 if
β1 q
β1 + β2 > 0. If β1 > 0 and β2 < 0 and −β
≥ 1, then ψ(β1 , β2 ) = β1 + β2 . If
2p
q
p
p−q
p−q
β1 q
β1 q
β1 q
β1 > 0 and β2 < 0 and −β
<
1,
then
ψ(β
,
β
)
=
β
+
β
.
2 −β2 p
1
2
1 −β2 p
2p
28
MEI YIN AND LINGJIONG ZHU
To summarize, we have
 q
p
p p−q

β1p−q
q
q p−q


−
q
 p
p
(−β2 ) p−q
ψ(β1 , β2 ) =
0



β + β
1
2
if (β1 , β2 ) ∈ {β1 q + β2 p < 0} ∩ {β1 > 0}
if (β1 , β2 ) ∈ {β1 + β2 < 0} ∩ {β1 ≤ 0}
otherwise
(4.63)
Acknowledgements
Mei Yin’s research was partially supported by NSF grant DMS-1308333.
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Department of Mathematics
University of Denver
2280 S. Vine St.
Denver, CO-80208
United States of America
E-mail address:
[email protected]
School of Mathematics
University of Minnesota-Twin Cities
206 Church Street S.E.
Minneapolis, MN-55455
United States of America
E-mail address:
[email protected]