The degree sequences and spectra of scale-free random graphs

The Degree Sequences and Spectraof Scale-Free Random Graphs

Jonathan JordanDepartment of Probability and Statistics, University of Sheffield, Hicks Building,

Sheffield S3 7RH, United Kingdom; email: [email protected]

Received 30 July 2004; accepted 11 May 2005Published online 12 December 2005 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/rsa.20101

ABSTRACT: We investigate the degree sequences of scale-free random graphs. We obtain a for-mula for the limiting proportion of vertices with degree d, confirming non-rigorous arguments ofDorogovtsev, Mendes, and Samukhin (Phys Rev Lett 85 (2000), 4633). We also consider a gener-alization of the model with more randomization, proving similar results. Finally, we use our resultson the degree sequence to show that for certain values of parameters localized eigenfunctions of theadjacency matrix can be found. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 29, 226–242, 2006

Keywords: scale-free random graphs; degree sequences; eigenvalues

1. INTRODUCTION

There has been considerable interest recently in random scale-free graphs as models ofvarious real-world phenomena. As defined in [1], these are growing graphs constructed by,at each stage, adding vertices that are joined to vertices already present in the graph usinga rule of preferential attachment, so that when an edge is added from a new vertex to anexisting vertex, that existing vertex is chosen with probability proportional to its degree. Inthe literature, including [1,4,7], it is suggested that this is a useful model for the growth ofnetworks found in various fields, including technological, social, and biological networks.One property that many real-world graphs possess is a power-law degree sequence: anda mean field argument is used in [1] that suggests that the preferential attachment rule as

Correspondence to: J. Jordan© 2005 Wiley Periodicals, Inc.

226

DEGREE SEQUENCES AND SPECTRA OF RANDOM GRAPHS 227

described in [1] leads to number of vertices with degree d following a power law withindex −3. Further analysis of growing networks and preferential attachment can be foundin [2, 3, 6, 7].

A survey of rigorous mathematical results on scale-free graphs appears as [9]. As pointedout in [9,10], the model as described in [1] does not actually make mathematical sense, and amathematically precise version of the model is defined in [10] where, for this mathematicallyprecise version, the result that the number of vertices with degree d follows a power lawwith index −3 is rigorously proved.

As a way of constructing graphs with degree sequences following power laws withindices other than −3, one possibility is, instead of choosing a vertex with probabilityproportional to its degree, to choose a vertex with probability proportional to its degreeplus some constant q. This is introduced in [14], where a non-rigorous argument showsthat the degree sequence follows a power law with index depending on q and on the otherparameters of the model. For some values of the parameters, this is made rigorous in [11],again for a mathematically precise version of the model of [14]. This is enough to giveexamples of power laws with all integer indices ≤−3.

In this paper, we use a slightly different mathematically precise version of the modelfrom those defined in [10,11], described in Section 2. For this model, we rigorously obtaina precise formula for the expected proportion of vertices with degree d, which correspondsto that in [14], and show that the proportion of vertices with degree d converges to this inprobability. This gives examples of power laws with all real indices ≤−2 and confirms theclaim of [14] on the degree sequence.

In the model described in [1], and also those considered in [10, 11], each vertex addedto the network initially has degree m, where m is a constant. A natural generalization is toallow the initial degree of new vertices to be independent and identically distributed randomvariables, and we consider this in Section 4, showing that, under fairly mild conditions onthe distribution of the random variables, a formula similar to that found in Section 3 givesthe limiting proportion of vertices with degree d.

In [12], a more general model is proposed, in which a new vertex can have a randomdegree, and also edges are allowed to be added between already existing vertices. Insteadof choosing vertices with probability proportional to their degree plus a constant q, thismodel chooses vertices uniformly at random with a fixed probability β and with probabilityproportional to their degree with probability 1−β. It is easy to show that these two methodsare asymptotically equivalent if β = q/(2m+q). For this general model, it is proved in [12]that the expected degree sequence asymptotically follows a power law with index dependingon the parameters of the model. Our results in Sections 3 and 4 provide more detailed resultsfor models similar to special cases of the model of [12] where edges are not added betweenalready existing vertices (that is, in the notation of [12], α = 0 and so procedure old is notused).

In [5], the spectra of adjacency matrices of various types of random graphs, includingscale-free graphs, are obtained using simulation. In [4], the spectrum of a molecular bio-logical network is analyzed, and the spectral properties appear to be similar to those ofscale-free graphs. Some further work on spectra of complex networks, including scale-freegraphs, appears in [13].

In Section 5 we apply our results on the degree sequence to obtain results on the spectra ofthe adjacency matrices of the graphs. Comparison with the simulation results obtained in [5]suggests that interesting differences in the spectra can result from varying the parametersof the scale-free model.

228 JONATHAN JORDAN

2. THE BASIC CONSTRUCTION

To construct random scale-free graphs, we use the following construction, based on thatin [1]. We take an integer parameter m. Starting from an initial connected graph G0, withm0 vertices and e0 edges, we then construct a sequence of graphs (Gn)n∈N.

To construct Gn+1 from Gn, we add a new vertex v and then add m edges between vand vertices of Gn. We choose m random vertices Wn+1,1, Wn+1,2, . . . , Wn+1,m according to apreferential attachment rule. If degGn

(w) is the degree of w in Gn, then we let Wn+1,i = wwith probability

degGn(w)∑

u∈V(Gn) degGn(u)

independently for each i ∈ {1, 2, . . . , m}.Note that this allows the possibility that we choose the same vertex more than once, and

hence the graphs may have multiple edges.As a generalization of this model, we set a constant q and, instead of choosing a vertex

with probability proportional to d, we choose it with probability proportional to d +q. Thatis, if degGn

(w) is the degree of w in Gn, then we let Wn+1,i = w with probability

q + degGn(w)∑

u∈V(Gn)(q + degGn(u))

independently for each i ∈ {1, 2, . . . , m}.This requires d +q > 0 for all possible vertex degrees d. If each vertex of G0 has degree

at least m then all vertices in all Gn will also have degree at least m, so, if we choose anappropriate G0, we can use any q ∈ (−m, ∞). The original model of [1] corresponds to thespecial case where q = 0.

The number of vertices in Gn is obviously n + m0. Because m edges are added at eachstage, the total number of edges in Gn is mn + e0, and, hence,

P(Wn+1,i = w) = q + degGn(w)∑

u∈V(Gn)(q + degGn(u))

= q + degGn(w)

2(mn + e0) + q(m0 + n).

We define

cn = 2(mn + e0) + q(m0 + n)

n,

so that

P(Wn+1,i = w) = q + degGn(w)

cnn,

and note that limn→∞ cn = 2m + q.This construction differs from those described in [9–11] in that a new vertex cannot be

connected to itself, so that there are no loops. In the context of this paper we also allow theconstruction to start from an unspecified connected graph G0, whereas the constructionsin [9–11] start from a specific graph. We also note that it is possible to consider the graphsconstructed as directed graphs, with edges considered as being from the added vertex to theolder vertices, and that this is how the graphs are considered in [11, 14].


3. THE DEGREE SEQUENCE

We define the random variable Ad,n to be the number of vertices of Gn with degree d, andwe set Ad,n to be the proportion of vertices of Gn with degree n, that is

Ad,n = Ad,n

n + m0,

and set αd,n = EAd,n, the expected proportion of vertices with degree d.We first prove a simple lemma on convergence of sequences, which we will use in our

proofs.

Lemma 1. For n ∈ N, let xn, yn, ηn, rn be real numbers such that

xn+1 − xn = ηn+1(yn − xn) + rn+1

and

(a) yn → x as n → ∞;(b) ηn > 0, and there exists N0 such that, for n > N0, ηn < 1;(c)

∑∞n=1ηn = ∞;

(d) As n → ∞, rn/ηn → 0.

Then xn → x as n → ∞.

Proof. First, we note that, using (d), we can assume rn = 0 for all n, replacing yn withyn + rn

ηn. So xn+1 = xn(1 − ηn+1) + ηn+1yn, so that, for n > N0, xn+1 is a weighted average

of xn and yn.Now, given ε > 0, consider Nε > N0 such that, for n ≥ Nε , |yn−x| < ε

2 . Hence, if xn > x − ε

for some n ≥ Nε , then xn+1 > x − ε, and similarly if xn < x + ε for some n ≥ Nε , then xn+1 <

x + ε.Finally, if xn < x − ε for all n ≥ Nε , then

∞∑n=Nε

(xn+1 − xn) ≥∞∑

n=Nε

ηn+1ε

2= ∞,

and so xn → ∞, which is a contradiction. Similarly if xn > x + ε for all n ≥ Nε , then∑∞n=Nε

(xn − xn+1) = ∞, which again gives a contradiction. So for n large enough we havethat |xn − x| ≤ ε.

Theorem 2. For each d ≥ m, the expected proportion αd,n satisfies

αd,n →((

2 + qm

)�

(3 + q

m + m + q)

(2 + q

m + q + m)�(m + q)

)�(d + q)

�(3 + q

m + d + q) as n → ∞.

In the q = 0 case, this becomes

αd,n → 2m(m + 1)

d(d + 1)(d + 2), (1)

as found in [10].

230 JONATHAN JORDAN

As in [14], we note that properties of gamma functions show that, as d → ∞, theformula in Theorem 2 is approximately a power law with index −(3 + q

m ). As we can haveany value of q ∈ (−m, ∞) (with conditions on G0, as described in Section 2, if q < 0), thisgives all indices in (−∞, −2).

For use in this proof, as well as later proofs, we define the σ -algebra Fn = σ(Gm; 0 ≤m ≤ n).

Proof. For 0 ≤ k ≤ m, the probability that Wn+1,i = w for exactly k values of i, and sothe degree of vertex w in Gn+1 is degGn

(w) + k, is(m

k

) (cnn − degGn

(w) − q

cnn

)m−k (degGn

(w) + q

cnn

)k

.

Hence, conditional on the value of Ad−k,n, the expected number of vertices with degree d inGn+1 and degree d − k in Gn is

Ad−k,n

(m

k

) (cnn − d − q + k

cnn

)m−k (d + q − k

cnn

)k

.

At each stage, one new vertex is added, with degree m. So,

E(Ad,n+1|Fn) =m∑

k=0

Ad−k,n

(m

k

) (cnn − d − q + k

cnn

)m−k (d + q − k

cnn

)k

+ Im(d)

(where Im(d) is a function of d that is 1 for d = m and 0 otherwise) and hence the proportionssatisfy

E(Ad,n+1|Fn) = n + m0

n + m0 + 1

m∑k=0

Ad−k,n

(m

k

) (cnn − d − q + k

cnn

)m−k

×(

d + q − k

cnn

)k

+ 1

n + m0 + 1Im(d), (2)

so the difference E(Ad,n+1|Fn) − Ad,n is

1

(n + m0 + 1)(cnn)m

[Ad,n((n + m0)(cnn − d − q)m − (n + m0 + 1)(cnn)m)

+ Ad−1,nm(n + m0)(d + q − 1)(cnn − d − q + 1)m−1

+m∑

k=2

Ad−k,n

(m

k

)(n + m0)(d + q − k)k(cnn − d − q + k)m−k + (cnn)mIm(d)

],

which, collecting terms in powers of n, is

1

(n + m0 + 1)cmn nm

[nmmcm−1

n

(Ad−1,n(d + q − 1) + cn

mIm(d) − Ad,n

(d + q + cn

m

))

+ mcm−1n

m∑j=1

nm−jj∑

k=0

Cd,j,k,nAd−k,n

],

where Cd,j,k,n is bounded in n.


So

E(Ad,n+1|Fn) − Ad,n

= 1

n + m0 + 1

(m

cn

) [ (d + q + cn

m

) (d + q − 1

d + q + cnm

Ad−1,n + cn/m

d + q + cnm

Im(d) − Ad,n

)

+m∑

j=1

n−jj∑

k=0

Cd,j,k,nAd−k,n

](3)

and so the expected proportions satisfy

αd,n+1 − αd,n

= 1

n + m0 + 1

(m

cn

) [ (d + q + cn

m

) (d + q − 1

d + q + cnm

αd−1,n + cn/m

d + q + cnm

Im(d) − αd,n

)

+m∑

j=1

n−jj∑

k=0

Cd,j,k,nαd−k,n

]

Because each new vertex has degree m, and, in the model, degrees of vertices can onlyincrease, there can be at most m0 vertices of degree d, for d < m (the m0 initial vertices). So

αd,n → 0 as n → ∞, for all d < m.

Now,

∞∑n=1

(1

n + m0 + 1

) (d + q + cn

m

) (m

cn

)= ∞,

and because the Cd,j,k,n are bounded in n and | αd−k,n |≤ 1, we have

n−jj∑

k=0

Cd,j,k,nαd−k,n → 0 as n → ∞,

for each d, m, j. So, starting with the d = m case, we can apply Lemma 1 with

ηn+1 =(

1

n + m0 + 1

) (m + q + cn

m

) (m

cn

),

rn+1 =(

1

n + m0 + 1

) (m

cn

) [m∑

j=1

n−jj∑

k=0

Cm,j,kαm−k,n

],

yn = 1

m + q + cnm

cn

m+ m + q − 1

m + q + cnm

αm−1,n,

which has limit

limn→∞ yn = 2 + q

m

m + q + qm + 2

232 JONATHAN JORDAN

to show that

αm,n → 2 + qm

m + q + qm + 2

as n → ∞.

For d > m, we proceed using induction, again using Lemma 1 but with yn = d+q−1d+q+ cn

mαd−1,n,

to find that

αd,n → 2 + qm

2 + qm + q + m

d∏l=m+1

l − 1 + q

2 + qm + q + l

as n → ∞,

so, simplifying the product, for d ≥ m,

αd,n →((

2 + qm

)�

(3 + q

m + m + q)

(2 + q

m + q + m)�(m + q)

)�(d + q)

�(3 + q

m + d + q) as n → ∞.

Theorem 3. The proportions Ad,n converge in L2 and in probability to the limitingexpectation ((

2 + qm

)�

(3 + q

m + m + q)

(2 + qm + q + m)�(m + q)

)�(d + q)

�(3 + q

m + d + q)

as n → ∞.

Proof. Add Ad,n to both sides of (3) and square both sides of the resulting equation. Thengroup terms that are O(n−2) as n → ∞, giving∣∣∣∣[E(Ad,n+1|Fn)]2 − A2

d,n − 2

(1

n + m0 + 1

) (m

cn

)

×[(

d + q + cn

n

) (d + q − 1

d + q + cnn

Ad−1,nAd,n − A2d,n

)]∣∣∣∣ < Kdn−2,

where Kd is a constant, for d > m and∣∣∣∣[E(Am,n+1|Fn)]2 − A2m,n − 2

(1

n + m0 + 1

) (m

cn

)

×[(

m + q + cn

n

) (cn/m

m + q + cnn

Am,n − A2m,n

)]∣∣∣∣ < Kmn−2.

Now, |Ad,n+1 − Ad,n| ≤ m, so |Ad,n+1 − Ad,n| ≤ m+1n+1 , so Var(Ad,n+1|Fn) ≤ (m+1

n+1 )2 and hence∣∣∣∣E(A2d,n+1|Fn) − A2

d,n − 2

(1

n + m0 + 1

) (m

cn

)

×[(

d + q + cn

n

) (d + q − 1

d + q + cnn

Ad−1,nAd,n − A2d,n

)]∣∣∣∣ < Kdn−2, (4)


where Kd is a constant, for d > m and∣∣∣∣E(A2m,n+1|Fn)

2 − A2m,n − 2

(1

n + m + 1

) (m

cn

)

×[(

m + q + cn

n

) (cn/m

m + q + cnn

Am,n − A2m,n

)]∣∣∣∣ < Kmn−2. (5)

Now, EAm,n → 2+ qm

m+q+ qm +2

, so taking expectations in (5) implies that we can use Lemma 1

with xn = EA2m,n and yn = 2+ q

mm+q+ q

m +2EAm,n to show that

EA2m,n →

(2 + q

m

m + q + qm + 2

)2

,

showing that Var(Am,n) → 0 as n → ∞.We now proceed by induction, assuming as our induction hypothesis that Var(Ad−1,n) →

0 as n → ∞. Then∣∣EAd−1,nEAd,n − E(Ad−1,nAd,n)| = | Cov(Ad−1,n, Ad,n)∣∣ → 0

as n → ∞ as Var Ad,n ≤ 1. So taking expectations in (4) implies that we can use Lemma 1again, with xn = EA2

d,n and

yn = d + q − 1

d + q + cnn

EAd,nAd−1,n

so that

yn → limn→∞

d + q − 1

d + q + cnn

EAd,nEAd−1,n

=(((

2 + qm

)�

(3 + q

m + m + q)

(2 + q

m + q + m)�(m + q)

)�(d + q)

�(3 + q

m + d + q))2

and hence Var(Ad,n) → 0 as n → ∞.As convergence in probability follows from L2 convergence [15, Section 7.2], this now

gives the result.

Using properties of gamma functions, this gives a degree sequence decaying approxi-mately as a power law with index −(3 + q

m ). With an appropriate choice of initial graph G0

and parameter q ∈ (−m, ∞) we can obtain a power law with any fixed index γ < −2.

4. A VARIATION WITH RANDOM m

In the model defined in Section 2, each new vertex is added to the graph with degree exactlym, where m is a constant. As a generalization, we now consider a model where the numberof old vertices the new vertex at stage n is connected to is a random variable Mn.

234 JONATHAN JORDAN

To define this model, we define a sequence of independent identically distributed positive-integer-valued random variables (Mn)n∈N, with Mn+1 independent of the σ -algebra Fn. Werestrict ourselves to the case q = 0, and define

Dn = 1

n

∑w∈V(Gn)

degGn(w),

so that

P(Wi = w) = degGn(w)

Dnn,

where the random variable Dn corresponds to the constant cn used in the deterministicmodel.

We now state and prove an analogue of Theorem 2 for this model.

Theorem 4. If the independent and identically distributed random variables Mn have amoment generating function that exists in a neighborhood of 0, then the expected proportionof vertices of Gn having degree d converges to

2

d(d + 1)(d + 2)E(M1(M1 + 1)I{M1≤d})

as n → ∞.(Here, for an event E, IE is defined to a random variable that is 1 on E and 0 otherwise.)

Proof. By the same method used to derive (2) in the proof of Theorem 2, we have

E

(Ad,n+1

n + 1

∣∣∣∣Fn, Mn+1

)

= n

n + 1

Mn+1∑k=0

Ad−k,n

n

(Mn+1

k

) (Dnn − d + k

Dnn

)Mn+1−k (d − k

Dnn

)k

+ 1

n + 1I{d=Mn+1}.

So

E

((Ad,n+1

n + 1− Ad,n

n

) ∣∣∣∣Fn, Mn+1

)

= 1

(n + 1)

[Ad,n

n(n(1 − d/Dnn)Mn+1 − (n + 1))

+Mn+1∑k=1

Ad−k,n

n

(Mn+1

k

)n(Dnn)k(d − k)k(1 − (d + k)/Dnn)Mn+1−k + I{d=Mn+1}

]

= 1

n + 1

Ad,n

n

Mn+1∑

r=2

(Mn+1

r

)D−r

n dr(−1)rn1−r − (Mn+1D−1

n d + 1) + I{d=Mn+1}

+Mn+1∑k=1

(Mn+1

k

)Ad−k,n

n

Mn+1∑r=k

(Mn+1 − k

r − k

)(−1)r−k(d − k)rD−r

n n1−r

(6)


We note that if we extract the r = 1, k = 1 term from the double sum in (6), the

Ad,n

n

Mn+1∑r=2

(Mn+1

r

)D−r

n dr(−1)rn1−r

term can be included in the sum as a k = 0 term, giving

1

n + 1

(I{d=Mn+1} + Mn+1

Ad−1,n

n(d − 1)D−1

n − Ad,n

n

(Mn+1D−1

n d + 1)

+Mn+1∑k=0

(Mn+1

k

)Ad−k,n

n

Mn+1∑r=min{2,k}

(Mn+1 − k

r − k

)(−1)r−k(d − k)rD−r

n n1−r

)

and we reverse the order of summation to give

E

((Ad,n+1

n + 1− Ad,n

n

) ∣∣∣∣Fn, Mn+1

)

= 1

n + 1

(I{d=Mn+1} + Mn+1

Ad−1,n

n(d − 1)D−1

n − Ad,n

n

(Mn+1D−1

n d + 1)

+Mn+1∑r=2

n1−rr∑

k=0

Ad−k,n

n

(Mn+1

k

)(Mn+1 − k

r − k

)(−1)r−k(d − k)rD−r

n

).

Note that Ad−k,n = 0 if k ≥ d, so

E

((Ad,n+1

n + 1− Ad,n

n

) ∣∣∣∣Fn, Mn+1

)

= 1

n + 1

(I{d=Mn+1} + Mn+1

Ad−1,n

n(d − 1)D−1

n − Ad,n

n

(Mn+1D−1

n d + 1)

+Mn+1∑r=2

n1−rD−rn

(Mn+1

r

) max{r,d−1}∑k=0

Ad−k,n

n

(r

k

)(−1)r−k(d − k)r

).

We now remove the conditioning on Mn+1, writing µ = EMn+1 and qm = P(Mn+1 = m):

E

((Ad,n+1

n + 1− Ad,n

n

) ∣∣∣∣Fn

)

= 1

n + 1

∞∑m=1

qm

(Im(d) + m

Ad−1,n

n(d − 1)D−1

n − Ad,n

n

(mD−1

n d + 1)

+m∑

r=2

n1−rD−rn

(m

r

) max{r,d−1}∑k=0

Ad−k,n

n

(r

k

)(−1)r−k(d − k)r

)

= 1

n + 1

(qd + Ad−1,n

nµD−1

n (d − 1) − Ad,n

n

(µD−1

n d + 1)

+∞∑

r=2

n1−rD−rn E

(M1

r

) max{r,d−1}∑k=0

Ad−k,n

n

(r

k

)(−1)r−k(d − k)r

). (7)

236 JONATHAN JORDAN

If µ is finite, then the Strong Law of Large Numbers shows that Dn → 2µ as n → ∞,almost surely, and

Ad−k,nn ≤ n+m0

n . So, for any ε ∈ (0, 2µ) we can, almost surely, find arandom Nε such that, if n ≥ Nε ,∣∣∣∣∣n1−rD−r

n E

(M1

r

) max{r,d−1}∑k=0

Ad−k,n

n

(r

k

)(−1)r−k(d − k)r

∣∣∣∣∣≤ n

∣∣∣∣∣n(2µ − ε)−rE

(M1

r

)(1 + ε)

r∑k=0

(r

k

)dr

∣∣∣∣∣≤ (1 + ε)n

∣∣∣∣(

n(2µ − ε)

2d

)−r

E

(M1

r

)∣∣∣∣≤ (1 + ε)nE

((n(2µ − ε)

2d

)−r (M1

r

))

and so ∣∣∣∣∣∞∑

r=2

n1−rD−rn E

(M1

r

) max{r,d−1}∑k=0

Ad−k,n

n

(r

k

)(−1)r−k(d − k)r

∣∣∣∣∣≤

∞∑r=2

(1 + ε)nE

((n(2µ − ε)

2d

)−r (M1

r

))

= (1 + ε)E

M1∑r=2

n

((n(2µ − ε)

2d

)−r (M1

r

))

= (1 + ε)n−1E

M1∑r=2

(n2−r

((2µ − ε)

2d

)−r (M1

r

)),

which will converge to 0 as n → ∞ if it is finite for some n.But

E

M1∑r=2

(n2−r

((2µ − ε)

2d

)−r (M1

r

))= n2

E

M1∑r=2

((n(2µ − ε)

2d

)−r (M1

r

))

≤ n2E

M1∑r=0

((n(2µ − ε)

2d

)−r (M1

r

))

= n2E

(1 + 2d

n(2EM1 − ε)

)M1

,

which will exist for n large enough as long as the moment generating function of M1 existsin a neighborhood of 0.

So, in (7), the last term converges to 0 as n → ∞ if the moment generating function ofM1 exists in a neighborhood of 0, so we use Lemma 1, with

yn = qd + EAd−1,n

n µD−1n (d − 1)

µD−1n d + 1

.


For d = 1, EA0,n = 0. As n → ∞, Dn → 2µ almost surely, so

EA1,n

n→ µq1

2µ+ µ = 2

3q1.

This gives us the result for d = 1, as

E(M1(M1 + 1)I{M1≤1}) = 2P(M1 = 1) = 2q1.

For d > 1, again using Lemma 1,

EAd,n

n→

(2

d + 2

)qd + d − 1

d + 2lim

n→∞ EAd−1,n

n. (8)

Now,

2

d(d + 1)(d + 2)E(M1(M1 + 1)I{M1≤d}) =

d∑l=1

ql2l(l + 1)

d(d + 1)(d + 2),

so, using induction on d, the limit in (8) becomes

2

d + 2qd + d − 1

d + 2

d−1∑l=1

ql2l(l + 1)

(d − 1)d(d + 1),

which is

d∑l=1

ql2l(l + 1)

d(d + 1)(d + 2)= 2

d(d + 1)(d + 2)E(M1(M1 + 1)I{M1≤d}).

As the proportion of vertices of Gn with degree d isAd,n

n+m0= n

n+m0

Ad,nn , this is enough to give

the result.

5. HIGH MULTIPLICITY OF EIGENVALUES

In this section we consider some properties of the eigenvalue spectra of the adjacencymatrices of our graphs. We will be particularly interested in the spectral density of theadjacency matrix of a graph, that is a measure on R with total weight 1 and equal weight oneach eigenvalue of its adjacency matrix and in limiting properties of the spectral densitiesof the adjacency matrices of a sequence of graphs Gn as n → ∞.

In [4, 5], the spectral densities of different models of random graph are discussed, withsimulation being used to obtain sample spectral densities for small-world and scale-freemodels. For the classical Erdös–Rényi model Gn,p, where we have n vertices and each pairof vertices is joined by an edge with probability p independently of other pairs of vertices,the normalized spectral density of the adjacency matrix converges to the Wigner semi-circlelaw as n → ∞ by general arguments for symmetric random matrices (see Theorem 14.12in [8]). The simulation results in [5] suggest that this does not hold for other models,and links between the spectral densities and the structure of the graphs are discussed, for

238 JONATHAN JORDAN

example, the link between the high third moment of the spectral density for small-worldgraphs and the high number of triangles in such graphs.

Eigenvectors of the adjacency matrix of a graph G can be considered functions from thevertex set of G to R. One type of eigenvector that will be of interest is a strictly localizedeigenvector, that is one which is zero on a large proportion of the vertices. Large numbersof localized eigenvectors can be associated with eigenvalues with high multiplicity andhence with singularities in limits of spectral densities. For example, as discussed in [13], asingularity at 0, due to localized eigenvectors, can occur in the spectral density of graphswith vertices of degree 1.

One of the examples investigated in [5] is spectra of adjacency matrices of scale-freegraphs, using the definition of [1], with m = 5 (and q = 0), the results suggesting that thecentral part of the scaled spectral density converges to a triangular shape with no singular-ities. We show that, in contrast, scale-free graphs with certain values of q and m, includingm = 2, q = 0, have strictly localized eigenvectors with eigenvalue 0, giving a singularityat 0 in the limiting spectral density.

Our results go further than those in [13] in that the localized eigenvectors can be asso-ciated with vertices of degrees other than 1. We begin with a lemma that gives a conditionunder that the adjacency matrix of a graph will have localized eigenvectors.

Lemma 5. If we can partition the vertex set of a graph (which may have multiple edges)G into sets A1, A2, . . . , Ar, B, and C with the properties that

(a) there are no edges of G connecting vertices in Ai and Aj for i �= j;(b) there are no edges of G connecting vertices in Ai with vertices in C for any i;(c) each Ai gives the same subgraph of G, that is the vertices in each Ai can be

labeled ai1, ai2, . . . , ais (where s is the number of vertices in each Ai) such that,for 1 ≤ p, q ≤ s, the number of edges in G between aip and aiq is the same for each i;

(d) r > |B|;

then, for each λ that is an eigenvalue of the s × s adjacency matrix of the subgraph givenby the Ai, we can find sets of r − |B| linearly independent eigenvectors of the adjacencymatrix of G with the same eigenvalue.

Proof. We let the adjacency matrix of G be M and let the adjacency matrix of the Ai

subgraphs be M and consider the equation Mx = λx.We take an eigenvector x of M with eigenvalue λ, and we consider vectors x on the

vertices of G satisfying

(a) xw = 0 for w ∈ B ∪ C;(b) for each p, 1 ≤ p ≤ s, we have xaip = zixp, for some fixed z1, z2, . . . , zr .

The condition on the sets Ai, x being zero on B and C and x being an eigenvector of Mwith eigenvalue λ ensure that vectors of this form satisfy

(a) (Mx)v = 0 for all v ∈ C;(b) (Mx)aip = ziλxp = λxaip .


So x is an eigenvector of M if and only if (Mx)w = λxw = 0 for each w ∈ B. But forw ∈ B the equation (Mx)w = 0 reduces to a linear equation in the unknowns z1, z2, . . . , zr

and hence the matrix equation reduces to |B| simultaneous linear equations in r unknowns,giving the result.

In what follows Lemma 5 will be used in the case where each Ai is a single vertex,and hence the eigenvalue involved is zero. This also applies in the case of the “dead-endvertices” (i.e., vertices with degree 1) discussed in [13]. In that case each Ai will be a singlevertex with degree 1, B will be the set of vertices with an Ai vertex as a neighbor, and Cwill be the set of remaining vertices.

Later in this section we will use Lemma 5 on Gn with each Ai consisting of a singlevertex of degree m in Gn. The following lemma shows that for n large enough there are noedges in Gn between vertices of degree m, and hence the number r in Lemma 5 can be equalto the number of vertices of degree m.

Lemma 6. For each d ≥ m, there almost surely exists a (random) Nd < ∞ such that, forn ≥ Nd, a vertex w of degree d has at most d − m neighbors of degree ≤ m.

Proof. Unless it was one of the m0 initial vertices, a vertex w was added to the graph atstage nw and has m edges to vertices Wnw ,1, . . . , Wnw ,m. Then each of these neighbors (otherthan those that were in the set of m0 initial vertices) has degree at least m + 1.

For one of the m0 initial vertices w, P(Wn,m = w) ≥ 1/n, so almost surely the degreeof w eventually grows to at least d + 1. So we can find Nd such that all m0 initial verticeshave degree at least d + 1. Hence, for n ≥ Nd a vertex of degree d has at least m edges tovertices with degree at least m + 1, giving the result.

Let γn be the expected multiplicity of the eigenvalue 0 in the spectrum of Gn. We will beparticularly interested in the limiting behavior of the normalized multiplicity of 0, in par-ticular whether lim infn→∞ γn/(n+m0) is non-zero, indicating the presence of a singularityin a limiting spectral measure.

Theorem 7. For the model described in Section 2, if

q < −m(m − 2)

m − 1

then the expected multiplicity of the eigenvalue 0 in the spectrum of Gn satisfies

lim infn→∞

γn

n + m0> 0

so that the that the limiting behavior of the spectral density has a singularity at 0.

Proof. If q < −m(m−2)

m−1 , then using the formula of Theorem 2 then the expected proportionof vertices with degree m in Gn, limn→∞ EAm,n > 1

2 , and so EAm,n is at least 12 + ε, for some

ε > 0, for n large enough.Then Lemma 6 applied with d = m implies that for n large enough we can use Lemma 5

on Gn, with each Ai consisting of a single vertex of degree m (which has adjacency matrix 0)and C empty, so that r = Am,n and |B| = n + m0 − Am,n.

240 JONATHAN JORDAN

Hence, the multiplicity from Lemma 5 is Am,n − (n + m0 − Am,n) = 2Am,n − (n + m0),which when divided by the number of vertices n + m0 gives 2Am,n − 1. This now gives theresult.

We now look at the case with m = 2 and q = 0, where q = −m(m−2)

m−1 exactly, and henceTheorem 7 does not show the existence of a singularity. However, a slight extension of theideas used to prove Theorem 7, using vertices of degree 3 with no degree 2 neighbor as theset C in Lemma 5, can be used to show that there is a singularity in this case.

To do this, we set N = max(N2, N3). Lemma 6 shows that for n ≥ N there are no edgesbetween degree 2 vertices. Let Zn be the number of the vertices that have degree 3 and haveone neighbor of degree 2 (Lemma 6 shows that for n ≥ N vertices of degree 3 cannot havemore than one degree 2 neighbor), and let zn = E(Zn|n ≥ N). We also let ζn be the expectedproportion of such vertices, conditional on n > N ,

ζn = zn

n + m0.

Lemma 8. In the case where m = 2 and q = 0, the expected proportion of vertices withdegree 3 and one degree 2 neighbor satisfies

E(Zn)

n + m0→ 1

7,

as n → ∞.

Proof. Given a vertex w of degree 3 in Gn, and its neighbor w′ of degree 2,

P(Wi �= w and Wi �= w′) = cnn − 5

cnn

and, given a vertex w of degree 2 in Gn, w has degree 3 with a degree 2 neighbor in Gn+1

with probability

22

cnn

(1 − 2

cnn

).

So

E(Zn+1|Fn, n > N) = Zn

(cnn − 5

cnn

)2

+ 4A2,n

cnn− 8A2,n

(cnn)2

and so

ζn+1 = n + m0

n + m0 + 1ζn

(cnn − 5

cnn

)2

+ 4(n + m0)

n + m0 + 1α2,n

(1

cnn− 2

(cnn)2

).

So the difference

ζn+1 − ζn = ζn((cnn − 5)2(n + m0) − (cnn)2(n + m0 + 1)) + 4α2,n(n + m0)(cnn − 2)

(n + m0 + 1)(cnn)2

= 1

n + m0 + 1

[4cnα2,n − (10cn + c2

n)ζn

c2n

+ D1α2,n + D2ζn

c2nn

+ D3α2,n + D4ζn

c2nn2

]

where D1, D2, D3, D4 are bounded in n.


We now use Lemma 1 again, with ηn = 10cn+c2n

c2n(n+m0+1)

and yn = 410+cn

α2,n. Using (1), yn → 17 ,

and we deduce

ζn → 1

7as n → ∞.

Using Lemma 6 the result follows.

Theorem 9. In the case where m = 2 and q = 0,

lim infn→∞

γn

n + m0≥ 1

35> 0

so that the limiting behavior of the spectral density has a singularity at 0.

Proof. We use Lemma 5 again. The Ai each consist of a single vertex of degree 2, so thatr is the number of vertices of degree 2, C is the set of vertices with degree 3 and no degree2 neighbor, and B consists of the remaining vertices. Then, using Theorem 2 in the caseq = 0, m = 2, d = 2, the expected value of r

n+m0converges to 1

2 , and using Lemma 8 togetherwith Theorem 2 in the case q = 0, m = 2, d = 3, the expected proportion of vertices that arein C, |C|

n+m0, converges to 1

5 − 17 = 2

35 .Hence, as the total number of vertices r + |B| + |C| = n + m0 and so r − |B| =

r − (n + m0 − (r + |C|)), the expected value of r−|B|n+m0

converges to

1

2−

(1 −

(1

2+ 2

35

))= 2

35

as n → ∞, giving the result.

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The degree sequences and spectra of scale-free random graphs