The Spectral Density of Scale-free Networks

Taro Nagao

(Graduate School of Mathematics, Nagoya University)

References:

[1] T. Nagao and G.J. Rodgers, J. Phys. A: Math. Theor. 41 (2008) 265002.

[2] G.J. Rodgers and T. Nagao, The Oxford Handbook of Random MatrixTheory (Oxford University Press, 2011) Chap. 43.

[3] T. Nagao, J. Phys. A: Math. Theor. 46 (2013) 065003.

1

Introduction

The theory of complex networks has dramatically been developed

since the end of the last century. It is based on the observation

that there are universal features in real biological and social net-

works.

One of such features is the scale-free property, meaning that the

degree (the number of edges directly connected to each node)

distribution function P(∆) obeys a power law P(∆) ∝ ∆−λ for

large ∆.

2

Barabasi and Albert argued that the scale-free property is caused

by the process of “preferential attachment”. For example, if the

Frankfurt airport has many flight connections to other airports,

then more airports like to have connections to the Frankfurt

airport.

As a result, the Frankfurt airport tends to have a huge number

of flight connections. In such a situation, the degree distribution

of the airline network tends to obey a power law.

3

An airport like the Frankfurt airport is called a “hub” in the air-

line network. In network theory, such special nodes connected

to singularly many edges are called hubs. The existence of hubs

makes the network structure inhomogeneous. Such an inhomo-

geneous structure is a typical feature of a scale-free network.

In this talk, the spectral densities of the adjacency and Laplacian

matrices describing scale-free networks are analyzed by using a

standard technique of random matrix theory.

4

Outline

1. Spectral density of networks

2. GKK model of scale-free networks

3. Replica method

4. Bipartite scale-free networks

5. Effective medium approximation

6. Summary

5

1. Spectral density of networks

Let us consider a network (graph) with N nodes and examine

the asymptotic behavior in the limit N → ∞.

Example: a network with N = 10

6

The connection pattern of the network is described by

the adjacency matrix A, which is an N × N symmetric matrix

with

Ajl =

1, if j−th and l−th nodes are directly connected,0, otherwise.

The number of edges attached to the j-th node

kj =N∑l=1

Ajl

is called the degree.

7

The degree distribution function P(∆) is defined as

P(∆) =

⟨1

N

N∑j=1

δ(∆ − kj)

⟩,

where the brackets stand for the average over the probability

density function of the adjacency matrices.

If the degree distribution function P(∆) obeys a power law

P(∆) ∝ ∆−λ, ∆ → ∞,

then the network is said to be scale-free.

8

In spectral theory of networks, the Laplacian matrix is of another

interest. The Laplacian matrix L is an N ×N symmetric matrix

defined as

Ljl =

kj, j = l,−Ajl, j = l.

The Laplacian matrix has non-negative eigenvalues and the small-

est eigenvalue is always zero.

The Laplacian matrix is important in physical applications. For

example, it appears in the equation

dujdt

= DN∑l=1

Ljlul,

which describes a diffusive transportation on the network. Here

D denotes the mobility parameter.

9

The spectral density of the adjacency (or Laplacian) matrix char-

acterizing the network is defined as

ρ(µ) =

⟨1

N

N∑j=1

δ(µ− µj)

⟩,

where µ1, µ2, · · · , µN are the eigenvalues of the adjacency matrix

A (or the Laplacian matrix L).

For scale-free complex networks, it is expected that the spectral

density of adjacency matrices also obeys a power law

ρ(µ) ∝ µ−γ, µ → ∞.

10

In order to evaluate the asymptotic behavior of the above quan-

tities in the limit N → ∞,

the generating function

G(tjl) = ln

⟨exp

−i N∑

j<l

Ajltjl

⟩

is useful. Here tjl are parameters independent of N .

11

2. GKK model of scale-free networks

Goh, Kahng and Kim introduced a simple model (GKK model)

of scale-free networks. Suppose that there are N nodes and that

the j-th node is assigned a probability

Pj =j−α∑Nj=1 j

−α ∼ (1 − α)Nα−1j−α, 0 < α < 1.

In each step two nodes are chosen with the assigned probabilities

and connected unless they are already connected. Such a step

is repeated pN/2 times.

12

Then the j-th and l-th nodes are connected with a probability

fjl = 1 − (1 − 2PjPl)pN/2 ∼ 1 − e−pNPjPl,

so that the adjacency matrix A of the network is distributed

according to a probability density function

Pjl(Ajl) = (1 − fjl)δ(Ajl) + fjlδ(Ajl − 1), j < l.

For the generating function G(tjl) of the GKK model, Kim,

Rodgers, Kahng and Kim proved a useful asymptotic formula in

the limit N → ∞:

G(tjl) ∼ pNN∑j<l

PjPl(e−itjl − 1).

13

As a special case, we find an estimate

Fj(t) ≡ ln⟨e−i

∑Nl=1Ajlt

⟩∼ pNPj(e

−it − 1).

Now let us calculate the degree distribution. One obtains

〈kj〉 =N∑l=1

〈Ajl〉 = i∂

∂tFj(t)

∣∣∣∣t=0

∼ pNPj,

so that the mean degree is

1

N

N∑j=1

〈kj〉 ∼ p.

14

Moreover it follows that the degree distribution function is

P(∆) =1

2πN

N∑j=1

∫dt ei∆t+Fj(t)

∼ 1

2π

∫dt∫ 1

0dx ei∆t+p(1−α)x−α(e−it−1).

Then, in the limit ∆ → ∞, we find

P(∆) ∼∫ 1

0dx δ

∆ − p(1 − α)x−α

=p(1 − α)1/α

α

1

∆1+(1/α).

Therefore the exponent λ of the GKK model is equal to 1+(1/α).

15

3. Replica method

In order to calculate the spectral densities of the adjacency and

Laplacian matrices of the GKK model, let us apply the replica

method in statistical physics.

To begin with, we introduce the partition function

Z(µ) =∫ N∏j=1

dφj exp

i2µ

N∑j=1

φ2j − i

2

N∑jl

Jjlφjφl

.

16

In terms of the partition function, we write the spectral density

ρ(µ) in the form

ρ(µ) =2

NπIm

∂

∂µ〈lnZ(µ+ iε)〉, ε ↓ 0.

Here J is the adjacency matrix A or the Laplacian matrix L.

Since there is a relation

〈lnZ〉 = limn→0

ln〈Zn〉n

,

we wish to evaluate the average 〈Zn〉.

17

Let us first evaluate the spectral density of adjacency matrices

A. Using the vector of replica variables

φj = (φ(1)j , φ

(2)j , · · · , φ(n)

j )

and the measure

dφj = dφ(1)j dφ

(2)j · · · dφ(n)

j ,

we find

〈Zn〉 =∫ N∏j=1

dφj ei2µ∑Nj=1

φ2j

⟨exp

− i

2

N∑jl

Ajlφj · φl⟩

=∫ N∏j=1

dφj exp

i2µ

N∑j=1

φ2j +G

φj · φl

2

in terms of the generating function G.

18

Using the asymptotic relation for the generating function G, we

obtain

〈Zn〉 ∼∫ N∏j=1

Dξj(φ) eS0+S1+S2.

The functional integrations are taken over the auxiliary functions

ξj(φ) satisfying ∫dφ ξj(φ) = 1.

19

The functionals S0, S1 and S2 are defined as

S0 = −N∑j=1

∫dφ ξj(φ) ln ξj(φ),

S1 =i

2µ

N∑j=1

∫dφ ξj(φ)φ

2

and

S2 =pN

2

N∑j,k=1

PjPk

∫dφ

∫dψ ξj(φ) ξk(ψ)

e−iφ·ψ − 1

.

20

The functional integrations over ξj(φ) are dominated by the sta-

tionary point satisfying

δ

S0 + S1 + S2 +

N∑j=1

θj

(∫dφ ξj(φ) − 1

) = 0,

where θj are the Lagrange multipliers.

21

Then we can derive the variational equations

ξj(φ) = Θj exp

i2µφ2 + pNPj

N∑k=1

Pk

∫dψξk(ψ)

(e−iφ·ψ − 1

) ,

where Θj are normalization constants.

In the limit of large mean degree p→ ∞, the variational equations

are satisfied by the Gaussian ansatz

ξj(φ) =1

(2πiσj)n/2exp

(−φ2

2iσj

).

This property simplifies the problem and enables us to evaluate

the asymptotic spectral density[2].

22

Putting the Gaussian ansatz into the variational equations, we

find

µ− 1

σj− pNPj

N∑k=1

Pkσk = 0,

which lead to the integral equation

Exα − xα

s(x)− (1 − α)2

∫ 1

0y−αs(y)dy = 0,

where E = µ/√p, s(x) =

√pσj (x = j/N).

23

Rodgers, Austin, Kahng and Kim solved the integral equation

and succeeded in deriving

ρ(µ) ∼ 2√p

(1 − α)1/α

α

1

E1+(2/α)

for large E. Therefore, in the limit p → ∞, the exponent γ is

1 + (2/α), so that the relation with λ = 1 + (1/α) is γ = 2λ− 1.

24

Let us moreover consider the spectral density of Laplacian ma-

trices. As shown by Kim and Kahng, it again follows from the

asymptotic relation of the generating function G that

ρ(µ) =p(1 − α)1/α

α

1

µ(1/α)+1Hµ− p(1 − α)

in the region µ = O(p) with p→ ∞. Here

H(x) =

0, x < 0,1, x > 0.

25

4. Bipartite scale-free networks

Let us next consider a bipartite network with N nodes of type A

and M nodes of type B (N ≥ M). Connections are made only

between the nodes of different types.

Example: a bipartite network with M = N = 5

26

Bipartite networks are applied to the analysis of human sexual

contacts, and of the connection patterns between collaborators

and collaboration acts, such as actors and movies, scientists and

papers.

We are interested in the asymptotic behavior of the network in

the limit

N → ∞ and M → ∞ with c =N

Mfixed.

27

Let us suppose that the nodes of type A and B are connected

according to the following procedure[3].

In each step we choose a node j of type A and a node k of type

B with probabilities Pj and Qk, respectively.

Then the nodes j and k are connected, unless they are already

connected.

28

After repeating such a step pN times, a node j of type A and a

node k of type B is connected with a probability

fjk = 1 − (1 − PjQk)pN ∼ 1 − e−pNPjQk.

Let us consider an N × M random matrix C (N ≥ M), where

Cjk = 1 if the node j of type A is directly connected to the node

k of type B, and Cjk = 0 otherwise.

Eacn matrix element Cjk is independently distributed with the

probability density function

Pjk(Cjk) = (1 − fjk)δ(Cjk) + fjkδ(1 − Cjk).

29

We assume that Pj and Qk are given by

Pj =j−αN∑l=1

l−α∼ (1 − α)Nα−1j−α, 0 < α < 1

and

Qk =k−βM∑l=1

l−β∼ (1 − β)Mβ−1k−β, 0 < β < 1.

There are thus two parameters α and β controlling the probability

distribution function of the matrix C.

30

We define the degree dj of the type-A node j as the number of

directly connected type-B nodes:

dj =M∑k=1

Cjk.

Then the type-A node degree distribution function is given by

P (A)(∆) =

⟨1

N

N∑j=1

δ(∆ − dj)

⟩,

where the brackets denote the average over the probability den-

sity function of the matrix C.

31

We can similarly introduce the degree ek of the type-B node k:

ek =N∑j=1

Cjk

and the type-B node degree distribution function

P (B)(∆) =

⟨1

M

M∑k=1

δ(∆ − ek)

⟩.

The generating function for this scale-free network is defined as

G(tjk) = ln

⟨exp

−i N∑

j=1

M∑k=1

Cjktjk

⟩ .

32

A useful asymptotic relation[3]

G(tjk) ∼ pNN∑j=1

M∑k=1

PjQk(e−itjk − 1)

can be derived in the limit N,M → ∞. Here tjk depends on

neither N nor M .

As special cases, we can derive asymptotic relations for

F(A)j (t) ≡ ln

⟨e−idjt

⟩, F

(B)k (t) ≡ ln

⟨e−iekt

⟩as

F(A)j (t) ∼ pNPj(e

−it − 1), F(B)k (t) ∼ pNQk(e

−it − 1).

33

Then we can see that

〈dj〉 = i∂

∂tF

(A)j (t)

∣∣∣∣t=0

∼ pNPj,

〈ek〉 = i∂

∂tF

(B)k (t)

∣∣∣∣t=0

∼ pNQk,

so that the mean degree m(A) of the type-A node is

m(A) =1

N

N∑j=1

〈dj〉 ∼ p,

while the mean degree m(B) of the type-B node is

m(B) =1

M

M∑k=1

〈ek〉 ∼ pc.

34

It can be seen that the type-A node degree distribution function

is written as

P (A)(∆) =1

2πN

N∑j=1

∫dt e

i∆t+F(A)j (t)

∼ 1

2π

∫dt∫ 1

0dx ei∆t+p(1−α)x−α(e−it−1).

Then in the limit ∆ → ∞ we find

P (A)(∆) ∼ p(1 − α)1/αα

1

∆(1/α)+1

and similarly obtain

P (B)(∆) ∼ pc(1 − β)1/ββ

1

∆(1/β)+1.

35

Thus we have seen that the network has the scale-free property.

The exponents of the power laws defined as

P (A)(∆) ∝ ∆−λA, P (B)(∆) ∝ ∆−λB, ∆ → ∞are found to be λA = (1/α) + 1 and λB = (1/β) + 1.

We can also study the adjacency and Laplacian matrices of this

scale-free network. The adjacency matrix A of this network is

defined as

A =

(ON C

CT OM

),

where CT is a transpose of C and On is an n × n matrix with

zero elements.

36

The Laplacian matrix L is an (N + M) × (N + M) symmetric

matrix with

Ljl =dj, j = l and 1 ≤ j ≤ N,ej−N, j = l and N + 1 ≤ j ≤ N +M,−Ajl, j = l.

Let us suppose that J is the adjacency matrix A or the Laplacian

matrix L. The spectral density of J is defined as

ρ(µ) =

⟨1

N +M

N+M∑j=1

δ(µ− µj)

⟩,

where µj, j = 1,2, · · · , N +M are the eigenvalues of J.

37

In order to calculate ρ(µ), we introduce the partition function

Z(µ) =∫ N+M∏

j=1

dΦj exp

i2µN+M∑j=1

Φ2j − i

2

N+M∑j=1

N+M∑l=1

JjlΦjΦl

.

Using the partition function Z, we can write the spectral density

as

ρ(µ) =2

(N +M)πIm

∂

∂µ〈lnZ(µ+ iε)〉

= limn→0

2

(N +M)nπIm

∂

∂µln〈Z(µ+ iε)n〉, ε ↓ 0.

Therefore we again need to evaluate the average 〈Zn〉.

38

Let us first consider the spectral density of the adjacency matri-

ces A. In terms of the replica variables

φj = (φ(1)j , φ

(2)j , · · · , φ(n)

j ), ψk = (ψ(1)k , ψ

(2)k , · · · , ψ(n)

k )

and

dφj = dφ(1)j dφ(2)

j · · ·dφ(n)j , dψk = dψ(1)

k dψ(2)k · · ·dψ(n)

k ,

we obtain

〈Zn〉 =∫ N∏j=1

dφj

∫ M∏k=1

dψk

× exp

i2µ

N∑j=1

φ2j +

i

2µ

M∑k=1

ψ2k + G

φj · ψk

2

.

39

It follows in the limit N,M → ∞ that

〈Zn〉 ∼∫ N∏j=1

Dξj(φ)M∏k=1

Dηk(ψ) eS0+S1+S2.

The functional integrations are taken over the auxiliary functions

ξj(φ) and ηk(ψ) satisfying∫dφ ξj(φ) =

∫dψ ηk(ψ) = 1.

40

The functionals S0, S1 and S2 are defined as

S0 = −N∑j=1

∫dφ ξj(φ) ln ξj(φ) −

M∑k=1

∫dψ ηk(ψ) ln ηk(ψ),

S1 =i

2µ

N∑j=1

∫dφ ξj(φ)φ

2 +i

2µ

M∑k=1

∫dψ ηk(ψ)ψ2

and

S2 = pNN∑j=1

M∑k=1

PjQk

∫dφ

∫dψ ξj(φ) ηk(ψ)

e−iφ·ψ − 1

.

41

The functional integrations over ξj(φ) and ηk(ψ) are dominated

by the stationary point satisfying

δ

S0 + S1 + S2 +

N∑j=1

θj

(∫dφ ξj(φ) − 1

)

+M∑k=1

ωk

(∫dψ ηk(ψ) − 1

) = 0,

where θj and ωk are the Lagrange multipliers.

42

Then we can derive the variational equations

ξj(φ) = Θj exp

i2µφ2 + pNPj

M∑k=1

Qk

∫dψηk(ψ)

(e−iφ·ψ − 1

) ,

ηk(ψ) = Ωk exp

i2µψ2 + pNQk

N∑j=1

Pj

∫dφξj(φ)

(e−iφ·ψ − 1

) ,

where Θj and Ωk are normalization constants.

In the limit of large mean degree p→ ∞, the variational equations

are satisfied by the Gaussian ansatz

ξj(φ) =1

(2πiσj)n/2exp

(−φ2

2iσj

), ηk(ψ) =

1

(2πiτk)n/2

exp

(−ψ2

2iτk

).

43

Putting the Gaussian ansatz into the variational equations, we

find

µ− 1

σj− pNPj

M∑k=1

Qkτk = 0,

µ− 1

τk− pNQk

N∑j=1

Pjσj = 0,

which lead to the integral equations

Exα − xα

s(x)− (1 − α)(1 − β)

∫ 1

0y−βt(y)dy = 0,

Eyβ − yβ

t(y)− c(1 − α)(1 − β)

∫ 1

0x−αs(x)dx = 0,

where E = µ/√p, s(x) =

√pσj (x = j/N) and t(y) =

√pτk

(y = k/M).

44

Let us consider the limit p→ ∞ with a scaling variable E = µ/√p.

Solving the integral equations, we find the asymptotic spectral

density of the adjacency matrix A as

ρ(µ) ∼ 2√p(1 + c)

c1/β(1 − β)1/β

βE(2/β)+1+c(1 − α)1/α

αE(2/α)+1

in the tail region E → ∞.

Thus the exponent γ is associated with λA = (1/α) + 1 and

λB = (1/β) + 1 as γ = 2min(λA, λB) − 1.

45

The asymptotic spectral density of the Laplacian matrix L is

similarly derived as

ρ(µ) =cp(1 − α)1/α

(1 + c)α

1

µ(1/α)+1H µ− p(1 − α)

+pc(1 − β)1/β

(1 + c)β

1

µ(1/β)+1H µ− pc(1 − β)

in the region µ = O(p) with p→ ∞. Here

H(x) =

0, x < 0,1, x > 0.

46

5. Effective medium approximation

We have so far dealt with the spectral density in the limit p→ ∞.

The calculation of the spectral density with a finite mean degree

p is a much more involved problem.

Here we briefly discuss a simple approximation method (effec-

tive medium approximation, EMA) for that problem applied to

bipartite scale-free networks.

47

In EMA, we put the Gaussian ansatz into the definitions of S0,S1 and S2 , and solve the stationary point equations

∂

∂σj(S0 + S1 + S2) = 0

and∂

∂τk(S0 + S1 + S2) = 0.

In the case of the adjacency matrix A, the above procedureresults in the EMA equations

µ− 1

σj− pNPj

M∑k=1

Qkτk1 − σjτk

= 0,

µ− 1

τk− pNQk

N∑j=1

Pjσj

1 − σjτk= 0.

48

As for scale-free networks with a single species of nodes, TN and

Rodgers [1] calculated the 1/p expansion of the spectral density

by using the corresponding EMA equation.

A similar analytical treatment could also be possible in the bi-

partite case. Here, however only results of numerical iterations

of the EMA equations are shown.

They are compared with the spectral densities of positive eigen-

values calculated by numerical diagonalizations of numerically

generated adjacency matrices (averaged over 100 samples).

49

0

0.02

0.04

0.06

0.08

0.1

1 2 3 4 5 6 7 8 9 10

p=1

p=5

p=10

ρ

µ

The EMA spectral densities (curves) and the spectral densities of numeri-cally generated adjacency matrices (histograms) with p = 1,5 and 10. Theparameters are N = 1000, M = 200 and α = 0.5, β = 0.2.

50

The EMA gives a better fit for a larger p, as expected from the

fact that the variational equations are satisfied by the Gaussian

ansatz in the limit p→ ∞.

When p = 1, the agreement significantly breaks down around the

origin, although it is still fairly good in the tail region with large

µ.

It is thus conjectured that the EMA (with a finite p) gives exact

asymptotic results in the limit µ→ ∞.

51

6. Summary

The generating function is a useful tool in the asymptotic anal-

ysis of the connection pattern of a network. It is known that

some typical network quantities, such as the average degree, the

degree distribution function and the eigenvalue densities of ad-

jacency and Laplacian matrices, can be evaluated by using the

generating function.

The asymptotic estimates of the generating functions in the limit

of large network size were presented for simple models (the GKK

model and a similar bipartite network) of scale-free networks and

applied to the analysis of those network quantities.

52