Description

Structure of shells in complex networks

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.

Related Documents

Share

Transcript

Structure of shells in complex networks
Jia Shao,
1
Sergey V. Buldyrev,
2,1
Lidia A. Braunstein,
3,1
Shlomo Havlin,
4
and H. Eugene Stanley
1
1
Department of Physics and Center for Polymer Studies, Boston University, Boston, Massachusetts 02215, USA
2
Department of Physics, Yeshiva University, 500 West 185th Street, New York, New York 10033, USA
3
Departamento de Física, Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR), Facultad de Ciencias Exactas y Naturales,Universidad Nacional de Mar del Plata–CONICET, Funes 3350, 7600 Mar del Plata, Argentina
4
Department of Physics and Minerva Center, Bar-Ilan University, 52900 Ramat-Gan, Israel
Received 11 March 2009; published 9 September 2009
We deﬁne shell
in a network as the set of nodes at distance
with respect to a given node and deﬁne
r
as the fraction of nodes outside shell
. In a transport process, information or disease usually diffuses from arandom node and reach nodes shell after shell. Thus, understanding the shell structure is crucial for the studyof the transport property of networks. We study the statistical properties of the shells of a randomly chosennode. For a randomly connected network with given degree distribution, we derive analytically the degreedistribution and average degree of the nodes residing outside shell
as a function of
r
. Further, we ﬁnd that
r
follows an iterative functional form
r
=
r
−1
, where
is expressed in terms of the generating function of the srcinal degree distribution of the network. Our results can explain the power-law distribution of thenumber of nodes
B
found in shells with
larger than the network diameter
d
, which is the average distancebetween all pairs of nodes. For real-world networks the theoretical prediction of
r
deviates from the empirical
r
. We introduce a network correlation function
c
r
r
/
r
−1
to characterize the correlations in thenetwork, where
r
is the empirical value and
r
−1
is the theoretical prediction.
c
r
=1 indicates perfectagreement between empirical results and theory. We apply
c
r
to several model and real-world networks. Weﬁnd that the networks fall into two distinct classes:
i
a class of
poorly connected
networks with
c
r
1,where a larger
smaller
fraction of nodes resides outside
inside
distance
from a given node than inrandomly connected networks with the same degree distributions. Examples include the Watts-Strogatz modeland networks characterizing human collaborations such as citation networks and the actor collaboration net-work;
ii
a class of
well-connected
networks with
c
r
1. Examples include the Barabási-Albert model andthe autonomous system Internet network.DOI: 10.1103/PhysRevE.80.036105 PACS number
s
: 89.75.Hc, 87.23.Ge, 64.60.aq
I. INTRODUCTION AND RECENT WORK
Many complex systems can be described by networks inwhich the nodes are the elements of the system and the linkscharacterize the interactions between the elements. One of the most common ways to characterize a network is to de-termine its degree distribution. A classical example of a net-work is the Erd
ő
s-Rényi
ER
1,2
model, in which the linksare randomly assigned to randomly selected pairs of nodes.The degree distribution of the ER model is characterized bya Poisson distribution
P
k
=
k
k
k
!
e
−
k
,
1
where
k
is the average degree of the network. Anothersimple model is a random regular
RR
graph in which eachnode has exactly
k
=
links, thus
P
k
=
k
−
. TheWatts-Strogatz model
WS
3
is also well studied, where arandom fraction
of links from a regular lattice with
k
=
are rewired and connect any pair of nodes. Changing
from 0 to 1, the WS network interpolates between a regularlattice and an ER graph. In the last decade, it has been real-ized that many social, computer, and biological networks canbe approximated by scale-free
SF
models with a broad de-gree distribution characterized by a power law
P
k
k
−
,
2
with a lower and upper cutoff,
k
min
and
k
max
4–9
. A para-digmatic model that explains the abundance of SF networksin nature is the preferential attachment model of Barabásiand Albert
BA
4
.The degree distribution is not sufﬁcient to characterize thetopology of a network. Given a degree distribution, a net-work can have very different properties such as clusteringand degree-degree correlation. For example, the network of movie actors
4
in which two actors are linked if they playin the same movie, although characterized by a power-lawdegree distribution, has higher clustering coefﬁcient com-pared to the SF network generated by the Molloy-Reed al-gorithm
10
with the same degree distribution.Besides the degree distribution and clustering coefﬁcient,a network is also characterized by the average distance be-tween all pairs of nodes, which we refer to as the network diameter
d
11
. The diameter
d
depends sensitively on thenetwork topology. Random networks with a given degreedistribution can be “small worlds”
2
,
d
ln
N
,
3
or “ultrasmall worlds”
8
,
d
ln
ln
N
.
4
Another important characteristic of a network is the struc-ture of its shells, where shell
is deﬁned as the set of nodesthat are at distance
from a randomly chosen root node
12
.The shell structure of a network is important for understand-ing the transport properties of the network such as the epi-demic spread
13,14
, information diffusion and synchroni-
PHYSICAL REVIEW E
80
, 036105
2009
1539-3755/2009/80
3
/036105
13
©2009 The American Physical Society036105-1
zation processes
15,16
, where the information or virusspread from a randomly chosen root and reach nodes shellafter shell. The structure of the shells is related to both thedegree distribution and the network diameter. The shellstructure of SF networks has been recently studied in Ref.
12
, which have introduced a new term “network tomogra-phy” referring to various properties of shells such as thenumber of nodes and open links in shell
, the degree distri-bution, and the average degree of the nodes in the exterior of shell
.Many real and model networks have fractal propertieswhile others do not
17
. Recently, Ref.
18
reported apower-law distribution of number of nodes
B
in shell
d
from a randomly chosen root. They found that a largeclass of models and real networks, although not fractals onall scales, exhibits fractal properties in boundary shells with
d
. Here we will develop a theory to explain these ﬁnd-ings.
II. GOALS OF THIS WORK
In this paper, we extend the study of network tomographydescribing the shell structure in a randomly connected net-work with an arbitrary degree distribution using generatingfunctions. Following Ref.
12
, we denote the fraction of nodes at distance larger than
as
r
1 −1
N
m
=0
B
m
5
and the nodes at distances larger than
as the exterior
E
of shell
. Similarly, we deﬁne the “
r
exterior,”
E
r
, as the
rN
nodes with the largest distances from a given root node. Tothis end, we list all the nodes in ascending order of theirdistances from the root node. In this list, the nodes with thesame distance are positioned at random. The last
rN
nodes inthis list which have the largest distance to the root are called
E
r
. Notice that
E
r
=
E
if
r
=
r
. Introducing
r
as a continuousvariable is a different step compared to Ref.
12
, whichallows us to apply the apparatus of generating functions tostudy network tomography.The behavior of
B
for
d
can be approximated by abranching process
19,20
. In shells with
d
, the network will show different topological characteristics compared toshells with
d
. This is due to the high probability to ﬁndhigh degree nodes
“hubs”
in shells with
d
, so there is adepletion of high degree nodes in the degree distribution in
E
with
d
. Indeed, the average degree of the nodes inshells with
d
is greater than the average degree in theshells with
d
12,18
.Here, we develop a theory to explain the behavior of thedegree distribution
P
r
k
in
E
r
and the behavior of the aver-age degree
k
r
as a function of
r
in a randomly connectednetwork with a given degree distribution. Further, we deriveanalytically
r
as a function of
r
−1
,
r
=
r
−1
, where
canbe expressed in terms of generating functions
20
of thedegree distribution of the network. Using these derived ana-lytical expressions, we explain the power-law distribution
P
B
B
−2
for
d
found in
18
. Further, based on ourapproach, we introduce the network correlation function
c
r
=
r
/
r
−1
to characterize the correlations in the net-work. We apply this measure to several model and real-worldnetworks. We ﬁnd that the networks fall into two distinctclasses:
i
a class of
poorly connected
networks with
c
r
1, where the network is less compact than its randomlyconnected counterpart with the same degree distribution;
ii
a class of
well-connected
networks with
c
r
1, in whichcase the network is more compact than its randomly con-nected counterpart.In a network with
c
r
1, more nodes reside in the ex-terior of shell
than in its randomly connected counterpart.Hence, the fraction of nodes residing inside and on shell
,1−
r
, is smaller than that of its randomly connected coun-terpart. Thus, the network is less compact than a randomlyconnected network with the same degree distribution, and wecall it poorly connected. The poorly connected networkshave high redundancy of their connections and high cluster-ing than their randomly connected counterpart. The well-connected networks are on the opposite side.In this paper we study RR, ER, SF, WS, and BA models,as well as several real networks including the actor collabo-ration network
Actor
4
, high energy physics citations net-work
HEP
21
, the Supreme Court citation network
SCC
22
, and autonomous system
AS
Internet network
DIMES
23
. As we will show below, WS, Actor, HEP, andSCC belong to the class of poorly connected networks
c
r
1
, while the BA model and DIMES network belongto the class of well-connected networks
c
r
1
.The paper is organized as follows. In Sec. III, we derive
analytically the degree distribution and average degree of nodes in
E
r
and test our theory on ER and SF networks. InSec. IV, we derive analytically a deterministic iterative func-
tional form for
r
. In Sec. V, we apply our theory to explain
the power-law distribution of number of nodes in shells. InSec. VI, we introduce the network correlation function and
apply it to different networks. Finally, we present a summaryin Sec. VII.
III. DEGREE DISTRIBUTION OF NODES IN THE
r
EXTERIOR
E
r
A. Generating function for
P
(
k
)
In this section, we deﬁne the generating functions for thedegree distribution which will be used extensively in ourderivations. The generating function of a given degree distri-bution
P
k
is deﬁned as
19,20,24,25
G
0
x
k
=0
P
k
x
k
.
6
It follows from Eq.
6
that the average degree of the net-work
k
=
G
0
1
. Following a randomly chosen link, theprobability of reaching a node with
k
outgoing links
thedegree of the node is
k
+1
is
P
˜
k
=
k
+ 1
P
k
+ 1
/
k
=0
k
+ 1
P
k
+ 1
.
7
Notice that
SHAO
et al.
PHYSICAL REVIEW E
80
, 036105
2009
036105-2
the growth process, as more and more nodes are connected tothe aggregate, the degree distribution of the remaining nodeschanges. In this section, we will present and solve the differ-ential equations describing these changes.Let
A
r
k
be the number of nodes with degree
k
in the
r
exterior
E
r
at time
t
. The probability to have a node withdegree
k
in
E
r
is given by
27
P
r
k
=
A
r
k
rN
.
11
When we connect an open link from the aggregate to a freenode
case
i
,
A
r
k
changes as
A
r
−1
/
N
k
=
A
r
k
−
P
r
k
k
k
r
,
12
where
k
r
=
P
r
k
k
is the average degree of nodes in
E
r
.In the limit of
N
→
, Eq.
12
can be presented in terms of the derivative of
A
r
k
with respect to
r
,
dA
r
k
dr
N
A
r
k
−
A
r
−1
/
N
k
=
N P
r
k
k
k
r
.
13
Differentiating Eq.
11
with respect to
r
and using Eq.
13
,we obtain−
r dP
r
k
dr
=
P
r
k
−
kP
r
k
k
r
,
14
which is rigorous for
N
→
.In order to solve Eq.
14
, we make the substitution
f
G
0−1
r
.
15
We ﬁnd by direct differentiation that
P
f
k
=
P
1
k
f
k
G
0
f
,
16
and
k
f
=
fG
0
f
G
0
f
17
is the solution satisfying Eq.
14
. Notice that
P
1
k
P
k
.Equations
16
and
17
are, respectively, the degree dis-tribution and the average degree in
E
r
as functions of
f
. Oncewe know the explicit functional form for
G
0
x
, we can in-vert
G
0
x
to ﬁnd
f
=
G
0−1
r
and ﬁnd analytically both
P
r
k
and
k
r
:
P
r
k
=
P
k
G
0−1
r
k
r
,
18
k
r
=
G
0−1
r
G
0
„
G
0−1
r
…
r
.
19
In a network with minimum degree
k
min
2, we ﬁnd by Tay-lor expansion that
k
r
=
k
min
+
P
k
min
+ 1
P
k
min
1+
r
+
O
r
2
,
20
where
1
/
k
min
.For ER networks, using Eqs.
10
and
17
, we ﬁnd
k
r
= ln
r
+
k
,
21
where
r
r
1. The value of
r
is presented in Eq.
33
.Note that
r
0 for ER networks. Equation
16
can be re-written as
P
r
k
=
P
k
ln
r
/
k
+ 1
k
r
=
e
−
k
r
k
r
k
k
!,
22
which implies that the degree distribution in the distantnodes remains a Poisson distribution but with a smaller av-erage degree
k
r
.Next, we test our theory numerically for ER networkswith
N
=10
6
nodes and different values of
k
. To obtain
P
r
k
, we start from a randomly chosen root node and ﬁndthe nodes in
E
r
and their degree distribution
P
r
k
. This pro-cess is repeated many times for different roots and differentnetwork realizations. The results are shown in Fig. 2
a
. Thesymbols are the simulation results of the degree distributionin
E
r
for
r
=1, 0.5, and 0.05. The analytical results
full lines
are computed using Eq.
22
. As can be seen, the theory
0 5 10 15 20 25 30
k
00.10.2
P
r
( k )
r=1 r=0.5 r=0.05
ER
<k>=6
(a) ER
10
0
10
1
10
2
10
3
k
10
-6
10
-4
10
-2
10
0
P
r
( )
r=1 r=0.5 r=0.1
(b) SF
λ=3.5
FIG. 2.
Color online
Comparison between the simulation re-sult and the theoretical prediction for the degree distribution,
P
r
k
,in
E
r
.
a
ER network with
N
=10
6
,
k
=6 and
r
=1, 0.5, and 0.05.The simulation results
symbols
agree very well with the theoreti-cal predictions
lines
of Eq.
22
.
b
SF network with
=3.5,
k
min
=2 and
N
=10
6
,
P
r
k
with
r
=1, 0.5, and 0.1. The simulationresults shown by symbols ﬁt well with the theoretical predictions of Eq.
16
. For a SF network, we compute Eq.
16
numerically usingthe
P
k
obtained from the generated network.SHAO
et al.
PHYSICAL REVIEW E
80
, 036105
2009
036105-4
agrees very well with the simulation results for both
r
=0.5and 0.05. We compared our theory with the simulations alsofor other values of
r
and
k
and the agreement is also excel-lent.For SF networks,
G
0
x
and
G
1
x
cannot be expressed aselementary functions
19
. But for a given
P
k
, they can bewritten as power series of
x
and one can compute the expres-sions in Eqs.
16
and
17
numerically. In order to reducethe systematic errors caused by estimating
P
k
, we write
G
0
x
and
G
1
x
based on the
P
k
obtained from the simu-lation results instead of using its theoretical form.We built SF networks using the Molloy-Reed algorithm
10
. In Fig. 2
b
, the symbols represent the simulation re-sults for
P
r
k
obtained for
E
r
of SF network with
=3.5 and
r
=1, 0.5, and 0.1. The lines are the numerical results calcu-lated from Eq.
16
. Good agreement between the simulationresults and the theoretical predictions can be seen in Fig.2
b
. Other values of
r
and
have also been tested with goodagreement.In Fig. 3
a
, we show the average degree
k
r
in
E
r
as afunction of
r
for ER networks with different values of
k
.Lines representing Eq.
21
agree very well with the numeri-cal results
symbols
even for very small
r
. We note that Fig.3
a
shows different value of lower limit cutoff
r
for
r
,when
k
r
is very small. As mentioned before,
r
is thefraction of nodes which are not connected to the aggregate atthe end of the process. In the next section, we will present anequation for
r
.In Fig. 3
b
, we present the numerical results of Eq.
17
for SF networks with different values of
. For a given
E
r
,
k
r
is computed from the simulated network and the re-sults are averaged over many realizations. Good agreementbetween the theory
lines
and the simulation results
sym-bols
can be seen.
IV. ITERATIVE FUNCTIONAL FORM OF
r
, THEFRACTION OF NODES OUTSIDE SHELL
In this section, we will derive a recursive relation between
r
of two successive shells of a randomly connected network,which is the main result of this paper. Let
L
t
be the numberof open links belonging to the aggregate at step
t
and
t
L
t
/
N
. The number of open links belonging to shell
of the aggregate is deﬁned as
L
t
and
t
L
t
/
N
.Afterwe ﬁnish building shell
and just before we start to buildshell
+1, all the open links in the aggregate belong to nodesin shell
, so at
t
=
t
, we have
t
=
t
28
. In the pro-cess of building shell
+1,
t
decreases to 0.Next we show that both
t
and
t
can be expressedas functions of
r
. In analogy with Eq.
9
, we deﬁne thebranching factor of nodes in the
r
exterior
E
r
as
k
˜
r
=
k
2
r
−
k
r
k
r
=
k
=0
k
2
P
r
k
k
r
− 1.
23
Using Eqs.
17
and
23
,
k
˜
r
can be rewritten as a functionof
f
as
k
˜
f
=
fG
0
f
G
0
f
.
24
Appendix A shows that
r
and
r
obey differentialequations
d
r
dr
= −
k
˜
r
+ 1 +2
r
r
k
r
,
25
d
r
dr
= 1 +
r
r
k
r
+
r
r
k
r
.
26
Equations
25
and
26
govern the growth of the aggre-gate. To solve them, we make the same substitution
f
=
G
0−1
r
Eq.
15
as before. The general form of the solu-tion for Eq.
25
is
f
= −
G
0
f
f
+
C
1
f
2
,
27
where
C
1
is a constant. At time
t
=0,
r
=
f
=1, and
1
=0.With this initial condition, we obtain
C
1
=
G
0
1
=
k
. UsingEq.
27
, the general solution of Eq.
26
is
f
=
G
0
1
f
2
+
C
2
f
,
28
where
C
2
is a constant. When
r
=
r
, the building of shell
isﬁnished. At that time, all the open links of the aggregate
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
r
0123456789
< k ( r ) >
<k>=4<k>=6<k>=8<k>=10(a) ER
0 0.2 0.4 0.6 0.8 1
r
23456
< k ( r ) >
λ=2.5λ=3.0λ=3.5λ=4.0
(b) SF
FIG. 3.
Color online
Comparison between the simulation re-sult and the theoretical prediction for average degree
k
r
of thenodes in
E
r
.
a
Four ER networks with different values of
k
and
b
four SF networks with
k
min
=2 and different values of
. Thesymbols represent the simulation results for ER and SF networks of size
N
=10
6
. The lines in
a
represent Eq.
21
. The lines in
b
arethe numerical results of Eq.
17
using the degree distribution ob-tained from the networks.STRUCTURE OF SHELLS IN COMPLEX NETWORKS PHYSICAL REVIEW E
80
, 036105
2009
036105-5

Search

Similar documents

Tags

Related Search

Synchronization in Complex NetworksQuality of Service in IP networksSpreading in Complex NetworksCommunity detection in complex networksCommunity Structure of Rotifers in Relation tImportance of Debt in Capital StructureStructure Of The EarthNames Of God In JudaismFreedom Of Speech In The United StatesSocial Structure of Science

We Need Your Support

Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks