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Journal ArticleDOI

Partially ordered sets in complex networks

07 May 2010-Journal of Physics A (IOP Publishing)-Vol. 43, Iss: 18, pp 185001
TL;DR: Most of the partially ordered set-based properties cannot be explained by the current well-known scale-free network models; therefore, it is required to propose more appropriate network models in the future.
Abstract: In this paper, a partial-order relation is defined among vertices of a network to describe which vertex is more important than another on its contribution to the connectivity of the network. A maximum linearly ordered subset of vertices is defined as a chain and the chains sharing the same end-vertex are grouped as a family. Through combining the same vertices appearing in different chains, a directed chain graph is obtained. Based on these definitions, a series of new network measurements, such as chain length distribution, family diversity distribution, as well as the centrality of families, are proposed. By studying the partially ordered sets in three kinds of real-world networks, many interesting results are revealed. For instance, the similar approximately power-law chain length distribution may be attributed to a chain-based positive feedback mechanism, i.e. new vertices prefer to participate in longer chains, which can be inferred by combining the notable preferential attachment rule with a well-ordered recommendation manner. Moreover, the relatively large average incoming degree of the chain graphs may indicate an efficient substitution mechanism in these networks. Most of the partially ordered set-based properties cannot be explained by the current well-known scale-free network models; therefore, we are required to propose more appropriate network models in the future.

Summary (3 min read)

1. Introduction

  • Over the past decade, characterizing and modeling complex networks attract many attentions from various areas [1–8].
  • This finding is of much interest because it suggests that motifs rather than vertices may be the basic cells to perform the higher level functions of the systems.
  • The vertex set together with such relation forms a partially ordered set in the network.
  • Measurements of chains and families in several real-world and artificial complex networks are carefully studied in sections 3 and 4, respectively.

2.1. Definitions

  • Thus, the relation ≼ can be called a partial order on the vertex set V, and the vertex set V together with the partial order ≼ is called a partially ordered set [24].
  • It should be noted that both the subsets {va, vb, vc} and {va, vb, vc, vd, vi} in the figure are not chains because the former is not a ‘maximum’ linearly ordered subset and the latter cannot be linearly ordered at all.
  • In fact, through the above definitions, the authors can get the following theorem: Theorem 1.
  • Removal of any vertex in a chain except the end-vertex will not influence the average shortest path length of the remaining subnetwork, that is, the end-vertex of a chain can completely replace the other vertices in the same chain on their contributions to the connectivity of the network.

2.2. The algorithm

  • Based on the above definitions, here, the authors will briefly introduce their method to find all the chains in a network.
  • In the procedure, the chain graph is constructed at first, then the chains are extracted by a depth first search algorithm in the chain graph, and the families can be obtained by simply combining these chains.
  • Particularly, the algorithm is summarized by the following three steps.
  • Replace each compressed vertex vk1 by a directed chain vk1 → vk2 → · · · → vkp ; at the same time, the incoming edges of the compressed vertex vk1 still point to the beginning vertex vk1 in the directed chain, while the outgoing edges of the compressed vertex vk1 are now originated from the last vertex vkp .
  • In the chain graph, there are three types of vertices: the original-vertices with no incoming edges, the end-vertices with no outgoing edges and the middle-vertices with at least one incoming edge and one outgoing edge.

2.3. Data sets

  • The authors mainly study the partially ordered sets in three kinds of real-world networks.
  • The first is the protein–protein interaction networks collected from BioGrid database [26], where each vertex represents a protein or a gene, and each edge denotes the interaction between these proteins or genes.
  • The authors use the giant components of four protein–protein interaction networks, corresponding to four species including Caenorhabditis elegans (CAE), Saccharomyces cerevisiae (SAC), Homo sapiens (HOM) and Drosophila melanogaster (DRO), derived by the two-hybrid experimental system.
  • The second is the autonomous system (AS) relationship networks collected from CAIDA database [27], where each vertex represents an autonomous system, and each edge denotes a certain relationship between two autonomous systems, such as provider–customer, peer-to-peer, or sibling-to-sibling.

3. Measurements of chains in complex networks

  • The vertices in a chain play similar roles in connecting other vertices in the network.
  • All the parameters are determined by fitting the preprocessed (e.g. logarithmized) data adopting the method of multiple linear regression using least squares (the same below).
  • That is, the PA rule with a well-ordered recommendation.

Freshman

  • Manner can explain the power-law chain length distribution revealed in the Douban friendship network.
  • Besides, a vertex in a network may appear in various chains, i.e. a vertex, like a multifunctional block, may have the ability to substitute different vertices in different chains when they break down.
  • Here, the authors would like to measure the multi-functional ability of a vertex by its incoming degree kin(i) and outgoing degree kout(i) in the corresponding chain graph.
  • In fact, the chain graph of each artificial network presented in this paper has a large number of isolated vertices (the chains with length equal to 1); as a result, the chain graphs of the artificial networks have extra small average incoming degree ⟨kin⟩.
  • The chain graph of the Douban friendship network has the largest ⟨kin⟩ in all the tested networks.

4. Measurements of families in complex networks

  • Denoting the diversity of a family φi by ψi representing the number of chains it contains, the number Nφ of families and the average family diversity of the tested networks are also recorded in table 2, where the authors can find that, compared with the three artificial networks, the real-world networks have much higher average family diversity.
  • Certainly, this vertex can be also connected to other hubs, i.e. it is also the end-vertex of many other chains, and thus leading to particularly high-diversity families.
  • Denoting d(lφ " L) as the average distance between families with their size larger than L, the relationships between d(lφ " L) and L for the real-world networks and the artificial networks are depicted in figures 7(a)–(d) respectively.
  • Similarly, the long chains as well as the quasi-star structure should be responsible for the relatively large〈.
  • Loφ 〉 of the real-world networks, because in these networks, there might be different vertices in the nucleus serving as the end-vertices of a same group of low-degree vertices, which leads to different but highly overlapping chains and therefore highly overlapping families.

5. Summary

  • The authors have defined a partially ordered relation between pairwise vertices by comparing their neighbor sets, and then proposed a series of new measurements based on predefined chains and families, such as chain length distribution and family diversity distribution, in order to evaluate different network models.
  • Unfortunately, most of these partially ordered set-based properties revealed in real-world networks cannot be explained by those well-known network models, such as the Barabási (BA) model, local-world (LW) model and Dorogovtsev, Mendes, and Samukhin (DMS) model, although, by comparing, the LW model and the DMS model behave a little better than the BA model.
  • Interestingly, it seems that long chains, high diversity, as well as strong overlap can be explained to a certain extent by a quasi-star structure with a densely connected nucleus which has already been revealed in the autonomous system relationship networks by Carmi et al adopting a k-shell decomposing algorithm.
  • So the authors believe that such new measurements can help us better understand the structure of real-world networks and further provide more appropriate models for them.

Acknowledgments

  • The authors would like to thank all the members in their research group in the Department of Control Science and Engineering, Zhejiang University at Yuquan Campus, for the valuable discussion about the ideas presented in this paper.
  • This work has been supported by China Postdoctoral Science Foundation (grant no 20080441256).

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IOP PUBLISHING JOURNAL OF PHYSICS A: MAT HEMAT IC AL AN D THEORETICAL
J. Phys. A: Math. Theor. 43 (2010) 185001 (13pp) doi:10.1088/1751-8113/43/18/185001
Partially ordered sets in complex networks
Qi Xuan
1
,FangDuandTie-JunWu
Department of Control Science and Engineering, Zhejiang University, Hangzhou 310027,
People’s R epublic of China
E-mail: crestxq@hotmail.com
Received 24 November 2009, in final form 5 March 2010
Published 15 April 2010
Online at stacks.iop.org/JPhysA/43/185001
Abstract
In this paper, a partial-order relation is defined among vertices of a network to
describe which vertex is more important than another on its contribution to the
connectivity of the network. A maximum linearly ordered subset of vertices is
defined as a chain and the chains sharing the same end-vertex are grouped as
afamily. Throughcombiningthesameverticesappearingindifferentchains,
adirectedchaingraphisobtained. Basedonthesedenitions,aseriesof
new network measurements, such as chain length distribution, family diversity
distribution, as well as the centrality of families, are proposed. By studying the
partially ordered sets in three kinds of real-world networks, many interesting
results are revealed. For instance, the similar approximately power-law
chain length distribution may be attributed to a chain-based positive feedback
mechanism, i.e. new vertices prefer to participate in longer chains, which
can be inferred by combining the notable preferential attachment rule with a
well-ordered recommendation manner. Moreover, the relatively large average
incoming degree of the chain graphs may indicate an efficient substitution
mechanism in these networks. Most of the partially ordered set-based properties
cannot be explained by the current well-known scale-free network models;
therefore, we are required to propose more appropriate network models in the
future.
PACS numbers: 89.75.Hc, 89.75.Fb, 02.10.Ab
(Some figures in this article are in colour only in the electronic version)
1. Introduction
Over the past decade, characterizing and modeling complex networks attract many attentions
from various areas [18]. Interestingly, it is found that lots of real-world complex networks
1
Author to whom any correspondence should be addressed.
1751-8113/10/185001+13$30.00 © 2010 IOP Publishing Ltd Printed in the UK & the USA 1

J. Phys. A: Math. Theor. 43 (2010) 185001 Q Xuan et al
share several common statistical properties [9], such as small-world [1], scale-free [2], power-
law clustering function [3], self-similarity [1012], symmetry [13, 14], etc. In order to explain
these properties, a large number of models have been proposed [13, 1518]. For instance,
Watts and Strogatz (WS) [1]proposedasimplesmall-worldnetworkmodelbyintroducing
randomness into a regular network through a rewiring process, and the first scale-free network
is proposed by Barab
´
asi and Albert (BA) [2]throughintroducingthepreferentialattachment
(PA) rule into the network growing process. Most of these traditional network properties are
based on the single-vertex or pair-vertices measurements, e.g. the degree and the clustering
coefficient of a vertex or the symmetry and the shortest path length between a pair of vertices,
etc. These types of properties are far from sufficient when the interaction between vertices is
especially emphasized in complex networks.
Recently, the basic structural motifs recurring frequently in complex networks were
carefully studied [19, 20], and it seems that the rank distributions of these motifs can be used
to classify networks [20], e.g. real-world networks always present significantly larger numbers
of these motifs compared with randomized networks. This finding is of much interest because
it suggests that motifs rather than vertices may be the basic cells to perform the higher level
functions of the systems. However, search of large motifs, if any, seems a little blindfold and
sometimes is very time-consuming because there must be a mass of different connected motifs
of large size. Therefore, in large-scale social networks, researchers are more likely to directly
reveal the modules defined by groups (communities) of vertices within which connections
are denser than among them [9, 21, 22]. It is widely believed [9]thatverticesinthesame
community are inclined to share common properties and dynamics. However, the ambiguous
definition [21]ofcommunityproducesaninevitableresultthateveryunionofcommunitiesis
also a community. In such a situation, a hierarchy among the communities has to be always
assumed apriorito overcome such limitation [9, 23].
In this paper, we would like to provide an optional method to rank and classify vertices
more precisely based on their contributions to the connectivity of the target network. Firstly,
we will introduce several definitions about relations in discrete mathematics [24]. There are
many kinds of relationships in the world. A network itself defines a relation among vertices,
i.e. two vertices are related if they are connected. And in many cases we are more interested in
order relations which could tell us when an element is ‘smaller than’ or ‘preceding’ another.
Apartiallyorderedsetthencouldbedenedasasettogetherwitharelationdescribingthat
one of the elements must precede the other for certain pairs of elements in the set. It should
be noted that, different from a linear order, in a partially ordered set some pairs of elements
may not be related to each other. A familiar real-world example of a partially ordered set is
acollectionofpeopleorderedbygenealogicaldescendancy[25]. Some pairs of people bear
the ancestor–descendant relationship, but other pairs do not.
Generally, in a network, vertices sharing several same neighbors could be compared with
each other, i.e. a vertex could be considered to precede another one if each of its neighbors
is also the neighbor of its posterior. In fact, a vertex can be completely replaced by its
posterior on its contribution to the connectivity of the network. The vertex set together with
such relation forms a partially ordered set in the network. A maximum linearly ordered
subset of vertices then is defined as a chain and the chains sharing the same end-vertex are
grouped as a family. Measurements of chains and families may provide ultra information to
understand the intrinsic mechanisms of real-world complex networks and thus are carefully
studied in this paper. The results are interesting and challenging because most of the common
quantitative and qualitative partially ordered set-based properties revealed in several real-
world networks, such as the similar approximately po wer-law chain length distribution,
cannot be explained by current well-known network models [2, 17, 18]. Therefore,
2

J. Phys. A: Math. Theor. 43 (2010) 185001 Q Xuan et al
more appropriate network models should be provided in the future to explain these new
properties.
The rest of the paper is organized as follows. In the next section, partially ordered set and
other related concepts, e.g. chain and family, are defined in a network and a brief algorithm
is provided to obtain them, at the same time, the tested data sets are also introduced here.
Measurements of chains and families in several real-world and artificial complex networks are
carefully studied in sections 3 and 4,respectively. Theworkissummarizedinsection5.
2. Partially ordered set in a network
2.1. Definitions
AnetworkisdenotedbyG = G(V , E),whereV is the set of vertices and E V × V is the
set of edges. Vertices v
i
and v
j
are adjacent if (v
i
,v
j
) E.AsubsetofverticesU
i
V is
named as the neighbor set of v
i
if each vertex in U
i
,whilenoneoftheverticesinV \U
i
,is
adjacent to v
i
.Bythesenotations,arelationR,denotedbythesymbol‘’, on the vertex set
V can be defined as
v
i
v
j
U
i
U
j
. (1)
Such a relation on the vertex set V is reflexive and transitive because equations (2)and(3)
must be always satisfied:
v
i
v
i
, (2)
v
i
v
j
,v
j
v
k
v
i
v
k
. (3)
Furthermore, if we consider that v
i
and v
j
are equal or, more specific, symmetric, denoted by
v
i
= v
j
,whenU
i
U
j
and U
j
U
i
are both satisfied, then the relation on the vertex set
V is also antisymmetric represented by
v
i
v
j
,v
j
v
i
v
i
= v
j
. (4)
It should be noted that here v
i
= v
j
just means that the two possible different vertices are equal
on their neighbor sets, i.e. U
i
= U
j
.Thus,therelation can be called a partial order on the
vertex set V,andthevertexsetV together with the partial order is called a partially ordered
set [24]. It is worthy of note that the partial order defined between two vertices completely
depends on the relationship between their neighbor sets; as a result, it is somewhat related
to the degree, i.e. the number of neighbors, of vertices, i.e. denoting the degree of v
i
by k
i
,
equation (5)mustbesatised:
v
i
v
j
k
i
! k
j
. (5)
A chain in such a partially ordered set is denoted by a maximum linearly ordered subset
θ
i
=
!
v
i
1
,v
i
2
,...,v
i
l
"
V satisfying equation (6):
v
i
1
v
i
2
···v
i
l
, (6)
at the same time, for each vertex v
j
in the set V \θ
i
,thevertexsubsetθ
i
v
j
cannot be linearly
ordered. For instance, the network in figure 1(a)hastwochainsv
a
v
b
v
c
v
d
and
v
i
v
j
v
k
v
l
.Itshouldbenotedthatboththesubsets{v
a
,v
b
,v
c
} and {v
a
,v
b
,v
c
,v
d
,v
i
}
in the figure are not chains because the former is not a ‘maximum’ linearly ordered subset and
the latter cannot be linearly ordered at all.
Naturally, each chain has an end-vertex, e.g. v
i
l
in equation (6); the end-vertex set of
anetworkisdenotedbyV
E
containing all the end-vertices of the chains in the network,
3

J. Phys. A: Math. Theor. 43 (2010) 185001 Q Xuan et al
a
b
c
i
j
d
k
l
Chains
a
b
c
d
e
i
j
k
Family
(a) Chain (b) Family
Figure 1. (a) Following the definition of the partial relation , the network has two chains
v
a
v
b
v
c
v
d
and v
i
v
j
v
k
v
l
. It should be noted that both the subsets {v
a
,v
b
,v
c
}
and {v
a
,v
b
,v
c
,v
d
,v
i
} are not chains here just because the former one is not a ‘maximum’ linearly
ordered subset and the latter one cannot be linearly ordered at all. (b) Two chains v
a
v
b
v
c
and v
e
v
d
v
c
sharing the same end-vertex v
c
are grouped as a family {v
a
,v
b
,v
e
,v
d
,v
c
}.
e.g. V
E
={v
d
,v
l
} for the network shown in figure 1(a). Furthermore, different chains in a
network may share the same end-vertex, and the chains θ
i
j
,j = 1, 2,...,ψ
i
,sharingthesame
end-vertex v
i
are grouped as a family φ
i
in the network denoted by
φ
i
=
ψ
i
#
j=1
θ
i
j
. (7)
So there will be totally |V
E
| families, and each family corresponds to an end-vertex in the
network. For instance, in figure 1(b), the chains v
a
v
b
v
c
and v
e
v
d
v
c
sharing the
same end-vertex v
c
can be grouped as a family {v
a
,v
b
,v
e
,v
d
,v
c
}.
The main contribution of a vertex in a network is determined by its hinge role in connecting
other vertices, which is especially remarkable for those end-vertices in the network. In fact,
through the above definitions, we can get the following theorem:
Theorem 1. Removal of any vertex in a chain except the end-vertex will not influence the
average shortest path length of the remaining subnetwork, that is, the end-vertex of a chain
can completely replace the other vertices in the same chain on their contributions to the
connectivity of the network.
Theorem 1 can be easily proven with the fact that, considering a chain denoted by equation (6),
U
i
j
U
i
l
must be satisfied for each j ! l,whichsuggeststhateachshortestpathpassingby
the vertex v
i
j
can change its route to pass through the vertex v
i
l
without increasing its length.
It should be noted that, if all the neighbors of v
i
are also the neighbors of v
j
, i.e. v
i
v
j
,
the betweenness centrality [9]ofv
i
,denotedbyC
i
,mustnotbelargerthanthatofv
j
,asis
represented by
v
i
v
j
C
i
! C
j
. (8)
Therefore, an end-vertex must have the largest betweenness centrality in its family.
2.2. The algorithm
Based on the above definitions, here, we will briefly introduce our method to find all the
chains in a network. In the procedure, the chain graph is constructed at first, then the chains
4

J. Phys. A: Math. Theor. 43 (2010) 185001 Q Xuan et al
a
b
c
d
e
f
g
(a)
(c)
(b)
a
g
c
d
e
b
f
a
b
c
d
e
f
g
Figure 2. (a)Asimplenetworkwithsevenvertices. (b) The partial-ordered relations among these
vertices. Each pair of ordered vertices are connected by a directed edge, and the dotted edges are
redundant. (c)Thechaingraphobtainedbydroppingthoseredundantedges.
are extracted by a depth first search algorithm in the chain graph, and the families can be
obtained by simply combining these chains. Particularly, the algorithm is summarized by the
following three steps.
(1) Build chain graph. For each pair of vertices v
i
and v
j
sharing at least one common
neighbor, determine if they could be ordered, i.e. v
i
v
j
or v
j
v
i
,bycomparing
their neighbor sets. Compress those symmetric vertices sharing exact same neighbors,
i.e. v
k
1
= v
k
2
··=v
k
p
(k
1
<k
2
< ··· <k
p
), as one vertex v
k
1
.Thenacompressed
digraph can be derived by considering that each pair of ordered vertices v
i
v
j
are
connected by a directed edge pointing from v
i
to v
j
.Revealallthetrianglemotifsinthe
compressed digraph, e.g. v
i
v
j
,v
j
v
k
,andv
i
v
k
,andremovealltheredundant
edges, e.g. v
i
v
k
,simultaneously. Replaceeachcompressedvertexv
k
1
by a directed
chain v
k
1
v
k
2
···v
k
p
;atthesametime,theincomingedgesofthecompressed
vertex v
k
1
still point to the beginning vertex v
k
1
in the directed chain, while the outgoing
edges of the compressed vertex v
k
1
are now originated from the last vertex v
k
p
.Thus,the
chain graph is constructed. A simple example is shown in figure 2.
(2) Extract chains. In the chain graph, there are three types of vertices: the original-vertices
with no incoming edges, the end-vertices with no outgoing edges and the middle-vertices
with at least one incoming edge and one outgoing edge. Then all the chains could be
revealed by a depth first search algorithm in the chain graph: begin with the original-
vertices, along the outgoing edges, till reaching the end-vertices.
(3) Combine to families. When all the chains are collected, each family can be obtained by
simply combining all the chains sharing the same end-vertex.
2.3. Data sets
We mainly study the partially ordered sets in three kinds of real-world networks. The first
is the protein–protein interaction networks collected from BioGrid database [26], where each
vertex represents a protein or a gene, and each edge denotes the interaction between these
proteins or genes. We use the giant components of four protein–protein interaction networks,
corresponding to four species including Caenorhabditis elegans (CAE), Saccharomyces
cerevisiae (SAC), Homo sapiens (HOM) and Drosophila melanogaster (DRO), derived by
the two-hybrid experimental system. The second is the autonomous system (AS) relationship
networks collected from CAIDA database [27], where each vertex represents an autonomous
system, and each edge denotes a certain relationship between two autonomous systems, such
as provider–customer, peer-to-peer, or sibling-to-sibling. We use six AS relationship networks
captured at different times from 2004 to 2009, i.e. 2004 (5 Jan.), 2005 (3 Jan.), 2006 (2 Jan.),
5

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33,771 citations

Journal ArticleDOI
TL;DR: This article proposes a method for detecting communities, built around the idea of using centrality indices to find community boundaries, and tests it on computer-generated and real-world graphs whose community structure is already known and finds that the method detects this known structure with high sensitivity and reliability.
Abstract: A number of recent studies have focused on the statistical properties of networked systems such as social networks and the Worldwide Web. Researchers have concentrated particularly on a few properties that seem to be common to many networks: the small-world property, power-law degree distributions, and network transitivity. In this article, we highlight another property that is found in many networks, the property of community structure, in which network nodes are joined together in tightly knit groups, between which there are only looser connections. We propose a method for detecting such communities, built around the idea of using centrality indices to find community boundaries. We test our method on computer-generated and real-world graphs whose community structure is already known and find that the method detects this known structure with high sensitivity and reliability. We also apply the method to two networks whose community structure is not well known—a collaboration network and a food web—and find that it detects significant and informative community divisions in both cases.

14,429 citations


"Partially ordered sets in complex n..." refers background in this paper

  • ...Therefore, in large-scale social networks, researchers are more likely to directly reveal the modules defined by groups (communities) of vertices within which connections are denser than among them [9, 21, 22]....

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Journal ArticleDOI
TL;DR: The major concepts and results recently achieved in the study of the structure and dynamics of complex networks are reviewed, and the relevant applications of these ideas in many different disciplines are summarized, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering.

9,441 citations

Journal ArticleDOI
TL;DR: This work states that rapid advances in network biology indicate that cellular networks are governed by universal laws and offer a new conceptual framework that could potentially revolutionize the view of biology and disease pathologies in the twenty-first century.
Abstract: A key aim of postgenomic biomedical research is to systematically catalogue all molecules and their interactions within a living cell. There is a clear need to understand how these molecules and the interactions between them determine the function of this enormously complex machinery, both in isolation and when surrounded by other cells. Rapid advances in network biology indicate that cellular networks are governed by universal laws and offer a new conceptual framework that could potentially revolutionize our view of biology and disease pathologies in the twenty-first century.

7,475 citations

Frequently Asked Questions (1)
Q1. What are the contributions mentioned in the paper "Partially ordered sets in complex networks" ?

In this paper, a partial-order relation is defined among vertices of a network to describe which vertex is more important than another on its contribution to the connectivity of the network. Most of the partially ordered set-based properties can not be explained by the current well-known scale-free network models ; therefore, the authors are required to propose more appropriate network models in the future. Ab ( Some figures in this article are in colour only in the electronic version )