# Partially ordered sets in complex networks

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)

Jump to: [1. Introduction] – [2.1. Definitions] – [2.2. The algorithm] – [2.3. Data sets] – [3. Measurements of chains in complex networks] – [Freshman] – [4. Measurements of families in complex networks] – [5. Summary] and [Acknowledgments]

### 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. Basedonthesedeﬁnitions,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 [1–8]. 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 [10–12], symmetry [13, 14], etc. In order to explain

these properties, a large number of models have been proposed [1–3, 15–18]. 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.

Apartiallyorderedsetthencouldbedeﬁnedasasettogetherwitharelationdescribingthat

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)mustbesatisﬁed:

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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39,297 citations

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TL;DR: A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.

Abstract: Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.

33,771 citations

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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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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

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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