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Network scienceFrom Wikipedia, the free encyclopedia

Network science is an interdisciplinary academic field which studies complex networks such astelecommunication networks, computer networks, biological networks, cognitive and semantic networks, andsocial networks. The field draws on theories and methods including graph theory from mathematics, statisticalmechanics from physics, data mining and information visualization from computer science, inferential modelingfrom statistics, and social structure from sociology. The United States National Research Council definesnetwork science as "the study of network representations of physical, biological, and social phenomena leading

to predictive models of these phenomena."[1]

Contents

1 Background and history

2 Department of Defense Initiatives3 Network Properties

3.1 Density3.2 Size

3.3 Average Degree3.4 Average Path Length

3.5 Diameter of a Network

3.6 Clustering Coefficient

3.7 Connectedness

3.8 Node Centrality4 Network Models

4.1 Erdős–Rényi Random Graph model

4.2 Watts-Strogatz Small World model

4.3 Barabási–Albert (BA) Preferential Attachment model5 Network analysis

5.1 Social network analysis5.2 Biological network analysis

5.3 Link analysis

5.3.1 Network robustness

5.3.2 Pandemic Analysis

5.3.2.1 Susceptible to Infected

5.3.2.2 Infected to Recovered

5.3.2.3 Infectious Period

5.3.3 Web Link Analysis5.3.3.1 PageRank

5.3.3.1.1 Random Jumping

5.4 Centrality measures

6 Spread of content in networks

6.1 The SIR Model

7 Interdependent networks

8 Network optimization

9 Network Analysis and Visualization Tools

10 See also

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11 References12 Further reading

13 External links

14 Notes

Background and history

The study of networks has emerged in diverse disciplines as a means of analyzing complex relational data. Theearliest known paper in this field is the famous Seven Bridges of Königsberg written by Leonhard Euler in 1736.Euler's mathematical description of vertices and edges was the foundation of graph theory, a branch ofmathematics that studies the properties of pairwise relations in a network structure. The field of graph theorycontinued to develop and found applications in chemistry (Sylvester, 1878).

In the 1930s Jacob Moreno, a psychologist in the Gestalt tradition, arrived in the United States. He developedthe sociogram and presented it to the public in April 1933 at a convention of medical scholars. Moreno claimedthat "before the advent of sociometry no one knew what the interpersonal structure of a group 'precisely' lookedlike (Moreno, 1953). The sociogram was a representation of the social structure of a group of elementaryschool students. The boys were friends of boys and the girls were friends of girls with the exception of one boywho said he liked a single girl. The feeling was not reciprocated. This network representation of social structurewas found so intriguing that it was printed in The New York Times (April 3, 1933, page 17). The sociogram hasfound many applications and has grown into the field of social network analysis.

Probabilistic theory in network science developed as an off-shoot of graph theory with Paul Erdős and AlfrédRényi's eight famous papers on random graphs. For social networks the exponential random graph model or p*is a notational framework used to represent the probability space of a tie occurring in a social network. Analternate approach to network probability structures is the network probability matrix, which models theprobability of edges occurring in a network, based on the historic presence or absence of the edge in a sampleof networks.

In 1998, David Krackhardt and Kathleen Carley introduced the idea of a meta-network with the PCANSModel. They suggest that "all organizations are structured along these three domains, Individuals, Tasks, andResources". Their paper introduced the concept that networks occur across multiple domains and that they areinterrelated. This field has grown into another sub-discipline of network science called dynamic networkanalysis.

More recently other network science efforts have focused on mathematically describing different networktopologies. Duncan Watts reconciled empirical data on networks with mathematical representation, describingthe small-world network. Albert-László Barabási and Reka Albert developed the scale-free network which is aloosely defined network topology that contains hub vertices with many connections, that grow in a way tomaintain a constant ratio in the number of the connections versus all other nodes. Although many networks, suchas the internet, appear to maintain this aspect, other networks have long tailed distributions of nodes that onlyapproximate scale free ratios.

Department of Defense Initiatives

The U.S. military first became interested in network-centric warfare as an operational concept based onnetwork science in 1996. John A. Parmentola, the U.S. Army Director for Research and LaboratoryManagement, proposed to the Army’s Board on Science and Technology (BAST) on December 1, 2003 thatNetwork Science become a new Army research area. The BAST, the Division on Engineering and Physical

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Sciences for the National Research Council (NRC) of the National Academies, serves as a convening authorityfor the discussion of science and technology issues of importance to the Army and oversees independent Army-related studies conducted by the National Academies. The BAST conducted a study to find out whetheridentifying and funding a new field of investigation in basic research, Network Science, could help close the gapbetween what is needed to realize Network-Centric Operations and the current primitive state of fundamentalknowledge of networks.

As a result, the BAST issued the NRC study in 2005 titled Network Science (referenced above) that defined anew field of basic research in Network Science for the Army. Based on the findings and recommendations ofthat study and the subsequent 2007 NRC report titled Strategy for an Army Center for Network Science,Technology, and Experimentation, Army basic research resources were redirected to initiate a new basicresearch program in Network Science. To build a new theoretical foundation for complex networks, some ofthe key Network Science research efforts now ongoing in Army laboratories address:

Mathematical models of network behavior to predict performance with network size, complexity, andenvironment

Optimized human performance required for network-enabled warfareNetworking within ecosystems and at the molecular level in cells.

As initiated in 2004 by Frederick I. Moxley with support he solicited from David S. Alberts, the Department ofDefense helped to establish the first Network Science Center in conjunction with the U.S. Army at the UnitedStates Military Academy (USMA). Under the tutelage of Dr. Moxley and the faculty of the USMA, the firstinterdisciplinary undergraduate courses in Network Science were taught to cadets at West Point. Subsequently,the U.S. Department of Defense has funded numerous research projects in the area of Network Science.

In 2006, the U.S. Army and the United Kingdom (UK) formed the Network and Information ScienceInternational Technology Alliance, a collaborative partnership among the Army Research Laboratory, UKMinistry of Defense and a consortium of industries and universities in the U.S. and UK. The goal of the allianceis to perform basic research in support of Network- Centric Operations across the needs of both nations.

In 2009, the U.S. Army formed the Network Science CTA, a collaborative research alliance among the ArmyResearch Laboratory, CERDEC, and a consortium of about 30 industrial R&D labs and universities in the U.S.The goal of the alliance is to develop a deep understanding of the underlying commonalities among intertwinedsocial/cognitive, information, and communications networks, and as a result improve our ability to analyze,predict, design, and influence complex systems interweaving many kinds of networks.

Today, network science is an exciting and growing interdisciplinary field. Scientists from many diverse fields areworking together. Network science holds the promise of increasing collaboration across disciplines, by sharingdata, algorithms, and software tools.

Network Properties

Oftentimes, Networks have certain attributes that can be calculated to analyze the properties & characteristicsof the network. These Network properties often define Network Models and can be used to analyze howcertain models contrast to each other. Many of the definitions for other terms used in network science can befound in Glossary of graph theory.

Density

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The density of a network is defined as a ratio of the number of edges to the number of possible edges,

given by the binomial coefficient , giving

Size

The size of a network can refer to the number of nodes or, less commonly, the number of edges whichcan range from (a tree) to (a complete graph).

Average Degree

The degree of a node is the number of edges connected to it. Closely related to the density of a network is theaverage degree, . In the ER random graph model, we can compute

where is the probability of two nodes being connected.

Average Path Length

Average path length is calculated by finding the shortest path between all pairs of nodes, adding them up, andthen dividing by the total number of pairs. This shows us, on average, the number of steps it takes to get fromone member of the network to another.

Diameter of a Network

As another means of measuring network graphs, we can define the diameter of a network as the longest of allthe calculated shortest paths in a network. In other words, once the shortest path length from every node to allother nodes is calculated, the diameter is the longest of all the calculated path lengths. The diameter isrepresentative of the linear size of a network.

Clustering Coefficient

The clustering coefficient is a measure of an "all-my-friends-know-each-other" property. This is sometimesdescribed as the friends of my friends are my friends. More precisely, the clustering coefficient of a node is theratio of existing links connecting a node's neighbors to each other to the maximum possible number of such links.The clustering coefficient for the entire network is the average of the clustering coefficients of all the nodes. Ahigh clustering coefficient for a network is another indication of a small world.

The clustering coefficient of the 'th node is

where is the number of neighbours of the 'th node, and is the number of connections between theseneighbours. The maximum possible number of connections between neighbors is, of course,

Connectedness

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This Erdős–Rényi model is generated

with N=4 nodes. For each edge in the

complete graph formed by all N

nodes, a random number is generated

and compared to a given probability.

If the random number is greater than

p, an edge is formed on the model.

The way in which a network is connected plays a large part into how networks are analyzed and interpreted.Networks are classified in three different categories:

Clique/Complete Graph: a completely connected network, where all nodes are connected to every

other node. These networks are symmetric in that all nodes have in-links and out-links from all others.Giant Component: A single connected component which contains most of the nodes in the network.

Weakly Connected Component: A collection of nodes in which there exists a path from any node to any

other, ignoring directionality of the edges.

Strongly Connected Component: A collection of nodes in which there exists a directed path from anynode to any other.

Node Centrality

Node centrality can be viewed as a measure of influence or importance in a network model. There exists threemain measures of Centrality that are studied in Network Science.

Closeness: represents the average distance that each node is from all other nodes in the network

Betweeness: represents the number of shortest paths in a network that traverse through that node

Degree/Strength: represents the amount links that a particular node possesses in a network. In a directed

network, one must differentiate between in-links and out links by calculating in-degree and out-degree.The analogue to degree in a weighted network, strength is the sum of a node's edge weights. In-strength

and out-strength are analogously defined for directed networks.

Network Models

Network models serve as a foundation to understanding interactions within empirical complex networks.Various random graph generation models produce network structures that may be used in comparison to real-world complex networks.

Erdős–Rényi Random Graph model

The Erdős–Rényi model, named for Paul Erdős and Alfréd Rényi, isused for generating random graphs in which edges are set betweennodes with equal probabilities. It can be used in the probabilisticmethod to prove the existence of graphs satisfying various properties,or to provide a rigorous definition of what it means for a property tohold for almost all graphs.

To generate an Erdős–Rényi model two parameters must bespecified: the number of nodes in the graph generated as and theprobability that a link should be formed between any two nodes as .A constant may derived from these two components with theformula .

The Erdős–Rényi model has several interesting characteristics in comparison to other graphs. Because themodel is generated without bias to particular nodes, the degree distribution is binomial in nature with regards to

the formula: . Also as a result of this characteristic, the

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The Watts and Strogatz model uses

the concept of rewiring to achieve its

structure. The model generator will

iterate through each edge in the

original lattice structure. An edge may

changed its connected vertices

according to a given rewiring

probability. in this

example.

clustering coefficient tends to 0. The model tends to form a giant component in situations where <k> > 1 in a

process called percolation. The average path length is relatively short in this model and tends to .

Watts-Strogatz Small World model

The Watts and Strogatz model is a random graph generation modelthat produces graphs with small-world properties.

An initial lattice structure is used to generate a Watts-Strogatz model.Each node in the network is initially linked to its closestneighbors. Another parameter is specified as the rewiring probability.Each edge has a probability that it will be rewired to the graph as arandom edge. The expected number of rewired links in the model is

.

As the Watts-Strogatz model begins as non-random lattice structure,it has a very high clustering coefficient along with high average pathlength. Each rewire is likely to create a shortcut between highlyconnected clusters. As the rewiring probability increases, theclustering coefficient decreases slower than the average path length. Ineffect, this allows the average path length of the network to decrease significantly with only slightly decreases inclustering coefficient. Higher values of p force more rewired edges, which in effect makes the Watts-Strogatzmodel a random network.

Barabási–Albert (BA) Preferential Attachment model

The Barabási–Albert model is a random network model used to demonstrate a preferential attachment or a"rich-get-richer" effect. In this model, an edge is most likely to attach to nodes with higher degrees. The networkbegins with an initial network of m0 nodes. m0 ≥ 2 and the degree of each node in the initial network should be

at least 1, otherwise it will always remain disconnected from the rest of the network.

In the BA model, new nodes are added to the network one at a time. Each new node is connected to existing nodes with a probability that is proportional to the number of links that the existing nodes already have.

Formally, the probability pi that the new node is connected to node i is[2]

where ki is the degree of node i. Heavily linked nodes ("hubs") tend to quickly accumulate even more links,

while nodes with only a few links are unlikely to be chosen as the destination for a new link. The new nodeshave a "preference" to attach themselves to the already heavily linked nodes.

The degree distribution resulting from the BA model is scale free, in particular, it is a power law of the form:

Hubs exhibit high betweenness centrality which allows short paths to exist between nodes. As a result the BAmodel tends to have very short average path lengths. The clustering coefficient of this model also tends to 0.While the diameter of Erdős Rényi random graph the diameter D, is proportional to log N (small world), for

scale free networks the due to the existence of hubs D~loglogN (ultr-small word).[4] Note that the average

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The degree distribution of the BA

Model, which follows a power law.

In loglog scale the power law

function is a straight line.[3]

opath length scale with N as the diameter.

Network analysis

Social network analysis

Social network analysis examines the structure of relationships between social entities.[5] These entities areoften persons, but may also be groups, organizations, nation states,web sites, scholarly publications.

Since the 1970s, the empirical study of networks has played a centralrole in social science, and many of the mathematical and statisticaltools used for studying networks have been first developed in

sociology.[6] Amongst many other applications, social networkanalysis has been used to understand the diffusion of innovations,news and rumors. Similarly, it has been used to examine the spread ofboth diseases and health-related behaviors. It has also been appliedto the study of markets, where it has been used to examine the role oftrust in exchange relationships and of social mechanisms in settingprices. Similarly, it has been used to study recruitment into politicalmovements and social organizations. It has also been used toconceptualize scientific disagreements as well as academic prestige.More recently, network analysis (and its close cousin traffic analysis)has gained a significant use in military intelligence, for uncovering insurgent networks of both hierarchical and

leaderless nature.[7][8]

Biological network analysis

With the recent explosion of publicly available high throughput biological data, the analysis of molecularnetworks has gained significant interest. The type of analysis in this content are closely related to social networkanalysis, but often focusing on local patterns in the network. For example network motifs are small subgraphsthat are over-represented in the network. Activity motifs are similar over-represented patterns in the attributes ofnodes and edges in the network that are over represented given the network structure.

Link analysis

Link analysis is a subset of network analysis, exploring associations between objects. An example may beexamining the addresses of suspects and victims, the telephone numbers they have dialed and financialtransactions that they have partaken in during a given timeframe, and the familial relationships between thesesubjects as a part of police investigation. Link analysis here provides the crucial relationships and associationsbetween very many objects of different types that are not apparent from isolated pieces of information.Computer-assisted or fully automatic computer-based link analysis is increasingly employed by banks andinsurance agencies in fraud detection, by telecommunication operators in telecommunication network analysis,by medical sector in epidemiology and pharmacology, in law enforcement investigations, by search engines forrelevance rating (and conversely by the spammers for spamdexing and by business owners for search engineoptimization), and everywhere else where relationships between many objects have to be analyzed.

Network robustness

The structural robustness of networks[9] is studied using percolation theory. When a critical fraction of nodes is

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The structural robustness of networks[9] is studied using percolation theory. When a critical fraction of nodes is

removed the network becomes fragmented into small clusters. This phenomenon is called percolation[10] and itrepresents an order-disorder type of phase transition with critical exponents.

Pandemic Analysis

The SIR Model is one of the most well known algorithms on predicting the spread of global pandemics within aninfectious population.

Susceptible to Infected

The formula above describes the "force" of infection fore each susceptible unit in an infectious population, whereβ is equivalent to the transmission rate of said disease.

To track the change of those susceptible in an infectious population:

Infected to Recovered

Over time, the number of those infected fluctuates by: the specified rate of recovery, represented by but

deducted to one over the average infectious period , the numbered of infecious individuals, , and the change

in time, .

Infectious Period

Whether a population will be overcome by a pandemic, with regards to the SIR model, is dependent on thevalue of or the "average people infected by an infected individual."

Web Link Analysis

Several Web search ranking algorithms use link-based centrality metrics, including (in order of appearance)Marchiori's Hyper Search, Google's PageRank, Kleinberg's HITS algorithm, the CheiRank and TrustRankalgorithms. Link analysis is also conducted in information science and communication science in order tounderstand and extract information from the structure of collections of web pages. For example the analysismight be of the interlinking between politicians' web sites or blogs.

PageRank

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PageRank works by randomly picking "nodes" or websites and then with a certain probability, "randomlyjumping" to other nodes. By randomly jumping to these other nodes, it helps PageRank completely traverse thenetwork as some webpages exist on the periphery and would not as readily be assessed.

Each node, , has a PageRank as defined by the sum of pages that link to times one over the outlinks or

"out-degree" of times the "importance" or PageRank of .

Random Jumping

As explained above, PageRank enlists random jumps in attempts to assign PageRank to every website on theinternet. These random jumps find websites that might not be found during the normal search methodologiessuch as Breadth-First Search and Depth-First Search.

In an improvement over the aforementioned formula for determining PageRank includes adding these randomjump components. Without the random jumps, some pages would receive a PageRank of 0 which would not begood.

The first is , or the probability that a random jump will occur. Contrasting is the "damping factor", or .

Another way of looking at it:

Centrality measures

Information about the relative importance of nodes and edges in a graph can be obtained through centralitymeasures, widely used in disciplines like sociology. Centrality measures are essential when a network analysishas to answer questions such as: "Which nodes in the network should be targeted to ensure that a message orinformation spreads to all or most nodes in the network?" or conversely, "Which nodes should be targeted tocurtail the spread of a disease?". Formally established measures of centrality are degree centrality, closenesscentrality, betweenness centrality, eigenvector centrality, and katz centrality. The objective of network analysisgenerally determines the type of centrality measure(s) to be used.

Degree centrality of a node in a network is the number of links (vertices) incident on the node.

Closeness centrality determines how “close” a node is to other nodes in a network by measuring the

sum of the shortest distances (geodesic paths) between that node and all other nodes in the network.

Betweenness centrality determines the relative importance of a node by measuring the amount of traffic

flowing through that node to other nodes in the network. This is done my measuring the fraction of paths

connecting all pairs of nodes and containing the node of interest.Eigenvector centrality is a more sophisticated version of degree centrality where the centrality of a

node not only depends on the number of links incident on the node but also the quality of those links. This

quality factor is determined by the eigenvectors of the adjacency matrix of the network.

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Katz centrality of a node is measured by summing the geodesic paths between that node and all

(reachable) nodes in the network. These paths are weighted, paths connecting the node with its

immediate neighbors carry higher weights than those which connect with nodes farther away from the

immediate neighbors.

Spread of content in networks

Content in a complex network can spread via two major methods: conserved spread and non-conserved

spread.[11] In conserved spread, the total amount of content that enters a complex network remains constant asit passes through. The model of conserved spread can best be represented by a pitcher containing a fixedamount of water being poured into a series of funnels connected by tubes . Here, the pitcher represents theoriginal source and the water is the content being spread. The funnels and connecting tubing represent the nodesand the connections between nodes, respectively. As the water passes from one funnel into another, the waterdisappears instantly from the funnel that was previously exposed to the water. In non-conserved spread, theamount of content changes as it enters and passes through a complex network. The model of non-conservedspread can best be represented by a continuously running faucet running through a series of funnels connectedby tubes . Here, the amount of water from the original source is infinite Also, any funnels that have been exposedto the water continue to experience the water even as it passes into successive funnels. The non-conservedmodel is the most suitable for explaining the transmission of most infectious diseases.

The SIR Model

In 1927, W. O. Kermack and A. G. McKendrick created a model in which they considered a fixed populationwith only three compartments, susceptible: , infected, , and recovered, . The compartments

used for this model consist of three classes:

is used to represent the number of individuals not yet infected with the disease at time t, or those

susceptible to the disease

denotes the number of individuals who have been infected with the disease and are capable of

spreading the disease to those in the susceptible category

is the compartment used for those individuals who have been infected and then recovered from the

disease. Those in this category are not able to be infected again or to transmit the infection to others.

The flow of this model may be considered as follows:

Using a fixed population, , Kermack and McKendrick derived the following

equations:

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Several assumptions were made in the formulation of these equations: First, an individual in the population mustbe considered as having an equal probability as every other individual of contracting the disease with a rate of

, which is considered the contact or infection rate of the disease. Therefore, an infected individual makes contactand is able to transmit the disease with others per unit time and the fraction of contacts by an infected with

a susceptible is . The number of new infections in unit time per infective then is , giving the

rate of new infections (or those leaving the susceptible category) as (Brauer &

Castillo-Chavez, 2001). For the second and third equations, consider the population leaving the susceptibleclass as equal to the number entering the infected class. However, a number equal to the fraction ( whichrepresents the mean recovery rate, or the mean infective period) of infectives are leaving this class per unit

time to enter the removed class. These processes which occur simultaneously are referred to as the Law ofMass Action, a widely accepted idea that the rate of contact between two groups in a population is proportionalto the size of each of the groups concerned (Daley & Gani, 2005). Finally, it is assumed that the rate of infectionand recovery is much faster than the time scale of births and deaths and therefore, these factors are ignored inthis model.

More can be read on this model on the Epidemic model page.

Interdependent networks

An interdependent network is a system of coupled networks where nodes of one or more networks depend onnodes in other networks. Such dependencies are enhanced by the developments in modern technology.Dependencies may lead to cascading failures between the networks and a relatively small failure can lead to acatastrophic breakdown of the system. Blackouts are a fascinating demonstration of the important role playedby the dependencies between networks. A recent study developed a framework to study the cascading failures

in an interdependent networks system.[12][13]

Network optimization

Network problems that involve finding an optimal way of doing something are studied under the name ofcombinatorial optimization. Examples include network flow, shortest path problem, transport problem,transshipment problem, location problem, matching problem, assignment problem, packing problem, routingproblem, Critical Path Analysis and PERT (Program Evaluation & Review Technique).

Network Analysis and Visualization Tools

Graph-tool and NetworkX, free and efficient Python modules for manipulation and statistical analysis of

networks. [2] (http://graph-tool.skewed.de/) [3] (http://networkx.lanl.gov/)igraph (http://igraph.sourceforge.net) , an open source C library for the analysis of large-scale complexnetworks, with interfaces to R, Python and Ruby.

Orange, a free data mining software suite, module orngNetwork(http://www.ailab.si/orange/doc/modules/orngNetwork.htm)

Pajek (http://pajek.imfm.si/doku.php) , program for (large) network analysis and visualization.Tulip, a free data mining and visualization software dedicated to the analysis and visualization of relational

data. [4] (http://tulip.labri.fr/)

See also

Complex network

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Collaborative innovation network

Dynamic network analysisGlossary of Graph Theory

Higher category theoryImmune network theory

Irregular warfarePolytelySystems theory

Erdos Renyi networksRandom networks

Constructal law[14]

Percolation

Network Theory in Risk AssessmentNetwork topology

Network analyzerSmall-world networks

Social circlesScale-free networksSequential dynamical system

References

"Network Science Center," http://www.dodccrp.org/files/Network_Science_Center.asf

"Connected: The Power of Six Degrees," http://ivl.slis.indiana.edu/km/movies/2008-talas-connected.movR. Cohen, K. Erez, D. ben-Avraham, S. Havlin, "Resilience of the Internet to random breakdown(http://havlin.biu.ac.il/Publications.php?

keyword=Resilience+of+the+Internet+to+random+breakdown&year=*&match=all) " Phys. Rev. Lett.85, 4626 (2000).

R. Cohen, K. Erez, D. ben-Avraham, S. Havlin, "Breakdown of the Internet under intentional attack(http://havlin.biu.ac.il/Publications.php?

keyword=Breakdown+of+the+Internet+under+intentional+attack&year=*&match=all) " Phys. Rev.Lett. 86, 3682 (2001)R. Cohen, S. Havlin, "Scale-free networks are ultrasmall (http://havlin.biu.ac.il/Publications.php?

keyword=Scale-free+networks+are+ultrasmall&year=*&match=all) " Phys. Rev. Lett. 90, 058701(2003)

Further reading

"The Burgeoning Field of Network Science," http://themilitaryengineer.com/index.php?option=com_content&task=view&id=88

S.N. Dorogovtsev and J.F.F. Mendes, Evolution of Networks: From biological networks to theInternet and WWW, Oxford University Press, 2003, ISBN 0-19-851590-1

Linked: The New Science of Networks, A.-L. Barabási (Perseus Publishing, CambridgeNetwork Science (http://www.nap.edu/catalog.php?record_id=11516) , Committee on NetworkScience for Future Army Applications, National Research Council. 2005. The National Academies Press

(2005)ISBN 0-309-10026-7Network Science Bulletin, USMA (2007) ISBN 978-1-934808-00-9

The Structure and Dynamics of Networks Mark Newman, Albert-László Barabási, & Duncan J.

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Watts (The Princeton Press, 2006) ISBN 0-691-11357-2Dynamical processes on complex networks, Alain Barrat, Marc Barthelemy, Alessandro Vespignani(Cambridge University Press, 2008) ISBN 978-0-521-87950-7

Network Science: Theory and Applications, Ted G. Lewis (Wiley, March 11, 2009) ISBN 0-470-33188-7

Nexus: Small Worlds and the Groundbreaking Theory of Networks, Mark Buchanan (W. W. Norton& Company, June 2003) ISBN 0-393-32442-7

Six Degrees: The Science of a Connected Age, Duncan J. Watts (W. W. Norton & Company,February 17, 2004) ISBN 0-393-32542-3

netwiki (http://netwiki.amath.unc.edu/) Scientific wiki dedicated to network theoryNew Network Theory (http://www.networkcultures.org/networktheory/) International Conference on'New Network Theory'

Network Workbench (http://nwb.slis.indiana.edu/) : A Large-Scale Network Analysis, Modeling andVisualization Toolkit

Network analysis of computer networks (http://www.orgnet.com/SocialLifeOfRouters.pdf)Network analysis of organizational networks (http://www.orgnet.com/OrgNetMap.pdf)

Network analysis of terrorist networks(http://firstmonday.org/htbin/cgiwrap/bin/ojs/index.php/fm/article/view/941/863)Network analysis of a disease outbreak (http://www.orgnet.com/AJPH2007.pdf)

Link Analysis: An Information Science Approach (http://linkanalysis.wlv.ac.uk/) (book)Connected: The Power of Six Degrees (http://gephi.org/2008/how-kevin-bacon-cured-cancer/)

(documentary)Influential Spreaders in Networks (http://havlin.biu.ac.il/Publications.php?

keyword=Identification+of+influential+spreaders+in+complex+networks++&year=*&match=all) , M.Kitsak, L. K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H. E. Stanley, H.A. Makse, Nature Physics 6,888 (2010)

A short course on complex networks (http://havlin.biu.ac.il/course4.php)A course on complex network analysis by Albert-László Barabási

(http://barabasilab.neu.edu/courses/phys5116/)

External links

Network Science Center at the U.S. Military Academy at West Point, NY

(http://www.westpoint.edu/nsc/SitePages/Home.aspx)http://press.princeton.edu/titles/8114.html

http://www.cra.org/ccc/NSE.ppt.pdfhttp://www.ifr.ac.uk/netsci08/GNET (http://www.fis.ua.pt/grupoteorico/gteorico.htm) — Group of Complex Systems & Random

Networkshttp://www.netsci09.net/

Cyberinfrastructure (http://cns.slis.indiana.edu/cyber.html)Prof. Nicholas A Christakis' introduction to network science in Prospect magazine

(http://www.prospectmagazine.co.uk/2010/02/let%E2%80%99s-all-be-friends/)Video Lectures on complex networks (http://havlin.biu.ac.il/videos1.php) by Prof. Shlomo Havlin

Notes

24/02/13 Network science - Wikipedia, the free encyclopedia

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1. ^ Committee on Network Science for Future Army Applications (2006). Network Science(http://www.nap.edu/catalog/11516.html) . National Research Council. ISBN 0309653886.http://www.nap.edu/catalog/11516.html.

2. ^ Albert, Réka; A.-L. Barabási (2002). "Statistical mechanics of complex networks"(http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/StatisticalMecha

nics_Rev%20of%20Modern%20Physics%2074,%2047%20(2002).pdf) . Reviews of Modern Physics 74: 47–97. arXiv:cond-mat/0106096 (http://arxiv.org/abs/cond-mat/0106096) . Bibcode 2002RvMP...74...47A(http://adsabs.harvard.edu/abs/2002RvMP...74...47A) . doi:10.1103/RevModPhys.74.47(http://dx.doi.org/10.1103%2FRevModPhys.74.47) .http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/StatisticalMechanics_Rev%20of%20Modern%20Physics%2074,%2047%20(2002).pdf.

3. ^ Albert-László Barabási & Réka Albert (October 1999). "Emergence of scaling in random networks"(http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/EmergenceRand

om_Science%20286,%20509-512%20(1999).pdf) . Science 286 (5439): 509–512. arXiv:cond-mat/9910332(http://arxiv.org/abs/cond-mat/9910332) . Bibcode 1999Sci...286..509B(http://adsabs.harvard.edu/abs/1999Sci...286..509B) . doi:10.1126/science.286.5439.509(http://dx.doi.org/10.1126%2Fscience.286.5439.509) . PMID 10521342(//www.ncbi.nlm.nih.gov/pubmed/10521342) .http://www.nd.edu/~networks/Publication%20Categories/03%20Journal%20Articles/Physics/EmergenceRandom_Science%20286,%20509-512%20(1999).pdf.

4. ^ R. Cohen, S. Havlin (2003). "Scale-free networks are ultrasmall" (http://havlin.biu.ac.il/Publications.php?

keyword=Scale-free+networks+are+ultrasmall&year=*&match=all) . Phys. Rev. Lett 90: 058701.http://havlin.biu.ac.il/Publications.php?keyword=Scale-free+networks+are+ultrasmall&year=*&match=all.

5. ^ Wasserman, Stanley and Katherine Faust. 1994. Social Network Analysis: Methods and Applications.Cambridge: Cambridge University Press.

6. ^ Newman, M.E.J. Networks: An Introduction. Oxford University Press. 2010

7. ^ "Toward a Complex Adaptive Intelligence Community The Wiki and the Blog"(https://www.cia.gov/library/center-for-the-study-of-intelligence/csi-publications/csi-studies/studies/vol49no3/html_files/Wik_and_%20Blog_7.htm) . D. Calvin Andrus. cia.gov.https://www.cia.gov/library/center-for-the-study-of-intelligence/csi-publications/csi-studies/studies/vol49no3/html_files/Wik_and_%20Blog_7.htm. Retrieved 25 August 2012.

8. ^ Network analysis of terrorist networks(http://firstmonday.org/htbin/cgiwrap/bin/ojs/index.php/fm/article/view/941/863)

9. ^ R. Cohen, S. Havlin (2010). Complex Networks: Structure, Robustness and Function(http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php) . Cambridge University Press.http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php.

10. ^ A. Bunde, S. Havlin (1996). Fractals and Disordered Systems(http://havlin.biu.ac.il/Shlomo%20Havlin%20books_fds.php) . Springer.http://havlin.biu.ac.il/Shlomo%20Havlin%20books_fds.php.

11. ^ Newman, M., Barabási, A.-L., Watts, D.J. [eds.] (2006) The Structure and Dynamics of Networks.Princeton, N.J.: Princeton University Press.

12. ^ S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, S. Havlin (2010). "Catastrophic cascade of failures ininterdependent networks" (http://havlin.biu.ac.il/Publications.php?

keyword=Catastrophic+cascade+of+failures+in+interdependent+networks&year=*&match=all) . Nature 464(7291): 1025–28. arXiv:0907.1182 (http://arxiv.org/abs/0907.1182) . Bibcode 2010Natur.464.1025B(http://adsabs.harvard.edu/abs/2010Natur.464.1025B) . doi:10.1038/nature08932(http://dx.doi.org/10.1038%2Fnature08932) . http://havlin.biu.ac.il/Publications.php?keyword=Catastrophic+cascade+of+failures+in+interdependent+networks&year=*&match=all.

13. ^ Jianxi Gao, Sergey V. Buldyrev3, Shlomo Havlin4, and H. Eugene Stanley (2011). "Robustness of a Networkof Networks" (http://havlin.biu.ac.il/Publications.php?keyword=Robustness+of+a+Tree-

like+Network+of+Interdependent+Networks&year=*&match=all) . Phys. Rev. Lett 107: 195701.arXiv:1010.5829 (http://arxiv.org/abs/1010.5829) . Bibcode 2011PhRvL.107s5701G(http://adsabs.harvard.edu/abs/2011PhRvL.107s5701G) . doi:10.1103/PhysRevLett.107.195701(http://dx.doi.org/10.1103%2FPhysRevLett.107.195701) . http://havlin.biu.ac.il/Publications.php?keyword=Robustness+of+a+Tree-like+Network+of+Interdependent+Networks&year=*&match=all.

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14. ^ [1](http://www.constructal.org/en/art/Phil.%20Trans.%20R.%20Soc.%20B%20%282010%29%20365,%201335%961347.pdf) Bejan A., Lorente S., The Constructal Law of Design and Evolution in Nature. PhilosophicalTransactions of the Royal Society B, Biological Science, Vol. 365, 2010, pp. 1335-1347.

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