Ebook Computational network science An algorithmic approach Part 1

(BQ) Computational network science seeks to unify the methods used to analyze these diverse fields. This book provides an introduction to the field of network science and provides the groundwork for a computational, algorithmbased approach to network and system analysis in a new and important way.

Computational Network Science

An Algorithmic Approach

Henry Hexmoor

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The days of the need for gurus and extensive libraries are behind us The Internet provides ready and rapid access to knowledge for all This book offers necessary and sufficient descriptions of salient knowledge that have been tested in traditional classrooms The book weaves founda-tions together from disparate disciplines including mathematical sociol-ogy, economics, game theory, political science, and biological networks.Network science is a new discipline that explores phenomena com-mon to connected populations across the natural and man-made world From animals to commodity trades, networks provide relationships among individuals and groups Analysis and leveraging connections provide insights and tools for persuasion Studies in this area have large-

ly focused on opinion attributes The impetus for this book is a need to examine computational processes for automating tedious analyses and usage of network information for online migration Once online, net-work awareness will contribute to improved public safety and superior services for all

A collection of foundational notions for economic and social works is available in Jackson (2008) A mathematical treatment of generic networks is present in Easly and Kleinberg (2010) A comple-mentary gap filled by this book is an algorithmic approach I provide a fast-paced introduction to the state of the art in network science Refer-ences are offered to seminal and contemporary developments The book uses mathematical cogency and contemporary computational insights

net-It also calls to arm further research on open problems

The reader will find a broad treatment of network science and review

of key recent phenomena Senior undergraduates and professional ple in computational disciplines will find sufficient methodologies and processes for implementation and experimentation This book can also

peo-be used as a teaching material for courses on social media and network analysis, computational social networks, and network theory and ap-plications Our coverage of social network analysis is limited and details are available in Golbeck (2013) and Borgatti et al (2013)

Whereas a teacher is a tour guide to the subject matter, this book is

a reference manual Chapters in each part are related and they progress

in maturity Chapters are semi-independent and a course instructor may choose any order that meets the course objectives Exercises at the end

of each chapter are students’ hands-on projects that are designed for covering learning activities during a semester Some code is provided

in appendices for prototyping and learning purposes only We do not provide a how-to guide to mainstream social media or codebook for application development that is available elsewhere

Henry HexmoorCarbondale, IL

2014

REFERENCES

Borgatti, S., Everett, M., Johnson, J., 2013 Analyzing Social Networks SAGE Publications Easly, D., Kleinberg, J., 2010 Networks, Crowds, and Markets Cambridge University Press Golbeck, J., 2013 Analyzing the Social Web Morgan Kaufmann Publications

Jackson, M., 2008 Social and Economic Networks Princeton University Press.

CHAPTER 1

Ubiquity of Networks

1.1 INTRODUCTION

Broadly speaking, a network is a collection of individuals (i.e., nodes)

where there are implicit or explicit relationships among individuals in

a group The relationships may be strictly physical as in some sort of physical formation (e.g., pixels of a digital image or cars on the road),

or they may be conceptual such as friendship or some similarity among pairs or within a pair In an implicit network, individuals are unaware

of their relationships, whereas in an explicit network, individuals are familiar with at least their local neighbors In certain implicit networks

called affinity networks, there is a potential for explicit connections from

relationships that account for projected connection such as homophilly (i.e., similarity) (McPherson et  al.,  2001) Biological networks capture relationships among biological organisms For instance, the human brain

neurons form a large network called a connectome (Seung, 2012) An ant

society is an example of a large biological network (Moffett, 2010) There are many examples of small-scale animal networks, including preda-tors and their prey, plant diseases, and bird migration Human crowds and network organizations (e.g., government or state agencies, honey grids in bee colonies) are other examples of natural networks Modern anonymous human networks have capacities for crowd solving problems (Nielsen, 2012), where a group of independently minded individuals pos-sess a collective wisdom that is available to singletons (Reingold, 2000) Social and political networks model human relationships, where social and political relations are paramount Economic networks are models

of parties related to economic relationships such as those among buyers (and consumers), sellers (and producers), and intermediaries (i.e., trad-ers and brokers) (Jackson,  2003) Beyond natural networks, there are myriads of synthetic networks The grid of a photograph is an example

of synthetic networks Nanonetworks are attempts to network

nanoma-chines for emerging nanoscale applications (Jornet and Pierobon, 2011)

A large class of networks is a complex engineered network (CEN) that is a

man-made network, where the topology is completely neither regular nor random A CEN supports evolving functionalities Examples of CENs are the Internet, wireless networks, power grids with smart homes and cars, remote monitoring networks with satellites, global networks of tele-scopes, and networks of instruments and sensors from battlefields to hos-pitals Time requirements in CENs range from seconds in cyber-attacks

to years in greenhouse gas emissions Data and control flow in CENs must be managed over connections that could span thousands of miles

A few synthetic network categories, including CENs, are created tionally Here, we list six types:

inten-1 Social networks through networking sites and services

2 Political networks as in parliamentary cabinets and political

committees

3 Computer networks that include computers as nodes and how they communicate over local, wide area, and wireless links (e.g., sensor networks)

4 Telecommunication networks as in switches for nodes and respective routing paths

networks for data mining and marketing Individuals sharing like votes

(or retweets) are part of an affinity network (or a hashtag) in the context

of what they liked (or tweeted)

Figure 1.1 depicts a taxonomy of network types Exchange networks are those in which a quantifiable entity is exchanged among the nodes whether or not the nodes are tangible (e.g., natural gas) or intangible

(e.g., trust) Relational networks are inert and merely reflect

juxtaposi-tion of nodes All CENs are exchange networks

Once a network emerges, we can explore interactions within the work Strategic interactions involve reasoning and deciding over selec-tion of strategies They can be modeled with game theory that will be our main focus in Chapter 3

net-Network theory is a set of algorithms that codifies relationships

among network topology and outcomes, which are meaningful to work inhabitants There is a movement afoot that codifies network phe-

net-nomena under the term network science These phenet-nomena and salient

algorithms will be discussed throughout this book

An Online Social Networking Services (OSNS) creates synthetic works among people The salient incentive for using an OSNS is to gain

net-social authority (i.e., legitimacy), which is a form of net-social power and

not generally a measure of vanity Social authority in social networks

is with respect to a group and with respect to specific topics Therefore, social authority is a relative measure and not an absolute quantity In Section 1.2, we review a few popular OSNSs from a rapidly growing list (Khare, 2012) Since they provide platforms to create, to share, and/or

Fig 1.1 A network taxonomy.

to exchange information and ideas in virtual communities, an OSNS

is considered to be a medium for social media There are quantitation

schemes over social media, such as Klout, which offers user scores (i.e.,

a number between 1 and 100) Klout calls influence, which is a measure

of a user’s ability to reach one other through an OSNS This measure is valuable for marketing products online

In Section 1.3, we review a few popular online bibliographic services (OBS) that house published articles We return to generic models of net-works in Section 1.4 This is followed by a review of popular models

of synthetic network generations in Section  1.5 A fully implemented NetLogo model (i.e., code and accompanying descriptions for use) of network generation models and analysis is available in the Appendix

1.2 ONLINE SOCIAL NETWORKING SERVICES

Facebook is an OSNS that connects people, organizations, friends, and

others who work or live around together Nodes in a Facebook network can be individuals or organizations Some of these may be entirely syn-thetic without real-world humans The main Facebook tool for connec-

tions is friendship Facebook is used largely for personal and recreational

functions As such, it has filled the social gaps created by physical and chological dispersion among traditional families and friends It also serves

psy-as a medium that creates relationships that would not otherwise exist

One Facebook’s feature known as sharing allows adjustments on

spread of information (i.e., selecting an audience) Sharing is used to limit who can view posts and photos It is a three-step process: (1) in-

dicates who you are (i.e., tagging), (2) tells where you live (i.e., adding a

location to a post), and (3) manages the privacy right for where you post

(i.e., the inline audience selector) Sharing gives users control over their

information diffusion, which in turn can yield a measure of social

au-thority Another Facebook’s like feature provides a directional

relation-ship (i.e., tie, connection, and link) that lends credibility to the item and

is proportional to the credibility (i.e., authority) of the endorser

Twitter is an OSNS that facilitates broadcasts of messages (i.e.,

tweets) The main twitter tool for connections is the explicit alignments

of ideas among people (i.e., following) Twitter can be used by small or large groups to form crowd sourcing For example, in the small network,

when a family stays organized about their travel itinerary, there are parate opinions In the large network, a large social project, such as a protest, can be planned Twitter can be used to work semi-anonymously

dis-with others Twitter’s hashtag (i.e., #) is a feature for labeling a topic

Any-one may introduce or reuse a hashtag to attract attention For example,

#flight1549 added to a tweet labels the tweet to be about “flight1549.” This hashtag labeling facilitates search related to specific topics Individ-uals who use specific hashtags form an implicit network in the context

of their hashtags This feature has been used for commercial marketing and anonymous coordination over social actions The range of poten-tial uses for hashtags is enormous, and they have been adopted by other OSNSs such as Facebook On the one hand, Twitter can be used for so-cial organizations of crime or dissent On the other hand, it can be used

to predict and mitigate violations of law enforcement Since Twitter vides democratization of opinion sharing and equal access for dissemi-nation, it is seen as a social equalizer and as such it might be feared by repressive systems (e.g., government regimes) Twitter’s social authority

pro-is composed of three components: (1) the retweet rate of users’ last few hundred tweets, (2) the recentness of those tweets, and (3) a retweet-

based model trained on users’ profile data Tagging someone shows the Twitter id to more people, whereas direct messaging someone just

puts spam in their inbox, which is generally undesirable Websites, such

as Klout.com, gauge the influence you have by monitoring things, for example, how active you are and how much you have been tagged on

Twitter Twitter’s lists are a way to organize others into groups When

you click on a list, you will retrieve a stream of tweets from all the users included in that group As a rule of thumb, if you want to develop re-lationships on Twitter, you should read other tweets, retweet good con-tents, tweet good contents, and stay on top of keywords and interests that you follow The same advice applies if you want to get retweeted

Linkedin is an OSNS that provides an online forum for professional

identity management The main tool for Linkedin’s connections is to

link people, who would like to support one another (i.e., connections)

Linkedin allows people to conduct a weak form of endorsement in gards to specific skills This creates directional links from endorsers to endorsees Linkedin allows a stronger directional endorsement through

re-recommendations Endorsed individuals’ profiles gain social authority

via Linkedin’s endorsements and recommendations Of course, the

gained authority is proportional to the authority of those endorsing and recommending.

Pinterest is an OSNS that allows users to create and manage

theme-based image collections Repining in Pinterest is the feature that creates

social authority

Started in 2011, Whisper.sh is a privately owned mobile OSNS that

allows anonymous posts including photographs It allows others to like posts, which creates a network of posts as nodes and directional links Since users are anonymous, the resulting network is implicit

1.3 ONLINE BIBLIOGRAPHIC SERVICES

DBLP is a Computer Science Bibliography database website hosted at

Universität Trier in Germany It houses a large collection of published articles and offers capabilities for browsing and searching The resulting database is a network of “author” nodes connected via coauthorship Through citations, papers are nodes of a separate network of paper, as nodes and citations are the links

Google Scholar is another bibliography database website released in

2004 by google.com It creates networks of authors and papers similar

to DBLP

Microsoft Academic Research is an OBS (with a corresponding

Win-dows app) that is supported by Microsoft.com that offers a similar vice to DBLP

ser-Research Gate is an independent privately owned online site founded

in 2008 for scientists and researchers to share papers, to ask questions,

to answer questions, and to find collaborators On the one hand, it is an OBS, even though it is far smaller than its rivals On the other hand, it is

an OSNS for professionals

1.4 GENERIC NETWORK MODELS

In this section, we review four of the most popular generic network models In contrast to descriptive models in this section, Section 1.5 will offer algorithms for artificially generating networks

1.4.1 Random Networks

G(n, p) is a random graph model with n nodes where the probability of a

pair of nodes in it being linked is denoted by p (Erdős and Rényi, 1959)

When p is small, the network is sparsely connected When p is close to

1/n, the network appears fully connected When p is almost 1.0, the

con-nectivity among nodes is very high and the network is said to be a giant

component The spread of node degrees for a random graph model (i.e.,

degree distribution) appears binomial in shape A closely related model

is the random geometric graph G(n, r), where there are n nodes and the

distance between a pair of nodes in the graph is less than or equal to r

(Penrose, 2003) Contrary to mathematical models, real-world networks

exhibit a degree distribution that is unevenly distributed In the

power-law distribution, the probability that a node has a degree distribution k

(i.e., the number of connected neighbors) is determined by P(k)  ≈ k g, where

parameter g is typically constrained between 2 and 3, that is, 2 ≤ g ≤ 3

Uneven distribution stems from preferential attachment, where the

prob-ability that a new node will attach to a node i is degree /( )i ∑jdegree ( )j

A node degree refers to the node’s number of neighbors Preferential

attachment is commonly found in nature as well as man-made networks

such as an economic network (Gabaix,  2009) Random networks are

mathematically the most well-studied and well-understood models

1.4.2 Scale-Free Networks

There is a model based on preferential attachment described by

Bara-basi and Albert (1999) In this model, a new node is created at each

time step and connected to existing nodes according to the

“prefer-ential attachment” principle At a given time step, the probability p

of creating an edge between an existing node u and the new node is

p [(degree( ) 1) / (| | | |)]u E V , where V is a set of nodes and E is the

set of edges between nodes The algorithm starts with some parameters

such as the number of steps that the algorithm will iterate, the

num-ber of nodes that the graph should start with, and the numnum-ber of edges

that should be attached from the new node to preexisting nodes at each

time step The Barabasi model of network formation produces a

scale-free network, a network where the node degree distribution follows a

power-law principle Scale-free networks produce small number of

com-ponents, small-diameter, heavy-tailed distribution, and low clustering

degree(i)/∑jdegree(j)

p=[(degree(u)+1)/(|E|+|V|)]

Many types of data studied in the physical and social sciences can be approximated with a Zipf distribution (Li, 1992), which is one of the families of discrete power-law probability distributions An implication

of the Zipf law is that the most frequent word will occur approximately twice as often as the second most frequent word, which occurs twice as often as the fourth most frequent word, etc

Unlike the growth model of Barabasi, Epstein and Wang’s (2002) steady-state model uses a rewiring scheme that results in power-law distribution This model evolves an initial graph according to Markov process, while maintaining constant size and density

Epstein and Wang’s algorithm has two major steps: (1) ize a sparse graph and (2) edit Markov edges To generate the sparse

initial-graph G, they randomly add an edge between vertices with probability

× −

m n n

2  / [   ( 1)], where m is the number of edges added and n is the

number of vertices If the number of edges in G is still less than m, they start adding edges with a probability of 0.5 until the graph G has m

edges The second step is to reiterate the algorithm in Figure 1.2 r times

on G, where r is a parametric value.

2m/[n×(n−1)]

Fig 1.2 Epstein and Wang’s (2002) algorithm.

1.4.4 Game Theoretic Models

Game theoretic model of network formation focuses on reasoning over

each node’s connection with others A strategy set of an agent i is a set

of strategies to connect each node in the network, that is, S i  = {s i1 , s i2,

…, s in }, where s ij is a strategy to connect a node i to a node j An agent

incurs a cost in a connection that is a combination of a fixed cost plus a

sum of distances between the node and all other nodes in the network

For example, cost( )s ij = + Σc j d i j( , ), where c is a fixed cost and d(i, j)

is the distance between nodes i and j in the number of links The cost is

shared if both parties choose the link Otherwise, it is incurred by one

agent Synergistic strategy selection will provide utility for agents that

are linked

Each strategy will have a payoff that is utility minus the link cost The

Nash equilibrium (Carmona, 2012) is achieved with a strategy profile

(i.e., a set of links) that minimizes cost for all agents, and no agent has

incentive to deviate from it

1.5 NETWORK MODEL GENERATORS

In this section, we review three of the most popular models for

generat-ing artificial networks

1.5.1 Kleinberg’s Small-World Model

A social network is called a small-world network if, roughly speaking,

any two of people in the network can reach each other through a short

sequence of acquaintances (Kleinberg,  2001) Milgram’s basic

small-world experiment is the most famous experiment that analyzed the

small-world problem (Milgram, 1967) The purpose of the experiment

was to determine whether most pairs of people in society were linked by

short chains of acquaintances So, individuals were asked to forward a

letter to a “target” through people whom they knew on a first-name basis

Watts and Strogatz (1998) proposed a small-world network model

that incorporated the features of Milgram’s experiment Kleinberg

(2001) proposed a variant of Watts and Strogatz’s basic model that can

be described as follows One starts with a p-dimensional lattice, in which

nodes are joined only to their nearest neighbors One then adds k-directed

long-range links out of each node v, for a constant k; the end point of

cost(sij)=c+∑jd(i,j)

each link is chosen uniformly at random (Kleinberg, 2001) Kleinberg studied the model from an algorithmic perspective and showed that, with a high probability, there will be short paths connecting all pairs

of nodes and the network will have the lattice-like structure Kleinberg model does not yield a heavy-tailed degree distribution

Kleinberg (2000) showed a simple greedy algorithm that can find paths between any source and destination using only O(log )2n expected edges Kleinberg’s algorithm that will be used in this study is based on two parameters: the lattice size and the clustering exponent Each node

u has four local connections, one to each of its neighbors and in addition

one long-range connection to some node v, where v is chosen randomly according to the probability proportional to d −a , where d is the lattice distance between u and v and a is the clustering exponent.

1.5.2 Barabási and Albert’s Scale-Free Network Generator

Barabási and Albert (1999) discussed the features of the scale-free works in detail and compared them with the features of other types of networks, for example, small-world networks Scale-free networks ex-pand continuously by the addition of new vertices, and new vertices at-tach preferentially to vertices that are already well connected Most of the real networks are free-scale networks, such as WWW and citation patterns of scientific publications, and both of them follow a power-law distribution (Barabási and Albert, 1999)

net-Albert and Barabási (2002) showed a comparison between their model and other previously proposed models They state that other network models start with a fixed number of vertices that are then randomly connected or reconnected without modifying the number of vertices However, the WWW as an example will grow exponentially in time by addition of any new web page Also, other network models assume that new edges are placed randomly, that is, the probability of connecting two vertices is independent of the vertices’ degree However, most of the real networks do not behave like that They exhibit preferential attachment, that is, connecting two vertices is dependent on the vertices’ degree (Albert and Barabási, 2002)

According to Albert and Barabási (2002), a new node is created

at each time step and connected to existing nodes according to the

O(log2⁡n)

“preferential attachment” principle At a given time step, the probability

p of creating an edge between an existing node u and the new node is

[(degree( ) 1) / (| | | |)] The algorithm starts with some

param-eters, such as the number of steps that the algorithm should iterate, the

number of nodes that the graph should start with, and the number of

edges that the new node should be attached to the preexisting nodes at

each time step

The hierarchical network model (HNM) is part of the scale-free model

family and shares its main property of yielding proportionally more

hubs among the nodes than by random network generation HNMs are

heavy-tailed, have small diameter, and have high clustering

1.5.3 Epstein and Wang’s Power-Law Network Generator

Epstein and Wang (2002) have proposed a graph model called the

steady-state model that results in power law by evolving a graph according to

Markov process while maintaining constant size and density The only

difference between their model and Barabási and Albert’s model is that

their model does not require incremental growth, whereas Barabási and

Albert’s model does Epstein and Wang’s algorithm can be viewed in two

steps: (1) initialize a sparse graph and (2) edit Markov process To

gener-ate the sparse graph G, the algorithm randomly adds an edge between

vertices with the probability 2  / [   (m n × −n 1)], where m is the number of

edges added and n is the number of vertices If the number of edges in G

is still less than m, the algorithm starts adding edges with a probability

of 0.5 until the graph G has m edges Then, we reiterate the algorithmic

steps, shown in Figure 1.2, r times on G, where r is a model parameter

(Epstein and Wang, 2002)

1.6 A REAL-WORLD NETWORK

In this section, we sketch essential components of a generic,

common-place exchange network applicable to package delivery and durable

products We are keeping this model simple in order to avoid

complexi-ties of supply chain management and economic networks Let C be a set

of consumers of a commodity (e.g., received packages or appliances)

and P be a set of producers of the same commodity (e.g., package

[(degree(u)+1)/(|E|+|V|)]

2m/[n×(n−1)]

senders) A node can be both a producer and a consumer at different

times T and locations L (i.e., nodes) The set C is strictly larger than P,

and it may subsume it entirely The production rate of a producer is a

function of time and location denoted by P(t, l) Similarly, the

consump-tion rate of a consumer is a funcconsump-tion of time and locaconsump-tion denoted by

C(t, l) After production but before consumption, commodities are in

transit at the rate P − C, that is, Transit(t, l1, l2) = P(t, l1) − C(t, l2)

Loca-tions of production and consumption must be distinct, that is, l1 ≠ l2 If these locations are the same, transit is null, that is:

∀ ∈t T l l, ,1 2 ∈L P t l, ( , ) 01 ≥ ∩C t l( , ) 02 ≥ ∩ = →l1 l2 Transit( , , )t l l1 2 = ∅

The Transit(·) function specifies the flow rate among nodes of the work If we could specify the maximum flow between all pairs of nodes

net-in the network, we could discover network capacity for transit usnet-ing the

standard graph theoretic flow network algorithm, for example, the Fulkerson algorithm (Kleinberg and Tardos,  2005) Transit/flow rates incur a cost corresponding to the amount of flow that needs to be paid

Ford-by the pair of a sender and a receiver It may be beneficial to share the transmission cost with neighbors, who form game theoretic coalitions that will be discussed in Chapter 3

In many scenarios, there is a need for intermediaries to facilitate transfer of commodities from producers to consumers For simplic-

ity, we assume intermediaries to be uniform handlers, who are neither

a producer nor a consumer of commodities they handle In economic networks, handlers are traders (discussed in Chapter 9) In produc-tion line networks, intermediaries are dealers In the mail carrier net-works, intermediaries are delivery personnel In electric networks, in-termediaries are switches In computer networks, intermediaries are routers

Handling capacity of agent i is a function of time and number of

items Let handleri (t, I) return a delay time in i’s ability to handle I items

at time t Delay time of zero is on time handling Typically, there are

more handlers than items in transit A property of interest is to find optimal number of handlers for the volume of items to be handled with

no delay

⁡t⁡T,l1,l2⁡L,P(t,l1)≥0∩C(t,l2)

≥0∩l1=l2→Transit(t,l1,l2)=∅

1.7 CONCLUSIONS

Networks are abundantly around us They are man-made or naturally occur They are implicit, hidden, explicit, or articulated They might be tangible and objectively quantified, or they might be subjective and dif-ficult to quantify They all tend to change in time, which is the subject of our future chapters on network dynamics

Epstein, D., Wang, J., 2002 A steady state model for graph power laws In: Proceedings of 2nd

International Workshop on Web Dynamics World Scientific Publishing Company.

Erdős, P., Rényi, A., 1959 On random graphs Publicationes Mathematicae 6, 290–297

Gabaix, X., 2009 Power laws in economics and finances Annu Rev Econ 1, 255–294

Jackson, M., 2003 A survey of models of network formation: stability and efficiency In: Demange, G., Wooders, M (Eds.), Group Formation in Economics: Networks, Clubs, and Coalitions Cambridge University Press

Jornet, J.M., Pierobon, M., 2011 Nanonetworks: a new frontier in communications In: Communications

of the ACM Vol 54, No 11 ACM, pp 84–89.

Khare, P., 2012 Social Media Marketing eLearning Kit For Dummies Wiley

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32nd ACM Symposium on Theory of Computing ACM, pp 163–170.

Kleinberg, J., 2001 Small-world phenomena and the dynamics of information In: Proceedings of

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McPherson, M., Lovin, L.S., Cook, J., 2001 Birds of a feather: homophily in social networks Annu Rev Sociol 27, 415–444

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Moffett, M., 2010 Adventures Among Ants University of California Press

Nielsen, M., 2012 Reinventing Discovery: A New Era of Networked Science Princeton University Press

Penrose, M., 2003 Random Geometric Graphs Oxford University Press

Reingold, H., 2000 The Virtual Community MIT Press

Seung, S., 2012 Connectome: How Brain’s Wiring Makes Us Who We Are Mariner Books Watts, D., Strogatz, S., 1998 Collective dynamic of small-world networks Nature 393 (6684), 440–442

1 Using examples, describe how animal swarms are networked

2 What are the salient characteristics of biological networks (e.g., brain cells and protein chains) that differentiate them from other types of networks?

3 What will be the role of network organizations in the year 2025? Give examples

4 How can social media be used to track cultural changes in a society?

CHAPTER 2

Network Analysis

There has been a long tradition of measuring qualities for network locations from both egocentric and global perspectives This is largely addressed with quantification attempts in mathematical sociology under

the theme of social network analysis (SNA) (Wasserman and Faust, 1994;

Knoke and Yang,  2007; Golbeck,  2013; Borgatti et  al.,  2013) There are also several popular software toolkits that perform analysis and visualization of social networks (i.e., sociograms) including UCINET

and NodeXL Tom Snijders’ SIENA is a program for the statistical

analysis of network data The NSF-sponsored visualization project is Traces (Suthers,  2011), which traces out the movements, confluences, and transformations of people and ideas in online social networks.The aim of this chapter is to review a selective subset of SNA measures that complement algorithmic descriptions explained in the remainder of this book For a glossary of SNA terms, readers are recommended to consult Golbeck (2013)

We will start with egocentric (i.e., node view) measures A degree-1

network of a node is the node and its immediate neighbor nodes A

degree-1.5 network of a node is the node’s degree-1 network and its links

among immediate neighbors (Golbeck,  2013) A degree-2 network of

a node is the node’s degree-1 network and all its immediate neighbors’

connections (Golbeck,  2013) A degree-n network of a node is the

degree-1 network of the node plus all the nodes and the corresponding

links that are no more than n links away from the starting node.

A path is a chain (i.e., succession) of nodes connected by links tween pairs of nodes Two nodes are connected if and only if (i.e., iff) there is a path between them A connected component is a set of nodes with connected paths among all pairs of nodes in the set A bridge is a link that connects two isolated connected components A hub is a node with many connections Reachability is whether two nodes are connected

be-or not by way of either a direct be-or an indirect path of any length

Geodesic distance, denoted by distance ij, is the number of links in the

shortest possible path from node i to node j Diameter of a network is the largest geodesic distance in the connected network Reverse distance,

denoted by RDij, is distanceij  − (1 + Diameter) Metrics in Equations 2.1

and 2.2 are adapted from Valente and Foreman (1998):

Structural centrality measures of a node are a host of measures

re-flecting the structural properties of the links surrounding a focal node For example, degree centrality of a node is the number of edges incident

on the node Closeness centrality of a node is the average of the

short-est path lengths from the node to all other nodes in the network It is a

rather small number in small-world networks (Watts and Strogatz, 1998)

Betweenness centrality of a node is a measure of the node’s importance

(and possibly influence as discussed in Chapter 7) and is computed using the algorithm shown in Figure 2.1

Eigenvector centrality measures the centrality of neighbor nodes and

has been used as a measure of influence and power, which are discussed

later in this book (Bonacich and Lu, 2012) Bonacich developed a beta

centrality measure CBC with a parameter a used for adjusting the tance of a node’s degree versus a parameter b for adjusting the impor-

impor-tance of the neighbor’s centrality This is shown in Equation 2.3:

Eigenvector centrality of a node at time t is computed with

Equa-tion 2.4, where C(t) is the vector of node centralities, A is the adjacency

matrix, and A t is the result of iterated multiplications of A:

=

C t( ) A C t t  ( ) (2.4)

As time approaches ∞, the dominant eigenvalue g will determine the

centrality vector value with the value  

γ ×t V    1, where  

V1 is the

eigenvec-tor corresponding to the dominant eigenvalue g (Chiang, 2012).

Let us consider a degree-1.5 network of a node and measure the ratio

of the actual number of links in that network over the total number of

possible links that could exist, which yields a measure called the local

Density of a network is the ratio of the actual number of links in that

network over the total number of possible links that could exist

Cohe-sion is the minimum number of edges that has to be removed before the

network is disconnected

Let us consider a cluster that is a subset of nodes s and each node may

count the ratio r as node r is the density of its neighbors in s versus the

total number of its neighbors In the set s, the node with the minimum

Whereas centrality is a microlevel measure, centralization is a

macro-level measure, which measures variance in the distribution of

central-ity in a network We show the most generic form of centralization in

Figure 2.2

Leadership (L) is a measure of network domination, computed

us-ing Equation 2.5, where dmax is the degree of the node with the highest

C(t)∈=AtC(t)∈ gt×V1∈

V1∈

Fig 2.2 Centralization algorithm.

degree and d i is the degree of node i (Freeman,  1978; Macindoe and

n

max 1

(2.5)

Richards, 2011) using Equation 2.6:

Diversity is a measure of the number of edges in a graph that are

dis-joint End vertices of such edges are not adjacent (i.e., disjoint dipoles) Diversity is shown in Equation 2.7:

Burt’s structural holes measure gaps among connected components

and as such are another measure of diversity (Burt, 1995)

2.1 CONCLUSIONS AND FUTURE WORK

Network analysis focuses on quantification (and statistical analyses) of qualities of relative nodes’ locations as well as entire network properties SNA has long been a stable tool for mathematical sociology (Borgatti

et  al.,  2013) An active direction of interest has been intelligence analysis of human networks to understand, predict, and mitigate law enforcement as well as understand geopolitical landscapes The recent debate over surveillance and monitoring of electronic communication metadata by the National Security Agency (NSA) is indicative of this fervent interest

A second direction of interest is marketing and branding on social media The interest is to understand human propensity for influence from network connections Marketers use these propensities to craft viral dissemination of consumption patterns and manipulation of economic

activities The documentary filmmaker, Morgan Spurlock, has publicly explored branding on social media His mission is to raise public aware-ness and to inform us about the changing landscape of cultural values in the society (e.g., supersize me app).

REFERENCES

Bonacich, P., Lu, P., 2012 Introduction to Mathematical Sociology Princeton University Press Borgatti, S., Everett, M., Johnson, J., 2013 Analyzing Social Networks SAGE Publications Ltd Burt, R., 1995 Structural Holes: The Social Structure of Competition Harvard University Press Chiang, M., 2012 Networked Life: 20 Questions and Answers Cambridge University Press Freeman, L., 1978 Centrality in social networks: conceptual clarification Soc Netw 1, 215–239 Golbeck, J., 2013 Analyzing the Social Web Morgan Kaufmann

Knoke, D., Yang, S., 2007 Social Network Analysis Sage Publications

Macindoe, O., Richards, W., 2011 Comparing networks using their fine structure Int J Soc Comput Cyber-Phys Syst 1 (1), 79–97, Inderscience Publishers.

Suthers, D., 2011 Interaction, mediation, and ties: an analytic hierarchy for socio-technical systems In: Proceedings of the Hawaii International Conference on the System Sciences (HICSS-44) January 4–7, 2011, Kauai, Hawai‘i.

Valente, T., Foreman, R., 1988 Integration and radiality: measuring the extent of an individual’s connectedness and reachability in a network Soc Netw 20 (1), 89–105

Wasserman, S., Faust, K., 1994 Social Network Analysis: Methods and Applications Cambridge University Press

Watts, D., Strogatz, S., 1998 Collective dynamics of ‘small-world’ networks Nature 393 (6684), 440–442