How can graph theory be used in social networks?

7/29/2019 How can graph theory be used in social networks?

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Topic: Graph Theory in Social Network Analysis

Research question: How can graph theory be used to better understand social

networks?

by Miles Peyton

IB Mathematics SL

Red Hook High SchoolMay 2013

Candidate 001327 078

Advisor: Mr. Nick Ascienzo

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Word Count: 3723

Contents

Abstract...........................................................................................................................3

Introduction: History & Clarification of Terminology

Graph Theory in Social Network Analysis...................................................5

ABrief History of Social Network Analysis.................................................6

Definitions..........................................................................................................7

Applications of Graph Theory in Social Network Analysis

Levels of Analysis.............................................................................................9

Centrality and Density ..................................................................................11

Identifying Subgroups: Cliques, n-Cliques, and Cohesiveness ..............14

Structural Equivalence .................................................................................20

Ramseys Theorem ........................................................................................24

A Case Study...............................................................................................................27

Conclusion ..................................................................................................................30

Works Cited................................................................................................................32

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Abstract

Social network analysis refers to a methodical analysis of social networks. (Reis, 4)

In its present condition, it is a somewhat disjointed field of knowledge, being comprised

of distinct (but related) analysis techniques. To date, no cohesive theories of social

network analysis have been developed. In lieu of indigenous theories, the field has

imported a number of theories from related areas of study. Particularly, it has drawn

heavily from the mathematical field of graph theory. (Kilduff, 38) This essay attempts to

illustrate some of the ways in which graph theory is an essential component of social

network analysis. Thus, the question: How can graph theory be used to better

understand social networks?

Several quantitative techniques are examined, most of which belong to an existing

canon of social network analysis approaches. In order, these topics are: levels of analysis,

centrality, density, cliques, n-cliques, cohesiveness, structural equivalence, and

Ramseys theorem. All are common in that they build on graph theory concepts,

therefore addressing the stated research question.

In practice, there is not one single approach to the rather daunting task of

analyzing social networks. Instead, the way in which one confronts an analysis varies

with the constraints of the experiment or study being conducted, and with the goals of

the researcher in conducting the study. This essay proposes a quantitative tool kit of

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sorts, that is, quantitative methods and concepts that are universally applicable to social

network analysis problems. This essay does not attempt to provide a comprehensive

overview of either social network analysis or the graph theory from which the former

inherits. It does, however, provide limited insight into the issues that a social network

analyst using graph theory might be concerned with.

Word count: 282

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Word count: 3723

Introduction: History & Clarification of Terminology

Graph Theory in Social Network Analysis

In popular culture, social network has come to be synonymous with social

networking service. This misnomer is further complicated by the fact that online social

networking services do indeed deal with social networks, as they are understood in the

context of social network analysis.

As their name suggests, social networking services facilitate the cultivation of

social networks: those structures that represent people and the connections between

them. (Scott, 8) Just as one can represent a 2-dimensional figure either as a series of

listed coordinates, or graphically on a cartesian plane, social networks can be

represented either in tabular form, or in graph form. (Scott 49)

Graphs represent an attractive alternative to the sociomatrix, that is, a social

network in matrix form. While the latter is useful for representing large sets of analyses,

it lacks the visual immediacy of graphs. How can graph theory be used to better

understand social networks?

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The emphasis on relationships in social network analysis, as opposed to isolated

properties, is highly profitable in the context of sociology. The former field, and the

graph theory that it makes use of, prioritizes the relations between entities. It is

assumed that social structure is more often determined by the connections between

entities than it is by the individual properties of entities. This preference is supported by

a study that categorized elite families in 15th-century Florence, who supported either

the Medici or oligarchic political factions. As the study showed, the marital and economic

relations between families more closely aligned with the familys political leanings than

did individual status attributes. (Scott 4)

A Brief History of Social Network Analysis

The Swiss mathematician Leonhard Euler laid the initial foundation for modern

social network analysis in earnest. (It must be pointed out that ancient Greek scholars

had discussed network analysis ideas well before the eighteenth century.) (History of

Social Network Analysis) In the renowned analysis of the Seven Bridges of Knigsberg

problem, Euler developed a mathematical notation of points and lines, from which he

derived proofs. Eulers formalization of paths across bridges was an important precursor

to modern graph theory. (Konigsberg Bridge Problem)

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Fig. 1

In the 1930s, ideas from diverse fields such as psychology, anthropology, and

mathematics coalesced into the field now known as social network analysis. (Scott 2)

While earlier sociology paradigms used cultural ideas to explain social patterns, social

network analysis instead placed attention on patterns of interaction and interconnection.

Similarly to Euler, pioneering social network theorists adopted a terminology of points

and lines to represent the webs of social structure. In the 1950s, researchers in the social

psychology specialism group dynamics began to use systematic mathematical arguments

to analyze and interpret group structure. Those researchers embraced the mathematical

approach known as graph theory, which enabled them to operationalize concepts about

social networks, like the centrality of individuals. (Bloomsbury Academic)

Definitions

Before proceeding in the investigation, it is necessary to clarify several graph

theory and social network analysis terms.

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The fundamental elements of any social network are actors and relations.

Actors may represent individuals persons or groups. An individual actor may be a

child in a daycare, or a group of construction workers. In graph theory, actors are

represented asvertices, or points on a graph. (Scott 45)

Relations are connections between actors. The relation between two actors in

conversation is conversing. In graph theory, relations are represented as lines between

vertices, commonly called edges. Relations can either bedirected or nondirected.

Where mutuality occurs, as in the example of two conversing persons, the relation is

considered nondirected. By contrast, the relation between a teacher and a student is

directed because the teacher teaches the student, and not the inverse. A directed graph

is known as a digraph. (Scott 51)

The degree of a vertex is the number of points to which a vertex is adjacent.

The degree of a vertex is otherwise known as itscentrality. In a social network,

centrality correlates directly with actor prestige. (Scott 62)

The order of a graph is the number of vertices in the graph.

A graph is connected when there exists a path from an arbitrary point to any

other arbitrary point on the graph. (Graph Theory Glossary)

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Asociomatrix is a tabular representation of a social network. (Scott 49)

Acomplete graph is an undirected graph in which every pair of vertices is

connected by an edge. That is to say, a connection exists between nodes where a

connection is possible. (Graph Theory Glossary)

Within a large enough graph, cliques will likely form. A clique is a complete

subgraph (a graph within a graph) with three or more constituent vertices. Cliques are

distinct from the larger network in that no other node in the network has a connection to

every node in the clique. If that were the case, then that node would be included in the

clique. (Scott 72)

Applications of Graph Theory in Social Network Analysis

Levels of Analysis

Network analysts have several options when beginning an analysis of structures

in a dataset. They choose out of four conceptually distinct levels of analysis; the

egocentric level, the dyadic level, the triadic level, and the complete level.

An egocentric network considers one actor (the ego), and the other actors with

which the ego has direct relations with. One would use an egocentric network to examine

the personal network of a middle school teacher: the co workers, family members, and

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friends she interacts with. The next level, the dyadic network, considers pairs of actors.

An analysis of a married couple a dyadic network might seek to explain the effect of

contrasting economic backgrounds on marital strength. Put in other words, the study

would seek the variation in dyadic relations as a function of pair characteristics. At the

third level, the triadic level, one searches for triadic structures within a survey of actors.

Triadic structures hold interesting transitive implications. For instance, if actor A

is friends with actor B, and B is friends with actor C, what is the likelihood that actor A is

friends with actor C? (It is worth noting that triadic structures are the simplest cases of

cliques, which will be elaborated on later.) Lastly, the complete network analysis takes

into consideration every possible relation in a set of actors.

Each level of analysis provides a distinct perspective. Suppose one initiates a

study of the 2009 global recession. On one hand, researchers may gain invaluable

insights from an ego network analysis, which is in this case the effect of the collapse of

the US housing market on the economies of other nations. But there is undoubtedly

value in examining relations on the dyadic, triadic, and complete levels. Analyzing the

interactions between various nations yields a particular class of insights, as does an

analysis of the interactions between citizens. The point being that phenomena may be

obvious on one level of analysis, and equivocal on another. (Scott 12)

Centrality and Density

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We now consider a network of friends, depicted on the following graph:

Fig. 2

Each connection represents a friendship between actors. Some actors contribute

more to the health of the network than others. The measure of an actors connectedness

is centrality, which is calculated as follows, where is the centrality of actor i, and g(N)C i

is the number of actors in the network. evaluates to 1 if a connection exists betweenxij

actor iandj, and to 0 if no connection exists. (Scott 63)

(N) (i = )C i = g

j = 1xij / j

Note that the summation excludes the actor in question, hence the i = ).( / j

We can calculate the centrality of each actor using the equation above.

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Sample calculation for Micah:

= 1 + 1 + 0 + 0(N)C i

= 2(N)C i

Actor Centrality

Connor 2

Erica 3Micah 2

Sandy 4

Devon 1

As Sandy is the actor with the most connections, her centrality is the highest. She

is therefore the most prestigious actor in the network. Sandy is what is known as a

cutpoint, insofar as the removal of her connections would disconnect the graph into

several constituent graphs. (Scott 49)

The removal of Sandys connections:

Fig. 3

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Recall that a connected graph contains a path from any point to any other point in

the graph. The graph, now disconnected via the cutpoint, cripples the communicative

ability of an actor like Devon.

While centrality quantifies connectedness of any one actor, how does one evaluate

the connectedness of the graph as a whole?

Density provides a concrete measure of a graphs connectedness. It is the ratio of

dyadic ties (ties between two actors) divided by the maximum possible dyadic ties,

whereL is the number of dyadic ties present, and is the maximum possible dyadicC2N

ties for a graph of N nodes. (Scott 53)

D = LC2N

In fig. 2, the graph density is calculated as follows:

= = .6 or 60%6C256

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In fig. 3, the graph density is:

= = .2 or 20%2C252

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The drop in density after the cutpoints ties were severed illustrates the

importance of locating central actors. In our case-study network, Sandys connections

accounted for 40% of the graphs density. As such, she occupies a perilous position

whereby her removal significantly compromises the connectivity of the graph.

Identifying Subgroups: Cliques, n-Cliques, and Cohesiveness

Fig. 4

It is of interest to social network analysts to identify sub-structures within a network.

Recall that a clique is (formally speaking) a complete subgraph, distinct from the

network in that no other node in the network is connected to every node in the clique.

(Scott 72) The above network contains 24 such structures.

Cliques in Fig. 4: {7,11,12,13,14}, {7, 12, 13, 14}, {11, 12, 13, 14}, {7, 11, 12, 13},

{7, 11, 13, 14}, {7, 11, 12, 14}, {1,2,3,4}, {1,2,4}, {2,3,4}, {1,2,3}, {1,3,4}, {5, 6, 10}, {6,7,

11}, {7,8,9}, {7,13,14}, {7,11,14}, {11,12,14}, {11, 12, 13}, {12,13,14}, {7, 12, 14}, {7, 12,

13}, {11,13,14}, {7, 11, 12}, {7,11,13}

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Infig. 4, many vertices belong to more than one clique. This alludes to the

tendency for social groups to have overlapping members.

The majority of the cliques in fig. 4 are triadic, meaning that they are comprised

of three connected actors. Of these triangle shaped cliques, 14 out of 17 (~ 82%) exist

within larger cliques. In fig. 4, the two larger cliques are {7, 11, 12, 13, 14} and {1,2,3,4}.

Fig. 5

The above figure shows the subgraph {7, 11, 12, 13, 14}, otherwise denoted byK5

a complete graph of 5 vertices. 10 three-node cliques exist within this subgraph. The

number of cliques within a clique by definition a complete subgraph can be calculated

by , whereJis the order of the sub-clique, andNis the order of the larger clique.CJN

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Cliques Within K5

J CJ5 Names of cliques

3 10 {7,13,14}, {7,11,14},

{11,12,14}, {11, 12, 13},{12,13,14}, {7, 12, 14}, {7,12, 13}, {11,13,14}, {7, 11,12}, {7,11,13}

4 5 {7, 12, 13, 14}, {11, 12, 13,14}, {7, 11, 12, 13}, {7, 11,13, 14}, {7, 11, 12, 14}

Cliques have implications in Ramseys theorem, which will be touched upon later.

In the case that one is dealt real world sociological data, the complete graph

requirement generally proves too stringent. Where one wishes to identify looser

groupings, that is, connected subgraphs that arent complete, the n-clique approach is

preferable.

The n-clique approach is concerned with distance, n,between any dyad in the

graph. A 2-clique requires that every node be at maximum two units of path length

away from every other node in the subgraph. A 3-clique requires that every node be at

maximum three units of path length away from every other node in the subgraph.

Notice thatfig. 5is a 1-clique, according to the n-clique definition. (Scott 74)

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Fig. 6

In order to classify the above n-clique, a table has been generated which lists the

geodesic distance (minimum distance) between all possible pairs of nodes .5C26 = 1

Pair name Geodesic distance between nodes

to 21 1

to 31 1

to 41 1

to 51 2

to 61 2

to 32 2

to 42 2

to 52 1

to 62 2

to 43 1

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to 53 2

to 63 2

to 54 1

to 64 1

to 65 1

Since in no pair does the geodesic distance exceed 2, fig. 6 shows a 2-clique.

A second subgraph of order 6 is shown below.

Fig. 7

Pair name Geodesic distance between nodes

to 21 1

to 31 3

to 41 2

to 51 2

to 61 3

to 32 2

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to 42 1

to 52 1

to 62 2

to 43 3

to 53 1

to 63 2

to 54 2

to 64 3

to 65 1

As shown in the table above, the maximum geodesic distance between nodes in

fig. 7is 3. So,fig. 7is a 3-clique.

fig. 8

Fig. 8 shows three graphs of order six from left to right, a 1-clique, 2-clique, and

3-clique. As the nvalue increases, the graph becomes less connected.Fig. 8 is meant to

illustrate the property of cohesion within subgroups, which is relatively intuitive with the

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graph representation. An n-clique that has a relatively high n is less cohesive than an

n-clique with a relatively lown. Note that a cohesive graph will necessarily be more

dense than an incohesive graph. (Scott 72)

Identifying subgroups in graphs cliques and n-cliques is of interest to social

scientists who wish to understand the way in which smaller structures form within a

larger network. Furthermore, classifying these smaller groups according to their

cohesiveness speaks to the degree to which actors trust each other, and to the amount of

social capital, or collective value, present in the group.

Structural Equivalence

While cliques naturally extend from ideas of cooperation within groups, structural

equivalence, generally speaking, refers to competition between actors in a network.

Actors that are structurally equivalent have identical sets of connections to other nodes.

In precise terms, node iandjare structurally equivalent if, for all nodes in the network

k, node ihas a tie with k if and only ifjhas a tie with k, andjhas a tie with k if and only if

ihas a tie with k.

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fig. 9

While the above example is perhaps facile, it serves to illustrate an elementary

case of structural equivalence. The digraph represents a school classroom, where every

student has identical connections with their teacher. Each student is structurally

equivalent vis-a-vis the equivalency of all student-teacher relationships. The structural

equivalence present in the above graph could indicate competition among students for

the limited resources and attention of the teacher.

But as with cliques, structural equivalence is better understood as a phenomenon

with a range of manifestations, as opposed to one that is more rigidly defined. In a real

world situation, two actors are more likely to be approximately structurally equivalent

than they are to be precisely structurally equivalent, as the formal definition prescribes.

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(Scott 76)

fig. 10

Sociomatrix Representation of Fig. 10

1 2 3 4 5 6 7 8

1 0 0 0 0 0 0 0 1

2 0 0 0 1 0 0 0 0

3 0 0 0 0 0 0 0 0

4 0 0 1 0 0 0 0 0

5 0 0 0 1 0 0 0 1

6 0 0 0 0 0 0 0 0

7 0 0 0 0 0 0 0 1

8 0 0 1 0 0 1 0 0

In a digraph, the degree to which two nodes are structurally equivalent can be

measured using a Euclidean distance formula. k iterates through all possible vertices,

excluding those being measured for structural equivalence. The first subscript refers to a

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row in the sociomatrix, and the second subscript refers to a column. (Scott 77)

(i = = )dij

=

[(x x ) ]

g

k=1 ik

jk

2 + (x x )ki

kj

2 / j / k

The structural equivalence of two points and are inversely proportional. Twodij

nodes that are structurally equivalent yield a value of 0, and as increases,dij dij

structural equivalence decreases .

For the above graph, the structural equivalence of nodes 4 and 8 (highlighted in

orange) is calculated as follows:

dij = [(0 ) ] [(0 ) ] [(1 ) ] [(0 ) ] 0 2 + (0 ) 1 2 + 0 2 + (1 ) 0 2 + 1 2 + (0 ) 0 2 + 0 2 + (1 ) 1 2

(0 ) [(0 ) ]+ [ 1 2 + (0 ) 0 2 + 0 2 + (0 ) 1 2

dij = 1 1 0 0 1 1+ + + + +

(reject )dij = 2 2

The distance formula measures the dissimilarities in the actors structural

relations. In this case, the formula yielded a distance greater than zero. Therefore, nodes

4 and 8 onfig. 10 are not structurally equivalent. That being said, other nodes on the

graph are more structurally equivalent than nodes 4 and 8, without being precisely

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structurally equivalent.

To illustrate this, the structural equivalence of nodes 1 and 3 is calculated below:

dij = [(0 ) ] [(0 ) ] [(0 ) ] 0 2 + (0 ) 0 2 + 0 2 + (1 ) 0 2 + 0 2 + (0 ) 0 2

[(0 ) ] [(0 ) ] [(0 ) ]+ 0 2 + (0 ) 0 2 + 0 2 + (0 ) 0 2 + 1 2 + (0 ) 1 2

dij = 2 1.414

Ramseys Theorem

Since social networks graphs are analogous to graphs in graph theory, most ideas

from graph theory have interesting sociological implications. One such idea was proposed

by the mathematician F.P. Ramsey. Broadly, Ramseys theorem makes statements

about the existence of complete subgraphs within a graph of a given size. (Ramsey

Theory)

Ramseys theorem states:

For each pair of positive integers k and lthere exists an integerR(k, l) such that

any graph withR(k, l) nodes contains a clique with at least k nodes or an independent

set with at least lnodes. [6] Stated differently,R(k, l), the Ramsey number, is the

minimum number of vertices a graph must have in order to contain either a clique of

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order k, or lunconnected vertices.

The party problem is Ramseys theorem is applied to a common social situation.

The problem asks the minimum number of guests,R(k, l) , that must be invited to a

party so that either at least k guests know each other, or at least lguests do not know

each other. In graph theory terms, to satisfy the party problem, either a clique of order

k, or an independent set oflverticesexists in the graph of orderR(k, l). (Ramseys

Theorem)

While a proof of Ramseys theorem is beyond the scope of this paper, the case of

R(3, 3) = 6 will be shown.

In the following graph, let blue edges represent friendships, and black edges

represent the relation between strangers.

fig. 11

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InR(3, 3),both k and lare 3, meaning that either a clique of order 3 or an

independent set of of 3 vertices must exist. Onfig. 11, therefore, the presence of a black

or blue triadic structure confirms the conclusion of Ramseys theorem, thatR(3, 3) = 6.

Indeed, by inspection, 7 black triangles exist: {1, 5, 6}, {2, 5, 6}, {1, 4, 5}, {2, 3, 5}, {3, 4,

5}, {4, 5, 6}, and {1, 4, 6}. Note that no blue triangles need appear on the graph, as the

condition of Ramseys theorem states that either a clique or an independent set must

exist, with the order prescribed byk and lrespectively.

In terms of social network analysis, Ramseys theorem has interesting sociological

implications. As shown above, in any group of six people, at least one or more groups of

three will be acquaintances, or one or more groups of three will be unacquainted. This is

significant because in either case, there is organization within the graph. Ramseys

numbers allow us to make specific predictions about the substructures of a graph of

arbitrary order.

Although the exact value of many Ramsey numbers is unknown, those that have

been found by mathematicians can provide insight into the dynamics of a

correspondingly sized group, represented in graph form. As proven by Greenwood and

Gleason in 1955, for example, any graph of 18 actors has at least one clique or order 4, or

one independent set of 4 vertices. (Greenwood 7) While previously described techniques

for social network analysis have relied on information about specific instances of graphs,

Ramseys theorem makes generalizations about graphs of a certain size. One gains

insights from known Ramsey numbers by applying inductive reasoning, moving from the

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general to the specific.

A Case Study

To summarize the previous sections, and to demonstrate how one might apply a

combination of the aforementioned techniques to a social network analysis, a case study

is presented.

fig. 12

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fig. 13

Levels of Analysis:

Fig. 12 shows a network at the egocentric level of analysis. In this case, the ego is

the actor named Nesbit, highlighted in orange. This study considers Nesbit and his

immediate circle of friends.

Cliques and Ramseys Theorem:

By virtue of the fact thatfig. 12 is a graph of order 6, Ramseys Theorem can be

used to make predictions about its relational content. As shown in the previous section

about Ramseys Theorem, 6 is the Ramsey number forR(3, 3). This means that in any

graph of order 6, there will be either a clique of order 3 or an independent set of 3

vertices. Currently, the network abides by these rules.Fig. 12 shows 3 order cliques, and

fig. 13,which superimposes the stranger relations on the previous graph, shows

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independent sets of order 3 as well. The insight to be gained from the application of

Ramseys Theorem on this network is that at any point in the future, either 3 actors out

of the set of Nesbit and his friends will remain friends, or 3 actors out of the set of Nesbit

and his friends will disband.

Centrality

In that the graph shows Nesbits egocentric network, it is not surprising that

Nesbit is the most central actor in the network. But note that the ego is not always the

most central actor; an egocentric analysis of a clique would yield actors with identical

centralities.

Actor Centrality

Nesbit 5

Kim 1

Drake 2

Lonnie 4

Norman 2

Kaitlyn 2

Density

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= = .5333 or about 53%8C268

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Nesbits egocentric network is just slightly more than 50% dense, indicating that

there is potential for the network to become significantly more connected, or cohesive.

By comparing this networks density to that of others, one might judge the relative

health of the network of Nesbit and his cohorts.

Structural Equivalence:

Let the subscript irepresent the actor Drake, and the subscriptjrepresent the

actor Norman.

dij = [(0 ) ] [(0 ) ] [(0 ) ] 0 2 + (0 ) 0 2 + 0 2 + (0 ) 0 2 + 0 2 + (0 ) 0 2

[(0 ) ] [(0 ) ] [(0 ) ]+ 0 2 + (0 ) 0 2 + 0 2 + (0 ) 0 2 + 0 2 + (0 ) 0 2

0dij =

Drake and Norman are structurally equivalent, because the Euclidean distance

between their shared connections is 0. We might conjecture that Drake and Norman

actively compete for the attention of their mutual connections, namely Lonnie, and the

ego Nesbit.

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Conclusion

Graph theorys emphasis on relationships is highly profitable in the context of

sociology. Social network analysis emphasizes the relationships between actors, as

represented by edges on a graph. In a given research task, one chooses to operate at a

certain level of analysis, which may entail a focus on the egocentric level, or on the dyadic

level.

In this essay, several analysis techniques were examined. Two of these measures

centrality and structural equivalence focused on the connections between individual

actors. Centrality refers to the prestige of an individual, while structural equivalence

alludes to competition for resources. Central actors that have the potential to break a

graph into several subgraphs are known as cutpoints. Removal of a cutpoint reduces the

graphs average connections, hence making it less dense overall.

Other techniques attempted to locate substructures within a graph. Cliques

identified those subgroups that were completely connected, or complete. N-cliques

relaxed the requirement somewhat, prescribing looser criteria for grouping. Although

cliques have mathematical definitions, they are also social structures, as understood by

the conversational usage of the word clique. Ramseys theorem made statements

regarding substructures as well: It predicted the size of a graph that contains either a

clique or an independent set of a given order.

In applying the above techniques to an actual network study, one gains deeper insight

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into the subtleties of social networks. One begins to form a coherent understanding of

networks in terms of the connectedness of actors, and dominant groupings.

Works Cited

"Konigsberg Bridge Problem." 2010. 3 Jan. 2013

Fox, Jacob. "Ramsey Theory." N.p., n.d. Web.

"Graph Theory Glossary." Graph Theory Glossary. N.p., n.d. Web. 1 Jan. 2013.

Greenwood, R. E. and Gleason, A. M. "Combinatorial Relations and Chromatic Graphs." Canad. J. Math.7, 1955.

Kilduff, Martin, and Wenpin Tsai. Social Networks and Organizations. London: SAGE, 2003. Print.

"Notes on the History of Social Network Analysis." History of Social Network Analysis. N.p., n.d. Web.10 Jan. 2013.

"Ramsey's Theorem." Wolfram MathWorld. N.p., n.d. Web.

Reis, Pinheiro Carlos Andre. Social Network Analysis in Telecommunications. Hoboken, NJ: Wiley,2011. Print.

Scott, John. "History of Social Network Analysis : What Is Social Network Analysis? : BloomsburyAcademic."History of Social Network Analysis : What Is Social Network Analysis? : Bloomsbury

Academic. N.p., n.d. Web. 1 Jan. 2013.

Scott, John. Social network analysis: A handbook. Sage Publications Limited, 2000.

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Weisstein, Eric W. "Ramsey Number." From MathWorld--A Wolfram WebResource.http://mathworld.wolfram.com/RamseyNumber.html
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