computing efficiency of complex networks

8/13/2019 Computing Efficiency of Complex Networks

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Supercomputer Education and ResearchCenter

Indian Institute of Science

Computing Efficiency of ComplexNetworks

8 week Project Report

for

Summer Research Fellowship Program 2013

Indian Academy of Sciences

Author:Sriniketh VijayaraghavanSupercomputer Education andResearch CenterIndian Institute of ScienceBangalore 560 012

Supervisor:Prof. N Balakrishnan

Associate DirectorIndian Institute of Science

Bangalore

July 17, 2013


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Contents

1 Social Network Analysis 6

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Representing the Network . . . . . . . . . . . . . . . . . . . . . . 61.3 Mathematics of networks . . . . . . . . . . . . . . . . . . . . . . 8

2 Centrality Measures 10

2.1 Degree Centrality . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Hubs and Authorities . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Closeness Centrality . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Betweenness Centrality . . . . . . . . . . . . . . . . . . . . . . . . 122.5 PageRank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Important Metrics in the Network 14

3.1 Clustering Coefficient . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Homophily and Assortative Mixing . . . . . . . . . . . . . . . . . 15

3.3 Degree Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Efficiency in the network 18

4.1 Attacks on the Network . . . . . . . . . . . . . . . . . . . . . . . 184.2 Vertex Attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Implementation 20

5.1 The Networks used . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2 Results of the Analysis . . . . . . . . . . . . . . . . . . . . . . . . 21

5.2.1 Betweenness Centrality . . . . . . . . . . . . . . . . . . . 215.2.2 Closeness Centrality . . . . . . . . . . . . . . . . . . . . . 235.2.3 Power Laws . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.3 Computing Efficiency in case of targetted and random attacks . . 275.3.1 Efficiency in the case of the NKN . . . . . . . . . . . . . . 275.3.2 Efficiency in the case of the Email Network . . . . . . . . 305.3.3 Efficiency in Zacharys karate club . . . . . . . . . . . . . 32

5.4 Analysis of all the sample Networks . . . . . . . . . . . . . . . . 34

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List of Figures

2.1 Histogram of closeness centralities of vertices on the Internet. . . 122.2 A low-degree vertex with high betweenness. . . . . . . . . . . . 12

3.1 Friendship network at a US high school[9]. . . . . . . . . . . . . . 153.2 A simple network . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 The degree distribution of the Internet[10]. . . . . . . . . . . . . 17

5.1 Betweenness of nodes in the National Knowledge Network . . . . 215.2 Betweenness of nodes in the Network Les Miserables . . . . . . . 215.3 Betweenness of nodes in Zacharys Karate Club . . . . . . . . . . 225.4 Plotting the Betweenness of nodes in the Email-Network . . . . . 225.5 Closeness Centrality of nodes in the National Knowledge Network 235.6 Closeness Centrality of nodes in the Les Miserables Network . . . 235.7 Closeness Centrality of nodes in Zacharys Karate Club . . . . . 245.8 Closeness Centrality of nodes in the Email Network . . . . . . . . 24

5.9 Values of Pk in the National Knowledge Network . . . . . . . . . 255.10 Values of Pk in the Les Miserables Network . . . . . . . . . . . . 255.11 Values of Pk in Zacharys Karate Club . . . . . . . . . . . . . . . 265.12 Values of Pk in the Email Network . . . . . . . . . . . . . . . . . 265.13 Efficiency of NKN in the case of Initial Degree vs Random Removal 275.14 Efficiency of NKN in the case of Initial Betweenness vs Random

Removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.15 Efficiency of NKN in the case of Recalculated Degree vs Random

Removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.16 Efficiency of NKN in the case of Recalculated Betweenness vs

Random Removal . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.17 Efficiency of the email network in the case of Initial Degree re-

moval vs Random Removal . . . . . . . . . . . . . . . . . . . . . 305.18 Efficiency of the Email Network in the case of Initial Betweenness

removal vs Random Removal . . . . . . . . . . . . . . . . . . . . 305.19 Efficiency of the Email Network in the case of Recalculated De-

gree Removal vs Random Removal . . . . . . . . . . . . . . . . . 315.20 Efficiency of Email network in the case of Recalculated Between-

ness vs Random Removal . . . . . . . . . . . . . . . . . . . . . . 315.21 Efficiency of Zacharys Karate club in the case of Inital Degree

removal vs Random Removal . . . . . . . . . . . . . . . . . . . . 325.22 Efficiency of Zacharys Karate club in the case of Inital Between-

ness removal vs Random Removal . . . . . . . . . . . . . . . . . . 32

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LIST OF FIGURES 3

5.23 Efficiency of Zacharys Karate club in the case of Recalculated

Degree removal vs Random Removal . . . . . . . . . . . . . . . . 335.24 Efficiency of Zacharys Karate club in the case of RecalculatedBetweenness removal vs Random Removal . . . . . . . . . . . . . 33


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Acknowledgements

I am extremely grateful to my research supervisor, Prof. N. Balakrishnan forhis invigorating guidance and valuable suggestions during the course of my re-search work. am also indebted to him for his complete support and inspiration

throughout the period of my work. I am thankful to him for making time forme despite his hectic schedule.

I would like to thank my guides Sudip Choudhury and Naimisha Kolli for con-stantly helping me through the course of this project. I would not have beenable to progress at all without their guidance and discipline. I am extremelythankful to them for taking some of their precious time and going out of theirway to help me.

I take this opportunity to thank the Indian Academy of Sciences for selectingme for the Summer Research Fellowship 2013. I also convey a special thanks toMs. Nagaratna, and all the other faculty and staff of SERC for their supportand also thank the professors at IISc, who make working during the summer

holidays this much fun.

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Abstract

Social network analysis (SNA) is the methodical analysis of social networks.Social network analysis views social relationships in terms of network theory,consisting of nodes (representing individual actors within the network) and ties

(which represent relationships between the individuals, such as friendship, kin-ship, organizational position, sexual relationships, etc.)[1][2]. These networksare often depicted in a social network diagram, where nodes are represented aspoints and ties are represented as lines.

This project focuses on the analysis of certain real world networks such asthe National Knowledge Network, etc. Analysis is done based on several mea-sures and metrics such as the centrality measures, homophily, the clustering anddegree distributions and the results are observed.

Once the measures are calculated, we compute the efficiency of the networksin case of targeted attacks and random attacks. Efficiency of a network is a

measure of how effectively the network exchanges information.

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Chapter 1

Social Network Analysis

1.1 Introduction

A social network is a social structure made up of a set of social actors (suchas individuals or organizations) and a complex set of the dyadic ties betweenthese actors. The social network perspective provides a clear way of analyzingthe structure of whole social entities[3].

Social scientists have used the concept of social networks since early in the20th century to connote complex sets of relationships between members of socialsystems at all scales, from interpersonal to international. In 1954, J. A. Barnesstarted using the term systematically to denote patterns of ties, encompassingconcepts traditionally used by the public and those used by social scientists:bounded groups (e.g., tribes, families) and social categories (e.g., gender, eth-nicity).

1.2 Representing the Network

An Adjacency Matrix is the simplest way of representing a network. It is anNxN matrix which is defined as follows:

Aij =

1 : if there is an edge between i and j0 : otherwise

The adjacency matrix is an elegant way of representing a network but it requiresthe allocation of a large amount of memory and is not memory efficient.

This leads to the need for better ways to represent large networks which aremostly sparse, i.e. the density of the network fades to 0 with the increase inthe total number of nodes in the network.

An adjacency list is a modification of the adjacency matrix which stores thelinks for each node. This way, the search time for a node increases but drasti-cally reduces the space required to store the network.

In principle these two representations are equivalent. Both include all the edges

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CHAPTER 1. SOCIAL NETWORK ANALYSIS 7

and either of them can be constructed from a knowledge of the other. When

creating computer programs, however, the crucial point is to have the informa-tion you need for your calculations easily available, so that the program runsfast. Different calculations require different information and some might needingoing edges while others need outgoing ones. The choice of which adjacencylist to use thus depends on the particular calculations being performed.

The adjacency list is, as we have said, probably the most commonly used for-mat for the storage of networks. Its main disadvantage is the comparativelylong time it takes to find or remove edges O(m/n) time, compared with theO(1) of the adjacency matrix.

So, we store the edge relations of the nodes in an adjacency list in the form of

a tree giving rise to an Adjacency Tree. An adjacency tree is identical to anadjacency list except that each row, the set of neighbors of each vertex, is storedin a tree rather than a simple array.

Time complexities for various operations

Operation Adjacency Matrix Adjacency List Adjacency TreeInsertion O(1) O(1) O(log(m/n))Deletion O(1) O(m/n) O(log(m/n))

Find O(1) O(m/n) O(log(m/n))Enumerate O(n) O(m/n) O(m/n)


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1.3 Mathematics of networks

Before we go deep into studying networks, we must understand the basic mathe-matics behind the network. In some situations, however, it is useful to representedges as having a strength, weight, or value to them, usually a real number.These networks are called weighted networks.

A directed network or directed graph, also called a digraph for short, is a net-work in which each edge has a direction, pointing from one vertex to another.Such edges are themselves called directed edges, and can be represented by lineswith arrows on them.

Aij =

1 : if there is an edge from j to i0 : otherwise

Thedegree of a node is defined as the number of edges that are connectedto it. We denote the degree of a node i by ki.

Every edge in an undirected graph has two ends and if there are m edges intotal then there are 2m ends of edges. But the number of ends of edges is alsoequal to the sum of the degrees of all the vertices, so

2m=n

i=1

ki,

So, we get the average mean as

c= 1n

ni=1

ki,

A path in a network is any sequence of vertices such that every consecutive pairof vertices in the sequence is connected by an edge in the network. Paths canbe defined for both directed and undirected networks.

In a directed network, each edge traversed by a path must be traversed in thecorrect direction for that edge. In an undirected network edges can be traversedin either direction.We can count the number of paths of length 3 such as,

N(3)ij =n

kl=1

AkiAklAlj = [A3]ij

Thus we can get the expression for the total number of paths of length r is,

N(r)

ij = [Ar]ij

From the concept of paths, we get to the concept of geodesic paths which areessentially the shortest path from one node to another. The diameter of the


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network can be hence defined as the longest geodesic path in the network.

A random walk is a path across a network created by taking repeatedrandom steps. Starting at some specified initial vertex, at each step of thewalk we choose uniformly at random between the edges attached to the currentvertex, move along the chosen edge to the vertex at its other end, and repeat.

Consider a random walk that starts at a specified vertex and takes t randomsteps. Let pi(t) be the probability that the walk is at vertex i at time t. Ifthe walk is at vertex j at time t 1, the probability of taking a step along anyparticular one of the kj edges attached to j is 1

kj, so on an undirected network

pi(t) is given by

pi(t) =j

Aijkj

pj(t 1),


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Chapter 2

Centrality Measures

Within the scope of graph theory and network analysis, there are various typesof measures of the centrality of a vertex within a graph that determine the rel-ative importance of a vertex within the graph.

Many of the centrality concepts were first developed in social network analysis,and many of the terms used to measure centrality reflect their sociological ori-gin[4]. We discuss the various centrality measures used to analyze networks.

2.1 Degree Centrality

The simplest centrality measure in a network is just the degree of a vertex, thenumber of edges connected to it. This is called thedegree centrality. The degreecentrality can be represented as follows,

C= 1

n2

ni=1

ki

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CHAPTER 2. CENTRALITY MEASURES 11

2.2 Hubs and Authorities

There are really two types of important nodes in these networks: authoritiesare nodes that contain useful information on a topic of interest; hubsare nodesthat tell us where the best authorities are to be found. An authority may alsobe a hub, and vice versa[5].

In Kleinbergs approach, the authority centrality of a vertex is defined to beproportional to the sum of the hub centralities of the vertices that point to it:

xi = j

Aijyj ,

whereis a constant and Similarly the hub centrality of a vertex is proportionalto the sum of the authority centralities of the vertices it points to:

yi = j

Aijxj ,

with another constant.

2.3 Closeness Centrality

An entirely different measure of centrality is provided by the closeness centrality,which measures the mean distance from a vertex to other vertices[6]. The mean

geodesic distance from i to j, averaged over all vertices j in the network, is

li = 1

n

j

dij ,

The inverse ofli is called the closeness centrality of the vertex i,

Ci = 1

li=

njdij

,

We can average the closeness centrality of each of the nodes in the network toobtain the global closeness centrality.


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CHAPTER 2. CENTRALITY MEASURES 13

2.5 PageRank

A natural extension of the simple degree centrality is eigenvector centrality. Notall neighbors are equivalent. In many circumstances a vertexs importance in anetwork is increased by having connections to other vertices that are themselvesimportant. This is the concept behind eigenvector centrality[7].

xi = 1j

Aijxj ,

which givesAx = 1x in matrix notation, where x is the right leading eigenvec-tor. There are still problems with eigenvector centrality on directed networks.Only vertices that are in a strongly connected component of two or more ver-tices, or the out-component of such a component, can have non-zero eigenvector

centrality.

One solution to the issues of the previous section is the following: we simplygive each vertex a small amount of centrality for free, regardless of its positionin the network or the centrality of its neighbors. In other words, we define

xi = j

Aijxj+ ,

where and are positive constants.

The Katz centrality has one feature that can be undesirable. If a vertex withhigh Katz centrality points to many others then those others also get highcentrality. So, in order to reduce this effect we divide the centrality obtained

from a node by the out-degree.

In mathematical terms this centrality is defined as

xi = j

Aijxj

koutj+.


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Chapter 3

Important Metrics in the

Network

There are other important measures which reveal a lot of information about thenetwork. They are:

3.1 Clustering Coefficient

The clustering coefficient measures the average probability that two neighborsof a vertex are themselves neighbors.The calculation of clustering coefficients isonly slightly more complicated than the calculation of degrees.

The overall clustering coefficient for the entire network is given by

C= 3(no.of)no.ofconnectedtriples

Most real world networks usually have a high clustering coefficient and lowdiameter.

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CHAPTER 3. IMPORTANT METRICS IN THE NETWORK 15

3.2 Homophily and Assortative Mixing

Assortative mixing or homophily is the tendency of vertices to connect to othersthat are like them in some way. Modularity or the assortativity coefficient isa measure of the homophily in the network[8]. The normalized value of themodularity is given by

Q

Qmax=

ij(Aij kikj/2m)(cicj)

2m

ij(kikj/2m)(cicj)

whereci and cj are the types of the vertex i and j and (i, j) is the Kroneckerdelta. The Kronecker delta is 1 when i=jand 0 otherwise.

Figure 3.1: Friendship network at a US high school[9].

Another way of representing the modularity is derived in terms of the fol-lowing quantities

ers =

1

2mi ki(ci, r),

which is the fraction of edges that join vertices of type r to vertices of type s, and

ar = 1

2m

i

ki(ci, r),

which is the fraction of ends of edges attached to vertices of type r. Then,fromthe following equation

(ci, cj) =r

(ci, r)(cj , r),


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We have a final expression for modularity as follows

Q= 1

2m

ij

(Aij kikj2m

)r

(ci, r)(cj, r) =r

(err a2r),

3.3 Degree Distributions

We look at one of the most fundamental of network properties, the frequencydistribution of vertex degrees.We define pk to be the fraction of vertices in sucha network that have degree k. Consider this network: It has n = 10 vertices, of

Figure 3.2: A simple network

which one has degree 0, two have degree 1, four have degree 2, two have degree3, and one has degree 4. Thus the values of pk for k = 0,..., 4 are

p0= 1

10, p1=

2

10, p2=

4

10, p3=

2

10, p4=

1

10,


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The quantities pk represent the degree distribution of the network. Degree

Figure 3.3: The degree distribution of the Internet[10].

distributions of the form

pk = C k,

whereC=ec is another constant, are called power laws. Networks which followthe power law are called scale free networks.


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Chapter 4

Efficiency in the Network

4.1 Attacks on the Network

Originated from studies of computer networks, attack vulnerability denotes thedecrease of network performance due to a selected removal of nodes or edges[12].

In general, this gives a measure of the decrease of network functionality under asinister attack. If one wants to protect the network by guarding or by temporaryisolation of some nodes (edges), the most important nodes (edges), breaking ofwhich makes the whole network malfunction, should be identified. Furthermore,one can learn how to build attack-robust networks, and also how to increase therobustness of vital biological networks[11].

4.2 Vertex Attacks

One may maximize the destructive effect at any fixed number of removed nodes(or edges). However, this requires the knowledge of the whole network structureand pinpointing the vertex to attack in this way makes a very time demandingcomputation.

A more tractable choice, used in the original study of computer networks, is toselect the vertices in the descending order of degrees in the initial network andthen to remove vertices one by one starting from the vertex with the highest

degree[12]. This attack strategy uses the initial degree distribution and thusis called ID removal throughout the current paper. The vertices with highbetweenness also play important roles in connecting vertices in the network[13].

The second attack strategy is called IB removal and uses the initial distributionof the betweenness. Both ID removal and IB removal use the information on theinitial network. As more vertices are removed, the network structure changes,leading to the different distributions of the degree and the betweenness than theinitial ones. The other two attack strategies are called recalculated degree(RD)and recalculated betweenness(RB), where we iteratively compute the degree andbetweenness of the remaining network.

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CHAPTER 4. EFFICIENCY IN THE NETWORK 19

We can compute the reduction in efficiency as the Nrm(The number of

nodes removed) increases. On the y-axis, we plot the size of the largest compo-nent(S) times the inverse of the average distance from every node to every othernode(l1). Intuitively, we can see that RB and ID will generally do better thanIB and ID, but these are more costly to evaluate.

We can evaluate efficiency as ,

E= S l1

Nrm/N

Where S denotes the size of the Giant Component, l1 denotes the averagecloseness centrality, Nrm denotes the number of nodes removed and N denotesthe total number of nodes in the network.


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Chapter 5

Implementation

5.1 The Networks used

We perform the analysis on the following network datasets.

The National Knowledge Network is a network containing 31 nodes and 159edges. It is aimed at creating a robust and strong internal Indian network ca-pable of providing secure and reliable connectivity.

The second network is the Zacharys karate club containing 34 nodes and 78edges. It is a more sparse network than the NKN network.

The third network is the relation between the characters of the novel Les Mis-erables set in the post French revolution 18th century. It consists of 78 nodesand 259 edges.

The fourth network is the E-mail network which contains the list of edges ofemail exchanges between members of University Rovira i Virgili. This networkcontains 1133 nodes and 10903 edges.

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CHAPTER 5. IMPLEMENTATION 21

5.2 Results of the Analysis

5.2.1 Betweenness Centrality

The Betweenness Centralities of the four networks are plotted.

Figure 5.1: Betweenness of nodes in the National Knowledge Network

Figure 5.2: Betweenness of nodes in the Network Les Miserables


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Figure 5.3: Betweenness of nodes in Zacharys Karate Club

Figure 5.4: Plotting the Betweenness of nodes in the Email-Network

From the plot, we can see that a majority of the nodes in the network havea very low betweenness and only a few nodes have high betweenness.


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5.2.2 Closeness Centrality

We now plot the closeness centrality for the 4 networks described.

Figure 5.5: Closeness Centrality of nodes in the National Knowledge Network

Figure 5.6: Closeness Centrality of nodes in the Les Miserables Network


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Figure 5.7: Closeness Centrality of nodes in Zacharys Karate Club

Figure 5.8: Closeness Centrality of nodes in the Email Network


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5.2.3 Power Laws

We can see that the figures 5.9,5.10,5.11,5.12 all follow the power law and liesbetween 2 and 3 from the formula

= 1 +n[

ni=1

ln xixmin

]1

Figure 5.9: Values of Pk in the National Knowledge Network

Figure 5.10: Values of Pk in the Les Miserables Network


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Figure 5.11: Values of Pk in Zacharys Karate Club

Figure 5.12: Values of Pk in the Email Network

The figures 5.9 and 5.10 depict the power laws in the Email and NKN net-works.We get the value of kmin as two in the NKN and seven in the EmailNetwork.

We notice that all these networks show similar graphs for varying values ofkmin.


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5.3 Computing Efficiency in case of targetted

and random attacksWe compute the efficiency for two networks, the NKN and the network repre-senting the coauthorship of papers in network science. The graph of Coauthor-ships in network science contains 1589 nodes. We compute the efficiency in thecase of random and targetted attacks.

We have computed the efficiency in the four cases of vertex removal, they areInitial Degree(ID), Initial Betweenness(IB), Recalculated Degree(RD) and Re-calculated Betweenness(RB).

We will show the effect of each of these.

The randomized removal results are averaged over 10 iterations.

5.3.1 Efficiency in the case of the NKN

Figure 5.13: Efficiency of NKN in the case of Initial Degree vs Random Removal


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Figure 5.14: Efficiency of NKN in the case of Initial Betweenness vs RandomRemoval

Figure 5.15: Efficiency of NKN in the case of Recalculated Degree vs RandomRemoval


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Figure 5.16: Efficiency of NKN in the case of Recalculated Betweenness vsRandom Removal

The figures 5.13-5.16 depict the degradation in the efficiency of the NKN inthe case of ID,IB,RD and RB.


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5.3.2 Efficiency in the case of the Email Network

We secondly compute the efficiency of the email network under attack of ID,IB,RDand RB and a random attack.

Figure 5.17: Efficiency of the email network in the case of Initial Degree removalvs Random Removal

Figure 5.18: Efficiency of the Email Network in the case of Initial Betweennessremoval vs Random Removal


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Figure 5.19: Efficiency of the Email Network in the case of Recalculated DegreeRemoval vs Random Removal

Figure 5.20: Efficiency of Email network in the case of Recalculated Betweennessvs Random Removal

We notice that the efficiency drops much more sharply in the case of RDand RB as compared to the other 3 methods.


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5.3.3 Efficiency in Zacharys karate club

Now, we finally compute the efficiency for Zacharys Karate Club.

Figure 5.21: Efficiency of Zacharys Karate club in the case of Inital Degreeremoval vs Random Removal

Figure 5.22: Efficiency of Zacharys Karate club in the case of Inital Betweennessremoval vs Random Removal


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Figure 5.23: Efficiency of Zacharys Karate club in the case of RecalculatedDegree removal vs Random Removal

Figure 5.24: Efficiency of Zacharys Karate club in the case of RecalculatedBetweenness removal vs Random Removal

We see from the results of the various graphs that the efficiency of targettedattacks induce damage much more quickly than the random attacks.


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5.4 Analysis of all the sample Networks

I have computed the various metrics and have displayed them in tabular form.The result of the analysis performed on each of the four networks is as follows:

Parameter Email Network Karate Club NKN Les Miserables No. of nodes 1133 34 31 77

Diameter 8 5 4 5 Avg. Degree 9.624007 4.58823 5.32280 6.59740

Closeness Centrality 0.28197 0.42647 0.49522 0.38881Modularity 0.968348 0.70281 0.49931 0.73710

Clustering Coefficient 0.2201760 0.57063 0.762655 0.57313 Average Betweenness 0.0023041 0.044006 0.037300 0.0218819

value of 2.29789 2.59000 2.28964 2.08812

We can see from the results of the table that lies between 2 and 3 statistically.We have taken one large network and 3 other small sample networks to performthe analysis.

We see from the graph that the Emailnetwork follows a power law and gives asmooth curve for Pk = C k

. We notice that the diameter of a network doesnot increase rapidly with the increase in the size of the network. It can be seen

that with the increase in the diameter of the network, the closeness centralityalso decreases.

We can see from the results that RB targeted attack is the most harmful to thenetwork as the efficiency comes down to zero the fastest.

In the email network, we see that attack strategies harm the network in theorder

RB > RD > ID > IB > RR

where RR stands for random removal. The other 2 networks are very small andresults are not conclusive but it can be seen that

RB and RD > ID and IB > RR


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Conclusion

The work done entails the study of complex networks by means of the variousmeasures and metrics available to us in order to study the properties of smalland large networks. From the results we see that the actual results correspond

to the results expected from such a network.

The work will extend into computing the efficiency of the network and improvingits robustness by means of a unique method known as a p-cycle. By extendingthis concept into social networks, we can bridge the gap between physical andlogical networks and find a common measure for the efficiency of a network incase of random or targeted attacks.

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[13] Attack vulnerability of complex networks Petter Holme and Beom Jun Kim2002

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