1 Comparative Study of Network Robustness of Different Complex Network Models subjected to Edge-Attack

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ABSTRACT

Robustness and resilience are extremely important parameters that decide overall

performance, reliability and safety of a complex network. In this project, we propose to

analyze how geometric characteristics and performance of the network is affected by

imposition of certain restrictions. We propose to measure and compare the attack

vulnerability of following complex network models with respect to edge attack:

(1) Erdos-Renyi model of random networks

(2) Watts Strogatz model of small world network

We shall be comparing the attack robustness of these complex network models.


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INTRODUCTION

A network is a set of items, which we will call vertices or sometimes nodes, with

connections between them, called edges. A set of vertices joined by edges is the simplest

type of network. A complex network is a graph (network) with non-trivial topological

features, which do not occur in simple networks such as lattices or random graphs but often

occur in real graphs.

Complex Network study and analysis is essentially falls under statistical studies.

Removal of a particular vertex or edge does not affect the network in any significant way;

rather, the percentage of vertices or edges removed is of relevance.

Most of the complex networks depend to a huge extent on their interconnectivity.

The paths connecting various nodes/ vertices of such complex networks form their lifeline.

These paths are called the edges of the complex network. On removal of these edges, at

first, any information flow will have to take alternate longer path; but ultimately, vertex

pairs will become disconnected. In such a scenario, communication between the nodes will

get disrupted, thereby hampering the performance of the network severely. A similar

hampering of communication results on removal of vertices. The extent to which network

remains unaffected by edge/ vertex removal is called the resilience of the network.

“Attack Vulnerability” denotes the decrease of network performance due to a

selected removal of vertices or edge. We measure the attack vulnerability of various

complex network models. We compare different ways of attacking the network and use

various ways of measuring the resulting damage. In general, this gives a measure of the

decrease of network functionality under a sinister attack. The meaningful purpose for attack

vulnerability studies is for the sake of protection: If one wants to protect the network by

guarding or by a temporary isolation of some edges, the most important edges, breaking of

which makes the whole network malfunctioning, should be identified. Thus, by spotting the

critical component of the network the security of the network can be improved.


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We will study the vulnerability of complex networks subject to edge attack. For

example, in computer networks, this can be interpreted as the malfunctioning or the loss of

the communication cables. In the context of social networks, the attack on an edge can be

interpreted as prohibition of contact between two individuals.

The edges in a complex network can be attacked in different ways:

1. ID Removal: This strategy uses initial distribution of degree for attacking edge. Edges are

selected in descending order of degrees in the initial network & then removing the

edges one by one starting from edge of highest degree. Note that degree of the edge

does not make a direct sense as degree of a vertex makes; however, it could be

understood as the importance of an edge in terms of the degree of two vertices it

connects.

2. IB Removal: This methodology of attack uses initial distribution of betweenness.

3. RD Removal: It uses re-calculated degree distribution for removal of edges.

4. RB Removal: It uses re-calculated distribution of betweenness for removing edges.

For calculations, edge degree (ke) can be found in following ways:

ke = kvkw

ke = kv + kw

ke = min(kv,kw)

ke = max(kv, kw)

It is noteworthy that while ID and RD removals are local strategies, the other two

strategies based on the betweenness are global ones, which makes the applications of ID

and RD O(N) algorithms, while IB and RB are O(NL) even with the best to the betweenness

of the corresponding edge known algorithm.

Both the degree and betweenness indicate the importance of an edge in a network.

While betweenness is based on the global information on paths connecting all pairs of

vertices, degree depends only on local information. Scale free networks show clear signs of


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correlation between the degree and betweenness. This correlation is not observed in WS

and ER models, because vertices with very high degrees are generally absent.

It is noteworthy that an edge does not necessarily become central just because it

connects to one central vertex; rather it has to be a bridge between two central vertices.

APPROACH & METHODS

In this project, we have analyzed and compared the impact of edge attack on

robustness of 2 models: (1) Erdos-Renyi model of random networks, and (2) Watts Strogatz

model of small world network.

In the Erd¨os-R´enyi (ER) model, we start from N vertices without edges.

Subsequently, edges connecting two randomly chosen vertices are added until the total

number of edges becomes L. It generates random networks with no particular structural

bias: The only restriction in the model is that no multiple edges are allowed between two

vertices. In this study, we choose the average degree <k>= 2L/N as a control parameter in

the ER model. The ER model graphs have a logarithmically increasing `, a Poisson-type

degree distribution, and a clustering coefficient close to zero.

In the WS model, a regular one-dimensional network is constructed with only local

connections of range r. Then, each edge is visited once, and with the rewiring probability p

is detached at the opposite vertex and reconnected to a randomly chosen vertex forming a

‘shortcut’. For p = 0 the network is a regular local network, with high clustering, but without

the small-world behavior: The average geodesic length in this case grows linearly with the

network size. In the opposite limit of P = 1, where every edge has been rewired, the

generated random graph has vanishing clustering, but shows a logarithmic behavior of the

average geodesic length α logN. In an intermediate range of P (typically P ~ O(1/N)), the

network generated by the WS model displays both high clustering and small world

behavior—the commonly found characteristics of real social networks.


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We began with sample adjacency matrix for both: ER and WS model. Edges were

attacked and removed one by one. Average path length of the network is an important

measure of its robustness. The lower the value of average path length, the more robust is

the complex network. After removing each edge, we evaluated the modified path length of

the network to see the impact of edge attack.

IMPLEMENTATION

We considered WS and ER models consisting of 8 nodes each (for the sake of

analysis). MATLAB codes were written to evaluate the average path length of the network.

Then we removed edges one by one, and evaluated the resultant average path length after

each step.

MATLAB Code for ER model of random networks:

W = [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ];

A = [6 1 2 2 3 4 4 5 5 6 1 1 7 8 6 7 ];

B = [2 6 3 5 4 1 6 3 4 3 5 7 2 5 8 8 ];

DG = sparse(A,B,W)

sum = 0;

UG = tril(DG + DG');

h = view(biograph(UG,[],'ShowArrows','off','ShowWeights','on'))

for i = 1:8

for j = 1:8

[dist,path,pred] = graphshortestpath(UG,i,j,'directed',false)

sum = sum+dist;

end

end

avg = sum/56;

W1 = [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ];

A1 = [1 2 2 3 4 4 5 5 6 1 1 7 8 6 7 ];

B1 = [6 3 5 4 1 6 3 4 3 5 7 2 5 8 8 ];

DG = sparse(A1,B1,W1)

sum1 = 0;


h1 = view(biograph(UG,[],'ShowArrows','off','ShowWeights','on'))

for i = 1:8

for j = 1:8


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[dist1,path1,pred1] = graphshortestpath(UG,i,j,'directed',false)

sum1 = sum1+dist1;

end

end

avg1 = sum1/56;

W2= [1 1 1 1 1 1 1 1 1 1 1 1 1 1 ];

A2 = [2 2 3 4 4 5 5 6 1 1 7 8 6 7 ];

B2 = [3 5 4 1 6 3 4 3 5 7 2 5 8 8 ];


sum2 = 0;



for i = 1:8

for j = 1:8


sum2 = sum2+dist2;

end

end

avg2 = sum2/56;

W3= [1 1 1 1 1 1 1 1 1 1 1 1 1 ];

A3 = [2 3 4 4 5 5 6 1 1 7 8 6 7 ];

B3 = [5 4 1 6 3 4 3 5 7 2 5 8 8 ];


sum3 = 0;



for i = 1:8

for j = 1:8


sum3 = sum3+dist3;

end

end

avg3 = sum3/56;

MATLAB Code for WS model of small world network:

W = [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1];

A = [ 1 1 1 1 2 2 2 3 3 4 4 5 5 5 6 8];

B = [ 2 3 6 8 3 4 8 4 7 5 6 6 7 8 7 7];

DG = sparse(A,B,W)

sum = 0;


h = view(biograph(UG,[],'ShowArrows','off','ShowWeights','on'))

for i = 1:8

for j = 1:8

[dist,path,pred] = graphshortestpath(UG,i,j,'directed',false)

sum = sum+dist;

end


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end

avg = sum/56;

W1 = [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1];

A1 = [ 1 1 1 2 2 2 3 3 4 4 5 5 5 6 8];

B1 = [ 3 6 8 3 4 8 4 7 5 6 6 7 8 7 7];


sum1 = 0;



for i = 1:8

for j = 1:8


sum1 = sum1+dist1;

end

end

avg1 = sum1/56;

W2 = [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1];

A2 = [ 1 1 2 2 2 3 3 4 4 5 5 5 6 8];

B2 = [ 6 8 3 4 8 4 7 5 6 6 7 8 7 7];


sum2 = 0;



for i = 1:8

for j = 1:8


sum2 = sum2+dist2;

end

end

avg2 = sum2/56;

W3 = [1 1 1 1 1 1 1 1 1 1 1 1 1];

A3 = [1 2 2 2 3 3 4 4 5 5 5 6 8];

B3 = [8 3 4 8 4 7 5 6 6 7 8 7 7];


sum3 = 0;



for i = 1:8

for j = 1:8


sum3 = sum3+dist3;

end

end

avg3 = sum3/56;


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RESULTS

ER Model:

(1) Initially: Average Path length = 1.4286

(2) After removing 1 edge: Average Path length = 1.4643


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(3) After removing 2 edges: Average Path length = 1.5000



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WS Model:

(1) Initially: Average Path length = 1.4286

(2) After removing 1 edge: Average Path length = 1.4643


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CONCLUSION

As expected, the average path length (l) increased on removing edges for both: ER as

well as WS complex network models. Increase in l signifies that the robustness of the

network is being compromised by each edge attack. Also, we observed that increase in path

length was more or less same for both WS and ER models for initial removal of edges. But as

the No. of edges removed increased, the increase in the value of path length for WS model

far exceeded the increase in value for ER model. This suggests that ER model is more

robust, as compared to WS model of complex network.


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Reka Albert and Albert Laszlo Barabasi, Statistical mechanics of complex networks,

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I Mishkovski, R Kojchev, D Trajanov, and L Kocarev, ICT Innovations, Ch 10, Springer.

Regino Criado, Julio Flores, Benito Hern´andez-Bermejo, Javier Pello and Miguel

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Complex networks: Comparison of network robustness

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