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Large Scale Network Growth Techniques

Valia Mitsou

Supervisor: Prof. Abbe Mowshowitz, City College

Network growth

Two major directions of research:

• Describe and explain how real-world networks grow

• Propose a new technique for building large networks with good properties

Introduction

• A natural way networks in the real world are built is by connecting smaller networks together to form a larger one

• It is useful to study binary graph operations that can be used in this concept (two smaller graphs are combined to create a larger one).

Outline

• Definitions of some basic graph operations that can be used for creating a larger network from two smaller ones.

• We will examine the properties of the produced network in terms of the smaller ones.

1. Diameter2. Vulnerability

Graph Operations

• Simple Operations: The size of the resulting graph is of the order of O(n1 + n2).

• Products: The size of the resulting graph is of the order of O(n1 · n2).

Simple Operations

• Coalescence. A coalescence of two graphs G1 and G2 is any graph obtained from the disjoint union of G1 and G2 by merging one vertex from G1 and one from G2. (Disjoint)

• Union. The (disjoint) union of two graphs G1(V1,E1), G2(V2,E2) is a graph G(V,E) where V = V1 U V2 and E = E1 U E2.

• Join. The join of two graphs G1(V1,E1), G2(V2,E2) is a graph G(V,E) where V = V1 U V2 and E = E1 U E2 U {(x,y) | x Є V1 & y Є V2}.

Union

Join

Coalescence

Products 1

• Cartesian Product. The cartesian product of two graphs G1(V1,E1) and G2(V2,E2) is a graph G(V,E)

with V = V1 x V2 and E = {(u1,u2)(v1,v2), where (u1

= v1 & u2v2 Є E2) or (u2 = v2 & u1v1 Є E1)}.

• Composition (or Lexicographic Product). The composition of two graphs G1(V1,E1) and G2(V2,E2)

is a graph G(V,E) with V = V1 x V2 and E = {(u1,u2)

(v1,v2), where (u1v1 Є E1) or (u1 = v1 & u2v2 Є E2)}.

Products 2

• Tensor Product (Direct Product, Categorical Product, Cardinal Product, or Kronecker Product). The tensor product of two graphs G1(V1,E1) and G2(V2,E2) is a graph G(V,E) with V = V1 x V2 and E = {(u1,u2)(v1,v2), where u1v1 Є E1 and u2v2 Є E2}.

• Co-normal Product (Disjunctive Product or Or Product). The co-normal product of two graphs G1(V1,E1) and G2(V2,E2) is a graph G(V,E) with V = V1 x V2 and E = {(u1,u2)(v1,v2), where u1v1 Є E1 or u2v2 Є E2}.

Products 3

• Compounding of Graphs. Roughly speaking, compounding of two graphs G(V,E) and H is obtained by taking V copies of H, indexed by the vertices of G, and joining two copies Hu, Hv of H by a single edge whenever (u,v) Є E.

• Rooted Product. A special case of compounding. The rooted product of a graph G(V,E) and a rooted graph H is defined as follows: take |V| copies of H and for every vertex vi Є V identify vi with the root node of the ith copy of H.

G1 G2

Cartesian

Composition

Rooted

Tensor

Co-normal

First Task“Compute the diameter”

Table (simple operations)

OperationNetwork

SizeNum. of Edges Diameter

Union n1 + n2 l1 + l2

Join n1 + n2 l1 + l2 + n1n2 min{2,max{d1,d2}}

Coalesence n1 + n2 - 1 l1 + l2

max{d1,d2 } ≤ d

≤ d1+ 2d2

Table (products)

Cartesian Product n1n2 n1l2 + n2l1 d1 + d2

Composition n1n2 l2n1 + l1 n22 max{d1, min{d2,2}}

Tensor n1n2 2l1l2 d ≥ max{d1, d2}

Co-Normal Product

n1n2 min{d1, d2}

Rooted Product n1n2 l1 + n1l2 d ≤ d1 + 2d2

Compounding n1n2 l1+n1l2

d ≤ (d1 + 1) d2

Operation Net. Size Num. of Edges Diameter

n12l2 + n2

2 l1 –l1l2

Second Task“Vulnerability of Large Scale Networks”

Parameters

• Knowledge of the adversary: The information an attacker has for the network. It is generally reasonable to assume no a priori knowledge of the network, however we will also examine the other case.

• Power of the adversary: cost of mounting an attack. Reasonable limitations:

– The attacker can undertake a small number of nodes.– The time complexity of the attack should be a relatively

small function of n.

Definition of Vulnerability

We say that a network is vulnerable for a specific attacker if he can address an attack that will isolate an at most logarithmic number of nodes or edges which if we remove then the network is disconnected into two pieces (not necessarily connected components) of the order of n and this procedure can be realized within at most linear time.

Vulnerability can be divided into three classes: node, edge, and mixed node and edge.

Node Vulnerability

A more formalized way to express node vulnerability would possibly be the minimum b-vertex separator problem: 

Definition: Given a graph G(V,E) and a number 0 < b ≤ ½, find a partition V into disjoint sets A, B, C such that min{A,B} ≥ b ∙ |V| and no node of A is connected to a node of B. We want to optimize the size of the separator, i.e |C|.

Results on Minimum b-Vertex Separator Problem

• NP-hard.

• Inapproximable within 1+ |V|1/2 – ε /OPT for any ε>0, even for graphs of maximum degree 3 (Bui and Jones)

• In planar graphs, a graph separator of size O(√|V|) can be found in polynomial time (Lipton and Tarjan)

Further Research

Examine the case of networks created using graph operations. Suppose that:– The adversary knows the graphs from which the

network was created.– The graphs are vulnerable

Can he take advantage of this information to destroy the network?

Further Research

Example: For the composition product Cn x Cn 2n nodes suffice to destroy the network. However it cannot be considered as “node vulnerable” since this amount is the sqrt of the total size of the whole network.

Thank you!