Large Scale Network Growth Techniques

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Large Scale Network Growth Techniques

• Valia Mitsou
• Supervisor Prof. Abbe Mowshowitz, City College

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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

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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).

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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.
• Diameter
• Vulnerability

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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).

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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 ? V1 y ? V2.

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Coalescence
Union
Join
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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 ? E2) or (u2 v2 u1v1 ? 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 ? E1) or (u1 v1
  u2v2 ? E2).

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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 ? E1 and u2v2 ?
  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 ?
  E1 or u2v2 ? E2.

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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) ? 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 ? V identify vi with
  the root node of the ith copy of H.

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G1
G2
Cartesian
Co-normal
Composition
Rooted
Tensor
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First TaskCompute the diameter
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Table (simple operations)
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Table (products)
n12l2 n22 l1 l1l2
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Second TaskVulnerability of Large Scale
Networks
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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.

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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.

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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 lt b ½, find a partition V into disjoint sets
  A, B, C such that minA,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.

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Results on Minimum b-Vertex Separator Problem

• NP-hard.
• Inapproximable within 1 V1/2 e /OPT for any
  egt0, even for graphs of maximum degree 3 (Bui and
  Jones)
• In planar graphs, a graph separator of size
  O(vV) can be found in polynomial time (Lipton
  and Tarjan)

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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?

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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.

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Thank you!