CALIFORNIA STATE UNIVERSITY SAN MARCOS THESIS SIGNATURE PAGE THESIS SUBMITTED IN PARTIAL FULLFILLMENT OF THE REQUIREMENTS FOR THE DEGREE MASTER OF SCIENCES IN MATHEMATICS THESIS TITLE: Communication Networks AUTHOR: Janina Silvana P. Patno DATE OF SUCCESSFUL DEFENSE: April 19,2011 THE THESIS HAS BEEN ACCEPTED BY THE THESIS COMMITTEE IN PARTIAL FULLFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCES IN MATHEMATICS. Andre Ki.indgen t;;f~/kJt{ THESIS COMMITTEE CHAIR SIGNATA~ DliTE I K. Brooks Reid ~~~R~c& THESIS COMMITTEE MEMBER SIGNATURE ~IE Marshall WhittleseJ:: 1/JJtl!Jbt. ?(_'1/1} THESIS COMMITTEE MEMBER SIGNATURE ' DATE Thesis Abstract We consider the problem of sending a message in a communication network. Our networks are modeled by graphs. In our graphs we want to send a message from a vertex s, the sender, to a vertex r, the receiver, via paths from s tor. One difficulty we may encounter if we try sending the message along every possible s, r-path is that the receiver r will be flooded, so we focus our attention on certain finite protocols for sending the message from s to r. We study the probability that such a protocol successfully sends a message from s to r if the edges of the network live with probability p. We present some results and some open problems on this topic. Keywords: network, communication, protocol, two-terminal graph, two-terminal reliability polynomial ii Contents 1 Introduction 1 1.1 Introduction . 1 1.2 Basic Definitions 4 2 The Deterministic Model 11 2.1 s, r-Graphs . 11 2.2 Protocols for Message Transmission . 11 2.3 A Forbidden Minor Characterization 17 3 The Probabilistic Model 22 3.1 Reliability of a Protocol . 22 3.2 Reliability of the Complete Forwarding Protocol 26 3.3 Reliability and Deficiency of a Partial Forwarding Protocol . 38 4 Maximum Protocol Reliability 49 4.1 Protocols with Reliability p 49 4.2 Using Deficiencies to find p . 50 5 Comparing Graphs 69 5.1 Path and Cycle Comparison . 70 5.2 Comparing Polynomials . 75 5.3 Comparing the 10 Forbidden Minors 79 5.4 The Shortest s, r-Paths . 80 6 Kelmans' Type Construction 87 6.1 Definitions . 87 6.2 Operations With s, r-Graphs . 88 6.3 Multiplying Constructions . 94 6.4 Reliable Pairs with Prescribed Crossing Profiles 96 CONTENTS iii 7 A Graph with a Piecewise Polynomial for p 100 100 701 Graph B 0 With Edge Replaced 702 Open Problems 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 108 1 Chapter 1 Introduction 1.1 Introduction A network is an "interconnected system of things or people" [14] and networks, such as the internet, enable people to share information with one another. Com­ munication in a network has been studied by many researchers in many different forms and under different assumptions, but the goal is typically to determine the reliability of such networks. In 1959, during the Cold War, an engineer named Paul Baran was interested in survivable communications and felt that a more reliable system of communica­ tion would help prevent unnecessary attacks during the war. Baran's first attempt with such a communication system would try to send a "Go" or "No go" message from the president to commanders via AM radio. After three years of development, he was able to create a better communication system that would send longer mes­ sages with better survivability using the idea of packet switching and distributive communication in a network [1]. CHAPTER 1. INTRODUCTION 2 In this thesis we represent networks as graphs, where the vertices are the nodes or processors and the edges are links between processors. Thus, Baran's idea for a communication network would have edges and vertices that can fail indepen­ dently without recovering, and there would be multiple distinguished vertices, the terminals, that send and receive information. The model we study can be described with graphs with no multiple edges or loops such that the vertices do not fail, but the edges can fail independently without recovering. We focus our attention on the case where we have only two specified terminals, a sender and a receiver. Paraphrasing Douglas West (see [13], page 149), in order for a network to be able to reliably transmit information, it needs to stay connected even when some of the edges or vertices fail. So one way to study the reliability of a network is to study the number of walks with working edges between terminals. We study the reliability of a network Gin terms of the reliability polynomial, p(G,p), which is the probability that some walk remains between every two terminals if all edges can fail independently with probability p. One can study this connectivity question for any number of terminals k, where k ~ 2. So given a graph on n vertices one can look at the all terminal ( k = n) or k-terminal reliability polynomial. We will primarily study the two-terminal case. In 1992, Brown and Colbourn [4] calculated the all terminal reliability poly­ nomial for several families of graphs, such as trees and cycles, and studied the roots and coefficients of the reliability polynomial for a connected graph. Colbourn [6] CHAPTER 1. INTRODUCTION 3 recognized that constraints on a network force restrictions on the coefficients of the reliability polynomial and that these restrictions can be more noticeable by writing the reliability polynomial in different forms. Our study of communication in a network combines Baran's idea of packet switching in network transmission with the idea of a reliability polynomial to arrive at the concept of a protocol reliability polynomial. Like Colbourn, we calculate the reliability of specific networks and interpret our findings by writing the reliability polynomial in one or two variables. In Chapter 2, we review the deterministic model of the network as studied by Fich, Kiindgen, Pelsmajer and Ramamurthi [8, 9]. We describe our model as a graph with a fixed sender s and a fixed receiver r, and discuss how messages are sent in our model using protocols. We also describe the main result from [8], a forbidden minor characterization. In Chapter 3, we change the model described in Chapter 2, by attaching a probability to the existence of every edge. We study how well a protocol will work in our model by determining its reliability. Chapter 4 discusses which protocol will give us the best reliability for each of the 10 forbidden minors from Chapter 2. Chapter 5 compares graphs of paths and cycles to determine which would work better as a communication network. We also give a result about polynomials that can be used to compare the reliabilities of protocols in different graphs. In Chapter 6, we use an idea from Kelmans [10] to construct a pair of graphs such that if their reliability polynomials were graphed, they would cross a CHAPTER 1. INTRODUCTION 4 specified number of times. We end the thesis with Chapter 7, in which we use our findings of paths and cycles from Chapter 5 to show that there are graphs such that the best protocol to use at low probability is different from that at high probability. We present a number of fundamental results and some open problems on this topic. 1.2 Basic Definitions In this section we review some basic graph theory, probability, and algebra defini­ tions and theorems we will be using in this thesis. For a more comprehensive graph theory review please refer to West [13] and Diestel [7]. A comprehensive probability review can be found in Pitman [11]. A graph G consists of a non-empty finite vertex set V(G), and an edge multiset E(G), where each edge is an unordered pair of two distinct vertices called the endpoints. If vertex vis an endpoint of edge e, then v and e are incident. When u and v are the endpoints of an edge, they are adjacent and are neighbors, and we will denote this edge by uv. A complete graph is a graph whose vertices are pairwise adjacent. A subgraph H of G is a graph such that the V(H) ~ V(G) and E(H) C E(G). His said to be induced if E(H) consists of all edges in E(G) whose endpoints are both in V(H). A walk of length k is a finite sequence v0 , v1 , v2 , ... , Vk-l, vk of vertices such that vi-l vi, for all i with 1 :<::; i :<::; k, is an edge in G. A trail is a walk such that if vi = Vj for i =/= j, then vi+l =/= Vj+l· Thus, each edge uv can be used at most twice CHAPTER 1. INTRODUCTION 5 in a trail: once as u, v and once as v, u. A walk (trail) is closed if the endpoints, v0 and vk, of the walk (trail) are the same. A path is a walk Vo, v1, v2, ... , Vk-1, vk such that v0, v1, ... , vk-l, vk are distinct vertices of G. Given two specified vertices u and v, au, v-walk (u, v-trail, u, v-path) is a walk (trail, path) vo, v1, ... , vk that starts at v0 = u and ends at vk = v. The length of a path is its number of edges, k. A walk contains a path (trail) if the sequence of vertices describing the path (trail) is a subsequence of the sequence of vertices describing the walk, and every pair of consecutive vertices in the sequence describing path (trail) also occurs as a consecutive pair in the sequence describing the walk.