搜尋散佈謠言者的數學模型

國 立 交 通 大 學

應用數學系

碩

士

論

文

搜尋散佈謠言者的數學模型

A Mathematical Model for Finding the Culprit Who

Spreads Rumors

研 究 生：李姿慧

指導老師：傅恆霖 教授

搜尋散佈謠言者的數

學模型

A Mathematical Model for Finding the Culprit Who

Spreads Rumors

研究生

：李姿慧

_{Student: Zi-Hui Lee}

指導教授：傅恆霖 教授 Advisor: Hung-Lin Fu

國 立 交 通 大 學

應 用

數

學 系

碩 士 論 文

A Thesis

Submitted to Department of Applied Mathematics

College of Science

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

Master

in

Applied Mathematics

June 2012

Hsinchu, Taiwan, Republic of China

中 華 民 國 一

一 年 六 月

搜

搜

搜尋

尋

尋散

散

散佈

佈

佈謠

謠

謠言

言

言者

者

者的

的

的數

數

數

學

學

學模

_模

_模型

_型

_型

研究生

：李姿慧

_{指導教授：傅恆霖 教授}

國 立 交 通 大 學

應 用

數

學 系

摘要

在這篇論文中，我們介紹謠言傳播模型，它的設計是根據一個在流

行

病學領域著名的易感–感染模型。我們描述在一個圖上散佈謠言

的源頭的最大概似估計值並計算預測到散佈謠言的源頭的機率。我

們發現：對於路徑的圖形，機率會隨著時間增加趨近到0，其關係

為t

；對於正則樹，機率有一個明確的範圍。當d = 3，其機率

值會隨著時間增加趨近到1/4，此結果已利用隨機圖模型得到 [6]。

中 華 民 國 一

一 年 六 月

A Mathematical Model for Finding the

Culprit Who Spreads Rumors

Student: Zi-Hui Lee

Advisor: Hung-Lin Fu

Department of Applied Mathematics

National Chiao Tung University

June, 2012

Abstract

In this thesis, we introduce a rumor spreading model based

on the common susceptible-infected (SI) model which is a well

known epidemiological model. We describe the maximum

likeli-hood estimators of graphs and we evaluate the detection

proba-bilities of finding the rumor source in d-regular trees. We observe

that: For paths, the detection probability of finding the rumor

source scales as t

, which approaches 0 as t approaches

infin-ity. For regular trees, we find an explicit bound of the detection

probabilities of finding the source in d-regular trees. As a

conse-quence, for d = 3, the detection probability approaches 1/4, this

result has been obtained earlier by using a random graph model

[6].

誌

誌

誌謝

謝

謝

在交大的兩年，謝謝你們。

謝謝我的指導教授傅恆霖老師，謝謝老師讓我成為他的學生，

跟著老師的

一年半我學到了許多，尤其是態度，老師總是期許

我們要多讀點書，多運動等等，所有的一切我會謹記在心。謝

謝與我同師門的學長姐：惠蘭學姐、貓頭學長、明輝學長、惠娟

學姐、施智懷學長、敏筠學姐、軒軒學長、小泡學姐還有瑩晏學

姐。謝謝你們在這兩年給我的支持、建議與鼓勵。因為過程中有

你們的幫助我才能夠還算順利的完成我的碩士論文。

謝謝系

上的老師們指導我完成學業，讓我順利的走到這。特別

感謝的還有

：謝謝資訊技術服務中心的蔡錫鈞主任提供我解決問

題的方法與建議；謝謝系上的資訊人員詹宗智還有麻將學長教我

使用系上的機器做運算；謝謝游森朋教授抽空看我的問題；謝謝

陳秋媛老師的提點，讓我的論文內容更加嚴謹。

謝謝我的好朋友

_{們：米奇、小雨、憶妏、沙發、惠雯、吉米、}

連

_{、DK、嗨派，有事情就幫助我，有玩的就找我。還要謝謝所}

有我在這裡認識的人，因為有你們我的碩士生活才能如此多采多

姿。

最後一定要謝謝我的父母，因為有他們的期許我才一路走到。

在外地念書不能常常回家，謝謝家人的體諒以及時不時打電話來

關心問候，我愛我的爸爸媽媽。

Contents

Abstract (in Chinese) i

Abstract (in English) ii

Acknowledgement iii

Contents iv

List of Figures v

1 Introduction 1

1.1 Preliminaries . . . 2

1.2 Rumor spreading (RS) model . . . 4

2 Rumor Source Estimator 5

2.1 The ML estimator of Gn in a regular tree . . . 5

2.2 Rumor centrality . . . 8

3 Main Result 11

3.1 Detection probabilities of d-regular trees, d ≤ 3 . . . 11

3.2 Detection probabilities of d-regular trees, d ≥ 4 . . . 17

List of Figures

1 Illustration of subtree T_uv . . . 3

2 Network with 4 infected vertices. . . 6

3 Network of calculating rumor centrality. . . 9

1

Introduction

Social network, internet and electrical power grid network are the common

networks everywhere in our life. We are surrounded by all kinds of networks

and are very easily subjected to the influence of network risks. Although

every network has its different structure, the common phenomenon is: an

isolated risk is enlarged because it can be spread by the network.

For example, in an electrical power grid network, an isolated failure can

lead to a rolling blackout. Computer viruses utilize the internet to infect

millions of computers everyday. The malicious rumors or misinformation

can be spread in the social networks quickly and the person concerned will

be deeply hurt and offended. In all of these situations, power network

operator, internet service provider or victim of a malicious rumor would

like to infer the source of risks as quickly as possible and then blockade the

spread of risks. All of these situations can be modeled as rumors spreading

through networks, where the goal is to find the source in order to control

and prevent these network risks based on limited information about the

network structure and the rumor infected vertices.

Prior work on rumor spreading has primarily focused on infectious

dis-eases in populations. The standard model of infectious disdis-eases is known

as the susceptible-infected-recovered (SIR) model [1]. In this model, there

are three types of vertices: (i) susceptible vertices that are capable of being

infected; (ii) infected vertices that can spread the virus further; and (iii)

recovered vertices that are cured and can’t be infected anymore. Research

infec-tion/cure [2, 3, 4, 5]. However, there has no idea of identifying the source

of an epidemic. Now, a mathematical model has been developed to identify

the rumor source in a network based on rumor infected vertices [6]. But,

not much is known if the network is getting more complicate. In this thesis,

we set forth to study the networks defined on d-regular trees.

1.1

Preliminaries

A graph G is a triple consisting of a vertex set V (G), and edge set

E(G), and a relation that associates with each edge two vertices called its

endpoints. The order of a graph G is the number of vertices in G and the

size of a graph G is the number of edges in G. A loop is an edge whose

endpoints are equal, and multiple edges are edges having the same pair

of endpoints. A simple graph G is a graph having no loops or multiple

edges. We consider a simple graph with a countably infinite vertex set.

A subgraph of graph G is a graph H such that V (H) ⊆ V (G) and

E(H) ⊆ E(G) and the assignment of endpoints to edges in H is the same

as in G. We then write H ⊆ G and say that “G contains H”. A path is a

simple graph whose vertices can be ordered so that two vertices are adjacent

if and only if they are consecutive in the list. A graph G is connected if

each pair of vertices in G belongs to a path.

In a graph G, the contraction of edge e with endpoints u, v is the

replacement of u and v with a single vertex whose incident edges are the

edges other than e that were incident to u or v. The resulting graph G · e

has one less edge e than G, and the new vertex {u, v}(or {v, u}).

Figure 1: Illustration of subtree Tv u

e = vu), they are adjacent and are neighbors. We write u ↔ v for ”u

is adjacent to v”. The neighborhood of v, written NG(v) or N (v), is the

set of vertices adjacent to v. The degree of vertex v in a graph G, written

dG(v) or d(v), is the number of edges incident to v. And G is regular if the

degrees of all vertices are the same. It is k-regular if the common degree

is k.

A graph with no cycles is acyclic. A tree is a connected acyclic graph. In

this thesis, A d-regular tree is a tree where every vertex has d neighbors.

Therefore, it is an infinite graph. A rooted tree Tr is a tree with one

vertex r chosen as root. For each vertex v, Let p(v) be the unique path

from r to v. The parent of v is its neighbor on p(v); its children are its

other neighbors. The set child(v) is the set of all children of v.

A branch of a tree is a subtree Tr

v of Tr induced by v and all its

de-scendants. In addition, tr

v denotes the number of vertices in Tvr. A d-regular

trees are branches of Tv and P_{u∈N (v)}tvu = tvv − 1. A simple example T is

shown in Figure 1. T1

7 is a branch of T1; T76 is a branch of T6. Clearly,

T1

7 = T76. And T can be decomposed into 6 subtrees (T21, T31, T41, T51, T61),

such that these trees are branches of T1.

These definitions about graph theory are cited from the book

”Intro-duction to Graph Theory” written by Douglas B. West. If this part is not

sufficient, please refer to the reference [8] for details.

1.2

Rumor spreading (RS) model

We consider a discrete time susceptible-infected (SI) model. Let S(t) be a

set of people who don’t know rumors yet at time t and let I(t) be a set of

people who have known rumors at time t in which the number of people of

S(t) and I(t) are denoted St and It, respectively. Using a fixed population,

St+ It= N . For convenience of study, we shall assume

   St+1 = St− 1, It+1 = It+ 1, S0 = N, I0 = 0.

Let each person be a vertex and the relationship between two people be an

edge, and these form the vertex set and edge set of a graph G. Let Gt be a

subgraph of order t of G. This graph is compose of t infected vertices which

are people who have known rumors at time t. The graph G1 is a vertex

which is called rumor source and in each discrete time-step t + 1, t > 0,

Gt+1 develops from Gt by adding a vertex z with an edge. We assume that

every vertex is chosen with the following probability distribution:

Pt+1(z) =

1 P

v∈V (Gt)d(v) − 2(t − 1)

2

Rumor Source Estimator

Consider a network G and a subgraph Gn of G that is a graph with n

infected vertices. The Maximum Likelihood(ML) estimator is the vertex v

which has the maximum of P (Gn|v). where P (Gn|v) is the probability of

observing Gn under the RS model if v is the source.

2.1

The ML estimator of G

in a regular tree

Continue from the paragraph above, if we want to know that what’s the

ML estimator of Gn, we would like to evaluate P (Gn|v) for all v ∈ Gn

and then choose v so that P (Gn|v) is maximum. In general, evaluating

every vertex’s P (Gn|v) is complicated. Let us consider a simple example

as shown in Figure 2 with n = 4. First, we suppose that the vertex 1 is the

source, and we would like to calculate P (G4|1). Then there are six ways

or vertex infection orders that a rumor can be spread to every vertex in

G4 with vertex 1 as the source: (1, 2, 3, 4), (1, 2, 4, 3), (1, 3, 2, 4), (1, 3, 4, 2),

(1, 4, 2, 3) and (1, 4, 3, 2). However, if we suppose that the vertex 2 is the

source, infection order (2, 3, 1, 4) is not possible, since 2 = 3 (2 and 3 are not adjacent in G4). Therefore, in general to evaluate P (Gn|v), we need

to find all such possible n-permutations of V (Gn) and their corresponding

probabilities.

Let S (v, Gn) be the set of all possible n-permutations of V (Gn) starting

with vertex v where v ∈ V (Gn), and σ = (v1 = v, v2, · · · , vn) for each σ ∈

S (v, Gn). The probability P (σ|v) is the probability of observing Gn under

the RS model if v is the source in the infected order σ. Let Gk(σ) be the set

Figure 2: Network with 4 infected vertices.

for 1 ≤ k ≤ n. Then we have

P (σ|v) = n−1 Y k=1 1 P vi∈Gk(σ)d(vi) − 2(k − 1) . (2)

For d-regular trees,

P (σ|v) = n−1 Y k=1 1 dk − 2(k − 1). (3)

From equation (3), we can see that every permutation σ has the same

probability and is independent of the source. Specifically, for any source v

and permutation σ, P (σ|v) is a constant. Thus

P (Gn|v) = X σ∈S(v,Gn) P (σ|v) = |S (v, Gn) | · n−1 Y k=1 1 dk − 2(k − 1) ∝ |S (v, Gn) |.

It follows that P (Gn|v) is proportional to |S (v, Gn) |. Let R(v, Gn) be the

number of distinct ways to spread a rumor to every vertex in Gn with v as

the source. Clearly, |S (v, Gn) | = R(v, Gn). Then

In summary, the ML estimator of Gn in a regular tree can be obtained by

finding the maximum of R (v, Gn) for all v. For the above example as shown

in Figure 2 the estimation of vertices are

P (G4|1) = X σ∈S(1,G4) P (σ|1) = P ((1, 2, 3, 4)|1) + P ((1, 2, 4, 3)|1) + P ((1, 3, 2, 4)|1) + P ((1, 3, 4, 2)|1) + P ((1, 4, 2, 3)|1) + P ((1, 4, 3, 2)|1) = 6 · 1 3· 1 4 · 1 5  = 1 10. P (G4|2) = X σ∈S(2,G4) P (σ|2) = P ((2, 1, 3, 4)|2) + P ((2, 1, 4, 3)|2) = 2 · 1 3· 1 4 · 1 5  = 1 30. P (G4|3) = X σ∈S(3,G4) P (σ|3) = P ((3, 1, 2, 4)|3) + P ((3, 1, 4, 2)|3) = 2 · 1 3· 1 4 · 1 5  = 1 30. P (G4|4) = X σ∈S(4,G4) P (σ|4) = P ((4, 1, 2, 3)|4) + P ((4, 1, 3, 2)|4) = 2 · 1 3· 1 4 · 1 5  = 1 30.

Clearly, we can now evaluate R (v, G4) for all v. R (1, G4) = 6, R (2, G4) =

R (3, G4) = R (4, G4) = 2. Thus vertex 1 is the probable rumor source in

2.2

Rumor centrality

In this paragraph, we shall explain how to calculate the rumor centrality

for each vertex in a given graph Gn. Moreover, the rumor center of Gn is a

vertex with the maximum of rumor centrality.

Let Gn be a rooted tree with root v and assume v has a rumor. The

next infected vertex must be one of the children of v. For all u ∈ child(v),

there are R(u, Tv

u) ways to spread a rumor in the branch Tuv with u as the

source. Thus R(v, Gn) = (n − 1)! Y u∈child(v) R(u, T_uv) tv u! . (5)

To understand the above expression, the number of ways to permute n − 1

steps from different subtrees is a permutation of the multiset {tv v1 · 1, t

v v2 · 2, · · · tv

vd · d}. If we continue this recursion (5) until we reach the leaves of the tree, we obtain

R(v, Gn) = (n − 1)! Y u∈child(v) R(u, T_uv) tv u! = (n − 1)! Y u∈child(v)   (tv u− 1)! tv u! · Y w∈child(u) R(w, Tv w) tv w!   = (n − 1)! Y u∈child(v) 1 tv u · Y w∈child(u) R(w, Tv w) tv w! = (n − 1)! Y u∈Gn−v 1 tv u .

Since a leaf vertex l has one vertex so R(l, Tv

l ) = 1. By the fact that tvv = n,

then R(v, Gn) = n! Y u∈Gn 1 tv u . (6)

Figure 3: Network of calculating rumor centrality.

Let us consider an example as shown in Figure 3.

R(1, G5) =

5!

5 · 3 · 1 · 1 · 1 = 8.

Indeed, there are 8 possible n-permutations of the network in Figure 4 with

vertex 1 as the source. They are

(1, 2, 4, 5, 3), (1, 2, 4, 3, 5), (1, 2, 3, 4, 5), (1, 3, 2, 4, 5)

(1, 2, 5, 4, 3), (1, 2, 5, 3, 4), (1, 2, 3, 5, 4), (1, 3, 2, 5, 4)

In order to find the rumor center of a given graph Gn, we have to find

the rumor centrality of every vertex in Gn respectively. In fact, the rumor

centrality of v can be deduced from rumor centrality of its neighbors. To

this end, consider two adjacent vertices u and v in Gn, we have

tv_u = n − tu_v. (7)

And tv_w = tu_w for each w ∈ Gn− {u, v}. Thus,

R(u, Gn) R(v, Gn) = t v u n − tv u . (8)

For example in Figure 4, we have

R(2, G5) R(1, G5) = 12 8 = t1 2 5 − t1 2 = 3 2.

It shows that for each vertex u, we can calculate its rumor centrality by

using its neighbor’s rumor centrality and tv u.

The following is an important property of the rumor center:

Theorem 2.2.1. [6] Given an n vertices tree, vertex v is the rumor center

if and only if

tv_u ≤ n 2

for all u 6= v. Furthermore, a tree can have at most 2 rumor centers.

3

Main Result

In this section, we shall explain the behavior of the detection probabilities

of finding the rumor source in different graphs. Let Et(G) be the event of

correct rumor source detection under the ML rumor source estimator after

time t on a graph G. If the graph G considered is prescribed, then we use Et

to denote Et(G). Then P(Et) is the correct detection probability of finding

the rumor source in a given graph G.

3.1

Detection probabilities of d-regular trees, d ≤ 3

In this paragraph, we shall present shorter proofs by using combinatorial

methods than the original proofs in reference [6]. We first consider the

detection probability of finding the source in a 2-regular tree. By (4), rumor

center is the ML estimator of Gn where Gn is a subtree of a d-regular tree.

Theorem 3.1.1. Suppose a rumor has spread in a 2-regular tree. Then we

have that P(Et) = O  1 √ t  . Proof:

Consider a path which compose of 2n (respectively 2n + 1) infected vertices.

By Theorem (2.2.1) and (6), v is a unique rumor center in G2n+1 and we

have R (v, G2n+1) = 2n n  ,

and there are two vertices v, v0 both are rumor centers in G2n then we have

R (v, G2n) = R (v0, G2n) =

2n − 1 n − 1

 .

Every graph Gn in a 2-regular tree is a path. The correct detection

proba-bility of finding the rumor source in a path is

P(source = vG_n) = P(Gn  source = v)P(source = v) P i∈GnP(Gn  source = i)P(source = i) .

The probabilities of vertices which are rumor sources in a d-regular tree are

equal possible. Thus

P(source = vGn) = P(Gn  source = v) P i∈GnP(Gn  source = i) . By (4), P(source = vG_n) = R (v, Gn) P i∈GnR (i, Gn) (9) Hence P(source = vG_2n+1) = 2n n  22n 0  +2n 1  + · · · +  2n n − 1  +2n n  =2n n  · 1 22n, and P(source = vG_2n) = 2n − 1 n − 1  22n − 1 0  +2n − 1 1  + · · · +2n − 1 n − 1  =2n − 1 n − 1  · 1 22n−1 = 2n n  · 1 22n. By Stirling’s formula, 2n n  · 1 22n ∼ √ 4πn 2n_e
2n 22n√_2πn n e
n√ 2πn n e
n = 1 √ πn.

Thus P(Et) =          P(source = vGt) = r 2 πt if t is even, 1 2P(source = v  G_t) + 1 2P(source = v 0 G_t) = s 2 π(t − 1) if t is odd. (10)

It follows that the path detection probability scales as t−1/2, which

ap-proaches 0 as t apap-proaches infinity. _

Now, we consider the detection probability of finding the source in a

d-regular tree, d ≥ 3.

Theorem 3.1.2. Suppose a rumor has spread in a regular tree. Then we

have that

0 < P (Et) ≤

1 2.

Proof:

Consider a graph Gn in a regular tree. We can regard the number of ways

to spread a rumor to every vertex in Gn with v as the source as the sum of

ways that rumor can be spread from v through u, where u is the neighbor

of v. Hence

R (v, Gn) =

X

u∈N (v)

R ({v, u}, Gn· vu) . (11)

Given any two adjacent vertices u and v, we have

R (u, Gn) =

X

w∈N (u)

R ({u, w}, Gn· uw) and

R (v, Gn) =

X

u∈N (v)

Since u is adjacent to v, there is at least one common term in above two

sums on the RHS. Thus we have

R (v, Gn) ≤ X i∈Gn,i6=v R (i, Gn) ⇒ R (v, Gn) ≤ 1 2 X i∈Gn R (i, Gn) . (12)

Since R (v, Gn) is positive, we have

0 < PR (v, Gn)

i∈Gn

R (i, Gn)

≤ 1

2.

This concludes that for every graph which composes of infected vertices,

the detection probability is greater than 0 and less than 1/2. _

For d-regular trees with d > 2, Theorem (3.1.2) states that the event

that positive detection probability happens is independent of the order of the

graph. In what follows, we shall evaluate the explicit detection probability

of finding the source in a d-regular tree.

Our goal is to calculate the detection probability of finding the rumor

source. Recall the Theorem (2.2.1), v is the unique rumor center if and only

if tv u <

n

2 for all u 6= v. So we consider the number of vertices of branches

of Tv, and these branches are rooted trees with root u, u ∈ N (v). Without

loss of generality, we assume tv

u ≥ 1 for all u ∈ N (v). Let Ad and Bdbe two

sets such that

Ad= {(a1, a2, · · · , ad)| 1 ≤ ai < n 2, d X i=1 ai = n − 1}, and Bd= {(b1, b2, · · · , bd)| bi ∈ N, d X i=1 bi = n − 1}.

Clearly, Ad⊆ Bd. Moreover, |Bd| = n − 1 − d + d − 1 d − 1  =n − 2 d − 1  .

Now, let Si = {(x1, x2, · · · , xd) ∈ Bd : xi ≥ n₂}. By principle of Inclusion

and Exclusion, |Ad| = |Bd| −   d [ i=1 Si  = |Bd| − d X i=1 |Si|. (13)

Let xj = yj+ 1, j 6= i and xi = dn₂e + yi. Pd_i=1yi = (n − 1) − (d − 1) − dn₂e =

bn 2c − d. Then we have |Si| = bn₂c−d+d−1 d−1
= bn₂c−1 d−1
. Hence |Ad| = n − 2 d − 1  − d · _bn 2c − 1 d − 1  .

For any vertex in a d-regular tree, say v, let (tv v1, t v v2, · · · , t v vd) denote the orders of branches (Tv v1, T v v2, · · · , T v

vd) which is the decomposition of Gn in a

d-regular tree in which P

vi∈N (v)vi = n − 1. Consider (a1, a2, · · · , ad) ∈ Ad, v is a unique rumor center of any graph that the orders of branches of it is

satisfied tv_v1 = a1, tvv2 = a2, · · · , t

v

vd = ad.

We want to calculate the total number of ways to spread a rumor to n

vertices from v. In addition, the order of breaches of this form graph Gn is

(tv v1, t

v

v2, · · · , t

v

vd). First, assume that the vertex v had spread a rumor to u where u ∈ N (v), and then it can be spread to u’s descendants only. There

are d − 1 choice of the next infected vertices since u has d − 1 children.

Now, there are two vertices (u and a child of u) have rumors, then there are

2d − 3 choice of the next infected vertices. Therefore, the number of ways

to spread a rumor to m vertices in that graph Tu is m

Y

i=1

Hence, given (tv v1, t

v

v2, · · · , t

v

vd), the total number of ways to spread a rumor is (n − 1)! d Y k=1 Qtvvk i=1((d − 2) (i − 1) + 1) tv vk! . (15)

Note that the number of ways to permute n − 1 steps from different subtrees

is a permutation of the multiset {tv v1 · 1, t

v

v2· 2, · · · t

v vd· d}.

Let Pd(n) be the ratio of the number of ways to spread a rumor to n

vertices such that (tv_v1, tv_v2, · · · , tv_vd) ∈ An to the number of ways to spread a

rumor to n vertices such that (tv v1, t v v2, · · · , t v vd) ∈ Bn. Thus Pd(t) ≈ P (Et). We have Pd(n) = X (tv v1,tvv2,··· ,tvvd)∈Ad d Y k=1 Qt v vk k=1((d − 2) (i − 1) + 1) tv vk! ! X (tv v1,tvv2,··· ,tvvd)∈Bd d Y k=1 Qtvu k=1((d − 2) (i − 1) + 1) tv vk! ! . (16)

Theorem 3.1.3. If G is a 3-regular graph, then we have that

lim

t→∞P(Et) =

1 4.

Proof:

Let the source be v. By (16),

P3(n) = X (tv v1,tvv2,tvv3)∈A3 d Y k=1 Qt v vk i=1 i tv vk! ! X (tv v1,tvv2,tvv3)∈B3 d Y k=1 Qt v vk i=1 i tv vi! ! = X (tv v1,tvv2,tvv3)∈A3 1 X (tv v1,tvv2,tvv3)∈B3 1 = |A3| |B3| = n − 2 2  − 3 · _bn 2c − 1 2  n − 2 2  = 1 − 3 (bn 2c − 1)(b n 2c − 2) (n − 2)(n − 3) .

This implies that P3(n) =        1 − 3 4 (n − 4) (n − 3) if n is even, 1 − 3 4 (n − 5) (n − 2) if n is odd. (17)

Hence, the proof follows. _

3.2

Detection probabilities of d-regular trees, d ≥ 4

In what follows, we use (16) to calculate the detection probability of finding

the source in a d-regular tree, d ≥ 4. Let v be the source and let wdn =

Qn

i=1((d − 2) (i − 1) + 1) (14). We can rewrite it by (13).

Pd(n) = 1 − d X i=1 X (x1,x2,··· ,xd)∈Si  wdx1 x1! wdx2 x2! · · ·wdxd xd!  X (tv v1,tvv2,··· ,tvvd)∈Bd _w dtv v1 tv v1! wdtv v2 tv v2! · · ·wdt v vd tv vd!  . Let xd ≥ n₂, Pd(n) = 1 − d X (x1,x2,··· ,xd)∈Sd  wdx1 x1! wdx2 x2! · · ·wdxd xd!  X (tv v1,tvv2,··· ,tvvd)∈Bd _w dtv v1 tv v1! wdtv v2 tv v2! · · ·wdt v vd tv vd!  .

Let f (x) be the exponential generating function for the sequence {wdn}∞n=1.

And we have (1 − ax)−1a = ∞ X n=0 −1 a n  (−ax)n = 1 + ∞ X n=1 Qn i=1(a(i − 1) + 1) n! x n_{. (18)}

From (18), we immediately know f (x) = (1 − ax)−1a − 1 where a = d − 2. Let Fk(x) = (f (x))k. We have Fk(x) =  wd1 x 1!+ wd2 x2 2! + · · · k =(1 − ax)−1a − 1 k = k X l=0 k l  (1 − ax)−al(−1)k−l = (−1)k+ k X l=1 (−1)k−lk l  ∞ X n=0 −l a n  (−ax)n = (−1)k+ k X l=1 (−1)k−lk l  1 + ∞ X n=1 −l a n  (−ax)n ! = k X l=0 (−1)k−lk l  + k X l=1 (−1)k−lk l  ∞ X n=1 Qn i=1(l + a(i − 1)) n! x n = ∞ X n=1 k X l=1 (−1)k−lk l  Qn i=1(l + a(i − 1)) n! ! xn. Let [xn_{]F (x) be the coefficient of x}n _{in F (x). Then,}

[xn]Fk(x) = k X l=1 (−1)k−lk l  Qn i=1(l + a(i − 1)) n! . (19) The coefficient of xn in F (x) is wdn1 n1! w_dn2 n2! · · · w_dnk nk! whose degree is n = n1+ n2+ · · · + nk. Therefore, Let a = d − 2, the detection probability of finding

the source in a d-regular tree is

Pd(n) = 1 − d n−d X m≥n₂ Qm i=1(a (i − 1) + 1) m! [x n−1−m_]F d−1(x)
xn−1_{ F} d(x) . (20)

Clearly, this is too complicate to simplify the right side of (20). In Figure

5, we use computer to obtain the general behavior of this term for several

d’s and n ≤ 100, 000. As a matter of fact, we have 1₂ ≥ Pd(n) ≥ Pd0(n) ≥ 1

4

if d ≥ d0 ≥ 3

4

Conclusion

In this thesis, we have obtained a mathematical model for finding the

cul-prit who spreads rumors in a network defined on a d-regular tree (countably

infinite graph). We are concerned with the detection probabilities of finding

the culprit. By using this model, we are able to give a shorter and more

explicit proof for the cases when d ≤ 3. See [6] for a comparison.

Further-more, we can estimate the detection probabilities of finding the source in

d-regular trees for d > 3 by an explicit formula, though it is quite

com-plicate. It will be better if we can simplify the formula by using certain

combinatorial identities. Moreover, if we can reduce the estimation error

for general graphs, and generalize the estimator to networks with different

rumor spreading rate, then we have a much better result than the known

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