On rumour propagation among sceptics

TL;DR

This paper investigates models of rumour spread among sceptical individuals who require multiple sources for transmission, extending coverage models in stochastic geometry to include at least double coverage of points.

Contribution

It introduces and analyzes models of rumour propagation with sceptical agents requiring multiple confirmations, extending existing coverage models in stochastic geometry.

Findings

Sceptical transmission reduces the spread of rumours.

Double coverage models generalize traditional single coverage models.

Results provide insights into the robustness of information dissemination.

Abstract

Junior, Machado and Zuluaga (2011) studied a model to understand the spread of a rumour. Their model consists of individuals situated at the integer points of the line $\N$. An individual at the origin $0$ starts a rumour and passes it to all individuals in the interval $[0,R_0]$, where $R_0$ is a non-negative random variable. An individual located at $i$ in this interval receives the rumour and transmits it further among individuals in $[i, i+R_i]$ where $R_0$ and $R_i$ are i.i.d. random variables. The rumour spreads in this manner. An alternate model considers individuals seeking to find the rumour from individuals who have already heard it. For this s/he asks individuals to the left of her/him and lying in an interval of a random size. We study these two models, when the individuals are more sceptical and they transmit or accept the rumour only if they receive it from at least two different sources. In stochastic geometry the equivalent of this rumour process is the study of coverage of the space $\N^d$ by random sets. Our study here extends the study of coverage of space and considers the case when each vertex of $\N^d$ is covered by at least two distinct random sets.

Equations

$$X_{\mathbf{i}}=\{\begin{array}{ll}1& \text{ with probability }p\\ 0& \text{ with probability }1-p.\end{array}$$

$$C$$

$$C_{\text{rev}}$$

$$D$$

$$\text{ and }x\in ([j,j+{\rho }_j]\cap [k,k+{\rho }_k])\},$$

$$D_{\text{rev}}$$

$$-1\le j\ne k<i\text{ with }{X}_j=X_k=1\text{ and }j,k\in [i-{\rho }_i,i]\}.$$

$$l:=\underset{j\to \infty }{\mathrm{liminf}}j\mathbb{P}(\rho \ge j)>1\text{ and }L:=\underset{j\to \infty }{\mathrm{limsup}}j\mathbb{P}(\rho \ge j)<\infty .$$

$$\mathbb{P}\{D\supseteq [t,\infty )\text{ for some finite }t\}=\{\begin{array}{ll}1& \text{ if }p>1/l\\ 0& \text{ if }p<1/L.\end{array}$$

$$\mathbf{C}:={\cup }_{\{\mathbf{i}:X_{\mathbf{i}}=1\}}(\mathbf{i}+[0,{\rho }_{\mathbf{i}}{]}^d)$$

$$\mathbf{D}$$

$$\text{ such that }x\in (\mathbf{i}+[0,{\rho }_{\mathbf{i}}{]}^d)\cap (\mathbf{j}+[0,{\rho }_{\mathbf{j}}{]}^d)\}.$$

$$l:=\underset{j\to \infty }{\mathrm{liminf}}j\mathbb{P}(\rho \ge j)>1\text{ and }L:=\underset{j\to \infty }{\mathrm{limsup}}j\mathbb{P}(\rho \ge j)<\infty .$$

$${\mathbb{P}}_p(\mathbb{N}\text{ is eventually doubly covered})=\{\begin{array}{ll}1& \text{if }p>1/l\\ 0& \text{if }p<1/L.\end{array}$$

$${\mathbb{P}}_p({\mathbb{N}}^d\text{ is eventually doubly covered})=\{\begin{array}{ll}1& \text{if }{\mathrm{liminf}}_{j\to \infty }j\mathbb{P}(\rho \ge j)>0\\ 0& \text{if }{\lim }_{j\to \infty }j\mathbb{P}(\rho \ge j)=0.\end{array}$$

$${\mathbf{C}}_{\mathrm{rev}}:={\cup }_{\{\mathbf{i}:X_{\mathbf{i}}=1\}}(\mathbf{i}+[-{\rho }_{\mathbf{i}},0{]}^d)$$

$${\mathbf{D}}_{\mathrm{rev}}:=$$

$$\text{ and there exist }\mathbf{j}\ne \mathbf{k},\mathbf{j},\mathbf{k}\in \left({\mathbb{N}}^d\cup \{-1,0\}\setminus \{\mathbf{i}\}\right)$$

$$\text{such that }{X}_{\mathbf{j}}=X_{\mathbf{k}}=1\text{ and }\mathbf{j},\mathbf{k}\in (\mathbf{i}+[-{\rho }_{\mathbf{i}},0{]}^d)\}.$$

$$A_i:=\{i\notin C\}\text{ and }{B}_i:=\{i\notin D\}.$$

$${\mathbb{P}}_p(A_i)=\prod _{l=0}^{i-1}{\mathbb{P}}_p\left(\{X_{i-l}=0\}\cup \{X_{i-l}=1\text{ and }{\rho }_{i-l}<l\}\right)=\prod _{l=0}^{i-1}{g}_p(l),$$

$${\mathbb{P}}_p(B_i)$$

$$=$$

$$\sum _i{\mathbb{P}}_p(B_i)<\infty ,$$

$$\sum _i\prod _{l=1}^{i-1}{g}_p(l)<\infty \text{ for }p>1/l.$$

$$\sum _{i=1}^{\infty }\sum _{k=1}^{i-1}G(k)\prod _{l\ne k,l=1}^{i-1}{g}_p(l)<\infty .$$

$$\sum _{k=1}^{i-1}G(k)\prod _{l\ne k,l=1}^{i-1}{g}_p(l)$$

$$+\prod _{l=1}^{i_0}{g}_p(l)\left\{\sum _{k=i_0+1}^{i-1}G(k)\prod _{l\ne k,l=i_0+1}^{i-1}{g}_p(l)\right\}.$$

$$\sum _{i=i_0+1}^{\infty }\sum _{k=i_0+1}^{i-1}G(k)\prod _{l\ne k,l=i_0+1}^{i-1}{g}_p(l)$$

$$<\infty .$$

$$b_k$$

$$=G(k)\prod _{l=i_0+1}^{k-1}{g}_p(l)\left\{1+\sum _{i=k+2}^{\infty }\prod _{l=k+1}^{i-1}{g}_p(l)\right\}.$$

$$\frac{b_{k+1}}{b_k}$$

$$\frac{G(k+1)}{G(k)}\le 1\text{ and }{g}_p(k)\le 1\text{ for all }k.$$

$$\sum _{i=k+3}^{\infty }\prod _{l=k+2}^{i-1}{g}_p(l)<0.1\text{ for all }k\text{ large enough.}$$

Full text

In stochastic geometry the equivalent of this rumour process is the study of coverage of the space $\mathbb{N}^{d}$ by random sets. Our study here extends the study of coverage of space and considers the case when each vertex of $\mathbb{N}^{d}$ is covered by at least two distinct random sets.

AMS Classification: 60K35

††journal: To appear in the Journal of Statistical Physics

1 Introduction

Gilbert (1961) introduced a model of transmission of information. This model consisted of a signal being transmitted through a relay of transmitters to its recipient. Variants of this process were later introduced and described in Maki and Thompson (1973). Two such versions are the Poisson Boolean model and the rumour processes. We briefly describe these processes here and indicate some literature connected with them.

Poisson Boolean model: The Poisson Boolean model consists of a homogenous Poisson point process $\Xi:=(\xi_{1},\xi_{2},\ldots)$ on $\mathbb{R}^{d}$ of intensity $\lambda$. Along with this is an independent collection of i.i.d. positive real valued random variables $\{\rho_{1},\rho_{2},\ldots\}$. The covered region of the Boolean model is defined to be the random set $C:=\cup_{i=1}^{\infty}B(\xi_{i},\rho_{i})$, where $B(\xi,\rho)$ is the closed ball centred at $\xi$ and of radius $\rho$ in the Euclidean norm.

This Boolean model is used to study coverage properties of space in stochastic geometry. Matheron (1968) used this Boolean model to study natural images and their occlusion. Later, other geometric properties were studied, see e.g., Hall (1988) and Chiu, Stoyan, Kendall and Mecke (2013) for further details. Kertesz and Vicsek (1982) picked up this model as a natural extension on the continuum space $\mathbb{R}^{d}$ of unoriented bond/site percolation in physics. The percolation parameter being the intensity $\lambda$ with the radius random variables $\rho_{1},\rho_{2},\ldots$ being either constants or of a fixed distribution. A review of the mathematical details of this percolation model and its variant, the random connection model, may be seen in Meester and Roy (1996) and Penrose (2003). Gupta and Kumar (1998) used this model to study questions of signal-to-interference-ratio (SINR) and other such problems in wireless transmission, see Franciscceti and Meester (2007) for a review.

Rumour process: Sudbury (1985) started the mathematical study of this variant of the information-transmission model introduced in Maki and Thompson (1973). Subsequently, Junior, Machado and Zuluaga (2011) christened this model as the ‘firework process’ and introduced a different variant the ‘reverse firework process’. Both these models are defined on a discrete graph as opposed to the Boolean model. For convenience we describe them for the non-negative integer line $\mathbb{N}$.

*Firework process: * The homogenous firework process consists of a sequence of non-negative integer valued i.i.d. random variables $\{R_{i}:i\geq 0\}$. At time [math] the origin starts a rumour and passes it onto all individuals in the interval $[0,R_{0}]$, and at time $t$, all individuals who heard the rumour for the first time at time $t-1$ spread the rumour, with the individual at site $j$ spreading it among all individuals in the region $[j,j+R_{j}]$. Note that allowing $\mathbb{P}\{R_{j}=0\}>0$ ensures that there are individuals who are inactive. The heterogeneous firework process removes the restriction on the random variables $\{R_{i}:i\geq 0\}$ of being identically distributed.

*Reverse firework process: * The reverse firework process consists of the origin who knows the rumour at time [math], and at time $t$ an individual located at site $j$ listens to individuals in the interval $[j-R_{j},j]$. If there is an individual at a site in this interval who has heard the rumour by time $t-1$, then the individual at site $j$ gets to know the rumour. Here the random variables $\{R_{i}:i\geq 0\}$ are as before, and accordingly we have a homogenous or heterogeneous version of the model.

Both these models have been extensively studied as models for spread of information on networks. Gallo, Garcia, Junior and Rodríguez (2014) exploit a relationship between the rumour processes and a specific discrete time renewal process to obtain various results. Junior, Machado and Zuluaga (2014) study a version of the firework process on homogeneous trees. Bertacchi and Zucca (2013) study these processes in a random environment. Junior, Machado and Ravishankar (2016) summarises the current state of the research on these topics.

In this paper, we study the spread of rumours among sceptics. Sceptics being those individuals who need at least $2$ different individuals from whom they receive the rumour before transmitting/accepting. The results are valid for sceptics who need the information from $k$ or more sources for acceptance or transmission, however for the sake of simplicity we restrict ourselves to $k=2$.

For Boolean models vis-à-vis coverage of space, Athreya, Roy and Sarkar (2004) introduced a notion of ‘eventual coverage’ which, for 1-dimension, is equivalent to the notion of percolation of the firework process as will be explained in the next section. This study was not restricted to the half-line, but more generally to a quadrant in $d$-dimensional space. Our paper extends this study to ‘double coverage’, i.e., the region in the quadrant which is covered by at least $2$ distinct random sets.

In the next section we present the formal set-up of the questions we study, and state the results. In Sections 3 and 4 we prove the propositions, and in Section 5 we state and explain in brief the version of the coverage result for the Poisson Boolean model.

2 The models and the statement of results

We present an alternate but equivalent formulation of the homogenous firework and the reverse firework processes. Let $X_{1},X_{2},\ldots$ be i.i.d. Bernoulli ($p$) random variables, i.e.,

[TABLE]

Also let $\{\rho_{i}:i\geq 1\}$ be a collection of i.i.d. $\mathbb{N}$ valued random variables, independent of the collection $\{X_{i}:i\geq 1\}$. Let $\rho$ denote a generic random variable with the same distribution as $\rho_{i}$. In addition, let $\rho_{0}$ be an independent $\mathbb{N}$ valued random variable, independent of the collections $\{X_{i}:i\geq 1\}$ and $\{\rho_{i}:i\geq 1\}$, with $\rho_{0}$ having the same distribution as $\rho$. Taking $X_{0}\equiv 1$, consider the regions

[TABLE]

The firework processes presented in the earlier section may be seen to be equivalent to this formulation by taking $p=\mathbb{P}\{R_{0}>0\}$ and $\rho$ to have the same distribution as that of $R_{0}|R_{0}>0$.

Remark 1

A simple argument using Kolmogorov’s [math]-$1$ law yields that $C$ (or $C_{\text{rev}}$) being an unbounded connected region with positive probability is equivalent to $C$ (or $C_{\text{rev}}$) containing a region $[t,\infty)$, for some $t\geq 1$ with probability $1$.

We study the spread of the rumour among sceptics. If individual at $i$ receives the rumour from at least two distinct sources, then s/he transmits it to all individuals in the region $[i,i+\rho_{i}]$, i.e.,

• (A)

for the firework process, there are two individuals at $j$ and $k$ (say) with $j\neq k$, $X_{j}=X_{k}=1$ and $-1\leq j,k<i$ such that $i\in[j,j+\rho_{j}]\cap[k,k+\rho_{k}]$; here we assume that there is an individual at location $-1$, who spreads the rumour in the region $[-1,-1+\rho_{-1}]$, where $\rho_{0}$ and $\rho_{-1}$ are two independent random variables, independent of all other random variables, and each having the same distribution and $X_{0}=X_{-1}\equiv 1$;

• (B)

for the reverse firework process, there are two individuals at $j$ and $k$ (say) with $j\neq k$, $X_{j}=X_{k}=1$ and $-1\leq j,k<i$ such that $j,k\in[{i}-\rho_{i},{i}]$, where $\rho_{0}$ and $\rho_{-1}$ are two independent random variables, independent of all other random variables, and each having the same distribution as $\rho$ and $X_{0}=X_{-1}\equiv 1$.

Towards this end we define the regions

[TABLE]

We look for conditions on the processes $\{X_{i}:i\geq 1\}$ and $\{\rho_{i}:i\geq 1\}$ such that the regions $D$ and $D_{\text{rev}}$ are unbounded connected regions with positive probability.

Remark 2

In the set-up (A) (or in (B)), as in the case of rumour propagation in $C$ (or $C_{\text{rev}}$), a simple argument using Kolmogorov’s [math]-$1$ law yields that $D$ (or $D_{\text{rev}}$) being an unbounded connected region with positive probability is equivalent to $D$ (or $D_{\text{rev}}$) containing a region $[t,\infty)$, for some $t\geq 1$ with probability $1$. Remark 5 given in the next section spells out the details.

Thus we define

Definition 1

The firework process (respectively, reverse firework process) percolates among sceptics if, with probability $1$, $D$ (respectively, $D_{\text{rev}}$) contains a region $[t,\infty)$, for some $t\geq 1$.

Remark 3

Using this equivalent definition obtained by the tail event properties allows us to study the model without considering the influence of the two initiators $X_{-1}$ and $X_{0}$. Indeed their influence will only be upto $\max\{\rho_{-1},\rho_{0}\}$.

Proposition 1

Let

[TABLE]

We have that the firework process given in (A) percolates among sceptics if $p>1/l$ and does not percolate if $p<1/L$, i.e.,

[TABLE]

For the reverse firework process we have

Proposition 2

For $p>0$, the reverse firework process given in (B) percolates among sceptics if and only if $\mathbb{E}(\rho)=\infty$, and, in case percolation occurs $\mathbb{P}\{D_{\text{rev}}=\mathbb{N}^{d}\}=1$.

We note here that these are the same conditions that Junior, Machado and Zuluaga (2011) obtain for percolation among ‘non-sceptical’ individuals. Indeed, the above propositions also go through among more radical sceptics, i.e. if individuals need to receive the rumour from $k\geq 1$ distinct sources before they transmit/believe the rumour.

As noted in the review article Junior, Machado and Ravishankar (2016), the above model is related to the study of coverage processes in stochastic geometry. Athreya, Roy and Sarkar (2004) introduce a notion of ‘eventual coverage’ which, for 1-dimension, is identical to the equivalent formulation of the percolation of rumour process as given in Remark 1. We state the model in brief here and present the results obtained. Their formulation considers $\{X_{\mathbf{i}}:{\mathbf{i}}\in\mathbb{N}^{d}\}$ to be a collection of i.i.d. Bernoulli ($p$) random variables and $\{\rho_{\mathbf{i}}:{\mathbf{i}}\in\mathbb{N}^{d}\}$ a collection of i.i.d. $\mathbb{N}$ valued random variables, independent of the collection $\{X_{\mathbf{i}}:{\mathbf{i}}\in\mathbb{N}^{d}\}$. Let $\rho$ denote a generic random variable with the same distribution as $\rho_{\mathbf{i}}$ and

[TABLE]

denote the covered region of $\mathbb{N}^{d}$; here and subsequently ${\mathbf{i}}+[0,\rho_{\mathbf{i}}]^{d}=[i_{1},i_{1}+\rho_{\mathbf{i}}]\times\cdots\times[i_{d},i_{d}+\rho_{\mathbf{i}}]$, where ${\mathbf{i}}=(i_{1},\ldots,i_{d})$.

Definition 2

$\mathbb{N}^{d}$ is eventually covered if there exists ${\mathbf{t}}\in\mathbb{N}^{d}$ such that ${\mathbf{t}}+\mathbb{N}^{d}\subseteq{\mathbf{C}}$.

Remark 4

From Remark 1, the above definition may be seen to be equivalent to percolation of the homogenous firework process for $d=1$, and in that sense, it extends the definition of percolation for a homogenous firework process in $\mathbb{N}^{d}$.

For our purposes we define

Definition 3

$\mathbb{N}^{d}$ is eventually doubly covered if there exists ${\mathbf{t}}\in\mathbb{N}^{d}$ such that ${\mathbf{t}}+\mathbb{N}^{d}\subseteq{\mathbf{D}}$, where

[TABLE]

We have

Proposition 3

(i) For $d=1$, let

[TABLE]

We have

[TABLE]

(ii) For $d\geq 2$ and $p>0$, we have

[TABLE]

Apropos the reverse firework process in higher dimensions, for $\{X_{\mathbf{i}}:{\mathbf{i}}\in\mathbb{N}^{d}\}$ and $\{\rho_{\mathbf{i}}:{\mathbf{i}}\in\mathbb{N}^{d}\}$ as earlier, let

[TABLE]

and for $-{\mathbf{1}}=(-1,-1,\ldots,-1)$, ${\mathbf{0}}=(0,0,\ldots,0)$, $X_{\mathbf{1}}=X_{\mathbf{0}}=1$,

[TABLE]

Proposition 4

For $p>0$, the reverse firework process on $\mathbb{N}^{d}$ percolates among sceptics if and only if $\mathbb{E}(\rho^{d})=\infty$, and, in case percolation occurs $\mathbb{P}\{D_{\text{rev}}=\mathbb{N}^{d}\}=1$.

Finally, because of the binomial approximation of the Poisson process, Proposition 3 has a natural extension to Poisson processes. This is relegated to the last section of this paper.

In stochastic geometry the notion of coverage of space has received extensive attention. In particular Hall (1988) and Chiu, et al (2013) provide a review of the topics studied. Our endeavour in this paper may be viewed as an effort to introduce a notion of ‘reinforced coverage’.

3 Proofs of Propositions 1 and 2

**Proof of Proposition 1: ** Let the random regions $C$ and $D$ be as in (2) and (2) respectively. In view of Remark 3, we may simplify the process to be defined only for the positive integers and ignore the individuals located at [math] and $-1$. Indeed, for any sample point $\omega$, if $D(\omega)$ contains an interval $[t,\infty)$ for some finite $t$, it will also contain the interval $[\max\{t,\rho_{0}(\omega),\rho_{1}(\omega)\},\infty)$, see also Remark 5 given later. Thus with a slight abuse of notation, we define $C$ and $D$ from only site $1$ onwards.

For $p>0$ and $i\geq 1$, let

[TABLE]

Taking $G(i)=\mathbb{P}(\rho\geq i)$ and $g_{p}(i)=1-pG(i)$, we observe that

[TABLE]

Now suppose $p>1/l$, where $l$ is as in Proposition 1. We will show that

[TABLE]

which, by Borel-Cantelli lemma yields $\mathbb{P}_{p}(B_{i}\text{ occurs finitely often})=1$, i.e. there is a random variable $T$, with $T<\infty$ almost surely, such that $\mathbb{P}_{p}\{D\supseteq[T,\infty)\}=1$.

From the proof of Proposition 3.1 (a) of Athreya, Roy and Sarkar (2004) we have that

[TABLE]

Thus, from expression (3), to show (5), we need to show

[TABLE]

Towards this end we note that since $p>1/l$ there exists $\eta>1$ such that $l>\eta/p$. Also fix $i_{0}$ such that $iG(i)>\frac{\eta}{p}$ for all $i\geq i_{0}$. Now

[TABLE]

From our observation (6) and since $\sum_{k=1}^{i_{0}}G(k)\prod_{l\neq k,l=1}^{i_{0}}g_{p}(l)$ is a constant for fixed $i_{0}$, to show (7) we only need to show

[TABLE]

In preparation for the ratio test we will use to show (3), let

[TABLE]

Also

[TABLE]

Since we will use such computations again in the next section, we elaborate the details here. For the numerator in the above expression, first note that

[TABLE]

From (6) we have $\sum_{i=k+3}^{\infty}\prod_{l=1}^{i-1}g_{p}(l)=\prod_{l=1}^{k+1}g_{p}(l)\sum_{i=k+3}^{\infty}\prod_{l=k+2}^{i-1}g_{p}(l)<\infty$ and thus, $\sum_{i=k+3}^{\infty}\prod_{l=k+2}^{i-1}g_{p}(l)<\infty$ for fixed $k\geq 1$, i.e.,

[TABLE]

Hence, till now we have

[TABLE]

Writing $\displaystyle\sum_{i=k+2}^{\infty}\displaystyle\prod_{l=k+1}^{i-1}g_{p}(l)=g_{p}(k+1)\left[1+\displaystyle\sum_{i=k+3}^{\infty}\displaystyle\prod_{l=k+2}^{i-1}g_{p}(l)\right]$, and noting that $G(k)\downarrow 0$ as $k\uparrow\infty$, we have

[TABLE]

Since $\displaystyle\sum_{i=k+3}^{\infty}\displaystyle\prod_{l=k+2}^{i-1}g_{p}(l)\geq 0$, we have from (12) and (13),

[TABLE]

The ratio test proves (3) which shows that (5) holds.

From Proposition 3.1 (b) of Athreya, Roy and Sarkar (2004), we know \mathbb{P}_{p}(C\supseteq[t,\infty)\text{ for some t finite})=0 for $p<1/L$ and $L:=\limsup_{j\to\infty}j\mathbb{P}(\rho>j)<\infty$. Now \{D\supseteq[t,\infty)\text{ for some t}\}\subseteq\{C\supseteq[t,\infty)\text{ for some t}\} which completes the proof of Proposition 1. $\square$

Remark 5

The Borel-Cantelli argument used in the proof gives that the random variable $T$ defined at the beginning of this section is finite almost surely. Thus, given $\epsilon>0$, we may obtain $m$ such that $\mathbb{P}(T\leq m)>\epsilon$. Now the probability that $X_{i}=1$ for $i=1,\ldots,m$ is $p^{m}>0$. Hence, an application of the FKG inequality yields

[TABLE]

This corroborates Remark 2.

Proof of Proposition 2: We present an argument here which also simplifies the proof of Proposition 2.3 (i) of Junior, Machado and Zuluaga (2011).

Fix $p>0$. Let $A_{n}:=\{X_{n}=1\text{ and }\rho_{n}\geq n+1\}$. Thus, if $A_{n}$ occurs, the individual at site $n$ can access the rumour from the two individuals at sites $-1$ and [math]. Noting that (i) $\{A_{n}:n\geq 1\}$ is a collection of independent events, (ii) $\mathbb{P}_{p}(A_{n})=p\mathbb{P}(\rho\geq n+1)$ and (iii) $\mathbb{E}(\rho)=\infty$ implies that $\sum_{n\geq 1}\mathbb{P}(\rho\geq n+1)=\infty$, we have from Borel-Cantelli lemma, $\mathbb{P}_{p}(A_{n}\text{ occurs infinitely often})=1$. Thus $\mathbb{P}_{p}(D_{\text{rev}}=\mathbb{N})=1$ whenever $\mathbb{E}(\rho)=\infty$.

To complete the proof of Proposition 2 we note that Proposition 2.3 (ii) of Junior, Machado and Zuluaga (2011) says $\mathbb{P}_{p}(C_{\text{rev}}\supseteq[t,\infty)\text{ for some }t)=0$ whenever $\mathbb{E}(\rho)<\infty$. Since $D_{\text{rev}}\subseteq C_{\text{rev}}$, the proof is complete. $\square$

Remark 6

The proof of Proposition 2 above reinforces our contention in Section 1 that our results are valid for any ‘radical’ sceptic. Indeed, whenever $\mathbb{E}(\rho)=\infty$ we have $\mathbb{P}\{$there is an infinite increasing sequence $n_{1},n_{2},\ldots$ such that for any $k$, the individuals $n_{j},\;j\geq k$ have heard the rumour from at least $k$ different sources$\}=1$. However for extending the proof of Proposition 1 a bit more work has to be done.

4 Proofs of Propositions 3 and 4

Proof of Proposition 3: As mentioned earlier Proposition 3 (i) is just Proposition 1 rephrased. Thus we need to prove Proposition 3 (ii).

First note that from Proposition 3.2(a) of Athreya, Roy and Sarkar (2004), we know that if $\lim_{j\longrightarrow\infty}j\mathbb{P}(\rho\geq j)=0$ then $\mathbb{P}_{p}(\mathbb{N}^{d}\text{ is eventually covered})=0$, and so

[TABLE]

We prove Proposition 3 (ii) for the case $d=2$; the proof carries through in a straightforward fashion for higher dimensions. Since the proof is technical and long, we break it up into various steps.

Step 1: Prelude

Fix $0<p<1$ and we assume that $\liminf_{j\longrightarrow\infty}j\mathbb{P}(\rho\geq j)>0$.

For $(i,j)\in\mathbb{N}^{2}$, define

[TABLE]

If we show that, for some $N\geq 1$,

[TABLE]

then Borel-Cantelli lemma guarantees that, with probability $1$, there exists $N_{0}\geq 1$ such that $(i,j)\in{\bf D}$ for all $i,j\geq N_{0}$, i.e. we have eventual coverage. Also an argument similar to that in Remark 5 shows that, in this case, percolation occurs among sceptics in $d=2$.

Step 2: Understanding the event $B_{i,j}$

Let $A_{i,j}:=\{(i,j)\not\in{\bf C}\}$. There are exactly three ways that $(i,j)\not\in{\bf D}$:

• (i)

$(i,j)\not\in{\bf C}$, see Figure 1(a). As noted earlier, the probability that this occurs is

[TABLE]

• (ii)

there exists exactly one vertex $(k,l)$ such that $(k,l)$ is open, $0\leq t:=\max\{i-k,j-l\}\leq j-1$ and $\rho_{(k,l)}\geq t$, see Figure 1(b). The probability that this occurs is

$b_{i,j}:=\prod_{k=1}^{i-j}g_{p}(k+j-1)^{j}\sum_{t=0}^{j-1}pG(t)g_{p}(t)^{2t}\prod_{l\neq t,l=0}^{j-1}g_{p}(l)^{2l+1}$.

• (iii)

there exists exactly one vertex $(k,l)$ such that $(k,l)$ is open, $1\leq k\leq i-j$, $1\leq l\leq j\}$ and $\rho_{(k,l)}\geq i-k$, see Figure 1(c). The probability that this occurs is

[TABLE]

Thus, for $i\geq j$, and from some elementary calculations,

[TABLE]

Step 3: Setting up the estimates

Before we proceed further we fix some quantities. Since $\displaystyle\liminf_{j\longrightarrow\infty}j\mathbb{P}(\rho\geq j)>0$ we may choose $\eta>0$ such that $\displaystyle\liminf_{j\longrightarrow\infty}j\mathbb{P}(\rho\geq j)>\eta$. Also, for this $\eta$ and our fixed $p\in(0,1)$ let $a$ be such $0<e^{-p\eta}<a<1$. Now we choose $N\geq 1$ such that for all $j\geq N$ the following hold:

[TABLE]

Note that (ii) above guarantees that for all $j\geq N$, we have $g_{p}(j)^{j}<a$.

From the proof of Proposition 3.2(b) of Athreya, Roy and Sarkar (2004) we know that, for $j\geq N$,

[TABLE]

Thus, from (4) if we show that, for $j\geq N$,

[TABLE]

then we will have, for fixed $j$,

[TABLE]

Step 4: The sum in (17)

Note that, $\displaystyle\sum_{t=0}^{i-1}\frac{G(t)}{g_{p}(t)}=\displaystyle\sum_{t=0}^{j-1}\frac{G(t)}{g_{p}(t)}+\displaystyle\sum_{k=1}^{i-j}\frac{G(k+j-1)}{g_{p}(k+j-1)}$. For fixed $j\geq N$,

[TABLE]

Taking

[TABLE]

as the inner sum in (4), and breaking the sum as $i=k+1$ and $i\geq k+2$, we have

[TABLE]

where

[TABLE]

Similarly, for $e_{k+1}$, as in the term in the first equality above, we have

[TABLE]

We see that

[TABLE]

because $0\leq\frac{G(k+1)}{G(k)}\leq 1$ and $0<g_{p}(k)\leq 1$ for all $k\geq 0$. Also, for fixed $j$,

• (i)

for all $k$ large enough, $0.9<g_{p}(k+1)^{j}\leq 1$;

• (ii)

for $j\geq N$ (as chosen for (4)), from equation (3.5) of Athreya, Roy and Sarkar (2004), we have $\sum_{i=k+3}^{\infty}\prod_{l=k+2}^{i-1}g_{p}(l)^{j-1}<\infty$ and hence

[TABLE]

which ensures that, for all $k$ large enough, $1\leq C(k,j)<1.1$.

Thus, for $j\geq N$ and all $k$ large enough, we have $\frac{e_{k+1}}{e_{k}}<\frac{1.1}{1.9},$ and so, by ratio test, $\sum_{k=1}^{\infty}e_{k}<\infty$. This shows, from (4), that (17) and thereby (18) hold.

Step 5: Understanding $\sum_{i,j\geq N}\mathbb{P}_{p}(B_{i,j})$

Now we show that, for $N$ as above, $\sum_{i,j\geq N}\mathbb{P}_{p}(B_{i,j})<\infty$. Towards this, we first observe that, for $i,j\geq 1$, by symmetry we have $\mathbb{P}_{p}(B_{i,j})=\mathbb{P}_{p}(B_{j,i})$, thus we need to show

[TABLE]

We will show separately that

[TABLE]

Noting that $i>j$ in the first the sum above and $i=j$ in the next sum, we have, from the argument leading to (4),

[TABLE]

From the proof of Proposition 3.2(b) of Athreya, Roy and Sarkar (2004) we know that,

[TABLE]

thus we need to show that

[TABLE]

Step 6: The first sum in (22)

For the first sum, interchanging the order of the summations we have

[TABLE]

Breaking up the inner sum according to the values taken by $t$ in the expression on the right side above, we have

[TABLE]

From the proof of Proposition 3.2(b) of Athreya, Roy and Sarkar (2004) we know that,

[TABLE]

and hence the first term in the right side of (4) is finite. Therefore, to show the first part of (22), we need to show that

[TABLE]

Now $\sum_{j=N}^{\infty}\sum_{i=\max(t+1,j+1)}^{\infty}\mathbb{P}_{p}(A_{i,j})=\sum_{m=0}^{\infty}\sum_{r=\max(t-N+1,m+1)}^{\infty}\mathbb{P}_{p}(A_{N+r,N+m})$, and breaking up the summation according to the values taken by $m$, we have

[TABLE]

where we have used the expression for $\mathbb{P}(A_{i,j})$ as given in (14).

To simplify the expressions we take $\sigma_{t}:=\frac{G(t)}{g_{p}(t)}$ and $s_{m}:=\displaystyle\prod_{l=0}^{N+m-1}g_{p}(l)^{2l+1}$. Using this notation, from the previous two equations we have

[TABLE]

We start with the first term on the right in the above equation. Reordering the sums, we have

[TABLE]

Let

[TABLE]

denote the summand. Observe that

[TABLE]

For the denominator above, we see

[TABLE]

While, for the numerator of (4), noting that our choice of $N$ and $0<a<1$ as in (4), implies $g_{p}(N+m)^{N+m}<a$, we have

[TABLE]

with the strict inequality above holding if

[TABLE]

Thus, if (27) holds, then, from (4), we have $\frac{\alpha_{m+1}}{\alpha_{m}}<a$; and so an application of the ratio test yields that the sum in (4) is finite.

To show (27) we again apply a ratio test. First we recall that from the proof of Proposition 3.2(b) of Athreya, Roy and Sarkar (2004) we have

[TABLE]

Let $\tau_{t}:=\sigma_{t}\sum_{r=t+1-N}^{\infty}\prod_{k=1}^{r-m-1}g_{p}(k+N+m)^{N+m+1}$. From (28) we have $\tau_{t}<\infty$.

Also

[TABLE]

From (28) we have $\displaystyle\sum_{r=t+2-N-m}^{\infty}\displaystyle\prod_{k=t+2-N-m}^{r}g_{p}(k+N+m)^{N+m+1}<\infty$ and so we may obtain a $t_{0}$ such that, for all $t\geq t_{0}$,

• (i)

$1\leq 1+\displaystyle\sum_{r=t+2-N-m}^{\infty}\displaystyle\prod_{k=t+2-N-m}^{r}g_{p}(k+N+m)^{N+m+1}<1.1$ and

• (ii)

$g_{p}(t+1)^{N+m+1}\left(1+\displaystyle\sum_{r=t+2-N-m}^{\infty}\displaystyle\prod_{k=t+2-N-m}^{r}g_{p}(k+N+m)^{N+m+1}\right)>0.9$.

This choice of $t_{0}$ ensures that for all $t\geq t_{0}$, (27) holds. Thus the first term on the right of (4) is finite.

A similar calculation and a use of ratio test shows that the second term on the right of (4) is finite. This shows that the first sum in (22) is finite.

Step 7: The second sum in (22)

Reordering the second sum in (22) and using the notation we introduced earlier, we have

[TABLE]

From the end of Section 3.1 of Athreya, Roy and Sarkar (2004) we know that with $N$ as above, $\sum_{i=N}^{\infty}\sigma_{t}\mathbb{P}_{p}(A_{i,i})<\infty$ and hence $\sum_{t=0}^{N-1}\sum_{i=N}^{\infty}\sigma_{t}\mathbb{P}_{p}(A_{i,i})<\infty$.

Expanding the term $\mathbb{P}_{p}(A_{i,i})$, we have

[TABLE]

Taking $a_{t}:=\sigma_{t}\sum_{i=t+1}^{\infty}\prod_{l=0}^{i-1}g_{p}(l)^{2l+1}$ and $l_{t}:=\sum_{r=t+2}^{\infty}\prod_{l=t+2}^{r}g_{p}(l)^{2l+1}$ , we see that

[TABLE]

from which we have

[TABLE]

Also, with $a$ as in (4),

[TABLE]

so, by the ratio test $l_{t}<\infty$ for $t\geq N$. Now, since $l_{t}\to 0$ as $t\to\infty$, from the remark following (4) we may obtain a $t_{0}$ such that for $t\geq t_{0}$, we have $g_{p}(t+1)^{2t+2}(1+l_{t})<a<1$. Also $\frac{G(t+1)}{G(t)}<1$ for all $t$, thus

[TABLE]

and by ratio test we have that the sum in (29) is finite.

This completes the proof of Proposition 3 (ii). $\square$

Proof of Proposition 4: Proposition 2 studied the 1-dimensional case of this proposition. Since the proof is along the same lines for $d\geq 2$, we provide a sketch of the argument for $d=2$.

Let $L_{n}:=\{(k,l):\text{ either (i) }k=n\text{ and }1\leq l\leq n\text{ or (ii) }l=n\text{ and }1\leq k\leq n\}$ and $A_{n}=\{-{\mathbf{1}},{\mathbf{0}}\in({\mathbf{i}}+[-\rho_{\mathbf{i}},0]^{2})\text{ for some }{\mathbf{i}}\in L_{n}\text{ with }X_{\mathbf{i}}=1\}$. Clearly $\mathbb{P}_{p}(A_{n})=1-\left(1-p\mathbb{P}(\rho\geq n+1)\right)^{2n-1}\sim p(2n-1)\mathbb{P}(\rho\geq n+1)$, so that $\sum_{n}\mathbb{P}(A_{n})=\infty$ if and only if $\sum_{n}(2n-1)\mathbb{P}(\rho\geq n+1)=\infty$, i.e. if and only if $\mathbb{E}(\rho^{2})=\infty$. Thus, by an application of the Borel-Cantelli lemma, we have $\mathbb{P}_{p}({\mathbf{D}}_{\text{rev}}=\mathbb{N}^{2})=1$ whenever $\mathbb{E}(\rho^{2})=\infty$.

Conversely, for $t\in\mathbb{N}$, let $\bar{{\mathbf{t}}}=(t,t)$ and $B_{m}:=\{\bar{{\mathbf{t}}}\in{\mathbf{C}}_{\rm{rev}}\text{ for all }t\geq m\}$. We will show that, if $\mathbb{E}(\rho^{2})<\infty$, the event $B:=\{\text{there exists }M\geq 1\text{ such that }B_{m}\text{ occurs for all }m\geq M\}$ is a null event, thereby implying that

[TABLE]

and hence the firework process does not percolate.

Towards this we first note that $B$ is a tail event vis-à-vis the collections of random variables $\{\chi_{k}\}_{k\geq 1}$ where $\chi_{k}=\{X_{\mathbf{i}}:\mathbf{i}=(i_{1},i_{2})\in\mathbb{N}^{2}\text{ with }i_{1},i_{2}\geq k\}$. Moreover, for any $\bar{\mathbf{t}}$ as in the calculations leading to the Borel-Cantelli lemma above, we have $\mathbb{P}\{\bar{\mathbf{t}}\not\in{\mathbf{C}}_{\rm{rev}}\}>0$ whenever $\mathbb{E}(\rho^{2})<\infty$. This completes the proof of Proposition 4. $\square$

5 Poisson Boolean model

A comparison argument with the discrete process, as in Athreya, Roy and Sarkar (2004) gives the us the following results for the Poisson Boolean model.

Let $x_{1},x_{2},\ldots$ be a Poisson process on $(0,\infty)^{d}$ of intensity $\lambda$ and $\rho,\rho_{1},\rho_{2},\ldots$ be i.i.d. random variables taking values in $(0,\infty)$. Let

[TABLE]

We say that $\mathbb{R}_{+}^{d}$ is eventually doubly covered if there exists ${\bf t}\in\mathbb{R}_{+}^{d}$ such that ${\bf t}+\mathbb{R}_{+}^{d}\subseteq D$.

Proposition 5

Let $(X,\lambda,\rho)$ be a Poisson Boolean model on $\mathbb{R}_{+}^{d}$ with

[TABLE]

(i) For $d=1$, suppose $0<l\leq L<\infty$. There exists $1/L\leq\lambda_{0}\leq 1/l$ such that

[TABLE]

(ii) For $d\geq 2$

[TABLE]

Acknowledgements.

The authors acknowledge the suggestions of the referees which led to a considerable improvement of the paper. Rahul Roy also acknowledges the grant MTR/2017/000141 from DST which supported this research.

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