Phase Transition for the Maki–Thompson Rumour Model on a Small-World Network

Abstract

We consider the Maki–Thompson model for the stochastic propagation of a rumour within a population. In this model the population is made up of “spreaders”, “ignorants” and “stiflers”; any spreader attempts to pass the rumour to the other individuals via pair-wise interactions and in case the other individual is an ignorant, it becomes a spreader, while in the other two cases the initiating spreader turns into a stifler. In a finite population the process will eventually reach an equilibrium situation where individuals are either stiflers or ignorants. We extend the original hypothesis of homogenously mixed population by allowing for a small-world network embedding the model, in such a way that interactions occur only between nearest-neighbours. This structure is realized starting from a k-regular ring and by inserting, in the average, c additional links in such a way that k and c are tuneable parameters for the population architecture. We prove that this system exhibits a transition between regimes of localization (where the final number of stiflers is at most logarithmic in the population size) and propagation (where the final number of stiflers grows algebraically with the population size) at a finite value of the network parameter c. A quantitative estimate for the critical value of c is obtained via extensive numerical simulations.

Appendices

Appendix A: Proof of Theorem 2.1

Our proofs are constructive and rely mainly on a comparison (coupling) between the Maki–Thompson rumour process and suitable defined branching processes. Then we apply well known results regarding survival and extinction of these processes to obtain conditions for major and minor outbreak in the rumour propagation. We refer the reader to [34] for a review of the theory of branching processes.

1.1 Subcritical Regime

In order to prove Theorem 2.1(i) we construct a subcritical branching process whose total progeny dominates the final number of informed individuals in the rumour process. Roughly speaking, there are two ways of spreading the rumour from a given vertex, namely, locally, i.e., through paths of vertices connected by local edges, that we call local clusters, or through shortcuts to other vertices on the graph. We will see that, for c small enough, the size of each new local cluster may be bounded above by a value that does not depend on n, and also, that the growth of new transmissions through shortcuts is related to the growth of the suitably defined subcritical branching process.

1.1.1 Local and Blocked Clusters

In what follows, consider the Maki–Thompson rumour spreading process on ${\mathcal G}(n,k,p)$, with $p=c/(n-2k+1)$, c a constant independent of n and $k\ge 1$. Before stating the first definitions, we give an order to the vertices of the set $\mathcal {R}(\tau _n)$, i.e., we write $\mathcal {R}(\tau _n)=\{v_0,v_{1},\dots ,v_{r-1}\}$, where $r:=R(\tau _n)$ and $v_{j}$, $j=1,\dots ,r$, is the jth informed vertex on [n] up to time $\tau _n$.

Definition 6.1

(The predecessor) We define the predecessor of a vertex $v_{j}\in {\mathcal R}(\tau _n)$, denoted by $pred(v_{j})$, as the vertex $u\in \mathcal {N}(v_{j})$ such that the event

$$\begin{aligned} \cup _{t=1}^{\tau _n}\big (\{v_{j}\in {\mathcal I}(t)\}\cap \{u\in {\mathcal S}(t)\}\cap \{U_t=u\}\cap \{U_u^m=v_{j}\}\big ), \end{aligned}$$

occurs. If such a vertex does not exist, we say that vertex $v_j$ has no predecessor.

In words, $pred(v_{j})$ is the first spreader vertex that informs $v_{j}$. Observe that there is not a predecessor for $v_0$, since this vertex is already a spreader at time $t=0$.

We are interested in identifying the set of those vertices influenced for a given vertex, in the sense of the transmission of the rumour, but only through a sequence of local edges. Such a set, that we call as local cluster, determines the local range of spreading from a spreader vertex on the graph.

Definition 6.2

(Local cluster) Given two vertices $v_i,v_j \in {\mathcal R}(\tau _n)$, $i<j$, we say that $v_j$ is informed from $v_i$ through local edges if, there exist a sequence of vertices $v_i=u_1,u_2,\ldots ,u_{\ell }=v_j$ such that $u_{i}\in \mathcal {N}_{\ell }(u_{i-1})$ and $pred(u_i)=u_{i-1}$, for all $i=1,2,\ldots ,\ell$. We denote this event as $v_i\rightarrow v_j$, and we define the local cluster for a given vertex $v_i\in {\mathcal R}(\tau _n)$, as the set

$$\begin{aligned} \mathcal {C}_{v_i}(\tau _n):=\{v\in {\mathcal R}(\tau _n): v_i\rightarrow v\}. \end{aligned}$$

Our purpose now, is to bound in a certain way the size of a local cluster. Before that, we need some additional definitions.

Definition 6.3

(Blocking vertices) For any vertex $v\in [n]$ we say that

• (1) v is a right blocking vertex (rbv) if $U_v^1 = v-1$;

• (2) v is a left blocking vertex (lbv) if $U_v^1 = v+1$.

Definition 6.4

(Blocked cluster) For any vertex $v\in [n]$ we define its blocked cluster as the random interval

$$\begin{aligned} B_v := v + \llbracket -J_v^{-}, J_{v}^{+} \rrbracket , \end{aligned}$$

(6.1)

where

$$\begin{aligned} J_v^{-} :=\left( \min \left\{ i>0: \bigcap _{j=0}^{k-1} \{v-i+j \text { is a }lbv\}\right\} \right) \wedge x(n), \end{aligned}$$

and

$$\begin{aligned} J_v^{+} :=\left( \min \left\{ i>0: \bigcap _{j=0}^{k-1} \{v+i-j \text { is a }rbv\}\right\} \right) \wedge x(n), \end{aligned}$$

where $x(n):=O(n^\beta )$, for some constant $\beta < 1/2$, and $a \wedge b$ denotes the minimum of a and b. See Fig. 6

Fig. 6

Representation of a blocked cluster for $k=2$. The set o vertices of $G_n$ may be partitioned in those vertices which are lbv ($\ominus$), in those which are rbv ($\oplus$), and others ($\bullet$). For simplicity we have not drawn the incident edges

Proposition 6.1

For any $v\in [n]$ fixed, we have that

$$\begin{aligned} J_v^{-}, J_v^{+} {\mathop {\rightarrow }\limits ^{\mathcal {L}}} X_k, \end{aligned}$$

as $n\rightarrow \infty$, where ${\mathop {\rightarrow }\limits ^{\mathcal {L}}}$ denotes convergence in law, and $X_k$ is the number of coin flips up to get k consecutive heads, assuming a coin with probability $\alpha$ of heads, given by

$$\begin{aligned} \alpha :=\alpha (k,c)= {\mathbb E}\left( \{X+2k\}^{-1}\right) , \end{aligned}$$

(6.2)

where X follows a Poisson distribution, Poisson(c).

Proof

Let $v\in [n]$. We prove the result for $J_v^{+}$. The proof of the corresponding result for $J_{v}^{-}$ is the same, and therefore omitted here. In order to study the distribution of $J_v^{+}$, we have to deal with the randomness coming from the shortcuts incidents in $I_v:=\llbracket v+1,\ldots v+x(n) \rrbracket$. First, we note that w.h.p. no pair of vertices inside this interval are connected through a shortcut. Indeed, for any $u\in I_v$ we have

$$\begin{aligned} {\mathbb P}\left( I_v \cap {\mathcal N}_s (u)\ne \emptyset \right) \le 1-\left( 1-\frac{c}{n-2k-1}\right) ^{x(n)}\sim 1- e^{-c\,x(n)/n}, \end{aligned}$$

and therefore

$$\begin{aligned}&{\mathbb P}\left( \cup _{u\in I_{v}} \left\{ I_v \cap {\mathcal N}_s (u)\ne \emptyset \right\} \right) \le x(n)\left[ 1-\left( 1-\frac{c}{n-2k-1}\right) ^{x(n)} \right] \sim \nonumber \\&\qquad x(n)\left( 1- e^{-c\,x(n)/n}\right) =o(1), \end{aligned}$$

(6.3)

where the o(1) term is due to $x(n)=O(n^{\beta })$, and $\beta <1/2$ is a constant.

By Definition 6.4 we know that $J_v^-\le x(n)$. Let $\ell < x(n)$ be fixed, and denote as $A_{nsc}$ the event that there is not shortcuts between a pair of vertices in $I_v$. Conditioning on $A_{nsc}$, we obtain

$$\begin{aligned} {\mathbb P}(J_{v}^{-}=\ell )=o(1)+{\mathbb P}(X_{k}^{x(n)}=\ell ), \end{aligned}$$

(6.4)

where $X_k^{{x(n)}}$ is the number of coin flips up to get k consecutive heads, in x(n) flips, assuming a coin with probability ${\mathbb P}(U_u^1=u-1|A_{nsc})$ of coming up heads, for $u\in I_v$. The o(1) term in Eq. (6.4) comes from (6.3), which implies ${\mathbb P}(A_{nsc})=1+o(1)$. On the other hand, the comparison with successive coin flips is possible because, on $A_{nsc}$, a vertex is a rbv independently of other vertices in $I_v$. Now, observe that

$$\begin{aligned} \begin{array}{ll} {\mathbb P}\left( U_{u}^1=u-1| A_{nsc}\right) &{}= \sum \nolimits _{j=0}^{n-x(n)}{\mathbb P}(U_{u}^1=u-1| |{\mathcal N}_{s}(u)|=j, A_{nsc})\,{\mathbb P}(|{\mathcal N}_{s}(u)|=j|A_{nsc})\\ &{} = \sum \nolimits _{j=0}^{n-x(n)}\left( \tfrac{1}{2k+j}\right) \,{\mathbb P}(|{\mathcal N}_{s}(u)|=j| A_{nsc})\\ &{}=(1+o(1)){\mathbb E}\left( \{X+2k\}^{-1}\right) , \end{array} \end{aligned}$$

(6.5)

where X follows a Poisson distribution, Poisson(c) (see [35, Theorem 6.1]), and the last equality is obtained by noting that, conditioned on $A_{nsc}$, the random variable $|{\mathcal N}_{s}(u)| \sim Binomial(n-x(n),c/(n-2k-1))$, and $x(n)=o(n)$. Therefore, we have from (6.4) and (6.5) that

$$\begin{aligned} {\mathbb P}(J_{v}^{-}=\ell )=o(1)+{\mathbb P}(X_{k}=\ell ), \end{aligned}$$

where $X_k$ is the number of coin flips up to get k consecutive heads assuming a coin with probability $\alpha$ of heads, and $\alpha = {\mathbb E}\left( \{X+2k\}^{-1}\right)$. $\square$

Corollary 6.1

For any $v\in [n]$ fixed, let $B_v$ be its blocked cluster. Then,

$$\begin{aligned} {\mathbb E}\left( |B_v|\right) = \left[ 1+ 2\, \left( \frac{\alpha ^{-k}-1}{1-\alpha }\right) \right] + o(1), \end{aligned}$$

where $\alpha$ is given by (6.2).

Proof

By Definition 6.4 and Proposition 6.1 we have that

$$\begin{aligned} {\mathbb E}\left( |B_v|\right) =1+2\, {\mathbb E}\left( J_v^{-}\right) =1+2\, (1+o(1)) \, {\mathbb E}\left( X_k\right) , \end{aligned}$$

where $X_k$ is the number of coin flips up to get k consecutive heads, assuming a coin with probability $\alpha$ given by (6.2). In addition, it is well known that

$$\begin{aligned} {\mathbb E}\left( X_k\right) =\frac{\alpha ^{-k}-1}{1-\alpha }, \end{aligned}$$

which completes the proof. To obtain the last expression, observe that one may apply successively the first step analysis (see for example [36, p. 116]), conditioning on the result of the first coin flip, to obtain

$$\begin{aligned} {\mathbb E}\left( X_k\right) = \frac{1+\alpha + \alpha ^2 + \cdots + \alpha ^{k-1}}{\alpha ^k}=\frac{1-\alpha ^k}{\alpha ^k(1-\alpha )}. \end{aligned}$$

$\square$

Lemma 6.1

$\displaystyle \bigcap _{v \in {\mathcal R}(\tau _n)} \left\{ {\mathcal C}_v(\tau _n) \subseteq B_v\right\}$ w.h.p.

Proof

Note that

$$\begin{aligned} \displaystyle {\mathbb P}\left( \bigcap _{v \in {\mathcal R}(\tau _n)} \left\{ {\mathcal C}_v(\tau _n) \subseteq B_v\right\} \right) =1- \displaystyle {\mathbb P}\left( \bigcup _{v \in {\mathcal R}(\tau _n)} \left\{ {\mathcal C}_v(\tau _n) \not \subseteq B_v\right\} \right) , \end{aligned}$$

(6.6)

and

$$\begin{aligned} \displaystyle {\mathbb P}\left( \bigcup _{v \in {\mathcal R}(\tau _n)} \left\{ {\mathcal C}_v(\tau _n) \not \subseteq B_v\right\} \right) \le \sum _{v \in {\mathcal R}(\tau _n)} {\mathbb P}\left( {\mathcal C}_v(\tau _n) \not \subseteq B_v\right) \le 2\, n\, {\mathbb P}\left( J_v^+ \ge x(n)\right) . \end{aligned}$$

(6.7)

Then, by Markov’s inequality we obtain the Chernoff bound

$$\begin{aligned} {\mathbb P}\left( J_v^+ \ge x(n)\right) \le e^{-x(n)} M_{J_v^+ }(1), \end{aligned}$$

where $M_{J_v^+ }(t)$ is the moment generating function of the random variable $J_v^+$. The previous inequality, together with Prop. 6.1 implies

$$\begin{aligned} \displaystyle \lim _{n\rightarrow \infty }n\, {\mathbb P}\left( J_v^+ \ge x(n)\right) =0, \end{aligned}$$

(6.8)

because $x(n)=O(n^\beta )$, for $\beta \in (0,1/2)$, and $M_{J_v^+ }(1) \rightarrow M_{X_k}(1)$ as $n\rightarrow \infty$ where $X_k$ is the number of coin flips up to get k consecutive heads assuming a coin with probability $\alpha$ of heads, and $\alpha = {\mathbb E}\left( \{X+2k\}^{-1}\right)$. By a straightforward calculation one can verify that $M_{X_k}(1)<\infty$; this may be accomplished by applying successively the first step analysis, conditioning on the result of the first coin flip, to obtain at the end and expression which only depends of $M_{X_1}(1)$. As $X_1$ follows a geometric law it follows that $M_{X_1}(1)<\infty$. The proof is completed by (6.6), (6.7) and (6.8). $\square$

1.1.2 Emergence of Local Clusters and a Subcritical Branching Process

The idea behind the proof of Theorem 2.1(i) is to compare the emergence and growth of local clusters with a subcritical branching process. This comparison allows us to dominate the final number of informed individuals in the rumour process by the corresponding set of individuals in the branching process. We will see that those vertices which are informed through shortcuts connections will have an important role in the spreading process. We start by labelling these vertices as $v_{I_1},v_{I_2},\ldots ,v_{I_{\ell }}$. That is, $v_{I_h}$ is the hth vertex in ${\mathcal R}(\tau _n)$, such that $pred(v_{I_h})\in {\mathcal N}_{s}(v_{I_h})$. Observe that $\ell < r$. Also, for the sake of simplicity, we denote ${\mathcal C}_0:={\mathcal C}_{v_0}(\tau _n)$, $B_0:=B_{v_0}$, and ${\mathcal C}_{j}:={\mathcal C}_{v_{I_j}}(\tau _n)$, $B_{j}:=B_{v_{I_j}}(\tau _n)$ for all $j=1,\ldots ,\ell$. The key ingredients for the proof are the following claims, which result as consequence of the construction of the process and Lemma 6.1.

• Claim 1. ${\mathcal R}(\tau _n) = \cup _{j=0}^{\ell } {\mathcal C}_{j}$.

• Claim 2. $\cup _{j=0}^{\ell } {\mathcal C}_{j} \subseteq \cup _{j=0}^{\ell }B_j$ w.h.p.

Claim 1 and 2 imply that by dominating the number of vertices in the respective blocked vertices we can dominate the number of removed vertices at the end of the process. In order to control the emergence of local clusters from above we use an exploration process of blocked clusters. We define the sets ${\mathcal U}_0:=[n]\setminus B_0, \mathcal {A}_0:=B_0, {\mathcal R}_0:=\emptyset$, and for $h\ge 0$ let

$$\begin{aligned} \begin{array}{ll} {\mathcal R}_{h+1}&{}={\mathcal R}_h \cup \mathcal {A}_h\\ \mathcal {A}_{h+1} &{}= \left\{ v\in {\mathcal U}_{h}:\cup _{u\in {\mathcal N}_s(\mathcal {A}_{h})} [v\in B_{u}] \right\} \\ {\mathcal U}_{h+1}&{}= {\mathcal U}_h - \mathcal {A}_{h+1}, \end{array} \end{aligned}$$

(6.9)

where, for any $A\subset [n]$ we denote by ${\mathcal N}_s(A)$ the set of vertices connected to a vertex in A through a shortcut, i.e.,

$$\begin{aligned} {\mathcal N}_s(A):=\{v\in [n]:\cup _{u\in A}\left[ I_{uv}=1\right] \}. \end{aligned}$$

(6.10)

Observe that ${\mathcal R}(\tau _n)\subseteq \cup _{h=0}^{\infty } \mathcal {A}_h$. For $h=0,1,\ldots$, let $Y_{h}: = |\mathcal {A}_{h}|$.

Lemma 6.2

The process $(Y_h)_{h\ge 0}$ is dominated by a branching process $(Z_h)_{h\ge 0}$ with mean offspring distribution equal to

$$\begin{aligned} m_n(k,c):=\left( 1+ 2\, \left\{ \frac{\alpha ^{-k}+1}{1-\alpha }\right\} \right) \,c\, +\, o(1), \end{aligned}$$

(6.11)

where $\alpha$ is given by (6.2).

Proof

We introduce two new independent collections of random variables, the set $\{X_{uv}^h:u,h\ge 1, v\in [n]\}$ of i.i.d. Bernoulli random variables of parameter p, and also the set $\{\tilde{X}_u^h :u,h\ge 1\}$ of i.i.d. random variables distributed as $|B_0|$. We define the branching process $(Z_{h})_{h\ge 0}$ as follows. Let $Z_0=|B_0|$, and for $h\ge 0$ define

$$\begin{aligned} Z_{h+1}=\sum _{u\in \mathcal {A}_h, v\in {\mathcal U}_h} \tilde{X}_v^{h+1} I_{u,v} + \sum _{u\in \mathcal {A}_h, v\in {\mathcal U}_h^{c}} \tilde{X}_v^{h+1} X_{u,v}^{h+1} + \sum _{u=n+1}^{n+Z_h - |\mathcal {A}_h|} \sum _{v=n+1}^{2n} \tilde{X}_v^{h+1} X_{u,v}^h.\nonumber \\ \end{aligned}$$

(6.12)

The idea is to compare individuals in the branching process with vertices of the exploration process of blocked clusters. Thus, the second term in (6.12) represents the set of additional births in the branching process due to $|{\mathcal U}_h|<n$, and the third term is the set of children of those individuals in the branching process that are not vertices in $\mathcal {A}_h$. On the other hand, observe from the first term that

$$\begin{aligned} \sum _{u\in \mathcal {A}_h, v\in {\mathcal U}_h} \tilde{X}_v^{h+1}\, I_{u,v} = \sum _{u\in \mathcal {A}_h, v\in {\mathcal U}_h} |B_v|\, I_{u,v} \,\ge \, |\mathcal {A}_{h+1}|. \end{aligned}$$

It follows from the construction that the process $(Z_{h})_{h\ge 0}$ is a branching process with mean offspring distribution given by $m_n(k,c):={\mathbb E}(|B_0|)\, c$, and $Z_h \ge Y_h$. Expression (6.11) for $m_n(k,c)$ is obtained by Corollary 6.1. $\square$

1.1.3 Proof of Theorem 2.1(i)

All the random variables, and processes defined in the previous sections depend of n, and we have suppressed this dependence for the sake of simplicity. Now, we want to compare our results for different values of n, and therefore we include n in the notation. By (6.9) we have that ${\mathcal R}(\tau _n)\subset \cup _{h=0}^{\infty }\mathcal {A}_h^n$, which implies

$$\begin{aligned} R(\tau _n)\,=\,|{\mathcal R}(\tau _n)|\,\le \, \sum _{h=0}^{\infty }Y_h^n, \end{aligned}$$

where $Y_h^n=|\mathcal {A}_{h}^n|$. Then, by Lemma 6.2 we obtain

$$\begin{aligned} R(\tau _n) \, \le \, \sum _{h=0}^{\infty } Z_{h}^{n} =:Z^n, \end{aligned}$$

where $Z^n$ denotes the total progeny of the branching process $(Z_{h}^n)_{h\ge 0}$, with offspring distribution given by

$$\begin{aligned} \sum _{i=1}^{X^n} \tilde{X}_i^n, \end{aligned}$$

where $X^n$ is a Binomial random variable, $Binomial\left( n-2k-1,c/(n-2k-1)\right)$, and $\tilde{X}_1^n, \tilde{X}_2^n, \ldots$ are a sequence of i.i.d. with the same distribution as $|B_0^n|$. By Definition 6.4, and Proposition 6.1 we have that $|B_0^n|$ converges in law, as $n\rightarrow \infty$, to $1+X_{-}^{k}+X_{+}^k$, where $X_{-}^{k}$, $X_{+}^{k}$ are independent copies of the number of coin flips up to get k consecutive heads assuming a coin with probability $\alpha$ of heads, with $\alpha = {\mathbb E}\left( \{X+2k\}^{-1}\right)$, and X is a Poisson random variable Poisson(c). This in turns, implies

$$\begin{aligned} \sum _{i=1}^{X^n} \tilde{X}_i^n \, {\mathop {\longrightarrow }\limits ^{\mathcal {L}}}\, \sum _{i=1}^{X} \tilde{X}_i^{\infty }, \end{aligned}$$

(6.13)

as $n\rightarrow \infty$, where $\tilde{X}_i^{\infty } {\mathop {=}\limits ^{\mathcal {L}}} 1+X_{-}^{k}+X_{+}^k$, for all i. In other words, we have a sequence of branching processes whose offspring distribution converges to the limit distribution defined by (6.13). Therefore, the total progeny of the process $(Z_{h}^n)_{h\ge 0}$, $Z^n$, converges in law, as $n\rightarrow \infty$, to the total progeny, Z, of a branching process with mean given by

$$\begin{aligned} m(k,c):=\left( 1+ 2\, \left\{ \frac{\alpha ^{-k}+1}{1-\alpha }\right\} \right) \,c. \end{aligned}$$

An explicit expression for the distribution of the total progeny in a branching process may be found in [37]. Finally, we can choose c sufficient small in such a way that $m(k,c)<1$. This can be justified by noting, after some calculations, that as a function of c, m(k, c) is right-continuous at $[0,\infty )$, non-decreasing, and $m(k,0)=0$. Then, the respective branching process is subcritical and ${\mathbb P}\left( Z < \infty \right) =1$. We complete the proof by noting that, for $k=1,2,\ldots$

$$\begin{aligned} \liminf _{n\rightarrow \infty }{\mathbb P}\left( R(\tau _n)\le \ln n\right) \ge \liminf _{n\rightarrow \infty }{\mathbb P}\left( Z^n\le \ln n\right) \ge \liminf _{n\rightarrow \infty }{\mathbb P}\left( Z^n\le k\right) = {\mathbb P}\left( Z\le k\right) , \end{aligned}$$

and letting $k\rightarrow \infty$, yields

$$\begin{aligned} \liminf _{n\rightarrow \infty }{\mathbb P}\left( R(\tau _n)\le \ln n\right) \ge \lim _{k\rightarrow \infty } {\mathbb P}\left( Z\le k\right) = {\mathbb P}\left( Z < \infty \right) =1. \end{aligned}$$

1.2 Supercritical Regime

In this section we shall prove Theorem 2.1(ii), which states that with positive probability there is a constant $c_2(k)\in (0,\infty )$, such that the total number of removed vertices at the end of the evolution of the rumour spreading process is at least of order $O(n^{\gamma })$, $0<\gamma \le 1$, provided $c>c_2(k)$. The idea in brief is to analyze the number of those removed vertices at time $\tau _n$ containing $v_0$, who were informed through a shortcut in a restricted version of the process. This restricted version is defined to consider that each spreader attempts to transmit the information at most twice. We will prove that the number of removed vertices in the restricted process, which is smaller than the total quantity of removed vertices at time $\tau _n$ in the original process, is related to the size of a supercritical Galton–Watson branching process provided c is sufficiently large.

Let $\kappa _{v_0}(\tau _n)$ denotes the cluster of removed vertices at time $\tau _n$ containing $v_0$ in the restricted process. We reveal $\kappa _{v_0}(\tau _n)$ as follows. Define the sets ${\mathcal V}_{0}:=[n]\setminus \{v_0\}$, ${\mathcal Y}_{0}:=\{v_0\}$, and for $r\ge 0$ let

$$\begin{aligned} \begin{array}{ll} {\mathcal Y}_{r+1} &{}= \left\{ U_{v}^i: U_{v}^i \in {\mathcal N}_s(v), i=1,2, v\in {\mathcal Y}_r \right\} \\ {\mathcal V}_{r+1}&{}= {\mathcal V}_r - {\mathcal Y}_{r+1}. \end{array} \end{aligned}$$

(6.14)

Then observe that

$$\begin{aligned} \kappa _{v_0}(\tau _n):=\cup _{r=0}^{\infty }{\mathcal Y}_{r} \subseteq {\mathcal R}(\tau _n). \end{aligned}$$

Our aim now is to study $\kappa _{v_0}(\tau _n)$. As in Sect. 6.1.2, we will look at the vertices which are informed through shortcuts connections, that is, at the set of vertices $\{v_{I_h}\}_{h\ge 1}^{\ell }$, $\ell < R(\tau _n)$, where as before $v_{I_h}$ denotes the hth vertex in ${\mathcal R}(\tau _n)$, such that, $pred(v_{I_h})\in \mathcal {N}_s(v_{I_h})$. Let $v_{I_0}:=v_0$.

Next we introduce some definitions and lemmas necessary to prove Theorem 2.1(ii).

Definition 6.5

(Truncated local cluster) Let $N:=N(n)\in {\mathbb N}$ with $N\le R(\tau _n)$. We define for each vertex $v_{I_h}$, $h=0,\dots ,N-1$, its truncated local cluster $(T\ell c)$ as the set

$$\begin{aligned} \mathcal {C}_{v_{I_h}}^T(t_N):=\Big \{u\in \{{\mathcal R}(t_N)\cup {\mathcal S}(t_N)\}: v_{I_h}\rightarrow u\Big \}, \end{aligned}$$

where $t_N$ is the time at which $v_{I_{N}}$ is informed, i.e.,

$$\begin{aligned} t_N:=\inf \{t\ge 0: v_{I_{N}}\in \mathcal {S}(t)\}. \end{aligned}$$

Observe that at time $t_N$ we have $N-1$ vertices informed through shortcuts, since $v_{I_0}$ is informed by construction. Moreover, note that the vertices inside $\mathcal {C}_{v_{I_h}}^T(t_N)$, $h=0,\dots ,N-1$, are connected through local edges, and they all are either removed or spreaders at time $t_N$.

Now for each $h=0,\dots ,N-1$, let $C_{v_{I_h}}$ be the set of all vertices connected by local edges in $\mathcal {C}_{v_{I_h}}^T(t_N)$. Note that $C_{v_{I_h}}$ may contain ignorant vertices at time $t_N$. Moreover, these sets of vertices $C_{v_{I_h}}, h=0,\dots ,N-1$, are not necessarily disjoint. For the next two lemmas we will take the following disjoint sets of vertices constructed from $\{C_{v_{I_h}}\}_{h=0}^{N-1}$ and the bloqued clusters $\{B_{v_{I_h}}\}_{h=0}^{N-1}$ of Definition 6.4. Let $\mathfrak {C}_{v_{I_0}}^T(t_N):=C_{v_{I_0}}$, $\mathfrak {B}_{v_{I_0}}:=B_{v_{I_0}}$ and for $h=1,\dots , N-1$, let

$$\begin{aligned} \mathfrak {C}_{v_{I_h}}^T(t_N):=C_{v_{I_h}}\setminus \Big (\cup _{j=0}^{h-1}C_{v_{I_j}}\Big ) \quad \text {and}\quad \mathfrak {B}_{v_{I_h}}:=B_{v_{I_h}}\setminus \Big (\cup _{j=0}^{h-1}B_{v_{I_j}}\Big ). \end{aligned}$$

(6.15)

Definition 6.6

We call a path of m vertices in $\{\mathfrak {B}_{v_{I_h}}\}_{h=0}^{N-1}$, $2\le m\le N$, a sequence of vertices such that two consecutive vertices in the sequence are connected by a local edge if they are inside of the same blocked cluster, say $\mathfrak {B}_{v_{I_h}}$, and connected by a shortcut if they are in different blocked clusters, say $\mathfrak {B}_{v_{I_h}}$ and $\mathfrak {B}_{v_{I_h'}}, h\ne h'$. In this definition no repetitions of vertices and edges are allowed other than the repetition of the starting and ending vertex, and in such situation we say that the path of vertices is a closed path.

Definition 6.7

We say that m disjoint blocked clusters in $\{\mathfrak {B}_{v_{I_h}}\}_{h=0}^{N-1}$ form a cycle, if there exists at least one closed path between them.

Recall that the vertices inside the disjoint blocked clusters are connected through local edges in $\mathcal {G}(n,k,p)$. Therefore, the previous definition tells us that m disjoint blocked clusters form a cycle, $2\le m\le N$, if for $m=2$ there exist at least 2 shortcuts between the 2 blocked clusters, while for $m>2$ there are at least m shortcuts between the m blocked clusters, each joint two different blocked clusters.

Let $X_N$ denote the random variable which counts the total number of cycles between the disjoint blocked clusters $\{\mathfrak {B}_{v_{I_h}}\}_{h=0}^{N-1}$. That is, the sum over m, for $m=2,3,\dots ,N$, of cycles between m disjoint blocked clusters in $\{\mathfrak {B}_{v_{I_h}}\}_{h=0}^{N-1}$.

Lemma 6.3

Let $0<\beta <1/2$, $\gamma :=\gamma (\beta )$ such that $\gamma < 1-2\beta$ and $N\le O(n^{\gamma })$. Consider $X_N$ as before. Then for $p=c/(n-2k+1)$, $c,k\in {\mathbb N}$,

$$\begin{aligned} {\mathbb P}(X_N>0)\rightarrow 0 \end{aligned}$$

as $n\rightarrow \infty$.

Proof

By Markov’s inequality,

$$\begin{aligned} {\mathbb P}(X_N>0)\le {\mathbb E}(X_N), \end{aligned}$$

and so it will be enough to show that ${\mathbb E}(X_N)\rightarrow 0$ as $n\rightarrow \infty$. Let $L_{ij}$ be the number of shortcuts between two disjoint blocked clusters of sizes i and j, with $i,j< O(n^{\beta })$ (see Definition 6.4). Note that $L_{ij}$ follows a Binomial distribution with parameters (ij, p). Hence,

$$\begin{aligned} {\mathbb P}(L_{ij}>0)=&1-(1-p)^{ij},\text { and} \end{aligned}$$

(6.16)

$$\begin{aligned} {\mathbb P}(L_{ij}\ge 2)=&1-(1-p)^{ij}-ijp(1-p)^{ij-1} . \end{aligned}$$

(6.17)

Using Taylor expansion for $e^x$ and $\ln (1-x)$, $|x|<1$, we get that for $a\ge 0$

$$\begin{aligned} (1-p)^{ij-a}=e^{(ij-a)\ln (1-p)}=1-(ij-a)p+O\Big (\Big (\frac{ij-a}{n-2k+1}\Big )^2\Big ). \end{aligned}$$

(6.18)

Replacing (6.18) in (6.16) and (6.17), we obtain

$$\begin{aligned} {\mathbb P}(L_{ij}>0)=\,&ijp+O\Big (\Big (\frac{ij}{n-2k+1}\Big )^2\Big )\le ijp+ O(n^{2(2\beta -1)}), \text { and}\end{aligned}$$

(6.19)

$$\begin{aligned} {\mathbb P}(L_{ij}\ge 2)=\,&O\Big (\Big (\frac{ij}{n-2k+1}\Big )^2\Big )\le O(n^{2(2\beta -1)}). \end{aligned}$$

(6.20)

Now let $\mathcal {S}_m$ be the set of all subsets of m disjoint blocked clusters, $2\le m\le N$, connected by shortcuts and ordered up to rotation and orientation of the cycle. Let $\mathcal {S}=\cup _{m=2}^N \mathcal {S}_m$. For each $S_m\in \mathcal {S}_m$, define $A_{S_m}$ to be the event that a cycle occurs between the m disjoint blocked clusters of $S_m$. As the expectation is linear,

$$\begin{aligned} {\mathbb E}(X_N)=\sum _{S\in \mathcal {S}}{\mathbb E}(1_{A_S})= & {} \sum _{m=2}^N\sum _{S_m\in \mathcal {S}_m}{\mathbb P}(A_{S_m}). \end{aligned}$$

(6.21)

Observe that for $m=2$, $A_{S_m}$ means that two disjoint blocked clusters are connected through two or more shortcuts, since the vertices inside the disjoint blocked clusters are connected through local edges in $\mathcal {G}(n,k,p)$. Otherwise, for $m\ge 3$, $A_{S_m}$ means that there is at least m independent shortcuts, each joint two disjoint blocked clusters. Therefore, by (6.20)

$$\begin{aligned} {\mathbb P}(A_{S_2})\le O(n^{2(2\beta -1)}), \end{aligned}$$

(6.22)

while for $m\ge 3$ and assuming that $S_m$ is formed by disjoint blocked clusters of sizes $i_1,i_2,\dots ,i_m$, we have by (6.19)

$$\begin{aligned} {\mathbb P}(A_{S_m})\le [i_1i_2p+ O(n^{2m(2\beta -1)})]\dots [i_mi_1p+ O(n^{2m(2\beta -1)})] \le&O(n^{m(2\beta -1)})). \end{aligned}$$

(6.23)

In addition, we need to find $|\mathcal {S}_m|$. The number of ordered sets of size m is $\left( {\begin{array}{c}N\\ m\end{array}}\right) m!$, which over-counts each $S_m\in \mathcal {S}_m$, $m\ge 3$, by 2m times (once for each starting position on the cycle $(\times m)$, and once for each direction of the cycle $(\times 2)$), while for $m=2$ over-counts only by 2 times (just for each direction of the cycle). Thus by (6.21), (6.22) and (6.23)

$$\begin{aligned} {\mathbb E}(X_N)\le & {} \left( {\begin{array}{c}N\\ 2\end{array}}\right) O(n^{2(2\beta -1)})+\sum _{m=3}^N \left( {\begin{array}{c}N\\ m\end{array}}\right) \frac{m!}{2m}O(n^{m(2\beta -1)})\\\le & {} \sum _{m=2}^N \Big [N O(n^{(2\beta -1)}))\Big ]^m. \end{aligned}$$

Since $N\le O(n^{\gamma })$, $\gamma <1-2\beta$ and $0<\beta <1/2$, then $NO(n^{2\beta -1})<1$ and goes to zero as $n\rightarrow \infty$. Hence, by the geometric series

$$\begin{aligned} {\mathbb E}(X_N)<\frac{[NO(n^{2\beta -1})]^2}{1-NO(n^{2\beta -1})}\rightarrow _{n\rightarrow \infty }0. \end{aligned}$$

$\square$

Lemma 6.4

Let $N\le O(n^{\gamma })$ and let $u,v\in {\mathcal R}(\tau _n)$, such that, $\{v\in \mathcal {N}_s(u)\}$ and for some $t\le {t_N}$ $\{u\in {\mathcal S}(t)\}\cap \{U_t=u\}$. Then either $\{v\in {\mathcal I}(t)\}$ or $\{v\in {\mathcal R}(t)\cup {\mathcal S}(t)\text { and } pred(v)=u\}$ w.h.p.

In words, what Lemma 6.4 tell us is that if at some time $t\le t_N$, a spreader u is chosen to inform, then w.h.p no neighbour v connected to u through shortcuts has already been informed by other spreader.

Proof

Let A be the event that the N disjoint $T\ell c$’s, $\mathfrak {C}_{v_{I_h}}^T(t_N)$, $h=0,\dots ,N-1$, are connected by at most $N-1$ shortcuts, each joint two distinct $T\ell c$’s, and E the event $\{\mathfrak {C}_{v_{I_h}}^T(t_N)\subseteq \mathfrak {B}_{v_{I_h}}, h=0,\dots ,N-1\}$.

By Lemma 6.1 we know that ${\mathcal C}_v(\tau _n) \subseteq B_v$ w.h.p, where $B_v$ is the blocked cluster of v ($|B_v| < O(n^{\beta })$, $\beta <1/2$ w.h.p by Definition 6.4 and Proposition 6.1). Then, it implies that for $h\ge 0$, $\mathfrak {C}_{v_{I_h}}^T(t_N)\subseteq \mathfrak {B}_{v_{I_h}}$ w.h.p. Thus, we have that E holds w.h.p.

Now note that if A holds, then up to time $t_N$, we see a tree of disjoint $T\ell c$’s. Therefore, if $u,v\in {\mathcal R}(\tau _n)$ such that, $\{u\in {\mathcal S}(t)\}\cap \{U_t=u\}$ and $\{v\in \mathcal {N}_s(u)\}$, $t\le t_N$, then up to time $t_N$, v may only be informed by u. Thus, A implies that

$$\begin{aligned} \{v\in {\mathcal I}(t)\}\cup \{v\in {\mathcal R}(t)\cup {\mathcal S}(t)\text { and } pred(v)=u\}. \end{aligned}$$

(6.24)

Moreover, since E holds w.h.p, then conditioning on this we get

$$\begin{aligned} {\mathbb P}(A)=\,&{\mathbb P}(A\mid E)+o(1)\nonumber \\ \ge \,&{\mathbb P}(X_N=0)+o(1), \end{aligned}$$

(6.25)

where $X_N$ is the number of cycles between the disjoint blocked clusters $\{\mathfrak {B}_{v_{I_h}}\}_{h=0}^{N-1}$.

Thus, by (6.24) and (6.25) we have

$$\begin{aligned} {\mathbb P}\Big (\{v\in {\mathcal I}(t)\} \cup \{v\in {\mathcal R}(t)\cup {\mathcal S}(t)\text { and } pred(v)=u\}\Big )\ge & {} {\mathbb P}(X_N= 0)+o(1). \end{aligned}$$

Finally, by Lemma 6.3 up to time $t_N$, ${\mathbb P}(X_N= 0)\rightarrow 1$ as $n\rightarrow \infty$. Hence, for any $t\le t_N$

$$\begin{aligned} {\mathbb P}\Big (\{v\in {\mathcal I}(t)\} \cup \{v\in {\mathcal R}(t)\cup {\mathcal S}(t)\text { and } pred(v)=u\}\Big )\rightarrow 1, \end{aligned}$$

as $n\rightarrow \infty$. $\square$

1.2.1 A Super Critical Branching Process

We start this section by exploring, up to time $t_N$, the set $\kappa _{v_0}(\tau _n)$. To pursue this aim we consider the following random times together with the next lemma. For any $v\in [n]$ let

$$\begin{aligned} t_v^s:=\inf \{t\ge 0: v\in \mathcal {S}(t)\};\,\,\, t_{v}':=\inf \{t> t_v^s:U_t =v\};\,\,\, t_{v}'':=\inf \{t> t_{v}':U_t =v\},\nonumber \\ \end{aligned}$$

(6.26)

and we say that these values are $\infty$, if the respective infimum is not attained. In words, $t_v^s, t_{v}'$ and $t_{v}''$ are the times at which the vertex v becomes a spreader, attempts to transmit the information for the first time, and attempts to transmit the information for the second time, respectively. Observe that $t_{v_0}^s=0$ and $t_{v_0}'$, $t_{v_0}''$ are finite. On the other hand, if $v\in {\mathcal R}(\tau _n)$ then $t_v^s$ and $t_{v}'$ are finite by definition and construction, but this is not necessarily true for $t_{v}''$ (for example, we could have $U_v^{1}\in {\mathcal R}(t_{v}')\cup {\mathcal S}(t_{v}')$).

Lemma 6.5

Let $t'_{v_{I_h}}$ and $t''_{v_{I_h}}$ be the first and the second time, respectively, at which $v_{I_h}$ attempts to transmit the information, and defined by (6.26). Then, $t''_{v_{I_0}}<\infty$, while for $h\ge 1$, and as long as $t'_{v_{I_h}}\le t_N$, $t''_{v_{I_h}}<\infty$ w.h.p.

Proof

Since at time $t'_{v_{I_0}}\equiv 1$, all the vertices except $v_{I_0}$ are ignorants, then $t''_{v_{I_0}}<\infty$. Moreover, for $v\in \mathcal {N}_s(v_{I_h})$, $h\ge 1$, and such that $t'_{v_{I_h}}\le t_N$, by Lemma 6.4 we have that either $\{v\in {\mathcal I}(t'_{v_{I_h}})\}$ or $\{v\in {\mathcal R}(t'_{v_{I_h}})\cup {\mathcal S}(t'_{v_{I_h}})\text { and } pred(v)=v_{I_h}\}$ w.h.p. Observe that this is the same to say $\{v\in {\mathcal I}(t'_{v_{I_h}})\}$ w.h.p, since $t'_{v_{I_h}}$ is the first time when $v_{I_h}$ is selected to inform. Therefore, what we have is that $t''_{v_{I_h}}<\infty$ w.h.p, as long as $t'_{v_{I_h}}\le t_N$. $\square$

Now we are ready to explore, up to time $t_N$, $\kappa _{v_0}(\tau _n)$:

• (1) Begin with $v_0$ and find all the vertices connected to $v_0$ by shortcuts, i.e, $\mathcal {N}_s(v_0)$. These vertices coincide with the set $\mathcal {E}_1$ defined in Sect. 2.1. Consider $t_{v_0}'$ and $t_{v_0}''$, defined by (6.26). At these times, either one or two of all possible vertices incident to $v_0$ are informed according to the uniform random variables $U_{v_0}^1$ and $U_{v_0}^2$, respectively. Take into account only the vertices informed at times $t_{v_0}'$ and $t_{v_0}''$ through shortcuts, and let $Z_{I_0}$ be the random variable corresponding to this number of vertices.

• (2) Given that $v_0$ informs a vertex in $\mathcal {N}_s(v_0)$ at time $t_{v_0}'$ or $t_{v_0}''$, e.g. $u^1$ and $u^2$, let $t_{u^j}'$ and $t_{u^j}''$ (assume these times are finite), $j=1,2$, be the first and the second time when $u^j$ attempts to transmit the information, respectively. If $t_{u^j}'\le t_N$, then find all the vertices connected to $u^j$ by shortcuts, but different of $v_0$, i.e., $\mathcal {N}_s(u^j)\backslash \{v_0\}$. Take into account only the vertices informed at times $t_{u^j}'$ and $t_{u^j}''$ through shortcuts.

• (3) Repeat step (2) for $u^j$, $j=1,2$, and so on, provided that the first time at which a vertex attempts to transmit the information, is less than $t_N$.

Denote this sub-cluster by $\kappa _{v_0}(t_N)$ and note that $|\kappa _{v_0}(t_N)|\le N$.

1.2.2 Proof of Theorem 2.1(ii)

We start by observing that the number of vertices at the end of each step in the previous exploration process of $\kappa _{v_0}(t_N)$, corresponds to the number of vertices informed by some $v_{I_h}$ through shortcuts, at the first and the second time at which $v_{I_h}$ attempts to transmit the information. For each $v_{I_h}$, $h\ge 0$ and as long as $t_{v_{I_h}}'\le t_N$, let $Z_{I_h}$ denote this number of vertices.

We define now a random variable $X_{I_h}$ such that it is stochastically dominated by $Z_{I_h}$, for each $h\ge 0$ and as long as $t'_{v_{I_h}}\le t_N$. That is, ${\mathbb P}(Z_{I_h}\ge z)\ge {\mathbb P}(X_{I_h}\ge z)$ for all z, with strict inequality at some z. To this aim we begin by observing that the event $\{Z_{I_h}=1\}$ holds, if and only if, any of the two following possibilities hold.

$P_1^h$::

At times $t'_{v_{I_h}}$ and $t''_{v_{I_h}}$, $v_{I_h}$ chooses the same neighbor and it belongs to $\mathcal {N}_s(v_{I_h})$, or

$P_2^h$::

at time $t'_{v_{I_h}}$, $v_{I_h}$ chooses a neighbor from $\mathcal {N}_l(v_{I_h})$ and at time $t''_{v_{I_h}}$, $v_{I_h}$ chooses a neighbor from $\mathcal {N}_s(v_{I_h})$.

Consider only the first possibility. Since at time $t'_{v_{I_0}}$ all the vertices except $v_{I_0}$ are ignorants, then for $h=0$,

$$\begin{aligned} {\mathbb P}(Z_{I_0}\!=\!1)> & {} \sum _{\ell \!=1}^{n-2k-1}{\mathbb P}\Big (\{U_{v_{I_0}}^1=\!U_{v_{I_0}}^2\!=\!v \}\cap \{v\!\in \! \mathcal {N}_s(v_{I_0})\}\cap \{v\!\in \!{\mathcal I}(t'_{v_{I_0}})\}\cap \{|\mathcal {N}_s(v_{I_0})|\!=\!\ell \}\Big )\nonumber \\= & {} \sum _{\ell =1}^{n-2k-1}\Big (\frac{\ell }{(\ell +2k)^2}\Big ){\mathbb P}(|\mathcal {N}_s(v_{I_0})|=\ell )\nonumber \\> & {} \sum _{\ell =1}^{n-2k-2}\Big (\frac{\ell }{(\ell +2k+1)^2}\Big ){\mathbb P}(|\mathcal {N}_s(v_{I_0})|=\ell ), \end{aligned}$$

(6.27)

where $|\mathcal {N}_s(v_{I_0})|$ follows a Binomial distribution with parameters $(n-2k-1,p)$.

On the other hand, observe that the event $\{Z_{I_0}=2\}$ holds, if and only if, at time $t'_{v_{I_0}}$, $v_{I_0}$ chooses a neighbor from $\mathcal {N}_s(v_{I_0})$ and at time $t''_{v_{I_0}}$, $v_{I_0}$ chooses another neighbor from $\mathcal {N}_s(v_{I_0})$ but different from the first one. Hence,

$$\begin{aligned} {\mathbb P}(Z_{I_0}=2)= & {} \sum _{\ell =2}^{n-2k-1}{\mathbb P}\Big (\{U_{v_{I_0}}^1=v,U_{v_{I_0}}^2=w, v\ne w\}\cap \{v,w\in \mathcal {N}_s(v_{I_0})\}\nonumber \\&\cap \{v\in {\mathcal I}(t'_{v_{I_0}})\}\cap \{|\mathcal {N}_s(v_{I_0})|=\ell \}\Big )\nonumber \\= & {} \sum _{\ell =1}^{n-2k-1}\Big (\frac{\ell (\ell -1)}{(\ell +2k)^2}\Big ){\mathbb P}(|\mathcal {N}_s(v_{I_0})|=\ell )\nonumber \\> & {} \sum _{\ell =1}^{n-2k-2}\Big (\frac{\ell (\ell -1)}{(\ell +2k+1)^2}\Big ){\mathbb P}(|\mathcal {N}_s(v_{I_0})|=\ell ). \end{aligned}$$

(6.28)

For $h>0$ and as long as $t'_{v_{I_h}}\le t_N$, by Lemma 6.5 we know that $t''_{v_{I_h}}<\infty$ w.h.p. Conditioning on this event, and following a similar reasoning to get (6.27) and (6.28), we obtain

$$\begin{aligned} {\mathbb P}(Z_{I_h}=1)> & {} \sum _{\ell =1}^{n-2k-2}\Big (\frac{\ell }{(\ell +2k+1)^2}\Big ){\mathbb P}(|\mathcal {N}_s(v_{I_h})|=l)+o(1), \end{aligned}$$

(6.29)

and

$$\begin{aligned} {\mathbb P}(Z_{I_h}=2)> & {} \sum _{\ell =2}^{n-2k-2}\Big (\frac{\ell (\ell -1)}{(\ell +2k+1)^2}\Big ){\mathbb P}(|\mathcal {N}_s(v_{I_h})|=l)+o(1), \end{aligned}$$

(6.30)

where $|\mathcal {N}_s(v_{I_h}|$ follows a Binomial distribution with parameters $(n-2k-2,p)$.

Using (6.27) and (6.28), let $X_{I_0}$ be a random variable with probability density given by

$$\begin{aligned} {\mathbb P}(X_{I_0}=1)=\sum _{\ell =1}^{n-2k-2}\Big (\frac{\ell }{(\ell +2k+1)^2}\Big ){\mathbb P}(|\mathcal {N}_s(v_{I_0})|=l), \\ {\mathbb P}(X_{I_0}=2)=\sum _{\ell =2}^{n-2k-2}\Big (\frac{\ell (\ell -1)}{(\ell +2k+1)^2}\Big ){\mathbb P}(|\mathcal {N}_s(v_{I_0})|=l)\text { and}\\ {\mathbb P}(X_{I_0}=0)=1-{\mathbb P}(X_{I_0}=1)-{\mathbb P}(X_{I_0}=2). \end{aligned}$$

In analogous way but using (6.29) and (6.30), define for each $h>0$ and as long as $t'_{v_{I_h}}\le t_N$, the random variables $X_{I_h}$’s. By definition we observe that each random variable $X_{I_h}$ is stochastically dominated by $Z_{I_h}$.

Observe that all the random variables $X_{I_h}$ and $Z_{I_h}$ depend on n. Furthermore, for each $h\ge 0$ and as long as $t'_{v_{I_h}}\le t_N$,

$$\begin{aligned} \lim _{n\rightarrow \infty }{\mathbb P}(X_{I_h}=1)= & {} {\mathbb E}\left( \frac{X}{(X+2k+1)^2}\right) ,\nonumber \\ \lim _{n\rightarrow \infty }{\mathbb P}(X_{I_h}=2)= & {} {\mathbb E}\left( \frac{X(X-1)}{(X+2k+1)^2}\right) \text { and}\nonumber \\ \lim _{n\rightarrow \infty }{\mathbb P}(X_{I_h}=0)= & {} 1-{\mathbb E}\left( \frac{X^2}{(X+2k+1)^2}\right) , \end{aligned}$$

(6.31)

where X follows a Poisson distribution with mean c, and where we have used the Poisson approximation to the Binomial distribution (see [35] Theorem 6.1).

Now, we construct a growth process in a similar way as we explored $\kappa _{v_0}(t_N)$ in Sect. 6.2.1, but using the random variables $X_{I_h}$ instead of $Z_{I_h}$. We denote this process by $\mathcal {Y}(t_N)$ and let $| \mathcal {Y}(t_N)|$ be its size.

Since for each $h\ge 0$ and as long as $t'_{v_{I_h}}\le t_N$, $Z_{I_h}$ stochastically dominates $X_{I_h}$, then we have that the random variable $| \mathcal {Y}(t_N)|$ is stocastically dominated by $|\kappa _{v_{I_0}}(t_N)|$, that is

$$\begin{aligned} {\mathbb P}\left( |\kappa _{v_0}(t_N)|\ge L\right) \ge {\mathbb P}\left( |\mathcal {Y}(t_N)|\ge L\right) , \end{aligned}$$

(6.32)

for all $L>0$, with strict inequality at some L. Furthermore, by construction, we have that

$$\begin{aligned} {\mathcal R}(\tau _n)\supseteq {\mathcal R}(t_N)\supseteq \kappa _{v_0}(t_N). \end{aligned}$$

(6.33)

Take $L\le N$ but of the same order. That is, $L:=L(N)=O(n^{\gamma })$. Then, by (6.32) and (6.33) we have that

$$\begin{aligned} {\mathbb P}(|{\mathcal R}(\tau _n)|\ge L)\ge & {} {\mathbb P}(|{\mathcal R}(t_N)|\ge L)\nonumber \\\ge & {} {\mathbb P}(|\kappa _{v_0}(t_N)|\ge L)\nonumber \\\ge & {} {\mathbb P}(|\mathcal {Y}(t_N)|\ge L). \end{aligned}$$

(6.34)

Finally, let $(Y_s)_{s\ge 0}$ be a branching process with offspring distribution given by the limit distribution of $X_{I_0}$. Denote by |Y| the size of the total progeny of $(Y_s)_{s\ge 0}$. Using (6.31), this branching process has mean offspring distribution

$$\begin{aligned} \tilde{m}(k,c):={\mathbb E}\left( \frac{X}{(X+2k+1)^2}\right) +2{\mathbb E}\left( \frac{X(X-1)}{(X+2k+1)^2}\right) . \end{aligned}$$

(6.35)

Therefore, by (6.34)

$$\begin{aligned} \liminf _{n\rightarrow \infty }{\mathbb P}(|{\mathcal R}(\tau _n)|\ge L)\ge \liminf _{n\rightarrow \infty }{\mathbb P}(|\mathcal {Y}(t_N)|\ge L)= & {} \lim _{n\rightarrow \infty }{\mathbb P}(|\mathcal {Y}(t_N)|\ge L)\\\ge & {} {\mathbb P}((Y_s)_{s\ge 0} \text { survives}). \end{aligned}$$

Using the theory of branching process, $(Y_s)_{s\ge 0}$ survives with positive probability if $\tilde{m}(k,c)>1$, see [34]. Finally, we can choose c sufficiently large in such a way that $\tilde{m}(k,c)>1$. This can be justified by noting, after some calculations, that as a function of c, $\tilde{m}(k,c)$ is right-continuous at $[0,\infty )$, non-decreasing, and $\tilde{m}(k,0)=0$.

Appendix B: Technical Details on Numerical Simulations

In this appendix we deepen and compare the two sources of stochasticity present in the process under consideration. In fact, on the one hand, the graph $\mathcal {G}(n,k,p)$ underlying the diffusion process is stochastic, since the nc new links to be inserted are drawn randomly. On the other hand, the dynamic process is intrinsically stochastic, since at each step the spreader and its neighbour are extracted randomly. Thus, one can in principle proceed in one of the following ways:

Dynamical noise:

• Fix k and take a k-ring of size n;

• Fix c and insert new edges with probability $p=c/(n-2k-1)$, hence getting the realization $G_n$ for the graph $\mathcal {G}(n,k,p)$;

• Perform M cycles of the Maki–Thomson dynamic, and for each of them, say the one labelled as i ($i=1,\ldots ,M$), calculate the final value of removed individuals $R^{(i)}(\tau _n,G_n)$;

• Compute the mean value over the M realizations of the dynamics as

  $$\begin{aligned} \langle R(c, k, G_n )\rangle _{dyn} =\sum _{i=1}^{M}\frac{R^{(i)}(\tau _n,G_n)}{M}. \end{aligned}$$

  (7.1)

• Repeat the procedure for different values of c and k.

Notice that this procedure does not take into account the noise due to the stochastic realization of the graph.

Topological noise:

• Fix k and take a k-ring of size n;

• Fix c and insert new edges with probability $p=c/(n-2k-1)$, hence getting the realization $G_n^{(i)}$ for the graph $\mathcal {G}(n,k,p)$;

• Perform the Maki–Thomson dynamic and calculate the final value of removed individuals $R(\tau _n,G_n^{(i)})$

• Build L independent realizations of the graph and for each of them, say the one labelled as i ($i=1,\ldots ,L$) and referred to as $G^{(i)}_n$, perform a cycle of the dynamic, and calculate the final value of removed individuals $R(\tau _n, G^{(i)}_n )$.

• Compute the mean value over the L realizations of the graph as

  $$\begin{aligned} \langle R(c, k) \rangle _{top}=\sum _{i=1}^{L}\frac{R(\tau _n, G^{(i)}_n)}{L}. \end{aligned}$$

  (7.2)

• Repeat the procedure for different values of c and k.

Notice that in this procedure each realization of the dynamics corresponds to a different realization of $\mathcal {G}(n,k,p)$.

We compared the outcomes from these experiments when the same number of items is averaged (i.e., $M=L$). First, we notice that, the average values stemming from both routes are comparable within the related errors. Moreover, the second route turns out to be more noisy; this result perfectly matches with the fact that, in a Monte Carlo-like simulation, means computed on different graphs are less accurate than the ones related to a single realization of network (see e.g., [38]).

Anyhow, the numerical path followed in Sect. 4 merges these two routes by first applying the former route and then repeating the procedure over different realizations of the underlying graph and finally averaging.