Bond Percolation in Small-World Graphs with Power-Law Distribution

\emph{Full-bond percolation} with parameter $p$ is the process in which, given a graph, for every edge independently, we delete the edge with probability $1-p$. Bond percolation is motivated by problems in mathematical physics and it is studied in parallel computing and network science to understand the resilience of distributed systems to random link failure and the spread of information in networks through unreliable links. Full-bond percolation is also equivalent to the \emph{Reed-Frost process}, a network version of \emph{SIR} epidemic spreading, in which the graph represents contacts among people and $p$ corresponds to the probability that a contact between an infected person and a susceptible one causes a transmission of the infection. We consider \emph{one-dimensional power-law small-world graphs} with parameter $\alpha$ obtained as the union of a cycle with additional long-range random edges: each pair of nodes $(u,v)$ at distance $L$ on the cycle is connected by a long-range edge $(u,v)$, with probability proportional to $1/L^\alpha$. Our analysis determines three phases for the percolation subgraph $G_p$ of the small-world graph, depending on the value of $\alpha$. 1) If $\alpha<1$, there is a $p<1$ such that, with high probability, there are $\Omega(n)$ nodes that are reachable in $G_p$ from one another in $O(\log n)$ hops; 2) If $1<\alpha<2$, there is a $p<1$ such that, with high probability, there are $\Omega(n)$ nodes that are reachable in $G_p$ from one another in $\log^{O(1)}(n)$ hops; 3) If $\alpha>2$, for every $p<1$, with high probability all connected components of $G_p$ have size $O(\log n)$. The setting of full-bond percolation in finite graphs studied in this paper, which is the one that corresponds to the network SIR model of epidemic spreading, had not been analyzed before.


Introduction
Given a graph G = (V, E) and a probability p(e) associated to each edge e ∈ E, the full-bond percolation process 1 on G is the construction of a random subgraph G p = (V, E p ) of G, called the percolation graph, obtained by selecting each edge e ∈ E to belong to E p with probability p(e), independently of the other edges. Often, the focus is on the homogeneous case in which all probabilities p(e) are equal to the same parameter p. The main questions of interest in this case are, depending on the choice of G and p, what is the typical size of the connected components of G p and the typical distances between reachable vertices. The study of the percolation process originates from mathematical physics [14,20,23] and it has several applications in parallel and distributed computing and network science [1,9,13,10,12,17]. For example, the study of network reliability in the presence of random link failures is equivalent to the study of the connectivity properties of the percolation graph of the network of links [12,16].
The independent cascade is a process that models the spread of information and the influence of individual choices on others in social networks, and it is equivalent to a percolation process in a way that we explain next. In the independent cascade, we have a network G = (V, E), an influence probability p(e) associated to each edge, and a set I 0 ⊆ V of network nodes that initially have a certain opinion 2 . The process evolves over time according to the following natural local rule: if a node u acquired the opinion at time t, the node v does not have the opinion, and the edge (u, v) exists in G, then node u will attempt to convince node v of the opinion, and it will succeed with probability p (u, v). All nodes that were successfully convinced by at least one of their neighbors at time t will acquire the opinion, and will attempt to convince their neighbors at time t + 1 and so on. The independent cascade is studied in [13], where the problem of interest is to find the most "influential" initial set I 0 . The resulting epidemic process is shown in [13] to be equivalent to percolation in the following sense: the distribution of nodes influenced by I 0 in the independent cascade process has the same distribution as the set of nodes reachable from I 0 in the percolation graph of G derived using the probabilities p(e).
The Reed-Frost process is one of the simplest and cleanest models of Susceptible-Infectious-Recovered (SIR) epidemic spreading on networks [22,24]. This process is identical to independent cascade with a fixed probability p(e) = p for all edges e. We can interpret nodes that acquired the opinion in the previous step as Infectious nodes that can spread the disease/opinion, nodes that do not have the disease/opinion as nodes Susceptible to the infection, and nodes that acquired the disease/opinion two or more steps in the past as Recovered nodes that do not spread the disease any more and are immune to it. The probability p corresponds to the probability that a contact between an infectious person and a susceptible one causes a transmission of the disease from the former to the latter. The set I 0 is the initial set of infectious people at time 0. This process, being equivalent to independent cascade, is also equivalent to percolation [13]: the distribution of nodes that are infected and eventually recover in the Reed-Frost process is the same as the distribution of the set of nodes reachable from I 0 in the percolation graph of G derived using the probabilities p(e). Furthermore, the distribution of nodes that are infectious at time t is precisely the distribution of nodes at distance t from I 0 in the percolation graph (see [8]).

C(α, n) ,
where d(·, ·) is the shortest-path distance in the ring and C(α, n) is the normalizing constant 3 C(α, n) = 2 The process of long-range percolation in the one-dimensional case is the variant of the percolation process applied to the generative model described above in which ring edges are preserved with probability one. The resulting percolation graphs are always connected, and the main question of interest is their diameter. Long-range percolation is well understood, and the one-dimensional case is studied in [4,15,25]. In particular, [4] provides bounds on the diameter and on the expansion of such graphs as a function of the power-law exponent α. Such results have been then sharpened and generalized to multi-dimensional boxes in [6]. The long-range percolation process, however, is not a realistic generative model for epidemiological processes, because even the most contagious diseases, including Ebola, do not have 100% probability of spreading through close contacts (see [18,21,22] for a discussion of this point).
Full-bond percolation in power-law small-world graphs has been studied in the case of infinite lattices, including the one-dimensional case that is the infinite analog of the model that we study in this paper. In the infinite case, the main questions of interest, which are studied in [5], are whether the percolation graph has an infinite connected component and, given two vertices, what is their typical distance in the percolation conditioned on them both being in the infinite component, as a function of their distance in the lattice. Although there are similarities, techniques developed to study infinite percolation graphs do not immediately apply to the finite case.

Roadmap
In this paper, we study full-bond percolation in power-law small-world graphs. The rest of this paper is organized as follows. Section 2 provides an overview of our results and their main consequences. Section 3 introduces notation and some key preliminary notions, while Sections 4-6 provide the full analysis of full-bond percolation in one-dimensional power-law, small-world graphs. Each of Sections 4-6 includes a preliminary, informal description of the main techniques we adopt for that particular regime and a comparison with previous approaches.
Due to lack of space, the proofs of some technical results and a more extensive review of previous work is given in the full version of this paper [2].

Our Contribution
We analyze the bond percolation process over the small-world graph G sampled according to the distribution SW(n, α). Consistently with previous results in long-range percolation models [4], our analysis shows that the process exhibits three different behaviors determined by different values of α. We formally state such results in the next three theorems.
▶ Theorem 2 (Case α ∈ (2, +∞)). Let α > 2 be a constant and p < 1 a percolation probability. Sample a graph G = (V, E) from the SW(n, α) distribution and let G p be the percolation graph of G with percolation probability p.
From an epidemiological point of view, this first regime is thus characterized by a negligible chance to observe an outbreak according to the Reed-Frost process, even in the presence of a large number (say some small root of n) of initially infected nodes (i.e. sources). In particular, the second claim of the theorem above strongly bounds the possible infected area of the ring graph.
The following case, determined by the range 1 < α < 2, shows the most interesting behavior.
▶ Theorem 3 (Case α ∈ (1, 2)). Let α ∈ (1, 2) be a constant and p a percolation probability. Sample a graph G = (V, E) from the SW(n, α) distribution and let G p be the percolation graph of G with percolation probability p. Then, constants p, p ∈ (0, 1) (with p ⩽ p) exist such that the following holds: 1. If p > p, w.h.p., there exists a set of Ω(n) nodes that induces a connected sub-component in G p with diameter O(polylog(n)); 2. If p < p, w.h.p. all the connected components of G p have size O(log n).
The first claim above implies that, if p is sufficiently large (but still a constant smaller than 1), then, there is a good chance that few source nodes are able to infect a large (i.e. Ω(n)) number of nodes and, importantly, this outbreak takes place at an almost exponential speed.
Finally, when α < 1, we show the emergence of a behavior similar to that generated by one-dimensional small-world models with bridges selected according to the Erdős-Rényi distribution [3,19].
▶ Theorem 4 (Case α ∈ (0, 1)). Let α ∈ (0, 1) and p a percolation probability. Sample a graph G = (V, E) from the SW(n, α) distribution and let G p be the percolation graph of G with percolation probability p. Then, constants p, p ∈ (0, 1) (with p ⩽ p) exist, such that the following holds: In this last case, the presence of a sparse subset of relatively-long random bridges implies, above the probability threshold, that a few (i.e. Ω(log n)) sources w.h.p. generate a large outbreak at exponential speed.
We conclude this section by observing that three phases above are characterized by sharply different distributions of the typical length (measured according to the ring metric) of the bridges. To gauge the difference, consider the expectation, for a fixed vertex, of the sum of the lengths (in ring metric) of the bridges incident on the vertex, and call this expectation BL α,n .
When α < 1, we have that BL α,n is linear in n. When 1 < α < 2, then BL α,n is of the form O(n 2−α ), going to infinity with n, but sublinearly in n. Finally, when α > 2, BL α,n is a constant that depends only on α and is independent of n. Nodes have, in expectation, only one bridge, so BL α,n is an indication of how much we can advance on the ring by following one bridge.
When α < 1, the bridges are basically as good as random edges, and we would expect a giant component to emerge even after full-bond percolation, if p is large enough. When α > 2, the bridges behave like a constant number of ring-edges, and we would not expect a large component when p < 1.
The case 1 < α < 2 is the one for which it is hardest to build intuition, and the fact that the bridges typically have length of the form n 1−Ω(1) might suggest that it would take n Ω(1) steps to reach antipodal nodes. Previous work on long-range percolation had established a polylogarithmic diameter bound in the model in which ring-edges are not subject to percolation. In that model, all pairs of nodes are reachable in a polylogarithmic number of steps even though the typical bridge has length n 1−Ω (1) , and this suggests that the shortest path structure is such that a small number of long bridges is used by several shortest paths. One thus would expect such a structure to be sensitive to full-bond percolation, and indeed the proof of [4] relies on the deterministic presence of the ring-edges. Instead, we prove that, when 1 < α < 2, w.h.p, most pairs of nodes are reachable from one another in a polylogarithmic number of hops after the full-bond percolation process.

Model and Preliminaries
In this paper, we study bond percolation of graphs sampled from SW(n, α), as formalized in Definition 1. In this section, we define notation, key notions and tools that will be used throughout the rest of this paper. Further notation used in the proofs of specific results is introduced wherever it is used. For the sake of completeness, we begin with the formal definition of bond percolation.
▶ Definition 5 (Bond percolation). Given a graph G = (V, E) and a real p ∈ [0, 1], its bond percolation graph G p is the random subgraph obtained from G by removing each edge e ∈ E independently, with probability 1 − p.

3:6 Bond Percolation in Small-World Graphs with Power-Law Distribution
is as in Definition 1). We also say that u is at ring-distance d (u, v) from v (and viceversa). Considered a bridge {u, v}, we say that {u, v} is on the clockwise (respectively, counter-clockwise) side of u if d(u, v) corresponds to moving clockwise (respectively, counter-clockwise) along the cycle (V, Given a graph G = (V, E) and s ∈ V , we denote by Γ G (s) the connected component containing s in G. We may omit G when clear from context, while with a slight abuse of notation, we simply write Γ p (s) for Γ Gp (s) when we refer to the percolation G p of some graph G, which will always be understood from context.
With a slight abuse of notation, we write diam p (S) when G is the percolation graph G p . Given G = (V, E) sampled from SW(n, α) and any subgraph In the sections that follow, unless stated otherwise, probabilities are always taken over both the randomness in the sampling of G from SW(n, α) and over the randomness of the percolation. We further remark that our choice of the normalizing constant C(α, n) in Definition 1 entails E [deg(v)] = 3, while the following, preliminary fact follows from a straightforward application of Chernoff bound:

Galton-Watson Branching Processes
Our analysis of the percolation process in part relies on a reduction to the analysis of appropriately defined branching processes.
▶ Definition 7 (Galton-Watson Branching Process). Let W be a non-negative integer random variable, and let {W t,i } t⩾1,i⩾1 be an infinite sequence of independent identically distributed copies of W . The Galton-Watson branching process generated by the random variable W is the process {X t } t⩾0 defined by X 0 = 1 and by the recursion All properties of the process {X t } t⩾0 (in particular, population size and extinction probability) are captured by the equivalent process {B t } t⩾0 , recursively defined as follows: where W 1 , W 2 , . . . are an infinite sequence of independent and identically distributed copies of W .
In the remainder, when we refer to the Galton-Watson process generated by W , we always mean the process {B t } t⩾0 . In particular, we define σ = min{t > 0 : B t = 0} (we set σ = +∞ if no such t exists). Note that, for any T < σ, The case α > 2 We recall that G p is the percolation graph of G, sampled from the SW(n, α) distribution. When α > 2, we show that each component of the percolation graph G p has w.h.p. at most O(log n) nodes. To prove this fact, we need to cope with the complex "connectivity" of G p yielded by the percolation of both ring-edges and the random bridges. To better analyze this structure, we introduce the notion of ℓ-graph G p is in turn a one-dimensional small-world graph of n/ℓ "supernodes". It is defined by any fixed partition of V of disjoint ring intervals, each of them formed by ℓ nodes: such n/ℓ intervals are the nodes, called super-nodes, of G p is formed by two types of random links: the super-edges that connect two adjacent super-nodes and the super-bridges connecting two non adjacent super-nodes of the n/ℓ-size ring (for the formal definition of G (ℓ) p see Definition 10). We then prove that, for any α > 2 and any p < 1, each super-node of this graph has: 1. Constant probability to have no neighbors (for every value of ℓ); 2. Probability O(1/ℓ α−2 ) to be incident to a super-bridge.
We then design an appropriate BFS visit of G (ℓ) p (see Algorithm 1) that keeps the role of super-edges and super-bridges well-separated. In more detail, starting from a single super-node s, we show that the number of super-nodes explored at each iteration of the visit turns out to be dominated by a branching process having two distinct additive contributions: one generated by the new, visited percolated super-edges and the other generated by the new, visited percolated super-bridges. Special care is required to avoid too-rough redundancy in counting possible overlapping contributions from such two experiments. Then, thanks to Claims 1 and 2 above, we can prove that, for ℓ = O(1) sufficiently large, w.h.p. this branching process ends after O(log n) steps, and this proves that the connected component of s in G (ℓ) p has O(log n) super-nodes. We also remark that our approach above also implies the following interesting result: with probability 1 − O(1/ℓ α−2 ), all super-nodes connected to v are at ring distance at most ℓ from v. This implies that, for each node s, the nodes in the connected component of s in G p are within ring distance O(ℓ 2 ) from s, with the same probability.
We now proceed formally with the proof of Theorem 2, which is a consequence of the two lemmas below.
The proof of Lemma 8 is given in Section 4.2, while the proof of Lemma 9 in Section 4.3. To prove these lemmas, we rely on the notion of ℓ-graph, defined in the following paragraph, together with supplementary notation that will be used in the remainder of this section.

ℓ-graphs
In what follows, we define a new graph on the vertex set V = {0, . . . , n − 1}, starting from any one-dimensional small-world graph G = (V, is a cycle, which defines a ring-metric, and E 2 is an arbitrary subset of bridges. In the remainder, we always assume n ⩾ 3. A generic element in V (ℓ) thus corresponds to ℓ consecutive nodes in the ring (V, E 1 ), it is called super-node, and, depending on the context, is denoted by v or by its corresponding interval I v in the partition {I (1) , . . . , I (n/ℓ) } of V . Fixed a partition, we denote with E The following, preliminary lemma is used in the proofs of Lemma 8 and Lemma 9. for each v ∈ V (ℓ) , we have: Pr v is incident to a super-bridge in G p Proof. Consider any super-node v ∈ V (ℓ) and, for the rest of this proof, denote by w 1 and w 2 the two boundary nodes of I v . Without loss of generality, we assume I v includes the ring-edges that we traverse if we move on the cycle (V, E 1 ) from w 1 to w 2 counter-clockwise. We first prove (3). The super-node v has no out-edges in G p (ℓ) if and only if i) each node in I v has no bridge to a node in V \ I v and ii) w 1 and w 2 share no-ring edges with nodes in V \ I v in G p . Condition i) above is equivalent to the following: for every x = 1, . . . , ℓ, the node u ∈ I v at distance d(w 1 , u) = x from w 1 has no bridge of length exceeding x on the clockwise side and of length exceeding ℓ − x on the counter-clockwise side. We have where the last inequality follows from the fact that C(α, n) ⩾ 1/2 α−1 . From the inequality above, we have We have: Then, let (E p ) 2 (ℓ) denote the set of super-bridges in G p (ℓ) , we use the above inequality to bound the expected number of super-bridges that are incident in v: Finally, the proof follows from

Proof of Lemma 8
To prove Lemma 8, we consider Algorithm 1 below that performs a BFS visit of the ℓ-graph of the percolation graph G p . Then, in Lemma 14, we prove that this algorithm terminates after visiting (only) O(log n) super-nodes, w.h.p., for a sufficiently large ℓ = O(1). Together with Claim 11, this proves Lemma 8. ▶ Remark 13. Note that, each time we add a super-node y to queue Q, we also also add the super-node y left to its left on the ℓ-ring if y left is connected y in G (ℓ) p . So, in each while loop at line 3, for each super-node w ∈ Q, at most one non-visited neighbor of w on the ℓ-ring will be added at line 7 since one of them has been already added to Q at the same while loop in which w has been added to Q (see line 10). Proof. For t = 1, 2, . . . , let Q t be the content of Q at the end of the t-th iteration of the while loop of Algorithm 1, and let W t denote the subset of super-nodes that were added to Q during the t-th iteration. We have |Q 0 | = 1 and Let X t and Y t denote the sets of super-nodes that were added to Q in the t-th iteration of the while loop, respectively at line 7 and at line 9. For t ⩾ 2, whenever a super-node is added to the queue, the queue also contains a super-node at distance 1 from it on the ℓ-ring, if the corresponding super-bridge is in E (ℓ) p . We thus have the following formula |W t | = |X t | + 2|Y t |, where |X t | ⩽ 1 for t ⩾ 2 and |X 1 | ⩽ 2 . Let δ = 1 − p, ϵ = 2 − α and note that, from (3) and (4), Moreover, note that for every t, |Q t | = 0, whenever 4 According to Definition 10. where |X i | are {0, 1} random variables and, if w is the node extracted from Q in the i-th iteration of the while loop, |Y i | is the number of super-bridge neighbors of w, so that it can also be written as the sum of random variables in {0, 1}. Note that {X t } t and {Y t } t are not independent random variables, because of the conditions that appear in lines 6, 8 and 11 of Algorithm 1. Still, it is easy to show that X t 's and Y t 's are dominated by independent copies of two random variables X and Y , such that Pr (|X| = 1) = 1−δ 2 e −2/ϵ and E [|Y |] ⩽ 1/(ϵℓ ϵ ). Hence, Chernoff bound and (5) imply that, for a certain t = O(log n) depending on δ and ϵ, As a result, with probability at least 1 − 1/n 2 , we have: Hence, we can choose ℓ = O(1) (depending only on ϵ and δ) large enough, so that with probability at least 1 − 1/n 2 ,

Proof of Lemma 9
Lemma 9 follows from Claim 11 and Lemma 15 below.
▶ Lemma 15. Assume the hypotheses of Lemma 9. For sufficiently large ℓ depending only on p and α, consider the ℓ-graph G p (ℓ) and the super-node s ∈ V (ℓ) such that s ∈ I s . Then, with probability at least all nodes in Γ Gp (ℓ) (s) are within ring-distance ℓ from s in the ring (V (ℓ) , E 1 (ℓ) ).
Proof. Without loss of generality, we assume s = 0 and let V (ℓ) = {0, . . . , n/ℓ} . Next, we consider nodes on the ring at increasing distance from s = 0 moving counter-clockwise, proving that nodes at distance exceeding ℓ (2−α)/2 are not part of Γ Gp (ℓ) (s), the connected component of s. To this purpose, denote by K be the random variable indicating the ring-distance of the super-node v closest to s, such that v has no incident super-edges in G p (ℓ) . From (3), and setting ϵ = 2 − α and δ = 1 − p, we have Pr v has no super-edges in G p (ℓ) ⩾ δ 2 e −2/ϵ , So, for k = ℓ ϵ/2 and for ℓ large enough Moreover, denote by B k be the event B k = {the k nodes nearest to s have no super-bridges}. From (4), from the independence of the edge percolation events and using a union bound, we have for the complementary event B C k that Pr B C k ⩽ k · 2(ϵℓ ϵ ) −1 . Iterating the same argument for nodes on the clockwise side of s, if k = ℓ ϵ/2 we have Pr there is a node at distance ⩽ k from s in Γ (ℓ) p (s) ϵℓ ϵ/2 , which completes the proof. ◀

3:12
Bond Percolation in Small-World Graphs with Power-Law Distribution 5 The case 1 < α < 2 The aim of this section is to prove Theorem 3. When 1 < α < 2, we make an analysis, based on a suitable inductive interval partitioning of the initial ring. In this case, our technique is inspired by the partitioning method used in [20] and [4]. However, though the high-level idea is similar, our analysis needs to address technical challenges that require major adjustments. In particular, the analysis of [20] only addresses and relies upon properties of infinite one-dimensional lattices with additional long-range links, applying Kolmogorov 0-1 law to the tail event of percolation. On the other hand, [4] borrows from and partly extends [20] to the finite case, proving a result similar to ours for finite one-dimensional lattices, but with important differences both in the underlying percolation model and in the nature of the results they obtain. As for the first point, in their model, ring-edges are deterministically present, while only long-range edges (what we call bridges in this paper) are affected by percolation. In contrast, in our model, all edges are percolated to obtain a realization of G p . Consequently, the graph is deterministically connected in [4], while connectivity is something that only occurs with some probability in our setting. In particular, we face a subtler challenge, since we need to show the existence of a connected component that both spans a constant fraction of the node and has polylogarithmic diameter 5 . Moreover, their result for the diameter of the percolated graph only holds with probability tending to 1 in the limit, as the number of nodes grows to infinity, while we are able to show high probability.
We next provide a technical overview of our approach, highlighting the main points of the proof in which our percolation model requires a major departure from the approach of [4].
Similarly to [4], we consider a diverging sequence {N k } k and for each k we consider in G p an interval of length N k consisting of adjacent nodes on the ring. Departing from [4], we then prove that each interval of size N k contains a constant fraction (1 − ε k )N k of nodes, that induces a connected component of diameter D k , with probability 1 − δ k , considering only edges internal to the interval and where ε k and δ k are suitable constants that only depend on k, p and α. To prove this fact, for each k, we consider an arbitrary interval I k of size N k and we proceed inductively as follows: 1. We divide I k in N k /N k−1 smaller intervals of size N k−1 ; 2. We assume inductively that, with probability 1 − δ k−1 , each of these intervals contains (1 − ε k−1 )N k−1 nodes that induces a connected component of diameter D k−1 ; 3. With concentration arguments, we prove that a constant fraction of the intervals in which we divide I k have the above property, with probability that increases with k (we call these intervals good); 4. We consider all good intervals of the previous point, and we prove that all these intervals are connected to each other with probability increasing in k.
The above reasoning therefore implies, for some ε k , the existence of a fraction (1 − ε k )N k of nodes in I k , that with increasing probability in k, induces a connected component of diameter D k = 2D k + 1. More details about the sequences N k , ε k and δ k can be found in the full proof. As mentioned earlier, our proof needs to specifically address percolation of both ring-edges and bridges. This in particular, means tackling points (2) -(4) above, which is a major challenge not present in [4] and is taken care of in the technical Lemma 18 and in I k+1 Figure 2 Visualization of the proof of Lemma 16. Red dotted boxes identify "good" intervals in H k+1 (intervals for which the event A k holds); for each good box, the included blue box contains the subset of nodes that induce a connected subgraph of size at least (1 − ε k )N k nodes and diameter D k . In the lemma, we have to prove the existence of at least a certain number of red boxes, and that blue boxes are all mutually connected (i.e. the existence of the red edges in the picture).
the proof of the main Lemma 16 itself. In particular, considering only edges internal to the interval is very important, since, in this way, the events denoting the connection of disjoint intervals are independent. The recurrence of the diameter derives from the fact that the path connecting two nodes u and v in I k consists of the following three sub-paths: 1. The first part has length at most D k , and it is a path in the aforementioned "good" interval of size (1 − ε k−1 )N k−1 containing u; 2. The second part consists of a single edge, connecting the good interval containing u with the good interval of v; 3. The third has length at most D k , and is a path of the good interval of size (1 − ε k−1 )N k− 1 containing v. Finally (Lemma 16), we consider k such that N k = Ω(n). For such k, we prove that so this implies the existence in G p of Ω(n) nodes inducing a connected component with diameter polylog(n), w.h.p.
We now proceed with the formal analysis which requires the two lemmas below. The proof of Lemma 16 is given in Section 5.1, while the proof of Lemma 17 follows an approach, based on Galton-Watson processes, similar to that of Lemma 8, and it is given in Section 5.2.

Proof of Lemma 16
The lemma that follows is our new key ingredient to apply an inductive approach similar to the one in [4] in the case of full-bond percolation. Let I k be an arbitrary interval of N k adjacent nodes in the cycle (V, E 1 ), D k any finite integer.
Assume further that, for suitable, real constants ε k , δ k and p k in (0, 1), Then, if we consider an interval I k+1 of N k+1 adjacent nodes, it holds: Proof. In words, A k is the event that there exists a subset of the nodes in I k that induces a connected component of G p of size ⩾ (1 − ε k )N k nodes and diameter ⩽ D k . Consider the interval I k+1 and divide it into C k+1 disjoint intervals of size N k (note that N k+1 = N k ·C k+1 ). Denote by H k+1 the set of sub-intervals of I k+1 of size N k for which event A k holds. In particular, for every i = 1, 2, . . . , C k+1 , we use the indicator variable A (i) k to specify whether or not A k holds for the i-th sub-interval, so that where independence of the A (i) k 's follows from disjointness of the sub-intervals of I k+1 . For the sake of brevity, let B k+1 denote the event {|H k+1 | ⩾ (1 − δ k )C k+1 − C 0.8 k+1 }. Application of Chernoff's bound then implies Note that by definition, each sub-intervals in H k+1 contains at least one subset (of the nodes) that induces a connected component in G p with at least (1−ε k )N k nodes and diameter at most D k . Our goal is to show that, with some probability, these intervals are all connected to each other: this in turn implies the existence of a set of nodes that induces a larger connected component, with diameter at most 2D k +1 and containing at least |H k+1 |(1−ε k )N k nodes, see see Figure 2 for a visual intuition of the proof. In the remainder of this proof, we denote by F k+1 the event that all connected components associated to intervals in H k+1 are mutually connected. In particular, we prove that which implies that A k+1 holds with probability at least 1 − δ k+1 , whenever we set and D k+1 = 2D k + 1.
Now, we estimate the probability that, given B k+1 , F k+1 holds. Two nodes in I k+1 have distance at most N k+1 on the cycle. Moreover, if we consider two sub-intervals of I k+1 both belonging to H k+1 , the corresponding connected components contain at least (1 − ε k )N k

3:16 Bond Percolation in Small-World Graphs with Power-Law Distribution
If we iteratively apply Lemma 18, we thus have for each k ⩾ 1 such that N k ⩽ n, , and ε k and D k are defined by the recurrencies: Now, we solve the recurrence for ε k , obtaining where, leveraging (6), we take p > σ, with σ > 0 such that 1 − σ e β h = 1/50. With this choice, for each k ⩾ 1 we obtain Moreover, for each i ⩽ k, we have that D k ⩽ 2 k−h+1 e β h . If we take m = log β (log n), then N m = e β m = n and C m = n (β−1)/β . We also have: Setting η = log β 2, we have η > 1, since β < 2. We thus have i.e., w.h.p. G p contains an induced subgraph of size at least (19/20)n nodes and diameter O(log η (n)). ◀

Proof of Lemma 17
Let p = 1/3 and consider an arbitrary node s ∈ V . We consider an execution of the BFS in Algorithm 2 with input the percolation subgraph G p = (V, E p ) of G = (V, E) and the source s. for each neighbor x of w in Gp such that x ̸ ∈ R do 8: We consider the generic t-th iteration of the while loop at line 4 and we denote by W t the number of nodes added to the queue Q at line 8. Moreover, B t is the set of nodes that are in R in the t-th iteration. By its definition, B t is a branching process described by the following recursion: Note that, from Definition 1, each node w ∈ V has expected degree E [deg(w)] = 3. So, since each edge in G is also in G p with probability p, we have E [W 1 ] = 3p and E [W t ] ⩽ 3p for t > 1. 8 Since p < p, there is a constant δ such that p = (1 − δ)/3 and, for each t ⩾ 0 We consider the T -th iteration of the while loop, where T = γ log n. Note that the random variables W 1 , . . . , W T are not independent as noted earlier but, as remarked in the proof of Lemma 14, it is easy to show that they are stochastically dominated by T independent random variables distributed as W 1 . For the sake of simplicity, we abuse notation, by using W 1 , . . . , W T to denote the T independent copies of W 1 in the remainder of this proof. Now, if then B T = 0. We notice that each W i can be written as a sum of n + 2 independent Bernoulli random variables. 9 Hence, applying Chernoff's bound to W = T i=1 W i we obtain: where the last inequality follows if we take γ ⩾ 4/(δ 2 (1 − δ)). This allows us to conclude that for some T = O(log n), which in turn implies that with the above probability, the connected component of which v is part contains at most O(log n) nodes. Finally, a union bound over all nodes in V concludes the proof.

6
The case α < 1 In this section, we prove Theorem 4. When α < 1, we first notice that every bridge-edges (u, v) is in G p with probability at least pc/n, for some constant c. This reduces our problem to the analysis of the percolation graph H p of a graph H, where H is the union of a ring and an Erdős-Rényi graph G n,q . [3] contains a detailed analysis of this case, and we use their results to prove that G p , for any sufficiently large p, contains a connected component with diameter O(log n). We finally remark that, for every value α, when p < 1/3 the connected components of G p have at most O(log n) nodes. This fact easy follows from concentration techniques, and the fact that every node in G p has expected degree at most 3.
The formal proof of Theorem 4 requires the two lemmas below. A key observation to prove Lemma 19 is that, when α < 1, the percolation graph of a graph sampled from SW(n, α) stochastically dominates a cycle with additional Erdős-Rényi random edges. To prove Lemma 19 we thus use a previous result in [3] for this class of random graphs. In particular, we first give an equivalent formulation of Lemma 5.1 in [3], stating that, with constant probability the sequential BFS visit (Algorithm 2) reaches Ω(log n) nodes within O(log n) rounds. We then consider a parallel BFS visit (see Algorithm 4 in Section 6.1) and show an equivalent formulation of Lemma 5.2 in [3], stating that, w.h.p., the parallel BFS visit starting with Ω(log n) nodes reaches a constant fraction of nodes within O(log n) rounds.
The full proof of Lemma 19 is given in Section 6.1, while the proof of Lemma 20 is omitted, since it proceeds along the very same lines as the proof of Lemma 17, which does not depends on the value of α.

Proof of Lemma 19
We first prove the following fact.

3:19
Hence, a constant c ∈ (0, 1) (depending only on α) exists such that Pr ((u, v) ∈ E p ) = p C(α, n)d(u, v) α ⩾ p C(α, n)(n/2) α ⩾ p 2 1−α (n/2) α + n/(1 − α) ⩾ p · c n . ◁ The above fact proves that, when α < 1, the percolation subgraph G p of a graph sampled from SW(n, α) stochastically dominates a cycle with additional Erdős-Rényi random edges. To prove Lemma 19 we thus use a previous result in [3] on such class of random graphs. In particular, we first give an equivalent formulation of Lemma 5.1 in [3], that states that, with constant probability the sequential BFS visit (Algorithm 3) reaches Ω(log n) nodes within O(log n) rounds. We then consider a parallel-BFS visit (Algorithm 4) and give an equivalent formulation of Lemma 5.2 in [3], that states that, w.h.p., the parallel-BFS visit starting with Ω(log n) nodes reaches a constant fraction of nodes within O(log n) rounds.
We also introduce a slightly different version of the BFS visit of Algorithm 4, where we have also a set R 0 of removed nodes in input. +ε, there are positive parameters k, β (depending only on c, p and ε) such that the following holds: For any pair of sets I 0 , R 0 ⊆ V , with |I 0 | ⩾ β log n and |R 0 | ⩽ log 4 n, in the parallel BFS-visit (Algorithm 4) with input G p , I 0 , and R 0 , with probability at least 1 − 1/n, a time τ 2 = O(log n) exists such that |(R \ R 0 ) ∪ Q| ⩾ n/k. Now we are ready to prove Lemma 19. . Let p > p + ε, for an arbitrarily-small constant ε > 0, and let β > 0 be the constant in Lemma 23 and k be the constant in Lemmas 22 and 23. We consider the following process, where we initialize R 0 = ∅ and τ 1 = O(log n) is as in Lemma 22.