3.1. Earlier work

The results of Section 2 are structurally similar to the main result in [Reference Janson11], which states that for the complete graph K = K_n on n nodes, the weighted distances (without boundary weights) satisfy

(8) [$${{{W_K}({u^*},{v^*})} \over {\log n/n}}\;\buildrel {\rm{p}} \over \longrightarrow \;{1 \over \lambda },$$

(9) [$${{\mathop {\max }\limits_v {W_K}({u^*},v)} \over {\log n/n}}\;\buildrel {\rm{p}} \over \longrightarrow \;{2 \over \lambda },$$

(10) [$${{\mathop {\max }\limits_{u,v} {W_K}(u,v)} \over {\log n/n}}\;\buildrel {\rm{p}} \over \longrightarrow \;{3 \over \lambda }.$$

The above results have more recently been extended to sparse random graphs. For a random graph G = G(n, d ⁽ⁿ⁾) satisfying the regularity conditions (2–4), the weighted distances (without boundary weights) satisfy

(11) [$${{{W_G}({u^*},{v^*})} \over {\log n}}\;\buildrel {\rm{p}} \over \longrightarrow \;{1 \over {\lambda (\nu - 1)}},$$

(12) [$${{\mathop {\max }\limits_v {W_G}({u^*},v)} \over {\log n}}\;\buildrel {\rm{p}} \over \longrightarrow \;{1 \over {\lambda (\nu - 1)}} + {1 \over {\lambda \delta }},$$

(13) [$${{\mathop {\max }\limits_{u,v} {W_G}(u,v)} \over {\log n}}\;\buildrel {\rm{p}} \over \longrightarrow \;{1 \over {\lambda (\nu - 1)}} + {2 \over {\lambda \delta }}.$$

Formulas (11–13) agree with (8–10) because ν ≈ n and δ ≈ n for the complete graph on n nodes. Formula (11) was proved in [Reference Ding, Kim, Lubetzky and Peres9] for degenerate degree distributions (random regular graph), in [Reference Bhamidi, van der Hofstad and Hooghiemstra7] for power-law degree distributions (when τ ∊ (2, 3)), and in [Reference Amini, Draief and Lelarge2] for general limiting degree distributions with a finite variance. Formulas (12–13) have been proved in [Reference Ding, Kim, Lubetzky and Peres9] for random regular graphs and in [Reference Amini, Draief and Lelarge2, Reference Amini and Lelarge3] for general limiting degree distributions with a finite variance. Sparse random graphs where the limiting degree distribution has infinite variance have in general a completely different behavior with typical passage times of order o(log n) [Reference Baroni, van der Hofstad and Komjáthy6, Reference Bhamidi, van der Hofstad and Hooghiemstra7] and they are not discussed further in this paper. The constant ν appearing in the above formulas can be recognized as the mean of the downshifted size biasing [Reference Leskelä and Ngo13] of the limiting degree distribution f, and ν is finite if and only if the second moment of f is finite.

Theorem 1 generalizes formulas (11–13) to the setting where nodes have nonnegative random weights X ₀(v) and X ₁(v) with exponential tail. The main qualitative findings are that the boundary weights have no effect on the typical passage time W(u*, v*), but they may affect the typical flooding time max_v W(u*, v) and the maximum flooding time max_u,v W(u, v). All boundary weight effects can be ignored on the log n time scale when the tails of the node weight distributions decay sufficiently fast (λ ₀, λ ₁ > λδ).

A notable feature of the results in Theorem 1 is that the leading role of the node weight distributions is the behavior of P(X_i(v) > t)as t →∞, whereas the leading role of link weight distribution is in many cases [Reference Baroni, van der Hofstad and Komjáthy6, Reference Janson11] governed by the behavior of P(W(e) > t)as t → 0.

3.2. Application: Broadcasting on random regular graphs

As an application, we discuss a continuous-time version of a message transmission and replication model operating in a push mode [Reference Aalto and Leskelä1, Reference Amini, Draief and Lelarge2, Reference Pittel17]. Let G be a random δ-regular graph on n nodes, where each node has a state in {0, 1, 2}. Initially one of the nodes called root is in state 1, and all other nodes are in state 0. Each node activates at random time instants according to a Poisson process of rate κ > 0, independently of other nodes and the underlying graph structure. When a node activates, it contacts a random target among its neighbors. The states of the nodes are updated in two ways:

• • 0 ↦ 1: If the initiator of a contact is in state 1 or 2, and the target node is in state 0, then the state of the target node changes from 0 to 1; otherwise nothing happens during the contact.

• • 1 ↦ 3: Having entered state 1, node v remains in this state for a random time period of length X ₁(v), and then the state of node v changes into 2.

We can interpret the model in the context of computer or biological viruses as follows: State 0 refers to nodes which are vulnerable to receiving a virus. State 1 refers to nodes carrying and spreading the virus but displaying no symptoms. State 2 refers to nodes carrying and spreading the virus and displaying symptoms. We denote by flood₁(G) the time until every node in the graph has received the virus, and by flood₂(G) the time until every node displays symptoms.

The above model can be analyzed using the weighted random graph where all links have a random exponentially distributed weight of rate parameter λ = κ/δ with X ₀(v) = 0, and X ₁(v) modeling the delay until an infected node displays symptoms. Then for a random root node u*,

[$$\matrix{ {{\rm{floo}}{{\rm{d}}_1}(G)\;\mathop = \limits^{\rm{d}} \;\mathop {\max }\limits_v {W_G}({u^*},v),} \hfill \cr {{\rm{floo}}{{\rm{d}}_2}(G)\;\mathop = \limits^{\rm{d}} \;\mathop {\max }\limits_v ({W_G}({u^*},v) + {X_1}(v)).} \hfill \cr } $$

Applying formula (6) in Theorem 1 with λ ₁ = ∞ corresponding to X ₁(v) = 0, we have, with high probability (w.h.p.),

(14) [$${\rm{floo}}{{\rm{d}}_1}(G)\; = \;({1 \over {\lambda (\nu - 1)}} + {1 \over {\lambda \delta }})\log n + o(\log n).$$

Note that the same formula can also be obtained from (12). Applying (6) again, we have, w.h.p.,

(15) [$${\rm{floo}}{{\rm{d}}_2}(G)\; = \;({1 \over {\lambda (\nu - 1)}} + {1 \over {\lambda \delta \wedge {\lambda _1}}})\log n + o(\log n).$$

These two formulas lead to the following results.

Corollary 1. For a random δ-regular graph G on n nodes with δ ≥ 3, when the distribution of X ₁(v) has an exponential tail of rate λ ₁ according to (1),

[$${\rm{floo}}{{\rm{d}}_1}(G)\; = \;{2 \over \kappa }({{\delta - 1} \over {\delta - 2}})\log n + o(\log n)$$

and

[$${\rm{floo}}{{\rm{d}}_2}(G)\; = \;({\delta \over {\kappa (\delta - 2)}} + {1 \over {\kappa \wedge {\lambda _1}}})\log n + o(\log n)$$

with high probability as n → ∞.

Proof. The results follow directly by substituting λ = δ/κ and ν = δ − 1 into(14) and (15). The coefficient in (14) simplifies by direct calculation into

[$${1 \over {\lambda (\nu - 1)}} + {1 \over {\lambda \delta }} = {\delta \over {\kappa (\delta - 2)}} + {1 \over \kappa } = {2 \over \kappa }({{\delta - 1} \over {\delta - 2}}).$$

Figure 1 illustrates how the limiting approximations of Corollary 1 relate to simulated values of the flooding times on 3-regular graphs. The sizes of the fluctuations around the theoretical values appear to be of constant order with respect to n. A constant order of fluctuations corresponds to the well-known fact in statistical extreme value theory that the maximum of n independent exponential random numbers is approximately Gumbel-distributed around a value of size log n. However, the additional randomness induced by the underlying random graph may cause the fluctuations to grow slowly with respect to n. Whether or not the fluctuations grow with n is not possible to detect from simulations of modest size, because the growth rate of the fluctuations is at most o(log n).

Figure 1:

Flooding times on random δ-regular graphs with δ = 3, κ = 1, and λ ₁ = 1/2. The blue circles represent simulated values of flood₁(G), and the red triangles represent simulated values of flood₂(G). The blue solid line and the red dashed line correspond to the limiting formulas of Corollary 1. (Color online.)

Figure 2 describes simulated trajectories of node counts in different states in a random 3-regular graph of 1000 nodes. The trajectories are approximately S-shaped, with random horizontal shifts caused by the initial and final phases of the process.

Figure 2:

Simulated trajectories of the number of nodes in state 1 or state 2 (blue, solid) and the number of nodes in state 2 (red, dashed) for a random δ-regular graph with n = 1000, δ = 3, κ = 1, and λ ₁ = 1/2. (Color online.)

4.1. Configuration model

A standard method for studying the random graph G = G(n, d ⁽ⁿ⁾) is to investigate a related random multigraph. A multigraph is a triplet G = (V, E, ϕ), where V and E are finite sets and [$$\phi {\kern 1pt} :{\kern 1pt} E \to \left( {\matrix{ V \cr 1 \cr } } \right) \cup \left( {\matrix{ V \cr 2 \cr } } \right)$$. Here, ϕ(e) refers to the set of one (loop) or two (non-loop) nodes incident to e ∊ E. A multigraph is called simple if ϕ is one-to-one (no parallel links) and [$$\phi (E) \subset \left( {\matrix{ V \cr 2 \cr } } \right)$$ (no loops). The degree of a node i is defined by [$$\sum\nolimits_{e \in E} ({\bf{1}}(i \in \phi (e)) + {\bf{1}}(\{ i\} = \phi (e)))$$ , that is, the number of links incident to i, with loops counted twice. A path of length k ≥ 0 from x ₀ to x_k is a set of distinct nodes {x ₀, x ₁,..., x_k} such that {x _j−1, x _j} ∊ ϕ (E) for all j. For a multigraph G weighted by W : E → (0, ∞), we denote

[$${W_G}(u,v)\; = \;\mathop {\inf }\limits_\Gamma {\kern 1pt} \sum\limits_{e \in \Gamma } W(e),$$

where Γ is the set of paths from u to v. When G is connected, the above formula defines a metric on G.

Let us recall the usual definition of the configuration model in [Reference Bollobás8]. Let n be a positive integer and d = (d ₁, d ₂,..., d_n) be a sequence of nonnegative integers. For each node i ∊ [n] we attach d_i distinct elements called half-edges. A pair of half-edges is called an edge. To obtain a random multigraph G*, it is required that the sum of half-edges d = (d ₁, d ₂,..., d_n) is even, [$$\sum\nolimits_{i = 1}^n {d_i} = 2m$$, where m refers to the number of edges. Let D_i be the set of half-edges of node i. Then the size of the set D_i is d_i and the sets D ₁, D ₂,..., D_n are disjoint. Let [$$D = \bigcup\nolimits_{i = 1}^n {D_i}$$ be the collection of all the half-edges and let E be a pairing of D (partition into m pairs) selected uniformly at random. The configuration model G* = G*(n, d) is the multigraph ([n], E,ϕ), where the function [$$\phi {\kern 1pt} :{\kern 1pt} E \to \left( {\matrix{ n \cr 1 \cr } } \right) \cup \left( {\matrix{ n \cr 2 \cr } } \right)$$ is defined by ϕ(e) = {i ∊ [n]: D_i ≠ e = Ø}.

A key feature of the configuration model is that the conditional distribution of G*(n, d) given that G*(n, d) is simple equals the distribution of the random graph G(n, d). Moreover, for a sequence of degree lists d ⁽ⁿ⁾ satisfying the regularity conditions (2) and (3), the probability that G*(n, d ⁽ⁿ⁾) is simple is bounded away from zero [Reference Janson and Luczak12]. Therefore, any statement concerning G*(n, d ⁽ⁿ⁾) which holds with high probability, also holds for G(n, d ⁽ⁿ⁾) with high probability. This is why, in the following, we write G in place of G* and the analysis of weighted distances will be conducted on the configuration model.

4.2. Notation

For a node u in the weighted multigraph, we denote by B(u, t) = {W_G(u, v) ≤ t} the set of nodes within distance t ∊ [0, ∞] from u. For an integer k ≥ 0, we define

[$${T_u}(k)\; = \;\min \{ t \ge 0{\kern 1pt} :{\kern 1pt} |B(u,t)| \ge k + 1\} ,$$

with the convention that min Ø = ∞. We also denote by S_u(k) the number of outgoing links from set B(u, T_u(k)). Then for any k less than the component size of u:

• • T_u(k) equals the distance from u to its kth nearest neighbor, and

• • S_u(k) equals the number of outgoing links from the set of nodes consisting of u and its k nearest neighbors.

Moreover, T_u(k) =∞ and S_u(k) = 0 for all k greater than or equal to the component size of u.

Throughout the following, we assume that G satisfies the regularity conditions (2–4). We introduce the scale parameters

[$$\matrix{ {{\alpha _n} = \mathop {\log }\nolimits^3 n,} \hfill \cr {{\beta _n} = 3\sqrt {{\textstyle{\mu \over {\nu - 1}}}n\log n} } \hfill \cr } $$

and, with high probability, [Reference Amini, Draief and Lelarge2, Proposition 4.2] (see alternatively [Reference Ding, Kim, Lubetzky and Peres9, Lemma 3.3] or [Reference Bhamidi, van der Hofstad and Hooghiemstra7, Proposition 4.9])

(16) [$${W_G}(u,v)\; \le \;{T_u}({\beta _n}) + {T_v}({\beta _n})$$

for all nodes u and v in the graph G. We will next analyze the behavior of T_u(β_n) and T_v(β_n)in typical (uniformly randomly chosen node) and extremal cases.

4.3. Upper bound on weighted distances

The following upper bound on the weighted distances is a sharpened version of [Reference Amini, Draief and Lelarge2, Lemmas 4.7, 4.12]. Below we assume that X ≥ 0 is an arbitrary random number and u* is a uniformly randomly chosen node, such that X, u*, and the graph G are mutually independent, and independent of the weights (W(e))_e∊E(G), where weights W(e) are exponentially distributed with rate λ > 0. We use [$${{\cal F}_{{S_{{u^*}}}}}$$ to denote the sigma-algebra generated by [$${S_{{u^*}}} = ({S_{{u^*}}}(0), \ldots ,{S_{{u^*}}}(n - 1))$$.

Lemma 1. Fix integers 0 ≤ a < b < n and numbers c ₁, c ₂ ≥ 0, and let ℛ be an [$${{\cal F}_{{S_{{u^*}}}}}$$ measurable event on which [$${S_{{u^*}}}(k) \ge {c_1} + {c_2}k$$ for all a ≤ k ≤ b − 1. For any 0 < θ < λ(c ₁ + c ₂a),

[$${\rm{E}}({{\rm{e}}^{\theta ({T_{{u^*}}}(b) - {T_{{u^*}}}(a) + X)}}|{\cal R})\; \le \;{M_X}(\theta )\exp ({\theta \over {{\theta _1} - \theta }} + {\theta \over {{\theta _0}}}({1 \over {a + 1}} + \log {{b - 1} \over {a + 1}})),$$

where [$${M_X}(\theta ) = {\rm{E}}({{\rm{e}}^{\theta X}})$$, [$${\theta _0} = \lambda {c_2} - {{{{(\theta - \lambda {c_1})}_ + }} \over {a + 1}}$$, and θ ₁ = λ(c ₁ + c ₂a).

Proof. A key property of the model is that, conditionally on [$${S_{{u^*}}}$$, the random numbers [$${T_{{u^*}}}(k + 1) - {T_{{u^*}}}(k)$$ are independent and exponentially distributed with rates [$$\lambda {S_{{u^*}}}(k)$$. On the event ℛ, we see that [$$\lambda {S_{{u^*}}}(a) \ge {\theta _1}$$, and for all a + 1 ≤ k ≤ b − 1,

[$$\lambda {S_{{u^*}}}(k) - \theta \; \ge \;\lambda {c_1} + \lambda {c_2}k - \theta \; \ge \;(\lambda {c_2} - {{{{(\theta - \lambda {c_1})}_ + }} \over k})k\; \ge \;{\theta _0}k.$$

As a consequence,

[$$\matrix{ {{\rm{E}}({{\rm{e}}^{\theta ({T_{{u^*}}}(b) - {T_{{u^*}}}(a) + X)}}|{{\cal F}_{{S_{{u^*}}}}})\; = \;{M_X}(\theta )\prod\limits_{k = a}^{b - 1} {{\lambda {S_{{u^*}}}(k)} \over {\lambda {S_{{u^*}}}(k) - \theta }}} \hfill \cr {\; = \;{M_X}(\theta )\prod\limits_{k = a}^{b - 1} (1 + {\theta \over {\lambda {S_{{u^*}}}(k) - \theta }})} \hfill \cr {\; \le \;{M_X}(\theta )\exp (\sum\limits_{k = a}^{b - 1} {\theta \over {\lambda {S_{{u^*}}}(k) - \theta }}).} \hfill \cr } $$

Separating the first term from the sum, and by the choice of the event ℛ, we obtain

[$${\rm{E}}({{\rm{e}}^{\theta ({T_{{u^*}}}(b) - {T_{{u^*}}}(a) + X)}}|{\cal R})\; \le \;{M_X}(\theta )\exp ({\theta \over {{\theta _1} - \theta }} + {\theta \over {{\theta _0}}}\sum\limits_{k = a + 1}^{b - 1} {1 \over k}).$$

By integration we have [$$\sum\nolimits_{k = m}^n {1 \over k} \le \log ({n \over {m - 1}})$$ for any integers 2 ≤ m < n. Hence, separating the first term again from the sum, we have the desired result,

[$${\rm{E}}({{\rm{e}}^{\theta ({T_{{u^*}}}(b) - {T_{{u^*}}}(a) + X)}}|{\cal R})\; \le \;{M_X}(\theta )\exp ({\theta \over {{\theta _1} - \theta }} + {\theta \over {{\theta _0}}}({1 \over {a + 1}} + \log {{b - 1} \over {a + 1}})).$$

4.4. Upper bounds on nearest neighbor distances

Proposition 1. For any 0 ≤ p ≤ 1, ε > 0, and any random variable X ≥ 0 independent of G,

[$${\rm{P}}({T_{{u^*}}}({\alpha _n}) + X > ({p \over {\lambda \delta \wedge {\theta ^*}}} + \varepsilon )\log n)\; = \;o({n^{ - p}}),$$

where θ* = sup{θ ≥ 0 : E(e^θX) < ∞} > 0.

Proof. Let

[$${t_n} = ({p \over {\lambda \delta \wedge {\theta ^*}}} + \varepsilon )\log n.$$

An upper bound for the event under study, [$${{\cal A}_n} = \{ {T_{{u^*}}}({\alpha _n}) + X > {t_n}\} $$, is obtained by

(17) [$$\matrix{ {{\rm{P}}({{\cal A}_n})} \hfill \amp {\; \le \;{\rm{P}}({{\cal A}_n}|{{\cal R}_1}) + {\rm{P}}({{\cal A}_n}|{{\cal R}_2} \cap {\cal R}_1^{\rm{c}}){\kern 1pt} {\rm{P}}({\cal R}_1^{\rm{c}}) + {\rm{P}}({\cal R}_2^{\rm{c}}),} \hfill \cr } $$

where

[$$\matrix{ {{{\cal R}_1}\; = \;\{ {S_{{u^*}}}(k) \ge \delta + (\delta - 2)k\;{\rm{forall0}} \le {\rm{k}} \le {\alpha _{\rm{n}}} - {\rm{1}}\} ,} \cr {{{\cal R}_2}\; = \;\{ {S_{{u^*}}}(k) \ge 1 + (\delta - 2)k\;{\rm{forall0}} \le {\rm{k}} \le {\alpha _{\rm{n}}} - {\rm{1}}\} .} \cr } $$

We will next analyze the conditional probabilities in (17).

(i) To obtain an upper bound of P(𝒜_n | ℛ ₁), by applying Lemma 1 with a = 0, b = α_n, c ₁ = δ, and c ₂ = δ − 2 and Markov’s inequality, we find that

[$${\rm{P}}({{\cal A}_n}|{{\cal R}_1})\; \le \;{M_X}(\theta )\exp ({\theta \over {{\theta _1} - \theta }} + {\theta \over {{\theta _0}}}(1 + \log {\alpha _n}) - \theta {t_n})$$

for all 0 < θ < θ ₁ ∧ θ*, where θ ₀ = λ(δ − 2) and θ ₁ = λδ. Now we may choose

[$$\theta \ge (1 - {{\lambda \delta \wedge {\theta ^*}} \over {2(p + \varepsilon (\lambda \delta \wedge {\theta ^*}))}}\varepsilon )(\lambda \delta \wedge {\theta ^*})$$

to have [$$\theta {t_n} \ge (p + {1 \over 2}\varepsilon ({\theta _1} \wedge {\theta ^*}))\log n$$ log n. Note that θ can be arbitrary close to its maximum value λδ ∧ θ* if we choose ε > 0 to be sufficiently small. Since the constant term [$${\theta \over {{\theta _1} - \theta }}$$ small compared to log α_n = Θ (log log n), we have, for large values of n,

[$${\theta \over {{\theta _1} - \theta }} + {\theta \over {{\theta _0}}}(1 + \log {\alpha _n})\; \le \;{1 \over 4}\varepsilon ({\theta _1} \wedge {\theta ^*})\log n.$$

These two inequalities imply that

(18) [$${\rm{P}}({{\cal A}_n}|{{\cal R}_1})\; \le \;{M_X}(\theta ){n^{ - (p + {1 \over 4}\varepsilon (\lambda \delta \wedge {\theta ^*}))}}\; = \;o({n^{ - p}}).$$

(ii) For an upper bound of [$${{\rm{P}}_{{{\cal R}_2}\backslash {{\cal R}_1}}}({{\cal A}_n})$$, we apply Lemma 1 with a = 0, b = α_n, c ₁ = 1, and c ₂ = δ − 2 and Markov’s inequality to conclude that

[$${\rm{P}}({{\cal A}_n}|{{\cal R}_2} \cap {\cal R}_1^{\rm{c}})\; \le \;{M_X}(\theta )\exp ({\theta \over {\lambda - \theta }} + {\theta \over {\lambda (\delta - 2)}}(1 + \log {\alpha _n}) - \theta {t_n})$$

for all 0 < θ < λ ∧ θ*. For any such θ, we see that [$$\theta {t_n} \ge {\varepsilon _1}\log n$$ with

[$${\varepsilon _1} = \theta ({p \over {\lambda \delta \wedge {\theta ^*}}} + \varepsilon ) > 0.$$

Because

[$${\theta \over {\lambda - \theta }} + {\theta \over {\lambda (\delta - 2)}}(1 + \log {\alpha _n})\; \le \;{1 \over 2}{\varepsilon _1}\log n$$

for all large n, it follows that

(19) [$${\rm{P}}({{\cal A}_n}|{{\cal R}_2} \cap {\cal R}_1^{\rm{c}})\; = \;O({n^{ - {\varepsilon _1}/2}}).$$

Note that our S_u(k) has the same distribution as the exploration process in [Reference Amini, Draief and Lelarge2, Section 4.1]. Hence, by [Reference Amini, Draief and Lelarge2, Lemma 4.6], [$${\rm{P}}({\cal R}_1^{\rm{c}}) = o({n^{ - 1}}\mathop {\log }\nolimits^{10} n)$$ and [$${\rm{P}}({\cal R}_2^{\rm{c}}) = o({n^{ - 3/2}})$$. Hence, by substituting the bounds (18) and (19) into (17), it follows that

[$${\rm{P}}({{\cal A}_n})\; \le \;o({n^{ - p}}) + O({n^{ - {\varepsilon _1}/2}})o({n^{ - 1}}\mathop {\log }\nolimits^{10} n) + o({n^{ - 3/2}})\; = \;o({n^{ - p}}).$$

4.5. Upper bounds on moderate distances

Proposition 2. For any ε > 0,

[$${\rm{P}}({T_{{u^*}}}({\beta _n}) - {T_{{u^*}}}({\alpha _n}) > ({1 \over {2\lambda (\nu - 1)}} + \varepsilon )\log n)\; = \;o({n^{ - 1}}).$$

Proof. Denote [$$c = {1 \over {2\lambda (\nu - 1)}}$$ and t_n = (c + ε) log n. Set c ₁ = 0 and c ₂ = 1/λc. Fix a number [$$\theta > {2 \over \varepsilon }$$, and set [$${\theta _0} = \lambda {c_2} - {\theta \over {{\alpha _n} + 1}}$$ and [$${\theta _1} = \lambda {c_2}{\alpha _n}$$. Then, for all sufficiently large values of n,we see that 0 < θ < θ ₁. When we apply Lemma 1 with X = 0, a = α_n, and b = β_n and Markov’s inequality, we find that on the event ℛ ₃ that [$${S_{{u^*}}}(k) \ge {c_2}k$$ for all α_n ≤ k ≤ β_n − 1,

[$$\matrix{ {{\rm{P}}({T_{{u^*}}}({\beta _n}) - {T_{{u^*}}}({\alpha _n}) > {t_n}|{{\cal R}_3})} \hfill \cr {\; \le \;\exp ({\theta \over {{\theta _1} - \theta }} + {\theta \over {{\theta _0}}}({1 \over {{\alpha _n}}} + \log {{{\beta _n}} \over {{\alpha _n}}}) - \theta {t_n})} \hfill \cr {\; = \;\exp ({\theta \over {\lambda {c_2}{\alpha _n} - \theta }} + {\theta \over {\lambda {c_2} - {\theta \over {{\alpha _n} + 1}}}}({1 \over {{\alpha _n}}} + \log {{{\beta _n}} \over {{\alpha _n}}}) - (c + \varepsilon )\theta \log n).} \hfill \cr } $$

Note that β_n/α_n ≤ n. Because α_n →∞, we see that

[$$\matrix{ {{\rm{P}}({T_{{u^*}}}({\beta _n}) - {T_{{u^*}}}({\alpha _n}) > {t_n}|{{\cal R}_3})\; \le \;\exp ((c + \varepsilon /2)\theta \log n - (c + \varepsilon )\theta \log n)} \cr {\; = \;\exp ( - {\varepsilon \over 2}\theta \log n).} \cr } $$

Due to our choice of θ, the right-hand side is o(n ⁻1). The claim follows from this because [$${\rm{P}}({\cal R}_3^{\rm{c}}) = o({n^{ - 3/2}})$$ by [Reference Amini, Draief and Lelarge2, Lemma 4.9].

4.6. Proof of Theorem 1: Upper bounds

Observe that [$${\rm{E}}({{\rm{e}}^{\theta {X_i}(v)}})$$ is finite for θ < λ_i and infinite for θ > λ_i due to our assumption on exponential tails (1). Hence, by applying Proposition 1 with p = 1,

[$${\rm{P}}({T_{{v^*}}}({\alpha _n}) + {X_i}({v^*}) > ({1 \over {\lambda \delta \wedge {\lambda _i}}} + \varepsilon )\log n)\; = \;o({n^{ - 1}}),$$

so that, by applying the generic union bound

(20) [$${\rm{P}}(\mathop {\max }\limits_v X(v) > t)\; \le \;\sum\limits_v {\rm{P}}(X(v) > t)\; = \;n{\rm{P}}(X({v^*}) > t),$$

it follows that

(21) [$$\mathop {\max }\limits_v ({T_v}({\alpha _n}) + {X_i}(v))\; \le \;({1 \over {\lambda \delta \wedge {\lambda _i}}} + \varepsilon )\log n\quad \quad {\rm{w}}{\rm{.h}}{\rm{.p}}{\rm{.}}$$

Furthermore, by applying Proposition 1 with p = 0, it follows that

(22) [$${T_{{v^*}}}({\alpha _n})\; \le \;{T_{{v^*}}}({\alpha _n}) + {X_i}({v^*})\; \le \;\varepsilon \log n\quad \quad {\rm{w}}{\rm{.h}}{\rm{.p}}.,$$

and by Proposition 2 and the generic union bound (20), w.h.p.,

(23) [$${T_{{v^*}}}({\beta _n}) - {T_{{v^*}}}({\alpha _n})\; \le \;\mathop {\max }\limits_v ({T_v}({\beta _n}) - {T_v}({\alpha _n}))\; \le \;({1 \over {2\lambda (\nu - 1)}} + \varepsilon )\log n.$$

By combining (21) and (22) with (23), we conclude that, w.h.p.,

(24) [$$\mathop {\max }\limits_v ({T_v}({\beta _n}) + {X_i}(v))\; \le \;({1 \over {\lambda \delta \wedge {\lambda _i}}} + {1 \over {2\lambda (\nu - 1)}} + 2\varepsilon )\log n$$

and

(25) [$${T_{{v^*}}}({\beta _n})\; \le \;({1 \over {2\lambda (\nu - 1)}} + 2\varepsilon )\log n.$$

To prove an upper bound for (5), observe that the distribution of X_i(v*) does not depend on the scale parameter n. Therefore, X_i(v*) ≤ ε log n with high probability. In light of (16) and (25), it follows that, w.h.p.,

[$$\matrix{ {W({u^*},{v^*})\; = \;{X_0}({u^*}) + {W_G}({u^*},{v^*}) + {X_1}({v^*})} \cr {\; \le \;{X_0}({u^*}) + {T_{{u^*}}}({\beta _n}) + {T_{{v^*}}}({\beta _n}) + {X_1}({v^*})} \cr {\; \le \;({1 \over {\lambda (\nu - 1)}}\log n + 6\varepsilon )\log n.} \cr } $$

To prove an upper bound for (6), observe that by applying (16), (24), and (25), with high probability,

[$$\matrix{ {\mathop {\max }\limits_v W({u^*},v)\; = \;\mathop {\max }\limits_v ({X_0}({u^*}) + {W_G}({u^*},v) + {X_1}(v))} \hfill \cr {\; \le \;\mathop {\max }\limits_v ({X_0}({u^*}) + {T_{{u^*}}}({\beta _n}) + {T_v}({\beta _n}) + {X_1}(v))} \hfill \cr {\; = \;{X_0}({u^*}) + {T_{{u^*}}}({\beta _n}) + \mathop {\max }\limits_v ({T_v}({\beta _n} + {X_1}(v)))} \hfill \cr {\; \le \;({1 \over {2\lambda (\nu - 1)}} + 3\varepsilon )\log n + ({1 \over {\lambda \delta \wedge {\lambda _1}}} + {1 \over {2\lambda (\nu - 1)}} + 2\varepsilon )\log n} \hfill \cr {\; = \;({1 \over {\lambda \delta \wedge {\lambda _1}}} + {1 \over {\lambda (\nu - 1)}} + 5\varepsilon )\log n.} \hfill \cr } $$

Finally, for an upper bound for (7), observe that, by (16), with high probability,

[$$\matrix{ {\mathop {\max }\limits_{u,v} W(u,v)\; = \;\mathop {\max }\limits_{u,v} ({X_0}(u) + {W_G}(u,v) + {X_1}(v))} \cr {\; \le \;\mathop {\max }\limits_{u,v} ({X_0}(u) + {T_u}({\beta _n}) + {T_v}({\beta _n}) + {X_1}(v))} \cr {\; = \;\mathop {\max }\limits_u ({X_0}(u) + {T_u}({\beta _n})) + \mathop {\max }\limits_v ({X_1}(v) + {T_v}({\beta _n})).} \cr } $$

And hence, by (24), it follows that, with high probability,

[$$\mathop {\max }\limits_{u,v} W(u,v)\; \le \;({1 \over {\lambda \delta \wedge {\lambda _0}}} + {1 \over {\lambda (\nu - 1)}} + {1 \over {\lambda \delta \wedge {\lambda _1}}} + 4\varepsilon )\log n.$$

The above inequalities are sufficient to confirm the upper bounds in Theorem 1 because ε> 0 can be chosen arbitrarily small.

4.7. Proof of Theorem 1: Lower bounds

The lower bounds are relatively straightforward generalizations of the analogous results (11–13) for the model without node weights, which imply that for an arbitrarily small ε > 0, the weighted graph distance W_G satisfies, w.h.p.,

(26) [$${{{W_G}({u^*},{v^*})} \over {\log n}}\; \ge \;{1 \over {\lambda (\nu - 1)}} - \varepsilon ,$$

(27) [$${{\mathop {\max }\limits_v {W_G}({u^*},v)} \over {\log n}}\; \ge \;{1 \over {\lambda (\nu - 1)}} + {1 \over {\lambda \delta }} - \varepsilon ,$$

(28) [$${{\mathop {\max }\limits_{u,v} {W_G}(u,v)} \over {\log n}}\; \ge \;{1 \over {\lambda (\nu - 1)}} + {2 \over {\lambda \delta }} - \varepsilon .$$

We first prove the following lemma and we apply it later in the proof of the lower bounds.

Lemma 2. For every integer n ≥ 1, let A_n(i) and B_n(i) be random numbers indexed by a finite set i ∊ I_n. Assume that [$${({A_n}(i))_{i \in {I_n}}}$$ are independent and identically distributed, that

[$$|{I_n}|{\kern 1pt} {\rm{P}}({A_n}(i) > {a_n})\; \to \;\infty ,$$

and that [$${B_n}({i^*}) > {b_n}$$ with high probability, where i* is a uniformly random point of I_n, independent of [$${({B_n}(i))_{i \in {I_n}}}$$. Assume also that A_n(i) and B_n(i) are independent for every i ∊ I_n. Then

[$$\mathop {\max }\limits_{i \in {I_n}} ({A_n}(i) + {B_n}(i))\; > \;{a_n} + {b_n}$$

with high probability.

Proof. Let

[$${M_n}\; = \;|\{ i \in {I_n}{\kern 1pt} :{\kern 1pt} {A_n}(i) > {a_n}\} |$$

and

[$${N_n}\; = \;|\{ i \in {I_n}{\kern 1pt} :{\kern 1pt} {A_n}(i) > {a_n},{\kern 1pt} {A_n}(i) + {B_n}(i) > {a_n} + {b_n}\} |.$$

Observe that M_n is binomially distributed with |I_n| trials and rate parameter p_n = P(A_n(i) > a_n). Then E(M_n) = |I_n|p_n and Var(M_n) ≤ E(M_n), and because |I_n|p_n →∞, it follows that [$${M_n} \ge {1 \over 2}|{I_n}|{p_n}$$ with high probability. Moreover,

[$$\matrix{ {{\rm{E}}({M_n} - {N_n})\; = \;\sum\limits_{i \in {I_n}} {\rm{P}}({A_n}(i) > {a_n},{\kern 1pt} {A_n}(i) + {B_n}(i) \le {a_n} + {b_n})} \cr {\; \le \;\sum\limits_{i \in {I_n}} {\rm{P}}({A_n}(i) > {a_n},{\kern 1pt} {B_n}(i) \le {b_n})} \cr {\; = \;\sum\limits_{i \in {I_n}} {\rm{P}}({A_n}(i) > {a_n}){\rm{P}}({B_n}(i) \le {b_n})} \cr {\; = \;|{I_n}|{p_n}{\kern 1pt} {\rm{P}}({B_n}({i^*}) \le {b_n}).} \cr } $$

Because P(B_n(i*) ≤ b_n) = o(1), Markov’s inequality implies that [$${M_n} - {N_n} \le {1 \over 4}|{I_n}|{p_n}$$ with high probability. We conclude that, with high probability,

[$${N_n}\; = \;{M_n} - ({M_n} - {N_n})\; \ge \;{1 \over 2}|{I_n}|{p_n} - {1 \over 4}|{I_n}|{p_n}\; = \;{1 \over 4}|{I_n}|{p_n}\; \ge \;1,$$

and [$$\mathop {\max }\nolimits_{i \in {I_n}} ({A_n}(i) + {B_n}(i)) > {a_n} + {b_n}$$.

(i) A suitable lower bound for (5) follows immediately from (26) because W(u*, v*) ≥ W_G(u*, v*) almost surely.

(ii) To prove a lower bound for (6), note that the exponential tail assumption (1) implies that

[$$n{\kern 1pt} {\rm{P}}\;({X_1}(v) > ({1 \over {{\lambda _1}}} - \varepsilon )\log n)\; \to \;\infty .$$

Then, by applying Lemma 2 (with A_n(v) = X ₁(v), B_n(v) = W_G(u*, v), and I_n = [n]), recalling (26), we find that, w.h.p.,

[$$\matrix{ {\mathop {\max }\limits_v W({u^*},v)} \hfill \amp {\; = \;\mathop {\max }\limits_v ({X_0}({u^*}) + {W_G}({u^*},v) + {X_1}(v))} \hfill \cr {} \hfill \amp {\; \ge \;\mathop {\max }\limits_v ({W_G}({u^*},v) + {X_1}(v))} \hfill \cr {} \hfill \amp {\; \ge \;({1 \over {\lambda (\nu - 1)}} + {1 \over {{\lambda _1}}} - 2\varepsilon )\log n.} \hfill \cr } $$

By noting that max_v W(u*, v) ≥ max_v W_G(u*, v) and applying (27), we also obtain

[$$\mathop {\max }\limits_v W({u^*},v)\; \ge \;({1 \over {\lambda (\nu - 1)}} + {1 \over {\lambda \delta }} - 2\varepsilon )\log n,$$

and hence, w.h.p.,

[$$\mathop {\max }\limits_v W({u^*},v)\; \ge \;({1 \over {\lambda (\nu - 1)}} + {1 \over {\lambda \delta \wedge {\lambda _1}}} - 2\varepsilon )\log n.$$

(iii) To prove a lower bound for (7), note that, for any u ≠ v,

[$$\matrix{ {{\rm{P}}\;({{{X_0}(u) + {X_1}(v)} \over {\log n}} > {1 \over {{\lambda _0}}} + {1 \over {{\lambda _1}}} - 2\varepsilon )} \cr {\; \ge \;{\rm{P}}\;({{{X_0}(u)} \over {\log n}} > {1 \over {{\lambda _0}}} - \varepsilon ,{{{X_1}(u)} \over {\log n}} > {1 \over {{\lambda _1}}} - \varepsilon )} \cr {\; = \;{\rm{P}}\;({{{X_0}(u)} \over {\log n}} > {1 \over {{\lambda _0}}} - \varepsilon ){\rm{P}}\;({{{X_1}(u)} \over {\log n}} > {1 \over {{\lambda _1}}} - \varepsilon ).} \cr } $$

Then the exponential tail assumption (1) implies that

[$$n(n - 1){\rm{P}}\;({{{X_0}(u) + {X_1}(v)} \over {\log n}} > {1 \over {{\lambda _0}}} + {1 \over {{\lambda _1}}} - 2\varepsilon )\; \to \;\infty .$$

Observe next that if u* and v* are independent uniformly random elements of [n], and i* is a uniformly random event in I_n ={(u, v) ∊ [n]² : u ≠ v}, then

[$$\matrix{ {{\rm{P}}({W_G}({u^*},{v^*}) \in F)\; = \;{1 \over {{n^2}}}\sum\limits_u \sum\limits_v {\rm{P}}({W_G}(u,v) \in F)} \cr {\; = \;{1 \over n}{\rm{P}}({W_G}({u^*},{u^*}) \in F) + {{n - 1} \over n}{\rm{P}}({W_G}({i^*}) \in F)} \cr } $$

for all measurable sets F ⊂ ℝ. Hence, (26) implies that, w.h.p.,

[$${W_G}({i^*})\; > ({1 \over {\lambda (\nu - 1)}} - \varepsilon )\log n.$$

Then we can apply Lemma 2 with A_n(u, v) = X ₀(u) + X ₁(v) and B_n(u, v) = W_G(u, v), to conclude that, w.h.p.,

[$$\mathop {\max }\limits_{u,v} W(u,v)\; \ge \;({1 \over {{\lambda _0}}} + {1 \over {\lambda (\nu - 1)}} + {1 \over {{\lambda _1}}} - 3\varepsilon )\log n.$$

We will next apply Lemma 2 again, this time with I_n = [n], A_n(v) = X ₁(v), and B_n(v) = max_u (X ₀(u) + W_G(u, v)), recalling (27), to conclude that, w.h.p.,

[$$\mathop {\max }\limits_{u,v} W(u,v)\; \ge \;({1 \over {\lambda \delta }} + {1 \over {\lambda (\nu - 1)}} + {1 \over {{\lambda _1}}} - 3\varepsilon )\log n.$$

By a symmetrical argument, we also find that, w.h.p.,

[$$\mathop {\max }\limits_{u,v} W(u,v)\; \ge \;({1 \over {{\lambda _0}}} + {1 \over {\lambda (\nu - 1)}} + {1 \over {\lambda \delta }} - 3\varepsilon )\log n.$$

By combining the above three inequalities with (28), we may conclude that, w.h.p.,

[$$\mathop {\max }\limits_{u,v} W(u,v)\; \ge \;({1 \over {\lambda \delta \wedge {\lambda _0}}} + {1 \over {\lambda (\nu - 1)}} + {1 \over {\lambda \delta \wedge {\lambda _1}}} - 3\varepsilon )\log n.$$