The contact process on scale-free geometric random graphs

We study the contact process on a class of geometric random graphs with scale-free degree distribution, deﬁned on a Poisson point process on R d . This class includes the age-dependent random connection model and the soft Boolean model . In the ultrasmall regime of these random graphs we provide exact asymptotics for the non-extinction probability when the rate of infection spread is small and show for a ﬁnite version of these graphs that the extinction time is of exponential order in the size of the graph


Introduction
In recent years the contact process has been studied extensively as a simple model for the spread of infection in a population or on a network.In this model each vertex of a given locally-finite graph has one of two states, 0 or 1 for each t ≥ 0, indicating whether the vertex is healthy or infected at time t.An infected vertex recovers at rate 1 and transmits the infection to a neighbouring vertex with rate λ > 0, independently of everything else.Precisely, the contact process on a locally-finite graph G = (V, E) is a continuous-time Markov process (ξ t ) t≥0 on the space {0, 1} V .By identifying ξ t with the subset {x ∈ V : ξ t (x) = 1} ⊂ V for each t ≥ 0 the transition rates are given by ξ t → ξ t \{x} for x ∈ ξ t at rate 1, and ξ t → ξ t ∪ {x} for x / ∈ ξ t at rate λ {y ∈ ξ t : x ∼ y} , where we denote by x ∼ y that x and y are connected by an edge.
Note that the contact process has a single absorbing state, corresponding to the configuration where all vertices are healthy.Thus, a natural question on the behaviour of the contact process is whether this state is reached in finite time, i.e. whether the extinction time of the contact process on G, defined by τ G := inf{t > 0 : ξ t = ∅}, is finite.
On the lattice Z d there exists a critical value λ c (Z d ) exhibiting a phase transition in whether the process dies out almost surely or not.If λ ≤ λ c (Z d ), the extinction time τ Z d is almost surely finite for any initial configuration where only finitely many sites are infected and we say it dies out, whereas if λ > λ c (Z d ) there is a positive probability that the extinction time is infinite even if the infection only starts in a single vertex, see [15].On finite graphs the extinction time is always almost surely finite and a more natural question in this setting is to ask how long the infection survives until it reaches the absorbing state.Interestingly, on the restriction of Z d to finite boxes the critical value λ c (Z d ) again exhibit a phase transition.If λ < λ c (Z d ) the extinction time is of logarithmic order in the volume of the box whereas if λ > λ c (Z d ) the contact process survives much longer and the extinction time is of exponential order in the volume of the box.In the later case the contact process is said to be in a metastable situation where it stabilizes for an exponentially long amount of time before it reaches the absorbing state where all vertices are healthy, see [15] for further details.The behaviour of the extinction time of the contact process has been studied on various different finite graphs including on finite regular trees by Stacey [21], Cranston et al. [6] and Mountford et al. [17], on regular graphs by Mourrat and Valesin [19], on Erdós-Renyi graphs by Bhamidi et al. [2] and for a general large class of finite graphs by Mountford et al. [17] and Schapira and Valesin [20].
The situation changes dramatically if we consider random graphs with a scale-free degree distribution such as the configuration model or preferential attachment networks.On these graphs the critical value of λ is zero, therefore for any choice of λ > 0 the extinction time is of exponential order in the size of the graph, see [5] and [17] for the configuration model and [1] for preferential attachment networks.For these models further results on the metastability are given by Mountford et al. [18] on the configuration model, where they provide estimates on the rate of decay of the density of infected vertices in terms of λ at a time when the infection has not yet reached the absorbing states, for λ small and graph large enough.This rate of decay solely depends on the power-law exponent τ of the scale-free degree distribution 1 , precisely it is given by 5  2 ] . (2) This result seems to have a universal character as the same rate of decay has been shown for hyperbolic random graphs by Linker et al. [16].The latter can be seen as a geometric variant of the configuration model, as in both cases the probability to form an edge between two given vertices depends on the product of independent weights which are assigned to each vertex in the graph.For further results on the density of infected vertices at a time when the contact process is still alive on the configuration model with τ ∈ (1, 2] and preferential attachment networks see [4] and [3].To obtain these estimates the analysis of the contact process on the corresponding limit graphs is important, i.e. the corresponding Galton-Watson process for the configuration model and the infinite hyperbolic model.In fact, for these models the rate of decay given by ρ τ coincides with the rate of decay of the probability that the extinction time is infinite on the limit graphs when λ goes to zero.Thus, the study of the non-extinction probability is a crucial step to obtain the stated metastability results. In this work we will study the contact process on a large class of geometric random graphs on a Poisson point process on R d , which have a scale-free degree distribution and allow the occurence of edges which span a large distance between two vertices, see Section The soft Boolean model In the Boolean model on a Poisson point process on R d each vertex x carries an independent, identically distributed random radius R x , which we assume to be heavy-tailed, i.e. there exists γ ∈ (0, 1) such that In the hard version of the model two vertices are connected by an edge if the balls centered at the vertices locations with associated radii intersect.We consider a soft version of this model, where an independent, identically distributed random variable X(x, y) is assigned to each unordered pair of vertices {x, y}.Then, an edge is formed between two vertices x and y if and only if The choice X(x, y) = 1, for all pairs {x, y}, corresponds to the hard version of the model.Writing X for an independent identically distributed copy of X(x, y), we assume X to be heavy-tailed with decay P(X > r) ≍ r −δd as r → ∞, for some δ > 1, leading to a relaxation of the condition to form an edge which can be interpreted in the following way.For any pair of vertices x and y we take a copy of their corresponding balls and expand those by multiplying both radii with X(x, y).Then, we form an edge between the two vertices if and only if the expanded balls intersect.
The age-dependent random connection model In the age-dependent random connection model each vertex carries a uniform on (0, 1) distributed birth time and two vertices x and y with birth times t and s are connected by an edge independently with probability where γ ∈ (0, 1), β > 0 and ϕ : (0, ∞) → [0, 1] is a non-decreasing function which we assume to satisfy ϕ(r) ≍ r −δ as r → ∞.This model emerges as a limit graph of a rescaled version of the age-based preferential attachment model introduced in [9].In this model vertices are added to the graph at rate of a Poisson process with unit intensity and placed on a torus of width one.A new vertex x added at time t forms an edge to each existing vertex y with probability given by (3), where s is the time vertex y has been added to the graph.As (t/s) γ is the asymptotic order of the expected degree at time t of a vertex with birth time s the age-based preferential attachment model mimics the behaviour of spatial preferential attachment networks introduced in [14].
This class of geometric random graphs has been studied recently and exhibits a phase transition in the parameters γ and δ such that these graphs are ultrasmall, i.e. two very distant vertices have a graph distance of doubly logarithmic order of their Euclidean distance, if and only if γ > δ δ+1 , as shown in [10].The same regime boundary depending on the parameters γ and δ is shown to appear in the existence of a subcritical percolation phase by Gracar et al. [9], giving a hint of a universal behaviour of these geometric random graphs that is remarkably different to the behaviour of spatial graph models investigated in [7].In this work we consider the class of geometric random graphs in their ultrasmall regime γ > δ δ+1 and in Section 2 we study the probability Γ(λ) that the contact process starting in a typical vertex does not go extinct.We prove that the critical value λ c is zero for these graphs and we give exact asymptotics on its rate of decay, when λ is small, see Theorem 2.1.In Section 3 we study a restriction of the these graphs to boxes , and show that the extinction time exhibits no phase transition, i.e. for any λ > 0 the extinction time is of exponential order in the volume of the boxes, see Theorem 3.1.

Framework
Let G be a general geometric random graph on a vertex set given by a Poisson point process X of unit intensity on R d × (0, 1).We write x = (x, t) for a vertex of the graph where we refer to x as the location and t as the mark of the vertex x.For x 1 , . . ., x n ∈ R d × (0, 1), denote by P x 1 ,...,xn the law of G given the event that x 1 , . . ., x n are points of the Poisson process.We consider geometric random graphs which satisfy the following assumption on the probability of the occurence of edges in the graph.This assumption is given in terms of two parameters γ ∈ (0, 1) and δ > 1.For two vertices x, y ∈ X we write x ∼ y if there exists an edge between them.Assumption 1.1.Given X , edges are drawn independently of each other and there exists α, κ 1 , κ 2 > 0 such that, for every pair of vertices x = (x, t), y = (y, s) ∈ X , it holds In some parts of this paper we will consider the Palm-version of G .More precisely, we add to X a vertex (0, T 0 ), where T 0 is an independent on the interval (0, 1) uniformly distributed random variable and denote by G (0,T 0 ) the resulting graph on X ∪{(0, T 0 )} determined by the connection rules satisfying Assumption 1.1.We denote the law of G (0,T 0 ) by P (0,T 0 ) and since T 0 is independent of the underlying Poisson point process X , it holds P (0,T 0 ) = P (0,t 0 ) dt 0 .We can think of P (0,T 0 ) of the law of G conditioned on the existence of a typical vertex, i.e. a vertex with typical mark, at the origin.
The parameter γ in Assumption 1.1 describes the influence of the vertices' marks on the connection probability and determines the degree distribution of a typical vertex in G .As edges are drawn independently of each other, the degree of a given vertex with mark t is Poisson distributed with parameter Λ(t), where by Assumption 1.1 there exist c, C > 0 such that it holds ct −γ ≤ Λ(t) ≤ Ct −γ for all t ∈ (0, 1).It is easy to see that the degree of a typical vertex (0, T 0 ) is scale-free with power-law exponent τ = 1+ 1 γ , see [9].The parameter δ controls the occurence of long edges in the graph, where small values of δ lead to more long edges.A natural example of such geometric random graphs is the weight-dependent random connection model, introduced in [11].One can understand Assumption 1.1 in the way that graphs satisfying the assumption are dominated by the weight-dependent random connection model with preferential-attachment kernel and dominate the model with the min kernel.
Remark 1.1.As we will see later, each of the individual proofs depends only on one of the two inequalities from Assumption 1.1.One could therefore separate this assumption into two separate assumptions and therefore make the statements of individual propositions used to prove the main results could be made stronger by assuming only the relevant bound from Assumption 1.1 holds.For simplicity and to help with the heuristic understanding of the proofs we omit this distinction.

The contact process and its graphical representation
The contact process on an arbitrary locally-finite graph G = (V, E) with parameter λ is a continuous time Markov process (ξ t ) t≥0 on the space {0, 1} V .At time t we say a vertex x ∈ V is infected if ξ t (x) = 1 and healthy if ξ t (x) = 0. Thus, we can also view ξ t as the subset {x : ξ t (x) = 1} of V of the infected vertices at time t.Infected vertices transmit the infection to each of their neighbours with rate λ and recover with rate 1, yielding the transition rates given by (1).We write (ξ A t ) t≥0 for the contact process with initial condition A ⊂ V , i.e. ξ A 0 = A and (ξ x t ) t≥0 if A = {x}.A very useful description of the contact process is its graphical representation given by a family of independent Poisson processes on [0, ∞).Assign to each vertex x ∈ V a Poisson process N x on [0, ∞) of rate 1.For each edge in G with endvertices x and y, assign to each of the pairs (x, y) and (y, x) a Poisson process N (x,y) , resp.N (y,x) , on [0, ∞) with rate λ.We can think of every element t ∈ N x as a recovery mark at x at time t, and every element t ∈ N (x,y) as a transmission arrow from x to y at time t.Hence, on V × [0, ∞) we assign a recovery mark at (x, t) for all t ∈ N x and x ∈ V and an arrow from (x, t) to (y, t) for all t ∈ N (x,y) and x, y ∈ V which are connected by an edge.An infection path in the graphical construction is a a function g : I → V for some interval I such that the process (g(t), t) t∈I on V × [0, ∞) which goes up in time never hits a recovery mark and only changes values in the first component by travelling along an arrow in its given direction.We write (x, t) → (y, s) if there exists an infection path g : [t, s] → V from x to y, i.e. an infection path with g(t) = x and g(s) = y.Then the contact process starting in A can be derived from infection paths of the graphical construction by see Figure 1 for an example.This graphical representation allows to derive important properties of the contact process easily, such as its monotonicity in the initial configuration, its additivity and its self-duality relation For further properties of the contact process we refer to [15].
Throughout the following sections we denote by P the joint law of the contact process and the underlying geometric random graph and with a slight abuse of notation denote by P x 1 ,...,xn and by P (0,T 0 ) the joint law of the contact process and the respective geometric random graph law under the conditions given in Section 1.1.

Non-extinction probability
In this section we consider the probability Γ(λ) that the contact process with parameter λ on G (0,T 0 ) starting in the origin (0, T 0 ) does not go extinct, i.e.
Theorem 2.1.Let G be a general geometric random graph which satisfies Assumptions 1.1 for γ > δ δ+1 .Then, as λ → 0, As stated in the introduction for the ultrasmall regime the non-extinction probability is positive for any λ > 0 and therefore the critical value when the contact process dies out is almost surely zero.To compare the result of Theorem 2.1 to the rates (2) given in [18], resp.[16], for the contact process on the configuration model and on hyperbolic random graphs, note that γ ∈ ( δ δ+1 , 1) implies τ ∈ (2, 3) since δ > 1.As τ = 1 + 1 γ the rate given in (5) matches the case τ ∈ ( 5 2 , 3] in (2).To see the reason why for geometric random graphs satisfying Assumption 1.1 only this case appears, it is helpful to look at the survival strategies of the infection leading to the two cases of (2) for which τ ∈ (2, 3].If τ ∈ (2, 5  2 ], the graph is so well connected that an infected vertex with a high degree, i.e. a small mark, has with high probability at least one neighbour with an even smaller mark to which the vertex transmits the infection.Thus, when the origin infects a relatively powerful vertex, i.e. a vertex with a small mark, with high probability the infection passes directly to more and more powerful vertices and therefore survives.This way of direct spreading does not work sufficiently well when τ ∈ ( 5 2 , 3] as the graph is not connected well enough.In this case the survival strategy relies on the observation that, when a vertex with sufficiently high degree is infected, the infection survives so long in the neighbourhood of the vertex which forms a star, that it reaches with high probability another vertex with similarly high degree from which this kind of delayed spreading can repeat, see [18,Section 3].For geometric random graphs satisfying Assumption 1.1 the strategy of direct spreading does not work, as for a vertex with mark t the expected number of neighbours with mark smaller than t does not increase when t becomes small, unlike in the configuration model or hyperbolic random graphs.Instead, two vertices with small mark are usually not connected directly but via a connector, a vertex with mark near one.This additional necessary step to transmit the infection to a vertex with smaller mark makes this strategy worse than the strategy of delayed spreading, which still works for the class of geometric random graphs, see Proposition 2.2, yielding that only the later one appears in Theorem 2.1.The same behaviour also holds for dynamical non-spatial preferential attachment with slow update rate, as studied in [12].
Remark 2.1.In recent work on a similar class of geometric random graphs the upper bound assumption could be relaxed by omitting the assumption of independent occurence of the edges given the Poisson point process, see [10].This is not possible here, as the proof of the asymptotic upper bound given in Proposition 2.5 requires not only the ability to control the occurence of self-avoiding paths in the graph, but also the occurence of stars, i.e. the neighbourhoods of vertices with high degree.An upper bound assumption on the existence of paths as in [10, Assumption 1.1] does not yield any meaningful bound on the size of stars.

Lower bound
We dedictate this section to proving a lower bound for Γ(λ) when λ is small.Namely, we will prove the following result.
Proposition 2.2.Let G be a general geometric random graph which satisfes Assumption 1.1 with γ > δ δ+1 .Then, there exists a constant c > 0 such that, for λ small, it holds For the proof of Proposition 2.2 we exploit the following observation.Let us denote by a star a connected graph where all but one vertex have degree one.Then, the contact process restricted to a subgraph isomorphic to a star survives for a constant time if the subgraph consists of at least order λ −2 vertices and survives even long enough to infect other neighbouring stars if the subgraph consists of order log(1/λ)λ −2 vertices, see [18].We denote by L r the graph consisting of the half-line N 0 , where to each even vertex m ∈ N 0 , r additional distinct neighbours are attached.Thus, L r forms a half-line of stars consisting of r+1 vertices, where two consecutive stars are connected via a path of two edges.Throughout this section we denote the vertex 0 ∈ N 0 as the origin of L r .Notice that if the size r of the stars is of order log(1/λ)λ −2 and only the origin of L r is infected, there exists a constant p > 0 such that the survival probability of the contact process on L r is at least p.This is direct consequence of Lemma [16, Lemma 2.4], since the stars are sufficiently large so that whether two stars are connected by a single edge or a path of bounded length makes no difference.Thus, as a first step we will show that such a half-line of stars exists in G (0,T 0 ) .For x ∈ R d , denote by H x the plane through x with normal vector x.Consequently, H x divides R d in two subsets and we denote by R d ≥x the subset that does not contain zero.As discussed in Section 1.1 the expected degree of a given vertex in G with mark t is of order t −γ .We therefore call vertices with small mark powerful vertices.For β > 0 and r := β log(1/λ)λ −2 , let T ⊛ := r −1/γ be the threshold such that vertices with smaller mark have an expected degree of order at least log(1/λ)λ −2 .Lemma 2.3.Let x = (x, t) ∈ R d × (0, 1) with t < T ⊛ .Then, given that x is a vertex of G (0,T 0 ) , we have with high probability as λ → 0 that there exists a subgraph of G (0,T 0 ) in R d ≥x × (0, 1) which is isomorphic to L r such that the origin of L r is identified with x.
To prove the existence of such a subgraph, we decompose R d ≥x × (0, 1) into in distinct parts, where areas with small marks represent potential midpoints of stars and areas with large mark represent either neighbours of the midpoints or vertices which are connected by an edge to two distinct midpoints.Choose θ > 0 such that 1 < θ < γ + γ/δ and note that his The anuli (A k ) k∈N centered around x and truncated with respect to H x , where the area Ã2 is shaded grey.
is always possible since γ > δ δ+1 .Set and R 0 = 0. Given the vertex x = (x, t) define, for k ∈ N, the anuli c and the sets k := Ãk × [1/2, 3/4) and S (2) where Ãk := A k ∩ R d ≥x , see Figure 2. Notice that all these sets are disjoint and therefore the point processes restricted to these sets are independent.For the proof of Proposition 2.3 it will be helpful to interpret • the vertices in S k as the potential midpoints of the k-th star of the subgraph, • the vertices in S (1) k as the potential neighbours of the midpoints forming a sufficiently large star as the potential connectors between consecutive midpoints, i.e. vertices which are connected to both midpoints.Before we prove Lemma 2.3, we state the following lemma as a corollary of [13], which follows with similar calculations as done in [10].With this lemma we can ensure that two vertices from consecutive areas S k and S k+1 are connected via a connector with high probability.
Lemma 2.4.Given two vertices x k ∈ S k and x k+1 ∈ S k+1 , the number of vertices in S (2) k which form an edge to x k and x k+1 is Poisson-distributed with parameter larger than where C > 0 is a constant not depending on k.
Proof.We consider the vertices z = (z, u) ∈ S (2) for sufficiently small λ.Then, the number of those vertices which form an edge to x k and x k+1 is Poisson-distributed with parameter bounded from below by where V d is the volume of the d-dimensional unit ball and α comes from Assumption 1.1.Thus, there exist a constant C > 0 sufficiently small such that (7) holds.
Proof of Lemma 2.3.Note that, for k ∈ N, the number of vertices in S k is Poisson-distributed with mean larger than for some c 1 > 0, which does not depend on k.Thus, S k is non-empty with probability larger than 1 − exp(−c 1 T −(γ+γ/δ−θ) ⊛ e k(γ+γ/δ−θ) ).As the boxes S 1 , S 2 , . . .are disjoint, the numbers of vertices in each box are independent from each other.Thus, it holds For k ≥ 2, on the event that S k is non-empty we denote by x k the vertex in S k with the smallest mark.For k = 1 we set x 1 := x and always treat S 1 as non-empty.To keep notation cleaner we treat without loss of generality the mark of x 1 as smaller than T 1 (this affects only the estimate in (8) below where γ + γδ becomes γδ which does not change the rest of the argument).
Given S k and S k+1 are non-empty, the Euclidean distance of the corresponding vertices x k and x k+1 is at most 2R k+1 and both vertices have a mark smaller T k .Thus, by Lemma 2.4 there exists c 2 > 0 such that, for all k ∈ N, the probability that x k and x k+1 are connected via one connector in S (2) k is larger than Given S k is non-empty, we now turn our attention to the number of neighbours of x k in S k .As in the proof of Lemma 2.4 we only consider the vertices z ∈ S (1) 2 d+1 of the ball volume itself for λ small enough.These vertices are connected to x k with probability bounded from below by α(1 ∧ κ 1 ).Thus, there exists c > 0 such that the number of neighbours of x k in S (1) is Poisson-distributed with mean larger than cT −γ k ≥ cre kγ .Thus, by a Chernoff-bound there exists c 3 > 0 such that, given S k is nonempty, the probability that the number of neighbours of x k is larger than r is at least 1 − exp(−cre kγ ), for all k ∈ N.
We choose 0 < ε < γ + γ/δ − θ ∧ (γ + γδ)(θ − 1) .Given all sets S k are non-empty, the two previously discussed events only depend on disjoint subsets of the Poisson-process.Thus, the probability that, for all k ≥ 2, • S k is non-empty, • x k , the vertex with smallest mark in S k , has at least r neighbours in S (1) k and which tends to one as λ → 0 since r and T −ε ⊛ tend to infinity in this case.
With Lemma 2.3 in hand we are now ready to complete the proof of Proposition 2.2.
Starting at the origin we explore the Poisson point process by expanding a sphere centered at the origin (0, T 0 ) until we find the nearest neighbour x of (0, T 0 ) with mark smaller T ⊛ .
As the number of neighbours of (0, T 0 ) with mark smaller than T ⊛ dominates a Poissondistributed random variable with parameter of order T 1−γ ⊛ , the probability that we find such a neighbour x and there is a transmission from (0, T 0 ) to x before (0, T 0 ) recovers is larger than for some c > 0, where λ λ+1 occurs as the probability that, given x is a neighbour of the origin (0, T 0 ), x got infected by the origin before it recovers.Given the nearest neighbour x = (x, t) with mark smaller T ⊛ , by Lemma 2.3, there exists with probability larger than 1 2 as λ is small a subgraph in the yet unexplored area R d ≥x which is isomorph to L r with origin in x.Conditioned on this subgraph being present and the origin of it being infected, the infection survives with a probability bounded away from zero, uniformly in λ, by [16,Lemma 2.4].Hence, Γ(λ) is up to a constant larger than the probability bound given in (9) which completes the proof of Proposition 2.2.

Upper bound
Proposition 2.5.Let G be a general geometric random graph which satisfies Assumption 1.1 for γ > 1 2 and δ > 1.Then, there exists a constant C > 0 such that To prove this we generalize and extend the arguments from [16] for the geometric random graphs characterized by the framework given in Section 1.
With that in mind, let T (n) = n −1/γ be the mark of a vertex with expected degree of order n.
Throughout the proof we classify the vertices by their expected degree into different groups.
Let n ⋆ = λ −2 and, for some constant θ > 0 to be specified later, n ⊛ = θ λ 2 log 1 λ .Vertices with degree larger than n ⋆ are the midpoints of stars on which the infection restricted to the star can survive for a constant time with a probability bounded away from zero, without necessarily surviving long enough such that it can reach other stars nearby.As we have seen in the proof of Proposition 2.2, this happens for stars with more than n ⊛ vertices.For σ > 0, set n σ = λ −2+σ .Vertices with degree smaller than n σ are centers of stars which are not sufficiently large, in the sense that the infection does not propagate through such stars and dies out when the graph is restricted to vertices with such small degree; see Lemma 2.13.We denote by the according mark of the vertices whose expected degree is of the corresponding order.
Proof of Proposition 2.5.We consider now the contact process (ξ (0,T 0 ) t ) t≥0 on G (0,T 0 ) which starts from the origin (0, T 0 ).For a vertex x, on the event that x and (0, T 0 ) are connected we denote by I x the event that x got infected from the origin (0, T 0 ) before (0, T 0 ) recovers.On I x we denote by τ x the time when x got infected by the origin and by (η x t ) t≥τx the contact process started at time τ x with a single infection at x determined by the same graphical construction as the original process (ξ For σ > 0, denote by E σ the event that each infection path g starting in the origin which jumps first to a vertex with mark larger than T σ is finite and never reaches a vertex with mark smaller than T σ .We will see later that when the mark of the origin is itself larger than T σ , the probability that E σ occurs goes much quicker to one, when λ goes to zero, than the rate given in (5).In fact, we show in Lemma 2.13 that for σ > 0 sufficiently small, it holds Let σ 0 > 0 such that it holds σ 0 > σ and let T σ 0 = T (n σ 0 ) be the associated boundary of the mark of vertices with expected degree of order λ −(2−σ 0 ) .Whereas T σ and T ⊛ will be used to distinguish the neighbours of the origin by their marks, for the survival of the contact process we consider whether T 0 < T σ 0 or not.Then, we have In fact, if the right-hand side is zero it holds that • every infection path starting in the origin which jumps in its first step to a vertex with mark larger than T σ is finite and never visits a vertex with mark smaller than T σ , • there exists no vertex with mark smaller than T ⊛ which is a neighbour of the origin and got infected by it, • there exists no vertex with mark inbetween T σ and T ⊛ , which is a neighbour of the origin and got infected by it and the infection emerging from this vertex survives.
As these three points imply that the infection (ξ (0,T 0 ) t ) t≥0 does not survive, the left-hand side is then also zero, from which the stated inequality follows.Note that this bound is especially not sharp in the second summand of the right-hand side, as we allow T 0 to take a larger range of values than needed.Taking expectation on both sides of (12) yields an upper bound for the probability of interest and it is therefore sufficient to establish upper bounds for the expectation of each of the four summands of the right-hand side.
First, by the definition of T σ 0 we have that for σ 0 and λ > 0 small enough and a bound for the second summand is given by ( 11), which will be proved in Lemma 2.13.
As mentioned beforehand, the third term on the right-hand side of (12) count the number of vertices with mark smaller than T ⊛ which got infected by the origin (0, T 0 ).We have seen in Section 2.1 that the contact process starting in such a vertex would ensure a spreading over a chain of infinitely many other stars with equally many neighbours.Thus, a sharp upper bound for the expected number of such vertices is crucial.By Mecke's equation, Assumption 1.1 and since T 0 is independent of X it holds that In fact, this is the dominant term of the right-hand side of ( 12) which contributes to the stated upper bound.
For the last summand of (12) we have by Mecke's equation, Assumption 1.1 and since T 0 is independent of X that In Lemma 2.12 we will show that there exist ε > 0 such that, for x = (x, t), y = (y, s) ∈ R d × (0, 1) with t > T ⊛ and s > T σ 0 , it holds where (ξ x t ) t≥0 is the contact process of rate λ which starts in x and only in x.By this inequality and the strong Markov-property of the contact process we have that for σ > 0 and λ > 0 sufficiently small which completes the proof.
We now proceed to establish the probability bounds ( 11) and ( 13) of the previous proof which have been left out.To show these bounds we need to have control over the occurence of infection paths which correspond to the both events, i.e. infection paths which jump from the origin to a vertex with mark larger than T σ and infection paths starting in a vertex with mark between T σ and T ⊛ .For these bounds we will rely on the arguments used in [16, Section 5], for which we will give a short overview.
We consider now a graph G = (V, E) with root 0 and let P be the set of all finite and infinite paths of vertices in the graph G. Instead of looking at infection paths themselves, we look at the paths of vertices which result from infection paths by capturing the visited vertices.
Precisely, for an infection path g : I → V we define its ordered trace p g ∈ P as the path of vertices in G given by the vertices visited by g in the same order.The following result by [16] shows the usefulness of this definition, as to control the occurence of a given class of infection paths, it is sufficient to control the number of ordered traces corresponding to this class.
We define the following subsets of P .Let A ⊂ V such that the root 0 is not in A and define A as the set of paths in G of length n starting in 0, where the first n vertices are distinct and not in A but the last vertex is in A, A as the set of paths in G of length n starting in 0, where the first n vertices are distinct and not in A but the last vertex is equal to a previous one.

We denote Q
Typically A is a set of vertices with small mark, for example smaller than T ⋆ or T ⊛ .These vertices are powerful as they have many neighbours and therefore ensure the survival of the infection for longer time.The set Q n A then describes all ordered traces which get to such a powerful vertex in its n-th step.The motivation for the second set R n A lies in the following result from [16].Given a vertex x ∈ V , the contact process (ξ t ) t≥0 starting in 0 is thin on x if there is no infection path g : [0, t] → V for some t ≥ 0 with g(0) = 0 where x appears more than once in the ordered trace p g of g.The contact process starting in 0 is thin on a set V ′ ⊂ V if it is thin on every vertex of V ′ .Lemma 2.7.([16, Lemma 5.4]) If V 0 ⊂ V is finite, then on the event that (ξ t ) t≥0 is thin on (V 0 ) c , the contact process almost surely dies out, that is, almost surely there is t ≥ 0 such that ξ t = ∅.
Ordered traces which are not in Q A ∪ R A do not connect to a vertex with high degree and visit each vertex at most once.Thus, by the previous lemma infection paths which have such ordered traces typically do not contribute to the survival of the contact process.A consequence is the following key result by [16] Lemma 2.8.([16, Lemma 5.5]) There exists c > 0 such that, for any λ < 1  2 , the following holds.Let G be a graph with root 0, A ⊂ V and let (ξ t ) t≥0 be the contact process on G starting in 0.Then, We consider again geometric random graphs given by our framework in Section 1.1 and proceed to establish bounds for the number of ordered traces in Q A and R A on G for a suitable choice of A. As A is a set of vertices typically defined in terms of their marks, the ordered traces in Q A and R A have restrictions on the marks of the vertices but not on their location.Thus, to control the number of such ordered traces we rely on the following definition as done in [10].Let ℓ ∈ (0, 1) be the truncation value and κ ≥ I ρ .We define, for n ∈ N and t 0 ∈ (0, 1), and set ν t 0 ℓ,0 (s) = δ 0 (t 0 − s).Note that as defined, ν x ℓ,n (s) can be written recursively as This allows us to establish an upper bound for ν x ℓ,n (s) analogous to the non-spatial case in [8, Lemma 1].This is a corollary of [10,Lemma 2.7].Lemma 2.9.Let ℓ ∈ (0, 1) and ν t 0 ℓ,n (s) be as defined in (14), where x = (x 0 , t 0 ) and s ∈ (0, 1).Then, there exists a constant c > 0 such that, for all n ≥ 2, where Proof.This follows analogously to the proof of [10,Lemma 2.7] by choosing a fixed truncation value ℓ instead of an arbitrary truncation sequence.
The following result gives an explicit upper bound for the sequence (α n ) n∈N .
Recall that For x, y ∈ R d ×(0, 1) with marks t > T ⊛ and s > T σ 0 we consider now the geometric random graph G under the law P x,y (• | x ∼ y).On the event that x and y are vertices in G , we declare x as the root of the graph and set as the set of vertices in G with mark smaller than T ⋆ together with y.
Lemma 2.11.Let x = (x, t), y = (y, s) ∈ R d × (0, 1) with t > T ⊛ and s > T σ 0 .Then, there exists ε > 0 not depending on the choice of θ such that for A given in (22) it holds For n ≥ 1, denote by Qn A be the set of paths in Q n A which do not visit y in its last step and denote by Qn the set of paths in Q n A with y as its last vertex.Note that by definition the paths in y never consist of the given edge between the vertices x and y.Thus, the occurence of paths in G which belong to ( y is independent of the edge between x and y by Assumption 1.1.Then, by Mecke's equation and Assumption 1.1 we have, for n ≥ 2, that where x i = (x i , t i ) for i = 1, . . ., n and x = (x 0 , t 0 ).Integration over the locations on the right-hand side and using (18) yields, for n ≥ 2, that ) for some constant C > 0, where we have used in the last step that t 0 > T ⊛ .As it is easy to see that this bound also holds for the case n = 1, there exists C > 0 such that, for λ small enough, we have By (17) there exists c > 0, such that α n > c(α n−1 + β n−1 ) for n ≥ 2, and therefore similarly to the previous calculation it holds that , where c > 0 changes throughout the lines and therefore there exists C > 0 such that, for λ small enough, we have Note that with the same calculation this bound also holds for ∞ n=3 (2λ By definition it directly follows that the first term is smaller than 2λ < λ ε .For the second term note that | Q2 y | is dominated by the number of neighbours of y which are not x.As the expectation of this number is smaller than Recall that there exists a constant C > 0 such that for any vertex x = (x, t) ∈ R d × (0, 1), its expected degree is smaller than Ct −γ .We now set θ in the definition of T ⊛ as θ := ε 8Cc , where ε > 0 is given by Lemma 2.11 and c > 0 by Lemma 2.8.The following result is then a consequence of Lemma 2.11 and follows with the same argumentation as [16,Proposition 5.8].It provides the bound (13) in the proof of Proposition 2.5.Lemma 2.12.Let x = (x, t), y = (y, s) ∈ R d × (0, 1) with t > T ⊛ and s > T σ 0 and (ξ x t ) t≥0 be the contact process on G with rate λ which only starts in x.Then, there exists ε > 0 such that Proof.As seen before the expected degree of a vertex with mark larger than T ⊛ is smaller than Cθ λ 2 log( 1 λ ).Thus, to find an upper bound for the survival probability of (ξ x t ) t≥0 we look at whether the degree of x is smaller than 2Cθ λ 2 log( 1 λ ) or not.Then, by Lemma 2.8 with T = λ −ε/2 and Lemma 2.11 it holds that As the number of neighbours of x different to y is Poisson distributed with parameter at most Cθ λ 2 log( 1 λ ), using a Chernoff bound yields for λ small enough, where c 1 > 0 is some constant.Thus, P x,y (ξ x t = ∅ ∀t ≥ 0 | x ∼ y) ≤ 3λ ε/4 which completes the proof.
As the last step to complete the proof of Proposition 2.5 we show inequality (11).Recall that, for σ > 0, E σ denotes the event that each infection path of the contact process (ξ (0,T 0 ) t ) t≥0 which jumps at first to a vertex with mark larger than T σ is finite and never reaches a vertex with mark smaller than T σ .Lemma 2.13.There exists ε > 0 and σ > 0, such that Proof.We denote by B σ = {x ∈ X : x = 0, t ≤ T σ } the set of vertices with mark smaller or equal to T σ and by Q n Bσ and R n Bσ the associated sets of paths, which either visit a vertex in B σ in its last step or whose vertices are not in B σ but the last vertex is equal to a previous one.We set Bσ and P 0 = (0, T 0 ), x, (0, T 0 ), y : x, y ∼ (0, T 0 ) .By [16,Lemma 5.10] the event that no infection path starting in (0, T 0 ) has an ordered trace in P 0 ∪ Q Bσ ∪ R Bσ implies E σ .In fact, if no infection path starting in (0, T 0 ) has ordered trace in P 0 ∪ Q Bσ ∪ R Bσ , then each infection path g : I → V which starts at (0, T 0 ) and jumps to a vertex x = (x, t) with t ≥ T σ never visits a vertex in B σ and never visits a vertex outside of B σ more than once.Thus, by Lemma 2.7 any such infection path is finite, as the contact process starting in (0, T 0 ) and restricted to B c σ is thin outside (0, T 0 ) and dies out.Hence, it then holds We will proceed to find upper bounds for the expected number of ordered traces corresponding to each of the three classes P 0 , Q Bσ and R Bσ .First, it holds by Mecke's equation and integration over the location of the vertices that for some small ε > 0 as 2/γ − 1 + σ(2 − 1/γ) is increasing in σ as γ > 1 2 .The positive constant C does not depend on λ and σ but may change throughout the lines.Using Lemma 2.9 and ( 18) with ℓ = T σ = λ 2−σ as done in the proof of Lemma 2.11 we have by Mecke's equation and Assumption 1.1, for n ≥ 2, that for some positive constant C > 0.Then, it follows that for σ > 0 sufficiently small.For the last summand Lemma 2.9 and (18) yield similarly that and we have therefore As all three summands of the righthand side of 23 are bounded by Cλ 2/γ−1+ε for some constants C > 0 and ε > 0 sufficiently small this completes the proof.

Exponential extinction time on finite restrictions
In this section we consider the graph sequence (G n ) n∈N , where G n is the spatial restriction of G on [− n 1/d 2 , n 1/d 2 ] d .As this is a sequence of finite graphs, the contact process with any potential initial condition will almost surely die out for all n ∈ N. Thus, the more natural question is to estimate the time the infection survives on these graphs when the infection starts with the best possible initial condition, i.e. when the graph is fully infected.We denote by τ n := inf{t > 0 : ξ Gn t = ∅} the extinction time of the contact process on G n .The main result of this section shows that for any choice of λ > 0 the extinction time is at least of exponential order in the number of vertices of G n with high probability as n becomes large.As seen in the proof of Proposition 2.2 the infection survives well on the neighbourhood of sufficiently powerful vertices, the so called stars.Our main contribution is to show that there exists a connected subgraph in G n which contains of order n stars.Then, with similar arguments as done in [17], it can be shown that the infection survives on this subgraph at least for a time of exponential order in n.
Proposition 3.2.Let S > 0 be given and (G n ) n∈N the restricted finite graph sequence of a general geometric random graph which satisfies Assumption 1.1 for γ > δ δ+1 .Then, there exists b > 0 and ε > 0 such that, for n sufficiently large, the probability that G n has a connected subgraph containing b • n disjoint stars of at least S vertices each is larger than 1 − exp(−n ε ).
Proof.We fix 0 < a < 1 log 2 and choose ε 1 > 0 small enough that log 2 > ε 1 +log 2 γ+γ/δ , which is possible since γ > δ δ+1 .Similar the proof of Lemma 2.3, the vertices with mark smaller than 1 2 will represent the potential midpoints of the stars of the subgraph, whereas the vertices with larger mark represent potential neighbours and connectors of the midpoints.For this proof it is not sufficient to use the arguments of Lemma 2.3 to show that a line of stars exists in G n , as such a subgraph would only consist of order log(n) many stars.Hence, we need to to break up the powerful vertices of G n with mark smaller than 1 2 more carefully into a system of boxes such that each of these boxes contains a midpoint of one potential star.Let n p = ⌊n (1−a log 2)/d ⌋ and k p = ⌊(a log n)/d⌋.For k = 0, . . .k p , we define 2 ] d into (n p 2 kp−k ) d cubes of volume 2 kd such that the finest tessellation is given for k = 0 and the coarsest for k = k p .Furthermore, the cubes are nested in each other in the sense that for each cube A k+1,v the cubes {A k,2v+e : e ∈ {0, 1} d } are a tessellation of A k+1,v .
Set θ > 0 such that log 2 > θ > ε 1 +log 2 γ+γ/δ and define We denote with the parameter k = 0, . . .k p the layer of the boxes {B k,v : v ∈ V k } which defines the range of the marks of points of X inside the boxes and the level of coarsness of the tessellation of the space.Thus, large values of k imply more powerful vertices and a coarser set of boxes to separate them.As an example, the boxes of the most powerful layer Since θγ < log 2, there exists c > 0 such that, for all k = 0, . . ., k p , on the event that B k,v is non-empty it holds ct 2 d of the volume of the ball itself.As in the proof of Lemma 2.3 this leads to two observations.First, given B k,v is nonempty, by the same arguments as in the proof of Lemma 2.4 the number of neighbours of x k,v in N k,v is Poisson-distributed with parameter larger than ce −kd(θγ∧δε 1 −ε 3 ) e kdθγ > ce kdε 3 for some constant c > 0 not depending on k.We denote by Star(k, v) the event that B k,v is non-empty and x k,v has at least S neighbours in N k,v .Then, by a Chernoff bound there exists c > 0 and k 0 sufficiently large and not depending on n such that for all k ≥ k 0 and v ∈ V k it holds Second, given the box B k+1,v and one of its children B k,2v+e are non-empty, note that x k+1,v and x k,2v+e have distance at most √ d2 k+1 and both have marks smaller 1 2 e −kθd .Thus, by the same argument as used in the proof of Lemma 2.3, there exists a constant c > 0 such that the number of vertices in C k,2v+e which form an edge to both x k+1,v and x k,2v+e is Poisson-distributed with parameter larger than ( With the structure of the boxes and the bounds (24)-( 26) at hand we will show that there exists a connected subgraph containing of order n distinct stars with at least S vertices each.This will be done in two steps.First, we show that the vertices in the most powerful layer k p form a connected subgraph containing n d p distinct stars, where each box B kp,v contains one of the midpoints of these stars.Second, recall that each box B kp,v for v ∈ V kp represents the root of a 2 d -regular tree.We will show that the trees resulting only from boxes contributing a star to the connected subgraph are percolated 2 d -regular trees with a depth of order k p containing of order 2 kpd distinct stars.As there are n d p many trees like this, this will lead to a connected subgraph of G n with of order n distinct stars.
To simplify notation we redefine the labeling of the boxes of layer k p .Let σ : {0, . . ., n d p − 1} → V k be a bijection such that B kp,σ(0) = B kp,0 and the boxes B kp,σ(i) , B kp,σ(i+1) are adjacent to each other for i = 0, . . ., n d p − 2. In the same way we relabel the vertices with the smallest mark in a box, i.e. on the event that B kp,σ(i) is non-empty we denote by x kp,σ(i) the vertex with the smallest mark in that box.We say B kp,σ(0) is good if and only if the box is nonempty and the vertex x kp,σ(0) has at least S neighbours in N kp,σ(0) .By ( 24) and (25), there exists c > 0 such that P(B kp,σ(0) is good) ≥ 1 − exp(−ce kpdε 2 ) 1 − exp(−ce kpdε 3 ) .
Thus we can deduce that P(B kp,v is good for all v ∈ V kp ) ≥ 1 − n d 0 exp(−ce kpdε 4 ) ≥ 1 − n 1−a log 2 exp(−cn aε 4 ).( 27) We continue the definition of good boxes on the other layers.For v ∈ N denote by ⌊ v 2 ⌋ the vector (⌊ v 1 2 ⌋, . . ., ⌊ v d 2 ⌋).Then, for each k = 0, . . ., k p − 1 and v ∈ V k the parent box of B k,v is given by B k+1,⌊ v 2 ⌋ .For k = 0, . . ., k p − 1 and v ∈ V k , we say that B k,v is good if (i) B k+1,⌊ v 2 ⌋ is good, (ii) B k,v is non-empty, (iii) x k,v has at least S neighbours in N k,v and (iv) x k,v and x k+1,⌊ v 2 ⌋ are connected via a connector in C k,v and otherwise we say that B k,v is bad.Note that again (24)-(26) implies that there exists c > 0 such that for all k = 0, . . ., k p − 1, v ∈ V k and e ∈ {0, 1} d , it holds and given B k+1,v is good, the events that B k,2v+e is good are independent of each other and any other box on this layer, since they depend on disjoint subsets of X and edges occur independently.Therefore, the number of good children of B k+1,v , given this box is good, is Binomial-distributed with parameters 2 With this observation we are able to give estimates on the number of good boxes in each layer and show that sufficiently many good boxes exists.For k = 0, . . ., k p − 1, we denote by | the event that layer k has sufficiently many good boxes in comparison to layer k + 1 of the next more powerful vertices and we denote by E kp the event that all boxes in layer k p are good.Then, the event E k ∩ . . .∩ E kp implies that where c = ∞ i=1 (1 − i −2 ).Thus, it is sufficient to show that there exists k 0 such that E k 0 ∩ . . .∩ E kp holds with high probability.We choose k 0 large enough that (25) still holds for all k ≥ k 0 and v ∈ V k and that 1 − exp(ce k 0 dε 4 ) > 1 − k −2 0 .Then, by a Chernoff bound for Binomial-distributed random variables it holds for all k ≥ k 0 .As a consequence, there exists c > 0 such that Hence, it follows together with (27) that

Figure 1 :
Figure 1: Sketch of the graphical representation of the contact process on Z. Arrows represent (potential) infection transmissions and crosses represent the recovery marks.The blue path is one potential infection path in this representation starting in the fourth vertex.

Theorem 3 . 1 . 1 . 3 . 1 .
Let (G n ) n∈N be the restricted finite graph sequence of a general geometric random graph which satisfies Assumption 1.1 for γ > δ δ+1 .For any λ > 0, there exists c > 0 such thatlim n→∞ P{τ n ≥ e cn } =Remark Note that the result of Theorem 3.1 also holds when we consider a graph sequence (G n ) n∈N , where each graph G n is defined on a Poisson process of unit intensity on the torus T n with volume n and satisfies Assumption 1.1, where the Euclidean metric is replaced by the torus metric.

2 ↔ 2 ↔
kdε 3  where the constant c > 0 changes through the steps but does not depend on k and v and we have used that θ > ε 1 +log 2 γ+γ/δ .Denote by x k+1,v x k,2v+e the event that the boxes B k+1,v and B k,2v+e are non-empty and the vertices x k+1,v and x k,2v+e are connected via a vertex in C k,2v+e .Then, we have for all k ∈ N and v ∈ V k that P(x k+1,v x k,2v+e | B k+1,v and B k,2v+e are non-empty) ≥ 1 − exp(−ce kdε 3 ).

d and p k > 1 −
exp(−ce kdε 4 ) and, given the good boxes on layer k + 1, the numbers of good children of each of those good boxes are independent of each other.Consequently, denote by |B k | the number of good boxes in layer k.Then, given |B k+1 |, |B k | is Binomial-distributed with parameters 2 d |B k+1 | and p k .

P 2 ↔ 2 ↔ 2 ↔
(E k 0 ∩ . . .∩ E kp ) ≥ P(E kp ) 1 − ⌊a log n⌋ exp − c n adε 4 (log n)2  ≥ 1 − exp(−n ε ) for some ε not depending on n.As E k 0 ∩. ..∩E kp implies the existence of up to a constant at least 2 −k 0 n good boxes and therefore the existence of a connected subgraph of G n containing bn distinct stars, for some b > 0, this completes the proof.Proof of Theorem 3.1.Given the subgraph G n = (V n , E n ) provided by Proposition 3.2 note that G n is a connected tree by construction.Denote by M n ⊂ V n the set of vertices which are the midpoints of the stars containing S vertices.For x, y ∈ M n , write x y if there exists a connector in G n which forms an edge to x and y.By definition all vertices x, y ∈ M n with x y have graph distance at most two in G n .The graph H n given by the vertex set M n and the edge setF n := {x, y} : x, y ∈ M n , x yis a connected tree with degree bounded by 2 d + 2 and for each pair x, y ∈ M n with x 2 ↔ y the connector is unique.Hence, for any λ > 0 and with S > 0 chosen sufficiently large depending on λ by the same arguments as used in the proof of[17, Theorem 1.4]  together with [17, Proposition 5.2] it holds lim n→∞ P{τ n ≥ e cn } = 1 for some constant c > 0.
1.1 for a formal definition.Two motivating examples of this class, which are shortly introduced in the following, are a soft version of the scale-free Boolean model introduced in [10] and the age-dependent random connection model introduced in [9] which emerges as a limit graph of a spatial preferential attachment network.