Rapid social connectivity

Given a graph $G=(V,E)$, consider Poisson($ |V|$) walkers performing independent lazy simple random walks on $G$ simultaneously, where the initial position of each walker is chosen independently with probability proportional to the degrees. When two walkers visit the same vertex at the same time they are declared to be acquainted. The social connectivity time $\mathrm{SC}(G)$ is defined as the first time in which there is a path of acquaintances between every pair of walkers. It is shown that when the average degree of $G$ is $d$, with high probability \[ c\log |V| \le \mathrm{SC}(G) \le C d^{1+5 \cdot 1_{G \text{ is not regular}} } \log^3 |V|.\] When $G$ is regular the lower bound is improved to $\mathrm{SC}(G) \ge \log |V| -6 \log \log |V| $, with high probability. We determine $\mathrm{SC}(G)$ up to a constant factor in the cases that $G$ is an expander and when it is the $n$-cycle.


Introduction
↔ w ′ , if there exists t 0 t such that they have the same position at time t 0 . After two walkers meet they continue their walks independently without coalescing. "Meeting by time t" is a symmetric relation. It induces the equivalence relation of having a path of acquaintances by time t, denoted by t ∼, defined as follows. Two walkers a and b have a path of acquaintances by time t iff there exist k ∈ N and walkers a = c 0 , c 1 , . . . , c k , c k+1 = b such that c i t ↔ c i+1 , for all 0 i k. Note that we are not requiring the sequence of times in which the walkers met to be non-decreasing, which is the main difference between the SN model and some existing models for spread of rumor/infection (e.g. the A + B → 2B model [12] and the Frog model [16,3,4,15,10], see § 2.6 for a description of these models and a discussion about their relationship to the SN model). Consequently, the SN model often evolves faster than such models. However, Conjecture 2.3 suggests one sense in which also theses models might evolve rapidly.
We are interested in the coalescence process of the equivalence classes of t ∼ (as t varies). In particular, we are interested in the social connectivity time SC(G), defined as the minimal time in which there is only one class (i.e. every pair of walkers have a path of acquaintances between them). Note that (2.2) is meaningful as long as r * |V | β−0.5 for some β > 0. Theorem 5.5 gives an extension covering in particular the case r * |V | β−1 for some β > 0. When G is a regular graph we have the following improved lower bounds on SC(G). Note that s * (G) > 1 32 log |V |. However, in some cases s * (G) C(log |V |) 2 . For instance, for the n-cycle C n we have that P 2i (v, v) = Θ( 1 √ i+1 ) and so s * (C n ) = Θ((log n) 2 ). 3 Hence by (2.4) P[SC(C n ) < c(log n) 2 ] = o(1). This is not surprising as initially there will be intervals of length Ω(log n) which will not be occupied by a single particle. This is why in terms of the lower bound on SC(C n ) it is more interesting to study the case that initially there is one walker per site (as done in Theorem 3). Conversely, there exists some C > 0 such that the probability that SC(C n ) > Cs * (C n ) C ′ (log n) 2 is at most C ′′ /n (see § 2.3). We conjecture that there exists some universal constant C > 0 such that for every connected finite vertex transitive graph G (Definition 8.2) 1 C E[SC(G)] s * (G)+1 C. 2 We use C, C ′ , C 0 , C 1 , K, M . . . (resp. δ, ε, c, c ′ , c 0 , c 1 , . . .) to denote positive absolute constants which are sufficiently large (resp. small) to ensure that a certain inequality holds. Similarly, we use C d (resp. c d ) to refer to sufficiently large (resp. small) positive constants, whose value depends on the parameters appearing in subscript. Different appearances of the same constant at different places may refer to different numeric values. 3 We write o(1) for terms which vanish as n → ∞ (or some other index, which is clear from context). We write f n = o(g n ) or f n ≪ g n if f n /g n = o (1). We write f n = O(g n ) and g n = Ω(f n ) if there exists a constant C > 0 such that |f n | C|g n | for all n. We write f n = Θ(g n ) if f n = O(g n ) and g n = O(f n ). We write f n O d (g n ) to indicate that the constant C above may depend on d.

Some remarks concerning our main results
Degree dependence: We now discuss the degree dependence in (2.1) and also a refinement of (2.1). Example 8.7 (resp. 8.8) describes for some absolute constants c 0 , c 1 , C 0 > 0 for all sufficiently large n and all C 0 d n c 0 n a family of graphs G n = (V n , E n ) such that n |V n | C 0 n which are d n -regular (resp. of average degree d n ) for which P[SC(G n ) c 1 d n log 2 (|V n |/d n )] > c 1 (resp. P[SC(G n ) c 1 d 2 n log 2 (|V n |/d n )] > c 1 ). This shows that the linear dependence on d in (2.1) is optimal for regular G and that we can have E[SC(G)] n for an n-vertex regular G and E[SC(G)] n 2 for non-regular G.
If G is d-regular and in addition for every set A ⊂ V such that |A| 2 3 (|V | + |V | 2/3 ), we have that for some absolute constants C, M (see Proposition 4.12, part (iii)). Now consider the case that G is a connected n-vertex Eulerian digraph of average degree d. Let ξ := 1 G is not regular . From the proof of Theorem 1 it follows that we can replace Cd 1+5ξ (log n) 3 in (2.1) by C ′ t * (G)d 1+3ξ log n, for some t * (G) C d 2ξ (log n) 2 . Namely, t * (G) := inf{t : max x,y |P t (x, y) − π y | d y /(3 · 2 9 d log n)}, where P is the transition matrix of the LSRW on G. By Fact 4.8 indeed t * (G) C d 2ξ (log n) 2 . For the cycle t * = Θ((log n) 2 ). This shows that (2.1) is sharp up to a factor O(log n), when d = O (1). We believe that the cycle is in some sense extremal for the SN model (see Remark 8.7, Theorem 3 and Conjecture 8.3 for more details on this point).
The next remark discusses the case in which the particle density is considered as a parameter of the model.
Allowing the particle density to vary: One may consider the case in which the density of walkers at each site is multiplied by some λ > 0 (i.e. for all v ∈ V the number of particles initially occupying it N v has a Poisson(λπ v ) distribution). Let ξ := 1 G is not regular . For (here we allow λ to depend on |V |, provided that λ C log |V | |V | d ξ ). With additional work this can be improved to w.h.p. SC(G) Cd 1+5ξ λ −2 (log |V |) 3 . When λ ∈ [|V | −1+ε , 1] for some ε ∈ (0, 1), this is optimal up to an O d,ε (log |V |) factor as described in Remark 2.1.
When G is vertex-transitive (see Definition 8.2) we conjecture that is monotone decreasing in λ (see Conjecture 8.4 for a slightly different formulation). It is not hard to extend the proofs of (2.2) and (2.3) to the case that the particle density at each site is multiplied by some λ ∈ [|V | −ε , 1] for some ε ∈ (0, 1/2). In this case, the lower bound on SC(G) is increased by a factor of λ −1 . Similarly, one can extend (2.4) as follows: if G = (V, E) is a connected n-vertex regular graph and λ ∈ [n −ε , 1] for some ε ∈ (0, 1/16) then w.h.p. SC(G) s * (G, λ) := min{t : t+1 κ t+1 > 1 32λ log n}. Other initial distributions: We may consider the case that initially each vertex is occupied by a single particle. Consider the case that G is a d-regular Eulerian graph. We outline in § 4.7 the necessary adaptations required in order to show that also in this setup we have w.h.p. 4 SC(G) Cd(log |V |) 3 . (2.5) Extremality of the complete graph: We will prove Theorem 2 for an arbitrary holding probability p ∈ [0, 1) for the walks (i.e. the probability of a walker staying put at each given step is p). Let K n be the complete graph on n vertices. Consider the SN model on K n with holding probability p n ∈ [0, 1/n]. It is not hard 5 to verify that SC(K n ) is concentrated around log n in a window whose width is of order 1. Thus, the complete graph is the regular graph with (asymptotically) the fastest social connectivity time (at least when the holding probability is between 0 and 1/(d + 1)).
Extensions: In § 8 we discuss some extensions of our results (e.g. allowing the walkers to belong to different communities, walking on different graphs with the same vertex set).

2.3
The n-cycle and higher dimensional tori Theorem 3. Let C n be the n-cycle. Consider the setup in which at time 0 there is exactly one walker at each vertex and that the walkers perform continuous-time independent SRWs (with jump rate 1). Then there exist some absolute constants c 1 , c 2 , α, C 1 , C 2 > 0 such that The proof of (2.6) is due to Gady Kozma. We thank him for allowing us to present his argument. Working in continuous-time simplifies the analysis of (2.6). The proof in the usual setup of Pois(1) walkers per site is in fact simpler (see Remark 6.1). As mentioned after Theorem 2, in the usual Poisson, discrete-time setup (2.7) follows from (2.4).
Remark 2.1. When initially the number of particles per site is Pois(λ) distributed (independently for different sites) it is possible to extend the analysis of Theorem 3 in order to show that w.h.p. c < C as long as λ ≫ 1 n (cf. [5] for the analysis of the frog model on the n-cycle for λ ≫ n −1 ). 4 We say that a sequence of events A n defined with respect to some probabilistic model on a sequence of graphs G n := (V n , E n ) with |V n | → ∞ holds w.h.p. ("with high probability") if the probability of A n tends to 1 as n → ∞. 5 The proof is similar to the proof that the connectivity threshold for G(n, p), the Erdős and Rényi random graph, occurs in a window of order 1/n around p n = (log n)/n (for the SN model the proof is still elementary, but it is somewhat tedious and thus omitted). Remark 2.2. Denote the d-dimensional torus of side length n by T d (n). This is the Cayley graph of (Z/nZ) d obtained by connecting by an edge each x, y ∈ (Z/nZ) d which disagree only in one coordinate, by ±1 mod n. Then T 1 (n) is the n-cycle. We comment that using similar ideas as in [5,Theorem 2] one can show that w.h.p. c < SC(T 2 (n)) log n log log n < C and c < This result appeared in an earlier version of this work. Note that SC(T 2 (n)) c log n log log n and SC(T d (n)) (1−o(1))d log n w.h.p. follow from (2.4)-(2.3).

Expanders
We denote by γ(G) the spectral-gap of LSRW on G, defined as the smallest non-zero eigenvalue of I − P , where P is the transition matrix of LSRW on G and I is the identity matrix. We say that a sequence of graphs G n is an expander family if inf n γ(G n ) > 0. We say that G is a γ-expander if γ(G) γ (we think of γ as being uniformly bounded away from 0, independently of the size of G, although our analysis remains valid even if this fails). The following result demonstrates that Theorem 2 is sharp.
Theorem 4. Let G be a connected d-regular n-vertex λ-expander. Then there exist absolute constants C, C ′ , C 0 such that if λ log d n C 0 , then SC(G) Cλ −1 log n w.p. at least 1−C ′ /n. Theorem 5. Let G be a connected d-regular n-vertex λ-expander. Then there exist some constants c, C 1 > 0 such that with probability at least 1 − e −cnd −4(s+1) , after s := ⌈C 1 /λ⌉ steps there exists a class of walkers of size at least n/6.
We refer to a class of walkers (in the SN model on some graph G = (V, E) at a certain time) of size Ω(|V |) as a giant class. We shall deduce Theorem 4 from Theorem 5 by bounding SC by the time it takes the giant class from Theorem 5 to "capture" the rest of the walkers. The requirement λ log d n C 0 in Theorem 4 is imposed in order to ensure that the error term e −cnd −4(s+1) from Theorem 5 is small.

Organization of this work
In § 3 we present some preliminaries. In § 4 we prove (2.1). In § 5 we prove (2.2) and Theorem 2. In § 6 we prove Theorem 3. In § 7 we study the SN model on expanders and prove Theorems 4-5. We conclude with some open problems, conjectures and concluding remarks in § 8.

Related work and connections to other models
The social network model on infinite graphs is investigated in [9]. The setup studied in [9] is as follows. Given an infinite connected regular graph G = (V, E), place at each vertex Pois(λ) walkers performing independent lazy simple random walks on G simultaneously. When two walkers visit the same vertex at the same time they are declared to be acquainted. It is shown in [9] that when G is vertex-transitive and amenable, for all λ > 0 a.s. every pair of walkers will eventually have a path of acquaintances between them. In contrast, it is shown that when G is non-amenable (not necessarily transitive) there is always a phase transition at some λ c (G) > 0. General bounds on λ c (G) are given and the case that G is the d-regular tree T d is studied in more details (it is shown that c λ c (T d )/ √ d C). Finally, it is shown that in the non-amenable setup there exists a finite time t such that a.s. there exists an infinite set of walkers having a path of acquaintances between them by time t. We note that the last result is the infinite setup counterpart of our Theorem 5.
We believe that, in spirit, the results in this paper should be true also for some other particle systems involving independent random walks. The A + B → 2B family of models [12,13], often interpreted as models for a spread of an epidemic/rumor, are defined by the following rule: there are type A and B particles occupying a graph G, say with densities λ A , λ B > 0. They perform independent SRW with holding probabilities p A ∈ [0, 1] and p B ∈ [0, 1). When a type B particle collides with a type A particle, the latter transforms into a type B particle. The frog model is the particular case of the above dynamics in which the type A particles are immobile (p A = 1) and λ A = λ B = λ.
We now consider a variant of the above model which is intimately related to the SN model. Consider the case in which initially only the particles at some vertex o are of type B and an additional B particle is planted at o (this is done to ensure that there is at least one B particle). Consider the case that each B particle performs only t steps of its walk before vanishing (after which it cannot become reactivated again). We call t the lifetime of the B particles. We say that the process dies out when there are no B particles left. One then defines the susceptibility, S(G) as the minimal lifetime which is sufficient so that all particles are transformed into type B particles before the process dies out (to make this definition formal, we think of the particles as first picking the infinite trajectory of their random walks, but then, once transforming into a B particle, performing only t additional steps of their walk). In the interpretation of the model as a model for a spread of a rumor/epidemic the susceptibility is meant to capture "how long should a virus live in order to infect the entire population" or "how interesting should a rumor be, so that eventually everybody will hear it". Similarly, one can consider the minimal lifetime required so that every vertex is visited by at least one B particle.
In [8,5] parallel results to Theorems 2,3,4 and 5 are proved for the above variant in the case of the frog model, where the susceptibility replaces the social connectivity time. We strongly believe that (certain variants of) Theorems 2,3,4 and 5 should hold even when the A particles are mobile. Moreover, we have the following conjecture concerning an analog of Theorem 1.

Conjecture 2.3.
Consider the aforementioned variant of the A + B → 2B model with λ A = λ B = λ and some fixed p A ∈ [0, 1] and p B ∈ (0, 1). Then there exist some C d,λ , ℓ > 0, such that for every sequence of finite connected graphs G n = (V n , E n ) with |V n | → ∞ of maximal degree at most d, we have that  Let G = (V, E) be an Eulerian digraph. Let P be the transition kernel of LSRW on G. We denote by P t (u, v) the t-steps transition probability from u to v. We now establish a certain independence property for walks in G, which in particular implies a certain stationarity property of the SN model. Recall that π v := d v /2|E| andπ = |V |π. For LSRW on a graph G reversibility is the property that for every v, u ∈ V and t 0, . Let Γ k be the collection of all walks of length k in G. For a walk γ = (γ 0 , . . . , γ k ) ∈ Γ k for some k 1, let Note that p(γ) is precisely the probability that a walker whose starting position is γ 0 performed the walk γ (i.e. her position at time i is γ i for all i k). Let γ ′ be the reversal of γ (i.e. γ ′ i = γ |γ|−i for all i |γ|). Reversibility is equivalent to the following condition: q(γ) = q(γ ′ ), for every walk γ.
We denote the number of walkers who performed a walk γ by X γ . We denote the number of walkers whose position at time t is v by Y v (t). Since π is the stationary distribution of the walks, , for every vertex v and time t > 0. Thus, the following fact follows easily from Poisson thinning.
Fact 3.1. Let G be an Eulerian digraph. Then for all t > 0 and γ ∈ Γ t , X γ ∼ Pois(q(γ)). Moreover, (X γ ) γ∈Γt are independent for each fixed t > 0. Consequently, for each fixed t > 0 Next we note that if the average degree of We shall use the following concentration estimate. Let Y ∼ Pois(µ). Then By taking an appropriate limit, this follows from Bernstein's inequality which says that if ξ 1 , . . . , ξ m are independent Bernoulli r.v.'s, Y = i∈ [m] ξ i , µ = E[Y ] and σ 2 = Var(Y ), then Another concentration estimate which we shall use in the proof of some of the lower bounds on SC is the McDiarmid's inequality, which we now state. Let f : X n → R. Let c i := sup |f (x) − f (y)|, where the supremum is taken over all x := (x 1 , . . . , x n ), y := (y 1 , . . . , y n ) ∈ X n such that x j = y j for all j = i. Let X 1 , . . . , X n be i.i.d. X valued r.v.'s. Then

Stochastic domination
Let G = (V, E) be a finite Eulerian digraph. The goal of this subsection is to formulate the intuitive notion that given a certain deterministic initial configuration of walkers, a time t > 0 and a vertex x, the more walkers there are at x at time t, the less walkers we expect to find at the other vertices of G. For a formal statement involving stochastic domination see The partial order such that a ∈ A we have that also b ∈ A. We say that a function f : Let X 1 := (X 1 (v)) v∈V and X 2 := (X 2 (v)) v∈V be random elements in Z V + . Denote their laws by µ 1 and µ 2 , respectively. We say that (the law of) X 1 stochastically dominates (the law of) X 2 if for every increasing set A we have that µ 2 (A) µ 1 (A). It is well-known that this is equivalent to the condition that f dµ 2 f dµ 1 for all increasing f and also to the condition that there exists a probability space (Ω, F , µ) which contains random vectors Y 1 := (Y 1 (v)) v∈V and Y 2 := (Y 2 (v)) v∈V such that for i ∈ {1, 2} the marginal distribution of Y i is µ i and Y 2 Y 1 µ-a.s..
We need to extend the definition of stochastic domination to random sets. Let A, B be two random subsets of some finite set F . We say that the law of A stochastically dominates the law of B if the law of (1 x∈A ) x∈F stochastically dominates the law of (1 x∈B ) x∈F .   [m] 1 X j =i be the number of X ℓ 's which take the value i. Then for all i and all k ∈ N we have that the law of (Y j ) j:j =i given Y i = k is stochastically dominated by its law given Y i = k − 1. In particular, the law of (Y j ) j:j =i stochastically dominates its conditional law given Y i > 0 and for all increasing (resp. decreasing) f 1 , . . . f n : Z + → R we have that (3.5) 4 A general upper bound on SC -proof of (2.1) In this section we prove (2.1), which is the harder and more interesting part of our main result, Theorem 1. Throughout the section we fix some connected n-vertex Eulerian digraph G = (V, E) of average degree d and set t := t d,|V | := ⌈Cd 1 G is not regular log n⌉ 2 (4.1) for some sufficiently large absolute constant C to be determined later.

An overview of the proof
We first describe an easy auxiliary baby model ( § 4.2) and analyze it (Lemma 4.2). We then ( § 4.3) define the notions of a marginally (α, t)-merging configuration, a globally (α, p, t)merging configuration and a (δ, t)-balanced configuration (Definition 4.3) and explain how the analysis of the SN model can be reduced to that of the auxiliary baby model, assuming one of the first two properties hold at each given time w.h.p. (Proposition 4.7). Finally, in § 4.4 we show that at each given time w.h.p. the configuration is (1/8, t)-balanced, and that this implies that it is marginally (α, t)-merging for some α = α d .

Auxiliary baby models
Consider an arbitrary discrete-time coalescent process starting with m distinct classes. Define the coalescence time CT as the minimal time in which there is only one class (if there is no such time it is defined to be ∞). We now describe several simple coalescence models, and derive upper bounds on the coalescence time in each model. We intentionally start with two simple models, each involving a deterministic condition, and then work our way towards a slightly more complicated model which is more faithful to the actual details of our argument for the SN model.
Clearly, if prior to CT in every time unit every class merges with at least one other class, then deterministically, it must be the case that CT ⌈log 2 m⌉. We now generalize this simple model, while introducing some terminology which will be used throughout the section.
Definition 4.1. Fix some t > 0. Let t k := kt. We call the time interval (t k , t k+1 ] the (k + 1)-th round. We say that the (k + 1)-th round is a p-merging round if either t k CT (this case is taken for technical reasons) or t k < CT and at least a p-fraction of the classes (w.r.t. time t k ) have merged with at least one other class by time t k+1 (including).
The last definition makes sense for the SN model, as well as for a general coalescence model. Note that if the number of classes at the beginning of a p-merging round is ℓ > 1, then the number of classes at the end of the round is at most (1 − p/2)ℓ. Assume that initially there are at most m classes. If all rounds are p-merging then, deterministically, CT ⌈C p t log m⌉ (for concreteness, C p := 1/ log( 2 2−p )). More generally, even if not all rounds are p-merging, there can be at most ⌈C p log m⌉ p-merging rounds prior to CT. This motivates the following simple model in which each round is p-merging with probability at least α, regardless of the history. We intentionally keep the description and analysis of this model somewhat informal, deferring a more rigorous statement to Lemma 4.2.
Simplified baby model: Start with m classes. At each round flip a coin with heads probability at least α (i.e. the heads probability in each round may depend on the history up to that time, but is guaranteed to be at least α). If it lands on heads then the round is p-merging. The list of pairs of classes which merge with one another in each round (given the evolution of the model up to the beginning of the round and the result of the coin flip corresponding to the round) is determined according to some arbitrary predetermined rule (satisfying that whenever the coin lands on heads the requirement of being p-merging is satisfied by the list of pairs of merged classes), perhaps using some additional independent source of randomness. The predetermined rule can be thought of as one decided by an adversary.
It is intuitively clear that there exists C α,p of the form C α,p = K 1 C p /α K 2 αp (for some sufficiently large absolute constants K 1 , K 2 , where C p is as above) such that the probability that the total number of rounds exceeds L = L m,α,p := ⌈C α,p log m⌉ tends to 0 as m → ∞. Informally, one can argue that the number of p-merging rounds in the first L rounds, stochastically dominates the Bin(L, α) distribution and then apply an appropriate large deviation estimate in order to bound P[Bin(L, α) < C p log m].
The auxiliary baby model: We now consider another small variation of the last model by introducing an error term ε m . We refer to this variation as the auxiliary baby model. Consider the case that at each round with probability we only have probability 1 − ε m , for some ε m = o(1/ log m), that the model evolves according to the above description (i.e. like the "simplified baby model"), while with the complement probability it stops evolving (i.e. the coalescence process is terminated). It is intuitively clear that the conclusion from the previous paragraph remains valid, with an additional error term of ε m L = o(1) for the probability that the number of rounds exceeds L. We will show that the analysis of the SN model on an Eulerian graph G = (V, E) of average degree d can be reduced to that of the auxiliary baby model (for a suitable choice of parameters).
Observe that above it was sufficient to consider the number of classes at the end of each round in order to bound CT from above. This motivates the following lemma, which formulates rigorously the assertions made in the previous paragraphs. One should think of J k +1 below as the number of classes at the end of the k-th round, and so T 0 := inf{k : J k = 0} corresponds to the coalescence time CT.
Let T 0 := inf{k : J k = 0}. Then there exists some constant C α,p K/(αp) such that The proof of Lemma 4.2 follows by noting that by (4.2) M k : The details are given in the appendix in § 9.4.

Reducing the SN model to the baby model via merging configurations
We are interested in reducing the SN model to the auxiliary baby model. Recall that Y v (s) is the number of walkers at vertex v at time s. It is beneficial to consider a notion of "independence of the history" which depends only on (Y v (s)) v∈V , whose distribution is stationary and hence convenient to work with, rather than also on the identity of the classes of walkers at time s, which might have a complex dependency structure. This motivates the following key definitions.
We identify Y with a configuration of walkers in which for each v ∈ V there are initially Y v particles at v. The walkers then perform a LSRW on G of length t, where t is as in (4.1).
· Let P be a partition of the walkers into (disjoint) sets A 1 , . . . , A m with m 2. We say that P respects vertices if walkers which initially occupy the same vertex all belong to the same set in the partition (i.e. for all v, each A i contains either all Y v walkers initially occupying v or none of them).
· We say that the configuration Y is marginally (α, t)-merging, if the following holds.
For every vertex-respecting partition of the walkers into sets, A 1 , . . . , A m with m 2, for every A i which contains less than half of the walkers, the probability that at least one walker from A i will meet some walker not belonging to A i by time t is at least α. We say such a configuration is globally (α, p, t)-merging if for every vertex-respecting partition of the walkers into sets, A 1 , . . . , A m with m 2, we have that with probability at least p, there are at least α(m − 1) i's such that at least one walker from A i met by time t some walker not belonging to A i .
Throughout, we take δ to be some constant in (0, 1/8]. We now record the fact that requirement (4.4) holds w.h.p. when (Y u ) u∈V are independent and Y u ∼ Pois(π u ).
. Finally, apply a union bound over all u ∈ V . Consider the SN model viewed only at integer multiples of t. Think of the time interval (kt, (k + 1)t] as the k-th round (for k = 0 we take [0, t] instead of (0, t]) Assume that the configuration at time kt is globally (α, p, t)-merging. Then the k-th round is α/2 merging in the sense of Definition 4.1 w.p. at least p. In light of Lemma 4.2, in order to prove (2.1), we need to show that for some absolute constant c 1 , at each given time the configuration is globally (c 1 /d 1+1 G is not regular , c 1 , t)-merging, with probability at least 1 − |V | −2 , where here t is as in (4.1). This observation is recorded in Proposition 4.7 below.
A priori, it may appear difficult to determine that a configuration is either marginally (α, t)-merging or globally (α, p, t)-merging (for some α, p ∈ (0, 1)), as the sets A 1 , . . . , A m from Definition 4.3 form an arbitrary vertex-respecting partition. On the other hand, the property of being (δ, t)-balanced is more tractable as it is stated in terms of a (lack of) a large deviation behavior for the expected number of walkers occupying each site t steps into the future, given the current configuration. We now explain the relations between the four properties of being (δ, t)-balanced, fully (δ, t)-balanced, marginally (α, t)-merging and globally (α, p, t)-merging.
To prove (2.1) we shall later show that (i) if C from the definition of t in (4.1) is taken to be sufficiently large then at each given time the configuration of walkers is fully ( 1 Before proving (ii) we show that (iii) a ( 1 8 , t)-balanced configuration is marginally (c 1 /d 1+1 G is not regular , t)-merging for some absolute constant c 1 > 0 and (iv) a marginally (α, t)merging configuration is globally (α/2, α/4, t)-merging (this is done in Corollary 4.6).
To prove (iv) we use the following simple lemma which asserts that a uniform lower bound on the marginal probabilities of certain m events, implies a corresponding lower bound on the probability that a certain fraction of them occur. The first equation from the lemma follows by a straightforward application of the Paley-Zygmund inequality (e.g. [11,Lemma 4.1]). The second equation is the one-sided Chebyshev's inequality.
In particular, if Cov(ξ i , ξ j ) 0 for all i = j then σ 2 µ and so P[S < 1 2 µ] 4 4+µ . In our setup, (4.5) implies the following. Assuming that the configuration at time t k = kt is marginally α-merging and A 1 , . . . , A m are the classes of walkers at time t k (the classes clearly form a vertex-respecting partition of the walkers), then with probability at least α/4, there exists an index set I of size at least ⌈α(m − 1)/2⌉ such that for all i ∈ I there is some walker from A i who met in the time interval (t k , t k + t] some walker not belonging to A i . We now record this observation. Corollary 4.6. A marginally (α, t)-merging configuration is globally (α/2, α/4, t)-merging.
Derivation of (2.1) assuming that at each given time the configuration is marginally or globally merging w.h.p.: Proposition 4.7. Let G = (V, E) be a connected Eulerian digraph. Fix some t > 0 and α ∈ (0, 1). Assume that for each time s w.p. at least 1 − ε n the configuration of walkers at time s is globally (α, p, t)-merging. Then there exists an absolute constant C > 0 such that In particular, if at each given time w.p. at least 1 − ε n the configuration of walkers is marginally (α, t)-merging then Proof. The second assertion follows from the first via Corollary 4.6. For the first assertion, if we think of each interval of length t as a round, we have that if at time kt the configuration of walkers is globally (α, p, t)-merging, then the (k + 1)-th round (corresponding to the time interval (kt, (k + 1)t]) is α 4 -merging, in the sense of Definition 4.1, with probability at least p (this holds regardless of the outcomes of the previous rounds). Since at the beginning of each round w.p. at least 1 − ε n the configuration of walkers is globally (α, p, t)-merging, we are precisely at the setup of the auxiliary baby model and the assertion of the proposition now follows from Lemma 4.2.
4.4 Proof of the validity of the marginally (α, t)-merging property via a reduction to the (δ, t)-balance condition Note that if (Y v (s)) v∈V is distributed like the configuration of walkers in the SN model on G at time s, then (by stationarity of the occupation measure, which allows one to consider only the case that s = 0) for every v ∈ V the sumZ v (s) := u P t (u, v)Y u (s) in (4.3) (whose mean isπ v ) is a linear combination (with non-negative coefficients) of independent Poisson r.v.'s. As demonstrated in Lemma 4.10 below, in order to obtain a large deviation estimate for such a linear combination (namely, for the eventZ v (s) , it suffices to control the maximal coefficient P t (u, v) appearing in the sum. We obtain such control using the following decay estimate for max x,y P t (x, y).
Recall that d y is the degree of vertex y.
In what comes, we shall assume that (*) 12d 1 G is not regular log |V | δ 2 |E|. Note that if (*) fails, then |V | has to be bounded (recall that δ is a constant and that |V |d = 2|E|, as d is the average degree). The following proposition, whose proof is deferred to § 4.5 implies (2.1) when (*) fails. Proposition 4.9. Let G = (V, E) be a connected Eulerian digraph. Then for some C > 0, Assume (*) holds for some fixed 0 < δ 1/8. By (4.8) we have that In light of Lemma 4.4 and the paragraph following (4.3), by combining (4.9) with the following lemma, it follows by a union bound over the vertices of G that the probability that the configuration is not (δ, t)-balanced at some fixed time s is at most C 1 /|V | 2 (assuming (*), when the constants are chosen as indicated above). (4.10) The proof of Lemma 4.10 is given in the appendix in § 9.3.
The following proposition concludes the proof of (2.1) when G is not regular, and provides a version of (2.1) also for a regular G which has a worse degree dependence than (2.1). Below we assume that the total number of walkers is at most |V | + |V | 2/3 . By (3.2), the probability that this fails is at most exp(− 1 3 |V | 1/3 ) and hence there is no harm in assuming this holds. Proposition 4.12. Let G = (V, E) be an n-vertex connected Eulerian digraph of average degree d. Let ξ := 1{G is not regular}. Let 0 < δ 1/8. Assume that (*) holds. Assume that a configuration of walkers is (δ, t)-balanced for t := ⌈12d ξ Mδ −2 (log n)⌉ 2 and that the total number of walkers it contains is smaller than n + n 2/3 . Then there exists some absolute constantc > 0 such that the following hold.
Proof. Consider some configuration (X v ) v∈V of walkers (i.e. for all v the number of walkers at vertex v is X v ) with at most n + n 2/3 walkers (i.e. v X v n + n 2/3 ) and a vertexrespecting partition of the walkers into disjoint sets A 1 , . . . , A m . Since both the properties of a configuration which we consider (being marginally (α, t)-merging and (δ, t)-balanced) are independent of the time, by stationarity of the occupation measure w.l.o.g. we may assume that this configuration corresponds to time 0. We start with part (i).
Let α d :=c/d 1+ξ , wherec > 0 shall be determined soon. We now fix some 1 i m such that |A i | (n + n 2/3 )/2 and show that the probability that at least one walker from A i meets some walker not from A i by time t is at least α d . Denote the collection of vertices occupied by the walkers from A j (at time 0) by V j (where 1 j m). Denote the expected number of walkers from A j (resp. not from A j , i.e. from ∪ i:i =j A i ) at vertex v at time t by As v∈V µ v (j) = |A j |, there can be at most one set A j with µ v (j) > (1 − 2δ)π v for all v (assuming n 8; where we have used δ 1/8 and v∈V X v n + n 2/3 ). Hence we only have to treat the case that µ v (i) (1 − 2δ)π v for all v and the case that µ v (i) (1 − 2δ)π v holds for some of the vertices, but not for all.
An elementary calculation (see Lemma 4.13) now shows that for all v the probability that there is a walker not from A i at v at time t is at least 1 − e −av(i) . Hence trivially every walker from the set A i has a chance of at least 1 − e −av (i) of meeting some walker not from A i at time t. In this case, the proof is concluded using (4.11).
Finally, assume that µ v (i) > (1 − 2δ)π v for some v but not for all v. We argue that in this case, there must be some vertex u such that for some absolute constant c 1 > 0 (4.12) Once this is establish, the proof is concluded as follows. Let u be as above. Since the configuration is (δ, t)-balanced we have that By (4.12) together with the elementary Lemma 4.13 below, the probability that u is occupied at time t by both a walker from A i and also by a walker not from A i is at least . We now verify the existence of such u satisfying (4.12). Let (4.14) We argue that if u / ∈ D i and (v, u) ∈ E for some v ∈ D i , then u satisfies (4.12) (by our assumption on A i the set D i is non-empty and is a proper subset of V ). This follows from Lemma 4.14 below (whose proof is deferred to the appendix § 9.5; See Remark 4.15 for the intuition behind this lemma), applied to the collection of walkers from A i (i.e. take Y y (0) in Lemma 4.14 to be 0 if the walkers which are at time 0 at vertex y do not belong to A i , otherwise take it to be the number of such walkers; Note that condition (4.4) from the definition of a (δ, t)-balanced configuration implies that the term Y exp(−c 1 t) from Lemma 4.14 is much smaller than the term E[Y v (t)]).
We now prove part (ii). We now assume G is regular, and soπ v = 1 for all v. We fix some i. The case that µ v (i) 1 − 2δ for all v is the same as before. We now treat the case that µ v (i) > 1 − 2δ for some v but not for all v. We may assume that |A i | 1 2 (n + n 2/3 ), as there can be at most one class which contains more than half of the walkers. Let D i be as in (4.14).
For each e = (v, u) let Z e be the indicator of the event that there is some walker from A i which was at v at time t − 1 and then moved to u at time t. Let Z := e∈∂ out E D i Z e . As in the proof of part (i), by Lemma  c 2 c 0 . Conditioned on Z e = 1 for e = (v, u) ∈ ∂ out E D i , as in the proof of part (i), the probability that there is a walker not from A i at u at time t is bounded from below (as u / ∈ D i and the configuration is (δ, t)-balanced). This concludes the proof in the case that |∂ out For each e = (v, u) let Z e be the indicator of the event that there is some walker from ∪ j:j =i A j which was at v at time t − 1 and then moved to u at time t. Let Z := e∈∂ in E D i Z e . As in the proof of part (i), by Lemma 4.14, there exists some c > 0 such that E[ Z e ] > c/d for all e ∈ ∂ in E D i . Hence E[ Z] c 1 c 0 . By Corollary 3.3 we have that ( Z e ) e∈E are negatively correlated. Thus as before, using the onesided Chebyshev's inequality (4.6), we get that P[ Z > 0] > c 2 c 0 > 0. Conditioned on Z e = 1 for e = (v, u) ∈ ∂ in E D i , as in the proof of part (i), the probability that there is a walker from A i at u at time t is bounded from below (as u ∈ D i ).
Lemma 4.14. Let G = (V, E) be a finite Eulerian graph. Let Y v (t) be the number of walkers which occupy vertex v at time t, where (Y v (0)) v∈V is some arbitrary deterministic initial configuration of walkers and different walkers perform independent LSRWs. Let (v, u) ∈ E.
Remark 4.15. The proof of Lemma 4.14 relies on laziness in a crucial manner (in fact, this is the only use of laziness in the proof of (2.1)). The idea is that typically a lazy walk of length t makes at least t/3 lazy steps (the error term Y exp(−c 1 t) corresponds to the total contribution to E[Y v (t)] of the paths that make less than t/3 lazy steps). Such a walk could be transformed into many walks of length t − 1 with the same end-point by deleting one of the lazy steps. The "cost" of this operation is at most a constant factor (this uses the fact that there are at least Ω(t) lazy steps). In fact, some algebra yields that it is at most
Hence for every pair of walkers with some arbitrary initial positions we have that the probability that they are at the same position at time t is at least x∈V (π x /2) 2 1 4|V | ( x π x ) 2 = 1 4|V | . We may assume that the total number of walkers Y is greater than 3|V |/4, as the probability that this fails is exponentially small in |V |. Under this assumption, it follows from Lemma 4.5 that there exists some constant α > 0 such that for every collection of 1 + 1 2 ⌈3|V |/4⌉ walkers with some arbitrary initial conditions, the probability that the first walker met at time t one of the other walkers in the collection is at least α. It follows that (if Y > 3|V | 4 ) the configuration of walkers at each fixed time s, deterministically, has the (α, t)-merging property. The proof is now concluded using Proposition 4.7.
4.6 Proof of (2.1) for regular G In this subsection we prove (2.1) when G is regular. This is done by combining Proposition 4.16 below, which refines Proposition 4.12, with Proposition 4.7.
Let G = (V, E) be a connected d-regular n-vertex Eulerian digraph. Let t * (G) := max{t * , ⌈ 2 c 1 log log n⌉}, wheret * := inf{t : max where c 1 > 0 is as in Lemma 4.14. Note that as the walk is lazy, the probability it stays put for ⌊log 2 (c log n)⌋ steps is at least 1 c log n and thus t * (G) Ct * . Recall that by Fact 4.8 The choice of the constant 3 · 2 8 was made so that by Corollary 4.11, provided that (*) holds for δ = 1/8, we have that the configuration of walkers is ( 1 8 , t * (G))-balanced w.p. at least 1 − 2n −2 . The choice of the constant 2 c 1 was made to ensure that for a ( 1 8 , t * (G))-balanced configuration we have in Lemma 4.14 . In this subsection we improve Proposition 4.12 as follows.
Proposition 4.16. Let G = (V, E) be a connected d-regular n-vertex Eulerian digraph satisfying (*) for δ = 1 8 . Assume that a configuration of walkers is fully ( 1 8 , t * (G))-balanced and that the total number of walkers it contains is smaller than n + n 2/3 . Then there exists some absolute constantc > 0 such that the following hold.
(ii) If |{(u, v) ∈ E : u ∈ A, v / ∈ A or u / ∈ A, v ∈ A}| c 0 d for every set A ⊂ V such that |A| 2 3 (n + n 2/3 ) then the configuration is marginally (cc 0 , t * (G))-merging. Proof. We first prove part (i). Consider some configuration (X v ) v∈V of walkers with at most n + n 2/3 walkers and a vertex-respecting partition of the walkers into disjoint sets A 1 , . . . , A m . Again, we may think of the current time as being time 0. Let a u (i) (resp. µ u (i)) be the expected number of walkers not from (resp. from) A i which occupy u at time t * (G). First consider the case that there is a set I ⊆ [m] of size at least (m − 1)/2 such that for all i ∈ I we have that µ u (i) 3 4 for all u. Then from the proof of Proposition 4.12 we see that for each i ∈ I the probability that some walker from A i has met by time t * (G) some walker not from A i is at least c 3 > 0. By Lemma 4.5 we have that with probability at least c 3 /4, there are at least c 3 |I| sets A i with i ∈ I such that some walker from A i has met by time t * (G) some walker not from A i . Now consider the case that there is a set J ⊆ [m] of size at least (m − 1)/2 such that for all j ∈ J we have that µ u (j) 3 4 for some u ∈ V but not for all u ∈ V . For each j ∈ J, as before, let D j := {v : µ u (j) > 3 4 }. Let For every j ∈ J fix some v j ∈ F j . Observe that since the configuration is fully ( 1 8 , t * (G))balanced we have that the sets (F j ) j∈J are disjoint, and so (v j ) j∈J are distinct.
Let Z j be the indicator of the event that some walker from A j and also some walker not from A j occupying v j at time t * (G). As in the proof of Proposition 4.12 we have that E[Z j ] c 3 /d. By Corollary 3.3 we have that (Z j ) j∈J are negatively correlated (indeed, for i = j knowing that Z i = 1 decreases the chances of both the requirements for Z j = 1). Hence by the one sided Chebyshev's inequality (4.6) P[ j∈J Z j > c 3 2d |J|] c 4 > 0 as desired. The proof of part (ii) is identical to the proof of part (ii) of Proposition (4.16).

Deterministic initial configuration
Let G = (V, E) be a connected d-regular n-vertex Eulerian digraph. Consider the case that initially there is exactly one walker at each site and that the walkers perform independent LSRWs. Let Y v (s) be the number of particles occupying v at time s. Let t * (G) be as in (4.16). Assume n is sufficiently large so that 1 n 1 3·2 9 log n (which by the definition of t * (G) implies that max x,y P t * (G) (x, y) 1 2 9 log n ). We show that at each given time the configuration is fully ( 1 8 , t * (G))-balanced w.p. at least 2n −4 + n − log log n . From this, as in the proof of (2.1), we get that w.h.p. SC(G) Cdt * (G) log n C ′ d(log n) 3 .
The proof of the following proposition is similar to that of Lemma 4.10. The main difference is that now the number of walkers at distinct vertices are not independent and thus one has to rely on (3.5) in order to bound E[exp(λZ v (s))] from above. Proof: Let v ∈ V and s 0 We first prove (4.17). Let λ ∈ [−2 9 log n, 2 9 log n] to be determined later. By (3.5) from Corollary 3.3 we have that Note that exp(λY u (s)P t * (G) (u, v)) = y∈V exp(λξ y (u, s)P t * (G) (u, v))], where ξ y (u, s) is the indicator of the event that the walker which initially occupied y is at u at time s. Then (4.20) Let p * := max{P t * (G) (x, y) : x, y ∈ V } 1 2 9 log n . As |λ|p * 1 and e x − 1 x + x 2 1+1 x 0 for x ∈ [−1, 1] we get that for all y, u ∈ V we have that . Using 1 + x e x , substituting the above in (4.20) and using (4.19), together with the fact that the uniform distribution is stationary (and thus y P s (y, u) = 1 = u P t * (G) (u, v)) we get that E[exp(λZ v (s))] exp(λ + λ 2 p * 1+1 λ 0 ).

General lower bounds on the social connectivity time
Clearly we can bound SC from below by the minimal time by which every walker has met at least one other walker. This motivates the following definition.
Definition 5.1. We say that a walker remained isolated by time t if this walker did not meet any other walkers by time t (including). We say that a walker is lazy by time t if this walker has not left her initial position by time t. We denote the event that there is an isolated lazy walker at vertex v by time t by IL v (t).
Recall that by elementary properties of the Poisson distribution, conditioned on the total number of walkers to be m, each of them independently start at a random initial position chosen according to the stationary distribution. We denote the corresponding probability and expectation for the aforementioned initial condition by P (m) and E (m) . In what comes, it will sometimes be convenient to condition on the total number of walkers.
In additional, if G is a regular graph then Proof. Let f (x 1 , . . . , x m ) be the number of walkers which remained isolated by time t when for all i ∈ [m] the i-th walker performs the walk x i . Let x := (x 1 , . . . , x m ) and let x ′ be x with its ith co-ordinate replaced by some other walk x ′ i . It is easy to see that |f (x)−f (x ′ )| t+2. In other words, given the walks performed by the rest of the walkers, the walk performed by the i-th walker can change the number of walkers which remained isolated by time t by an additive term whose absolute value is at most t + 2. Thus (5.1) follows by the McDiarmid's inequality (3.4).
Let A := {v :π v 2}. The second inequality in (5.2) follows by noting that |A| |V |/2 and so π(A) |A| min v∈V π v d min 2d . We now prove the first inequality in (5.2). As each of the m walkers w.p. π{v :π v 2} has its initial position in A, it suffices to show that for each v ∈ A the conditional probability that IL v (t) = 1 given that the first walker's initial position is v is at least 2 −t (1 − 2(t+1) |V | ) m−1 . Indeed this holds as the probability of the first walker staying put up to time t is 2 −t , while the probability that no other walker visits v by time t is where (X i ) i 0 is a LSRW with X 0 ∼ π and P π and E π are the corresponding probability and expectation (where we have used π v 2/|V | since v ∈ A). We now prove (5.3). Let X := (X i ) i 0 and ( X i ) i 0 be independent LSRWs with X 0 ∼ π and X 0 ∼ π. Let N(s) := |{i s : X i = X i }| be the number of times the two walks collide by time s. Since conditioned on the walk that a certain walker performs, the events that each of the other m − 1 walkers does not collide with it by time t are independent with the same probability (used in the first equality in (5.4)), by Jensen's inequality we get that where we have used regularity to argue that π is the uniform distribution and hence E[N(2t)] = 2t i=0 P[X i = X i ] = (2t + 1) v∈V π 2 v = (2t + 1)/|V |, and reversibility to argue that Recall the notation from § 3 (in particular thatπ := |V |π that p((γ 0 , · · · , γ t )) := t i=1 P (γ i−1 , γ i ) and that q(γ) :=π γ 0 p(γ) and that Γ t is the collection of all walks of length t). The following proposition follows straightforwardly from Fact 3.1 (about Poisson thinning).
Proposition 5.4. Let G = (V, E) be a finite connected Eulerian digraph. Let γ = (γ 0 , . . . , γ t ) ∈ Γ t . Then, given that X γ = 1, the conditional distribution of the number of walkers that the walker who performed the path γ met by time t is Poisson(a γ ), where In particular, . Proof: For the inequality, note that the probability that there is one walker at v at time 0 and that this walker is lazy by time t isπ v 2 −t e −πv . The conditional probability that no other walker visited v by time t is P[Pois( γ∈Γt:v∈γ,γ 0 =v q(γ)) = 0] P[Pois(tπ v ) = 0] e −tπv . We now prove the second line of (5.8). Let 1, 2, 3). Then b 1 = m u,v (t) and by the above reasoning Theorem 5.5. Let G be an n-vertex connected graph. Let µ := v:πv 2π v (recall that π = nπ) and a t := µ2 −t e −2(t+1) . Then for all t 0 we have that Hence, if there exists some δ > 0 such that µ c 1 n β for some β > 0, then for s β := max{⌊c β log n − 2 log log n⌋ − 1, 0}, wherec β := β 2 log(2e 2 ) , we have that Proof. We may assume that 7t For every a ∈ A we have that b∈A ν t (a, b) t and so This (together with 7t √ a t , which implies that t √ a t > 1) implies that there exists some D ⊆ A such that a∈Dπ a µ/(1 + 2t √ a t ) µ/(3t √ a t ) and (5.10) π a ν t (a, b) < max{π a ,π b }/(2 √ a t ), for all distinct a, b ∈ D. (5.11)

By (5.8) and (5.10), E[Z]
(a t /µ) a∈Dπ a √ a t /(3t), where Z := a∈D Z a and Z a := 1 ILa(t) . For all distinct a, b ∈ D, we have that 2π a tν t (a, b) 4t/ √ a t 4/7. Thus by (5.11) and so by (5.8) The next example demonstrates that the assertion of Theorem 5.5 is quite sharp.
Example 5.6. Denote L n := ⌈log 10 n⌉. Consider a ⌈n/L n ⌉-clique, such that each vertex in the clique is the center of a star of size L n (where all stars are disjoint). Then one can show that SC C log log n w.h.p. (and if the walkers perform non-lazy SRWs we would get that SC C w.h.p.). The reason for this is that the expected number of walkers whose initial position is not in a center of a star is at most L 2 n , while at each vertex of the clique there are Pois(µ n ) walkers for some µ n = L n (1 ± o(1)). An easy calculation shows that w.h.p. after one step all of the walkers whose initial positions are in the clique are in the same class.
Proof of Theorem 2: Equation (2.4) follows from Corollary 5.3. We now prove (2.3). Let G = (V, E) be a connected n-vertex d-regular graph. Consider the SN model on G in which the walks have some fixed holding probability p ∈ [0, 1). Denote the transition matrix of the corresponding walks by P . Recall that Γ t is the collection of all walks on G of length t and that for γ = (γ 0 , . . . , γ t ) ∈ Γ t , p(γ) := t−1 i=0 P i (γ i , γ i+1 ). Fix t = t n = ⌊log n − 6 log log n⌋ − 1. Assume t 0 as otherwise there is nothing to prove. For every Γ ′ ⊆ Γ t let m(Γ ′ ) := γ∈Γ ′ p(γ). For every γ ∈ Γ t and 0 i t let Clearly |B γ,i | (t + 1) 4 for all γ ∈ Γ t and i ∈ {0, 1, . . . , t}, and thus Fix some order ≺ on Γ t . Recall that X γ denotes the number of walkers who performed the walk γ. We sequentially expose X γ for some paths γ ∈ Γ t according to the following procedure. Assuming that we have already exposed A ⊂ Γ t so that the following holds: for every γ ∈ A with X γ > 0 we have that Bad γ ∩ {γ ′ ∈ A : γ ≺ γ ′ } = {γ} (where γ ≺ γ ′ indicates that γ ′ is larger in the ordering than γ). In the next stage we expose X γ ′ for the minimal γ ′ ∈ Γ t \ B(A), where B(A) := ∪ γ∈A:Xγ >0 Bad γ . In the following stage we apply the same rule, with the set A replaced by the set A ∪ {γ ′ }. At the end of this procedure we obtain a collection W = {γ 1 , . . . , γ |W| } (resp. N ) of all γ's for which we exposed that X γ > 0 (resp. = 0), where the indices are taken so that i < j iff γ i is before γ j in the ordering. The collection of all γ ∈ Γ t for which X γ was not exposed is precisely Let N i be the collection of all γ ∈ N which were exposed in between γ i−1 and γ i (and N 1 is the collection of all γ's exposed prior to γ 1 ). For every 1 i |W|, the probability that m(N i ) > (t + 1) 5 is at most P[Pois((t + 1) 5 ) = 0] = e −(t+1) 5 . Hence by (5.12) We now condition on W = W and N = N for some arbitrary W, N ⊂ Γ t so that (i) |W| n/[2(t + 1) 5 ] and (ii) Bad γ ∩ {γ ′ ∈ W : γ ≺ γ ′ } = {γ}, for all γ ∈ W . Observe that, given (W, N ) = (W, N), the joint distribution of (X γ ) γ∈Γt\(W ∪N ) is the same as their unconditional joint distribution. For γ = (γ 0 , . . . , γ t ), By construction γ ′ / ∈ Bad γ , for all γ, γ ′ ∈ W such that γ ≺ γ ′ and so t j=0 P |i−j| (γ j , γ ′ i ) < (t + 1) −3 for all 0 i t. Hence by reversibility Finally, for each γ ∈ W set Z γ to be the indicator of the event that γ∈Γγ X γ = 0. Set Z = γ∈W Z γ . Observe that (using m γ t + 1 for all γ ∈ W ) which by Chebyshev implies that P[Z = 0] < 1 E[Z] + δ n .

The cycle
In this section we consider the case that G is the n-cycle C n . In this section we assume that at time 0 there is precisely one walker at each site and that the walkers perform independent continuous-time random walks with jump rate 1. We first prove (2.6). We fix s = s n := C 1 log 2 n. By a union bound over the n edges of the cycle, in order to deduce that P[SC(C n ) s] C/n it suffices to show that for every pair of neighboring vertices u, v, the probability that the walkers who started at those vertices do not have a path of acquaintances by time s is at most Cn −2 .
Consider some fixed edge e = {u, v}. It partitions the vertices into 2 sides (segments) A u , A v of size roughly n/2, according to the identity of the end-point to which they are closer to (brake a tie arbitrarily). Let e ′ be the other edge, apart from e, which connects A u and A v . A walker starting from either u or v is extremely unlikely to cross e ′ in s time units.
Let CROSS s (e, w) be the event that the walker who started at vertex w did not cross e ′ by time s and that her position at time s is at the opposite side of e compared to w.
We now argue that for every (w, w ′ ) ∈ A u × A v the event CROSS s (e, w) ∩ CROSS s (e, w ′ ) "forces" the walkers who started at u and v to have a path of acquaintances by time s (which uses only walkers whose starting positions are in {u, v, w, w ′ }, possibly a subset of this set). Indeed, the continuous-time setup eliminates the possibility of two walkers swapping positions without meeting. This implies that on the event CROSS s (e, w) ∩ CROSS s (e, w ′ ) it must be the case that by time s the walkers whose starting positions are w and w ′ , resp., have met, and that each of the walkers whose starting positions are u and v, resp., must have met at least one of the previous pair of walkers (in fact, this is the case for any pair of vertices u ′ , v ′ lying in the segment between w and w ′ which contains e).
This reduces the proof of (2.6) into the following simple calculation: If C 1 is taken to be sufficiently large then P[CROSS s (e, w)] 0.4 for every edge e = {u, v} and vertex w of distance at most 10 log n from e. Indeed, starting from such a vertex w, if C 1 is sufficiently large then the probability of crossing e by time s is arbitrarily close to 1/2 (in fact it is arbitrarily close to 1), and for all sufficiently large n the probability that e ′ is crossed by time s is at most 1/20 (this probability tends to 0 as n to infinity). Hence (using ( 12  This concludes the proof of (2.6). We now prove (2.7). We fix t = t n = c 1 log 2 n for some constant c 1 to be determined later. Observe that if there are at least 2 edges which no walker crossed by time t, then deterministically SC(C n ) > t. Let NC t (e) be the event that the edge e ∈ E(C n ) was not crossed by any walker by time t. We argue that if c 1 is taken to be sufficiently small, then for all sufficiently large n and every edge e it holds that where F is the event that no walker performed at least ( √ n/2) − 2 steps by time t. We now explain how (2.7) can be derived from (6.1). Denote m := ⌊ √ n⌋. Let (e i ) m i=1 be a collection of edges which are all at distance at least √ n from one another. Then (by a union bound over the n walkers) 1 − P[F] C 2 ne −cn 1/4 C ′ 2 e −c ′ n 1/5 . Finally, note that conditioned on F, the events (NC t (e i )) m i=1 become independent, and so by (6.1) the conditional probability that none of them occur is at most (1 − c 3 n −1/4 ) m e −c ′ 3 n 1/4 . We now prove (6.1). It is not hard to see that in order to prove prove (6.1) it suffices to show that for every k if we put one particle at each site of N := {1, 2 . . .} and the particles perform independent continuous-time SRW on Z with jump rate 1, then the probability that no walker reached the origin by time k is at least e −M √ k , for some absolute constant M. Denote the probability of the previous event by p k . Let (S i r ) r 0 be a SRW on Z starting at i ∈ N. Then by the refection principle (see e.g. [14, § 2.7]; the proof in continuous-time is analogous) We argue that there exist constants c 5 , c 6 , C 3 > 0 such that The first line follows from the local CLT. The first inequality in the third line follows from the fact that 1 − a i (k) P[Pois(k) i]. The last inequalities in the second and third lines follows from the fact that for every 0 < c < 1 there exists C > 0 such that 1 − x e −Cx , for all x c (applied to x = 2e −i 2 /(8k) for i > b). The first inequality in the second line is obtained by noting that for discrete-time SRW (S r ) r∈Z + starting at the origin we have that E[e λSr ] = 1 2 e λ + 1 2 e −λ r e λ 2 r/2 (by comparing Taylor expansion coefficients). Hence, We are now in a position to conclude the proof. By (6.3) and Stirling's approximation Denote b ′ := ⌊ √ k⌋. Finally, using (6.4) it is not hard to show that i>b a i (k) √ k (we leave the details to the reader; Alternatively, since a i (k) are uniformly bounded away from 0 for i > b, we get that i>b a i (k) e −C i>b (1−a i (k)) and the reasoning in the following remark yields that i>0 (1 − a i (k)) = i>0 P i [T 0 k] C ′ √ k). We are done as p k := i>0 a i (k). is similar to the case of one walker per site at time 0. However, as we now explain, the proof of the lower bound on SC(C n ) becomes much simpler. While this was already noted in the introduction after Theorem 2, we present below an alternative argument, as it will be used in the analysis of Examples 8.7 and 8.8. For simplicity, we present the argument in the discrete-time setup. Let v be a vertex in C n . Denote ν t := t i=0 P i (v, v) (this sum is independent of v). For t n 2 we have that ν t = Θ( √ t + 1). By Fact 3.1, the total number of walkers to reach vertex v by time t has a Poisson distribution with mean (by reversibility) Thus if t + 1 < ( log n 8C ) 2 we get that the probability that no walker reached v by time t is at least e −µt n −1/4 . This implies the lower bound on SC(C n ) as in the proof above.

Expanders
In this section we study the case that G is a d-regular λ-expander. It is not difficult to extend the results to the case G is an expander of maximal degree d. LSRW on a regular λ-expander G = (V, E) mixes rapidly in the following sense max x,y∈V |P t (x, y) − π y | (1 − λ) t , for all t. (7.1) Indeed max x∈V |P t (x, x) − π x | (1 − λ) t follows from the spectral decomposition of P t (x, x) along with the non-negativity of the eigenvalues of P . However, for LSRW on a regular graph max x,y∈V |P t (x, y) − π y | = max x∈V |P t (x, x) − π x |, since in general (cf. [7, (2.2)] for even t and [14, p. 135] for moving from even t to odd t using laziness)

Proof of Theorem 4
We argue that Theorem 4 follows from Theorem 5. This follows at once from Corollary 7.2 below. Indeed, after a ("giant") class of walkers A of size at least n/6 emerges, by Corollary 7.2 (using a union bound over the walkers not in A, together with the concentration of the total number of walkers around n), w.p. at least 1 − C 2 n −1 , the additional amount of time until every walker not from A will meet some walker from A is at most ⌈Cλ −1 log n⌉.
Lemma 7.1. Let G be a connected d-regular n-vertex λ-expander. Then there exists C > 0 such that for t := ⌈Cλ −1 log n⌉ and every v, u 1 , u 2 , . . . , u t sequence of vertices, the probability that a LSRW started at v visits some u i at time i for some 1 i t is at least 1 8 min( λt n , 1).
Proof. Let (X s ) s 0 be a LSRW on G started at v. Let Y i = 1 X i =u i and Y := t i=1 Y i , where t := ⌈Cλ −1 log n⌉, for some constant C, to be determined shortly. By (7.1 Corollary 7.2. Let G = (V, E) be a connected d-regular n-vertex λ-expander. Let ℓ := ⌈n/6⌉. Let (v i ) ℓ i=0 be an arbitrary sequence of vertices. Denote a v := |{0 i ℓ : v i = v}|. Assume that at time 0 there are a v particles at vertex v for all v ∈ V and that the particles perform independent LSRWs on G. Assume that a v 0 = 1. Let A s be the event that the particle that started at v 0 met (i.e. visited the same vertex at the same time) at least one other particle by time s. Then there exist some constants C, n ′ (independent of G, d, λ and the sequence (v i ) ℓ i=0 ) such that P[A t ] 1 − n −2 for t := ⌈Cλ −1 log n⌉, as long as n n ′ . Proof. First, condition on the walk that the particle started at v 0 performed by time t, denoted by γ. Then use Lemma 7.1 and independence to obtain the desired estimate for the conditional probability (uniformly, for all possible values of γ). Finally, average over γ.

Proof of Theorem 5
Proof: We adapt a technique from [2, Proposition 3.1] to our setup. Let t n to be determined later. Set s = s(t, λ) := ⌈8(t + 2)/λ⌉. Call a class of walkers at time s good if the collection of walkers from this class occupy at time s at least t vertices. By Corollary 7.4 below (if t is sufficiently large), w.p. at least 1 − exp(−c 1 n/d 4(s+1) ) at time s there exists a collection containing at most n/(2t) good classes such that the walkers belonging to the union of these classes occupy at least n/2 vertices at time s. We conditioned on this event and on the identity of the good classes from this collection and on the positions at time s of the corresponding walkers. We claim, that if t is taken to be sufficiently large, then w.p. at least 1 − exp(−c 2 n) there is no way of splitting this collection into 2 (disjoint) collections, A, B, such that (i) The walkers belonging to the union of the classes in A (resp. B), occupy at time s at least n/6 vertices.
(ii) No walker from (the union of the classes in) A met a walker from B by time s + r, where r := ⌈Cλ −1 ⌉ for some sufficiently large constant C.
Indeed, this follows from Lemma 7.5 below which asserts that for any choice of A, B satisfying (i), the probability that (ii) holds is at most e −c 2 n , by a union bound over all 2 n/(2t) such partitions (and picking t such that t ⌈1/c 2 ⌉; Recall that in the considered collection of good classes there are at most n/(2t) classes). Proof. We have that P[|W u (s)| > 0] = 1 − e −1 . Throughout the proof we condition on |W u (s)| > 0. Moreover, we fix some walker w ∈ W u (s) and condition on the walk w = (w(0), . . . , w(s) = u) that she performed by time s. By averaging, it suffices to show that the conditional probability that A u,t (s) fails is at most e −ct for some constant c > 0 independent of w. Below, all probabilities and expectations are the conditional ones, given w.
Let (X k ) k 0 be a LSRW on G. Let τ w := inf{k ∈ [0, s] : X k = w(s − k)}. By Fact 3.1 and reversibility, the (conditional) distribution (given w) of the number of walkers not from W u (s) that w met by time s has a Poisson distribution with mean µ w : We argue that µ w 4t. Indeed, let b v,w := s i=0 P i (v, w(s − i)). By reversibility and (7.1) Again by (7.1) and reversibility, for all v  Proof. Use Lemma 7.3 together with Azuma inequality on an appropriate Doob's sequence, by exposing the value of indicators on the l.h.s. sequentially, one at a time. The absolute value of the increments are bounded by the maximal size of a ball of radius 2s ( d 2(s+1) ).

Summing over all
Lemma 7.5. Let G = (V, E) be a connected d-regular n-vertex λ-expander. Let 0 < δ 1/2. Let A, B ⊂ V be two disjoint sets of size at least δn. Put one walker at each vertex of A ∪ B and let them perform independent LSRWs. There exists an absolute constant c > 0 such that the probability that no walker from A met a walker from B by time r := ⌈λ −1 log(4/δ 2 )⌉ is at most exp[−cδ 2 n].
Proof: Let W A , W B be the collection of walkers occupying A and B, respectively, at time 0 (as indicated in the statement of the lemma, at each vertex there is exactly one walker at time 0). Let D t be the set of vertices which are occupied at time t by at least one walker in W B . Using Lemma 7.6 below and the fact that P[u ∈ D t ] = P t (u, B) 1 for all u, it is easy to verify that E[|D t |] |B|(1 − e −1 ) for every t. Thus by applying Azuma inequality to an appropriate Doob's sequence (by exposing the positions at time t of the walkers in W B sequentially, one at a time), we get that P[|D t | < δn/4] e −c 1 δn (7.2) (the absolute values of the increments of this martingale are bounded by 1). Denote the uniform distribution on A by π A and by P r π A the distribution at time r of LSRW with X 0 ∼ π A . If some walker from W A is at D r at time r then this walker meets a walker from W B at time r. Given that D r = D, the conditional probability that no walker from W A is at D r at time r is at most To conclude the proof, we show that P r π A (D) δ/8 for all D ⊆ V such that |D| δn/4, which implies the assertion of the Lemma using (7.2).

Concluding remarks, conjectures and open problems
The acquaintances graph AG t = (W, E t ) at time t is the graph whose vertex set is the set of all walkers and whose edge set satisfies that {w, w ′ } ∈ E t iff w and w ′ have met by time t. Note that SC is inf{t : AG t is connected}. While for different underlying graphs AG SC can have very different typical structures (e.g. when the underlying graph is an expander of size n we expect the order of the diameter of AG SC to be between log n/ log log n and log n, while for the n-cycle it is between n/ log n log log n and n/ log n) one feature of AG SC shared by all bounded degree size n underlying graphs is that, w.h.p. the average and maximal degrees are polylogarithmic. In fact, in all of the (bounded degree) examples in this paper for which we determine the order of SC, the average (resp. maximal) degree in AG SC is Θ(log n) (resp. O(log 3/2 n)).

Some extensions of (2.1) beyond the LSRW setup
We now discuss some extensions (2.1). Our proof of (2.1), apart from the independence of the walks, used only the following properties of LSRW (X t ) t 0 : (1) It is ergodic. Hence it has a stationary distribution π. Consequently, when the density of walkers at each vertex is normalized so that it is proportional to π the expected number of walkers occupying a certain fixed vertex at time t does not depend on t.
(2) The Markov property, along with the existence of a uniform decay estimate (w.r.t. t) for sup u,v∈V,s 0 |P[X t+s = u | X s = v] − π u |. This together with (1) is enough in order to establish that the configuration of walkers at each fixed time is (δ, t)balanced w.p., say at least 1 − |V | −3 , if t = t |V | is taken to be sufficiently large, so that sup u,v∈V,s 0 |P[X t+s = u | X s = v] − π u | c/ log |V |.
(3) For all x ∈ V and s 0, for all sufficiently large t and all neighboring y, y ′ ∈ V we have that where d is the average degree and y δ(t, x, y) is small for all x. This is the only place in which we used the laziness of the walks.
Thus the proof could be extended to more general (but still independent) walks. The walks can even be time-inhomogeneous Markov chains, as long as we have (1)-(3) above. It is easy to see that in order to derive a bound on SC for general walks (2) and (3) are both necessary. For (2) consider the case of continuous-time SRW with jump rate 1/t. Clearly any bound on SC should depend on t. For (3), consider (non-lazy) weighted nearest neighbor random walk on some bipartite graph with one loop of weight δ added at a single vertex. Clearly, SC depends on δ.
Open Problem 8.1. Assuming (1) and (3), find a more general condition than (2) which implies an upper bound on SC (without assuming the Markov property).
It is not hard to see that the proof of (2.1) can also be extended to the following setup in which there are m "communities" of walkers. Let m n. Let k 1 , . . . , k m > 0 be such that m i=1 k i = n. Let G 1 , . . . , G m be connected graphs on the same vertex set V of size n. Denote the stationary distribution of LSRW on G i by π i . At time 0 there are Pois(k i π i v ) walkers at vertex v performing independent LSRWs on G i (for all 1 i m and v ∈ V , independently). As usual, when two walkers reach the same vertex at the same time they become acquainted, even if they walk on different graphs. The bound on SC depends in this case on max i m d (i) , where d (i) is the average degree of G i .  There exists some C > 0 such that for every connected n-vertex graph G of average degree d, we have SC(G) (log n) 2 C(d 2 1 G is non-regular + d1 G is regular but not vertex-transitive + 1) w.p. at least 1 − C/n. We believe that in the setup of the previous conjecture it is often the case that [SC(G n ) − τ 1 (G n )]/E[SC(G n )] → 0 in distribution (e.g. we believe this is the case for transitive expanders). However, this might not be the case for the n-cycle. It is plausible that Conjecture 8.6 holds even without the transitivity assumption as long as the graphs G n are of bounded degree. Conversely, if G n is two disjoint n-cliques connected by a single edge, then SC(Gn)

E[SC(Gn)]
converges in distribution to the Exponential distribution with parameter 1.
In the last example we have that E[SC(G n )] = Θ(n). This shows E[SC(G n )] may increase with the maximal degree. Below are two families of graphs, generalizing the previous example, which we believe to be extremal. Example 8.7. In general, E[SC(G)] may grow linearly in d, even when G is regular, as the following example demonstrates. Consider two disjoint n-cliques. Delete an edge from each clique and connected the two cliques by two edges so that the obtained graph is (n−1)-regular. By considering the time required until some walker visits both cliques it is not hard to show that E[SC] cn. We now describe a similar construction of d-regular graphs which satisfy E[SC] cd log 2 (n/d).
Fix some 2 d, n such that 2d n. Let J k be a graph obtained from the complete graph on k vertices by deleting a single edge. Consider ⌈n/d⌉ disjoint copies of J d : I 0 , . . . , I ⌈n/d⌉−1 , where for all 0 j < ⌈n/d⌉, I j is connected to I j+1 (j + 1 is defined mod⌈n/d⌉) by a single edge that connects two degree d − 1 vertices. This can be done so that the obtained graph is d-regular. It is not hard to verify that the obtained graph satisfies E[SC] cd log 2 (n/d) =: s d,n = s (in fact, if d ≪ n we have that w.h.p. SC s). To see this, observe that similarly to the analysis of the n-cycle it suffices to show that provided that c from the definition of s is taken to be sufficiently small, with probability bounded from below (in fact, w.h.p. when d ≪ n) there will be multiples j's such that no walker crossed the edge connecting I j with I j+1 by time s. To prove this, one can imitate the analysis from Remark 6.1. The claimed bound becomes intuitive when one considers the fact that the projection of the walk to the index of the copy of J d to which it belongs, behaves similarly to a random walk on the ⌈n/d⌉ cycle with holding probability 1 − 1 d 2 . Thus the expected number of walkers to cross an edge connecting two copies of J d by time s is comparable to d times the expected number of walkers which cross a certain edge by time s/d for the n-cycle (one has to multiply by d as J d has d vertices).
We now consider a non-regular family of examples.
Example 8.8. Consider two disjoint n-cliques connected by a path of length n. Using the fact that the number of walkers which initially occupy the n-path has a Poisson distribution with a constant mean, it is not hard to show that E[SC] cn 2 .
We now describe a similar construction whose maximal degree is d 8|V | which satisfy E [SC] c[d log(|V |/d)] 2 . Take ⌈n/d⌉ copies of J d : I 0 , . . . , I ⌈n/ℓ⌉−1 and for all 0 k < ⌈n/d⌉ − 1 connect I k to I k+1 via a path of length d.
To see that E[SC] c[d log(n/d)] 2 = t d,n , observe that with positive probability (in fact, w.h.p. if d ≪ n) there will be many vertices lying in the middle of a length d path (connecting some I k to I k+1 ) which will not be visited by a single walker up to time t d,n . We leave the details to the reader (hint: use the idea from Remark 6.1).

Additional Open Problems
The following conjecture concerns the existence of infinite classes of walkers after a constant number of steps in the setup of infinite graphs. Conjecture 8.9. Let G be a connected infinite bounded degree graph such that the critical density for independent percolation on G is strictly less than 1. If the number of walkers at time 0 at each vertex v has a Pois(λd v ) distribution, independently for different v's (let P λ be the corresponding probability measure), then for every λ > 0 there exists t c (λ, G) > 0 such that for all t > t c (λ, G), P λ [there exists an infinite class of walkers at time t] = 1.
Let K n be the complete graph on n vertices. When the holding probabilities of the walks are taken to be either q n = 0 or q n = 1/n, one can show that after a single step the SN model (with Pois(1) walkers per site) behaves in some sense like a critical Erdős Rényi random graph, G(n, 1/n), while after two steps it behaves like the super-critical random graph G(n, 2/n) (hence w.h.p. a giant class of walkers emerges after precisely 2 steps). We believe that also in the continuous-time setup there is a phase transition.
Open Problem 8. 10. Consider the SN model on K n in continuous-time (each walker has jump rate 1). What is the critical time t c for the emergence of a giant class of walkers?
Proof: Denote the complement of A k by B k . Let τ := inf{k : 1 B k = 1}. Define I k = J k for k < τ and I k = 0 for k τ . Denote L := ⌈C α,p log m⌉, where C α,p shall be determined soon.