The size of the largest component below phase transition in inhomogeneous random graphs
aa r X i v : . [ m a t h . P R ] J un The size of the largest component below phasetransition in inhomogeneous random graphs.
TATYANA S. TUROVA Mathematical Center, University of Lund, Box 118, Lund S-221 00, Sweden.
Abstract
We study the ”rank 1 case” of the inhomogeneous random graph model. In thesubcritical case we derive an exact formula for the asymptotic size of the largest con-nected component scaled to log n . This result is new, it completes the correspondingknown result in the supercritical case. We provide some examples of application of anew formula. Mathematics Subject Classification : 60C05; 05C80.
We consider here a subclass of a general inhomogeneous random graph model G V ( n, κ ) witha vertex space V = ( S, µ, ( x , . . . , x n ) n ≥ )introduced in [1]. Here S is a separable metric space and µ is a Borel probability measureon S . Recall the basic definitions and assumptions from [1]. For each n the set of verticesof the graph G V ( n, κ ) is a deterministic or random sequence x , . . . , x n of points in S , suchthat for any µ -continuity set A ⊆ S { i : x i ∈ A } n P → µ ( A ) . (1.1)Given the sequence x , . . . , x n , we let G V ( n, κ ) be the random graph on these vertices, suchthat any two vertices x i and x j are connected by an edge independently of the others andwith a probability p x i ,x j ( n ) = min { κ n ( x i , x j ) /n, } , (1.2) Research was supported by the Swedish Natural Science Research Council. κ n is a symmetric nonnegative measurable function on S × S . We assume also thatfor all x ( n ) → x and y ( n ) → y in Sκ n ( x ( n ) , y ( n )) a.s. → κ ( x, y ) (1.3)as n → ∞ , where the kernel κ is graphical on V , which means that(i) κ is continuous a.s. on S × S ;(ii) κ ∈ L ( S × S, µ × µ );(iii) 1 n E e (cid:16) G V ( n, κ ) (cid:17) → Z S κ ( x, y ) dµ ( x ) dµ ( y ) , where e ( G ) denotes the number of edges in a graph G .It was observed in [6] that random graphs can be naturally related to a certain branchingprocess underlying the algorithm of revealing a connected component in a graph. This ideawas extended in [7] for some inhomogeneous graph model, where a multi-type branchingprocess was introduced to study the connectivity of the graph. But it was not until [1] thata comprehensive theory of inhomogeneous random graphs was developed, which provided aunified approach to many models studied previously.Already in [6] it was shown that in the classical random graph model G n,p with p = c/n the size of the largest connected component scaled to n asymptotically equals the survivalprobability of the associated branching process. A correspondent result was established in[1] for a general model G V ( n, κ ) described above. We shall recall this result here. Let C (cid:16) G (cid:17) denote the size (the number of vertices) of the largest connected component in a graph G .Then Theorem 3.1 from [1] states that C (cid:16) G V ( n, κ ) (cid:17) n P → ρ κ := Z S ρ κ ( x ) dµ ( x ) , (1.4)where ρ κ ( x ) is the survival probability of a multi-type Galton-Watson process B κ ( x ) definedas follows. The type space of B κ ( x ) is S , and initially there is a single particle of type x ∈ S .Then at any step, a particle of type x ∈ S is replaced in the next generation by a set ofparticles where the number of particles of type y has a Poisson distribution with intensity κ ( x, y ) dµ ( y ). It was also proved in [1] that ρ κ ( x ) is the maximum solution to ρ κ ( x ) = 1 − e − R S κ ( x,y ) ρ κ ( y ) dµ ( y ) . Whether ρ κ is zero or strictly positive depends only on the norm of an integral operator T κ defined as ( T κ f )( x ) = Z S κ ( x, y ) f ( y ) dµ ( y ) (1.5)2ith norm k T κ k = sup {k T κ f k : f ≥ , k f k ≤ } . Then according to Theorem 3.1 from [1] the survival probability ρ κ (cid:26) > , if k T κ k > , = 0 , if k T κ k ≤ . (1.6)Hence, while (1.4) describes rather accurate the size of the largest connected componentabove the phase transition, i.e., when k T κ k >
1, all what we can get from (1.4) when k T κ k ≤ C (cid:16) G V ( n, κ ) (cid:17) = o P ( n ) . Only under an additional assumptionsup x,y,n κ n ( x, y ) < ∞ (1.7)Theorem 3.12 in [1] proves in the case k T κ k < G V ( n, κ ) = O (log n ) whp (which means”with high probability”, i.e., with probability tending to one as n → ∞ ).In the case of a homogeneous random graph G n,p with p = c/n the following convergencein probability (and even more precise result) was derived already in [3]: if c < C ( G n,c/n )log n P → c − | log c | (1.8)as n → ∞ . However, the method used in [3] is not applicable for an inhomogeneous model. Our aim here is to derive the asymptotics of the size of the largest component scaled to log n (similar to (1.8)) for inhomogeneous random graph model in the case k T κ k <
1. We show,that this is also directly related to the parameters of the introduced branching process B κ .Assume from now on that S ⊆ R + is finite or countable, µ is a probability on S , and agraphical kernel κ on S × S has a form κ ( x, y ) = cψ ( x ) ψ ( y ) , (1.9)where ψ is a positive function on S and c is a positive constant. We consider a graph G V ( n, κ ) on the vertex space V which satisfies condition (1.1), and given x , . . . , x n , theedges are independent and have probabilities (1.2) with κ n = κ , i.e., p x i ,x j ( n ) = min (cid:26) cψ ( x i ) ψ ( x j ) n , (cid:27) .
3n this case operator T κ defined in (1.5) has rank 1 (giving the name ”the rank 1 case” ofinhomogeneous random graph model, see Chapter 16.4 in [1]), and k T κ k = c X S ψ ( x ) µ ( x ) . (1.10) Assumption 1.1.
Let function ψ satisfy one of the following conditions: either sup x ∈ S ψ ( x ) < ∞ , (1.11) or for some monotone increasing unbounded function ψ and positive constants A ≤ A A ψ ( x ) ≤ ψ ( x ) ≤ A ψ ( x ) , (1.12) for all large x , and X S e aψ ( x ) µ ( x ) < ∞ (1.13) for some positive a . Also we shall assume that for any ε > q > P (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) { i : x i = k } n − µ ( k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ εe qψ ( k ) µ ( k ) , k ∈ S (cid:27) → n → ∞ . Notice that when S is finite, convergence (1.14) trivially follows by (1.1).Let B κ ( x ) be a branching process defined as above: it starts with one particle of type x ∈ S , and then at any step, a particle of type x ∈ S produces P o (cid:16) κ ( x, y ) µ ( y ) (cid:17) number ofoffspring of each type y ∈ S . Denote X ( x ) the size of the total progeny of B κ ( x ), and let r ( c ) = sup { z ≥ X S µ ( x ) ψ ( x ) E z X ( x ) < ∞} . (1.15) Theorem 1.1.
Let κ ( x, y ) = cψ ( x ) ψ ( y ) and set c cr := X S ψ ( x ) µ ( x ) ! − . (1.16) Under Assumption 1.1 and (1.14) we have C (cid:16) G V ( n, κ ) (cid:17) log n P → r ( c ) (1.17)4 s n → ∞ , where r ( c ) > , if c < c cr , = 1 , if c ≥ c cr . (1.18)Observe that due to (1.10) one has k T κ k < ⇔ c < c cr . (1.19)Hence, the statement of Theorem 1.1 is exactly complementary to (1.4) (under the conditionsof Theorem 1.1), since r ( c ) > ρ κ = 0 , as well as ρ κ > r ( c ) = 1. Notice, however, that when c = c cr then both r ( c ) = 1and ρ κ = 0, and none of statements (1.4) or (1.17) provides substantial information.It is rather obvious that one can extend Theorem 1.1 for the case of non-countable S undersimilar assumptions, replacing sum by the integral with respect to µ . It is less apparent, butone may conjecture as well, that the statement similar to (1.17) is not restricted to the rank1 case only.The rank 1 case is proved to be versatile for applications. One may interpret ψ ( x ) asan ”activity” of a vertex of type x . One particularly often seen choice of ψ is ψ ( x ) = x on S = { , , . . . } . Here ”type x ” can represent a degree of a node as in [2] or a size of amacro-vertex as in [9] (see also Chapter 16.4 in [1] on other examples). A special featureof the rank 1 case is that it allows one to compute r ( c ) in a rather closed form as we statebelow. Theorem 1.2.
Assume, the conditions of Theorem 1.1 are satisfied. Let X be a randomvariable in S with probability function µ . There exists a unique y > which satisfies y = 1 c E ψ ( X ) E ψ ( X ) exp { cψ ( X ) ( E ψ ( X )) ( y − } E ψ ( X ) exp { cψ ( X ) ( E ψ ( X )) ( y − } . (1.20) Then r ( c ) = 1 c E ψ ( X ) exp { cψ ( X ) ( E ψ ( X )) ( y − } . (1.21)Notice that Theorems 1.1 and 1.2 immediately yield (1.8). Indeed, in the case of ahomogeneous model G n,c/n we have ψ ( X ) ≡ y = 1 /c ,which together with (1.21) and (1.17) gives (1.8).The result (1.17) is new for the inhomogeneous random graphs. (In fact, even in thecase of G n,c/n the role of r ( c ) was not disclosed previously.) In the subcritical case of G n,c/n L denote a component of a graph G V ( n, κ ). In some applications one studies afunction of L in the form P x i ∈ L ψ ( x i ), which is natural to call an ”activity of a component”if we call ψ ( x ) an activity of a vertex of type x . A similar characteristic of a graph, called a”volume” was also treated in [4].Consider again the branching processes B κ ( x ) defined above, which starts with one parti-cle of type x ∈ S . Denote e X ( x ) the set of all the offspring of the branching processes B κ ( x ).(Recall that previously we set | e X ( x ) | = X ( x ).) Let Φ( x ) be the sum of ”activities” of allthe offspring of the branching processes B κ ( x ):Φ( x ) = X v ∈ e X ( x ) ψ ( v ) . This implies that Φ( x ) satisfies the following equality in distributionΦ( x ) d = ψ ( x ) + X y ∈ S N x ( y ) X i =1 Φ i ( y ) , (1.22)where N x ( y ) ∈ Po( cψ ( x ) ψ ( y ) µ ( y )), independent for different x and y ; random variables Φ( x )and Φ i ( x ) , i ≥ , are i.i.d. , and also independent for different values of x ; and a sum overempty set is assumed to be zero. Define also similar to (1.15) α ( c ) = sup { z ≥ X S ψ ( x ) E z Φ( x ) µ ( x ) < ∞} . (1.23)Now we are ready to state another result similar to Theorem 1.1. Theorem 1.3.
Let L denote a set of all connected components in G V ( n, κ ) . Under theconditions of Theorem 1.1 we have max L ∈L P x i ∈ L ψ ( x i )log n P → α ( c ) (1.24)6 s n → ∞ , where α ( c ) > , if c < c cr , = 1 , if c ≥ c cr . (1.25)One can also find formula for α ( c ) similar to the one in Theorem 1.2. Let G N ( p, c ) be a graph with the set of vertices B ( N ) := {− N, . . . , N } d in Z d , d ≥
1, withtwo types of edges: the short-range edges connect independently with probability p eachpair u and v if | u − v | = 1, and the long-range edges connect independently any pair oftwo vertices with probability c/ | B ( N ) | . By this definition there can be none, one or twoedges between two vertices in graph G N ( p, c ), and in the last case the edges are of differenttypes. Assume, that 0 ≤ p < p c ( d ), where p c ( d ) is the percolation threshold in dimension d . As it is shown in [9], this graph is naturally related to the described above rank 1 case.Consider the subgraph of G N ( p, c ) induced by the short-range edges only, which is a purelybond percolation model. Let K N denote the number of open clusters (i.e., connected by theshort-range edges only), and let X = { X , X , . . . , X K N } denote the collection of all openclusters X i ⊆ B ( N ). Let also C denote an open cluster containing the origin. Recall that K N | B ( N ) | → E | C | (1.26)a.s. and in L as N → ∞ (see, e.g., [5]). Call each set X i a macro-vertex of type | X i | .Now given a collection of clusters X , introduce another graph e G N ( X , p, c ), whose verticesare macro-vertices X , X , . . . , X K N . The probability that two (macro-)vertices X i and X j with | X i | = x and | X j | = y are connected is derived from the original model G N ( p, c ), whichis e p xy ( N ) = 1 − (cid:18) − c | B ( N ) | (cid:19) xy =: κ K N ( x, y ) K N . (1.27)Clearly, the size of the largest connected component in G N ( p, c ) has the following represen-tation C ( G N ( p, c )) = max L X X i ∈ L | X i | (1.28)where the maximum is taken over all connected components L in e G N ( X , p, c ).It was shown in [9] that graph e G N ( X , p, c ) fits the definition of an inhomogeneous randomgraph. In particular, κ K N ( x, y ) a.s. → κ ( x, y ) := c E ( | C | − ) xy (1.29)7s N → ∞ , and { ≤ i ≤ K N : | X i | = k } K N a.s. → E ( | C | − ) P {| C | = k } k =: µ ( k ) (1.30)as N → ∞ . (We refer to [9] for the details.) It follows also from the results of [9], thatmodel e G N ( X , p, c ) satisfies the conditions of Theorem 1.1 with S = { , , . . . } , κ ( x, y ) = c E ( | C | − ) xy , (we set here ψ ( x ) = x ), and µ defined in (1.30). In this case according to (1.5) k T κ k = c E ( | C | − ) X S ψ ( x ) µ ( x ) = X S x E ( | C | − ) P {| C | = x } x = c E | C | , implying that | T κ k < c < E | C | . Hence, taking into account (1.28) and alsoconvergence (1.26) we readily get the result on G N ( p, c ) model. Corollary 1.1.
Assume, that d ≥ and ≤ p < p c ( d ) . Then C ( G N ( p, c ))log | B ( N ) | P → γ ( p, c ) as N → ∞ , where γ ( p, c ) = α ( c E ( | C | − )) and α ( c ) is defined by (1.23) with ψ ( x ) = x , and γ ( p, c ) > , if c < E | C | , = 1 , if c ≥ E | C | . (1.31) ✷ This result was conjectured in [9], where it was proved that for any ε > N →∞ P (cid:26) C ( G N ( p, c ))log | B ( N ) | > γ ( p, c ) + ε (cid:27) = 0 . We shall also refer to [9] on more exact description of γ ( p, c ) which is similar to the derivationof (1.21). Recall that X ( x ) denote the total number of the particles (including the initial one) producedby the branching process B κ ( x ), and Φ( x ) is the total activity as defined in (1.22). Let for8 ≥ h z ( x ) = E z X ( x ) , g z ( x ) = E z Φ( x ) . Define also H z = X S ψ ( x ) E z X ( x ) µ ( x ) . Then we rewrite (1.15) r ( c ) = sup { z ≥ H z < ∞} . First we shall prove the following lemma, which in particular yields (1.18) and (1.25).
Lemma 2.1.
Let µ be a probability on S , and let function ψ be positive on S and satisfy(1.13). Write (as in (1.16)) c cr := X S ψ ( x ) µ ( x ) ! − . Then(I) r ( c ) (cid:26) > , if c < c cr , = 1 , if c ≥ c cr ; (2.32) (II) α ( c ) (cid:26) > , if c < c cr , = 1 , if c ≥ c cr ; (2.33) (III) for all n ≥ sup { z ≥ h z ( n ) < ∞} = r ( c ) , (2.34) and sup { z ≥ g z ( n ) < ∞} = α ( c ) . (2.35) Proof.
Note that function h z ( k ) (as a generating function for a branching process) satisfiesthe following equation h z ( k ) = z exp (X x ∈ S κ ( k, x ) µ ( x )( h z ( x ) − ) = z exp ( cψ ( k ) X x ∈ S ψ ( x ) µ ( x ) ( h z ( x ) − ) . X denote a random variable in S with distribution µ . Then we can rewrite the lastformula as follows h z ( k ) = z exp { cψ ( k )( H z − E ψ ( X )) } . (2.36)Multiplying both sides by ψ ( k ) µ ( k ) and summing up over k we find for all z < r ( c ) H z = X k ∈ S ψ ( k ) µ ( k ) z exp { cψ ( k )( H z − E ψ ( X )) } (2.37)= z E ψ ( X ) exp { cψ ( X )( H z − E ψ ( X )) } . Notice, that H = E ψ ( X ) , and clearly, H z is an increasing function of z . Hence, equation (2.37) has a finite solutionfor some z > y = 1 E ψ ( X ) z E ψ ( X ) exp { cψ ( X ) ( E ψ ( X )) ( y − } . (2.38)(for the same value z ) has a root y >
1. It is easy to see taking into account assumption(1.13), that at least for some y > z > f ( y, z ) := zf ( y ) := z E ψ ( X ) E ψ ( X ) exp { cψ ( X ) ( E ψ ( X )) ( y − } (2.39)is increasing in both variables, it has all the derivatives of the second order, and ∂ ∂y f ( y, z ) >
0. Compute now ∂∂y f ( y, z ) | y =1 ,z =1 = c E ψ ( X ) = cc cr . (2.40)Hence, if c ≥ c cr then for any z > y ≥ c < c cr then there exists z > ≤ z ≤ z there is a finite solution y ≥ z > H z .This proves (2.32). The statement (2.34) follows immediately by (2.36).Exploring formula (1.22) it is easy to derive that function g z ( k ) satisfies the followingequation g z ( k ) = z ψ ( k ) exp ( cψ ( k ) X x ∈ S ψ ( x ) µ ( x ) ( g z ( x ) − ) . Then with a help of random variable X we can rewrite the last formula as follows g z ( k ) = z ψ ( k ) exp { cψ ( k )( G z − E ψ ( X )) } , (2.41)10here G z = X S ψ ( x ) µ ( x ) g z ( x ) . Multiplying both sides of (2.41) by ψ ( k ) µ ( k ) and summing up over k we find for all z < α ( c ) G z = X k ∈ S ψ ( k ) µ ( k ) z ψ ( k ) exp { cψ ( k )( G z − E ψ ( X )) } (2.42)= E z ψ ( X ) ψ ( X ) exp { cψ ( X )( G z − E ψ ( X )) } . The rest of the proof of (2.33) and (2.35) is identical to that of (2.32) and (2.34). ✷ We shall find r ( c ) = z as the (unique!) value for which function y is tangent to f ( y, z ) (see(2.39) and (2.38)) if y ≥ z > y is tangent to f ( y, z ), and let y be the tangencypoint. Hence, z and y satisfy the following equations z f ′ ( y ) = 1 ,z f ( y ) = y , which implies that y is the unique solution to y = f ( y ) f ′ ( y ) = 1 c E ψ ( X ) E ψ ( X ) exp { cψ ( X ) ( E ψ ( X )) ( y − } E ψ ( X ) exp { cψ ( X ) ( E ψ ( X )) ( y − } , and then r ( c ) = z = 1 f ′ ( y ) = 1 c E ψ ( X ) exp { cψ ( X ) ( E ψ ( X )) ( y − } . (2.43)This proves formula (1.21). ✷ We shall assume here that ψ satisfies (1.12) and (1.13). In the case of (1.11) when inf x ∈ S ψ ( x ) > ψ ( x ) = const in (1.12), and the proof will follow by the same argument.When (1.11) holds with inf x ∈ S ψ ( x ) = 0, it is easy to construct an upper and a lowerapproximations for the kernel κ (consult also [1] on approximations), so that the proof willagain be reduced to the previous case. 11 .3.1 The lower bound. First we shall prove that for any δ > P (cid:26) C (cid:16) G V ( n, κ ) (cid:17) > (cid:18) r ( c ) + δ (cid:19) log n (cid:27) → N → ∞ .Recall the usual algorithm of finding a connected component in a random graph. Con-ditionally on the set of vertices V := { x , . . . , x n } , take any vertex x i ∈ V to be the root.Find all the vertices { v , v , ..., v m } connected to this vertex x i in the graph G V ( n, κ ), andthen mark x i as ”saturated”. Then for each non-saturated but already revealed vertex, wefind all the vertices connected to it but which have not been used previously. We continuethis process until we end up with a tree of saturated vertices.Denote τ n ( x i ) the set of the vertices in the tree constructed according to the abovealgorithm with the root at a vertex x i .First we shall prove the following intermediate result. Lemma 2.2. If c < c cr then lim n → P n C (cid:16) G V ( n, κ ) (cid:17) > n / o = 0 . (2.45) Proof.
Let constant a be the one from the condition (1.13). Then for any0 ≤ q < a/ S : µ q ( k ) = m q e qψ ( k ) µ ( k ) , (2.47)where the normalizing constant m q := X S e qψ ( k ) µ ( k ) ! − > . Notice that µ ( k ) = µ ( k ) for all k ∈ S , and m q is continuous on [0 , a/
2] with m = 1. Fix ε > < q < a/ B n = (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) { ≤ i ≤ n : x i = k } n − µ ( k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ εµ q ( k ) , k ∈ S (cid:27) . (2.48)12et | τ n ( x ) | denote the number of vertices in τ n ( x ). Then we easily derive taking intoaccount the assumption (1.14) that P n C (cid:16) G V ( n, κ ) (cid:17) > n / o ≤ P (cid:26) max ≤ i ≤ n | τ n ( x i ) | > n / | B n (cid:27) + o (1) (2.49) ≤ n X k ∈ S ( µ ( k ) + εµ q ( k )) P (cid:8) | τ n ( k ) | > n / | B n (cid:9) + o (1)as n → ∞ .To approximate the distribution of | τ n ( k ) | we shall use the following branching processes.Let B c,q be a process defined similar to B κ , but with the distribution of the offspring P o ( cψ ( x ) ψ ( y ) µ q ( y ))instead of P o ( cψ ( x ) ψ ( y ) µ ( y )). Notice, that B c, is defined exactly as B κ . We set c cr ( q ) = X S ψ ( x ) µ q ( x ) ! − . Clearly, c cr ( q ) is a continuous function of q on [0 , a/
2] with c cr (0) = c cr .Let further X c,q ( k ) denote the total number of the particles (including the initial one)produced by the branching process B c,q starting with a single particle of type k . Proposition 2.1.
For any c < c cr one can find q > and c < c ′ < min { c cr ( q ) , c cr } arbitrarilyclose to and c , correspondingly, such that for all k ≥ and all large n P (cid:8) | τ n ( k ) | > n / | B n (cid:9) ≤ e b ( log n ) P n X c ′ ,q ( k ) > n / o , (2.50) where b is some positive constant independent of k and n . Proof.
Observe that at each step of the exploration algorithm which defines τ n , the numberof the type y offspring of a particle of type x has a binomial distribution Bin ( N ′ y , p xy ( n ))where N ′ y is the number of the remaining vertices of type y .We shall explore the following relation between the binomial and the Poisson distribu-tions. Let Y n,p ∈ Bin ( n, p ) and Z λ ∈ P o ( λ ), where 0 < p < / λ >
0. Then for all k ≥ P { Y n,p = k } ≤ (1 + γp ) n P { Z n p − p = k } , (2.51)where γ is some positive constant (independent of n , k and p ).13otice that conditionally on B n we have N ′ y ≤ { ≤ i ≤ n : x i = y } ≤ n ( µ ( y ) + εµ q ( y )) (2.52)for each y ∈ S . The last inequality implies that for any y such that { ≤ i ≤ n : x i = y } > n ( µ ( y ) + εµ q ( y )) ≥ . (2.53)By the assumptions (1.13), (1.12) and (2.46) we have for all large yµ ( y ) + εµ q ( y ) ≤ e − aψ ( y ) + εm q e ( q − a ) ψ ( y ) ≤ b e ( q − a ) A ψ ( y ) ≤ b e − aA ψ ( y ) / , where 0 < b < ε for all small q >
0. Combining this with (2.53) we obtain for all large n and y such that { ≤ i ≤ n : x i = y } > n ≤ µ ( y ) + εµ q ( y ) ≤ b e − aA ψ ( y ) / . This implies that conditionally on B n max x ∈{ x ,...,x n } ψ ( x ) ≤ A log n for some constant A , and thus conditionally on B n p x i x j ( n ) ≤ c ( A log n ) n (2.54)for all x i , x j ∈ V . This and (2.52) together with the continuity of m q allow us for any fixedpositive ε to choose ε and q in (2.48) so that conditionally on B n we get N ′ y p xy ( n )1 − p xy ( n ) ≤ ( µ ( y ) + εµ q ( y )) np xy ( n )1 − p xy ( n ) ≤ (1 + ε ) µ q ( y ) cψ ( x ) ψ ( y ) =: µ q ( y ) c ′ ψ ( x ) ψ ( y ) (2.55)for all large n . In other words, for any q > c ′ > c arbitrarily close to 0 and c ,respectively, and such that c < c ′ < min { c cr ( q ) , c cr } , (2.56)bound (2.55) holds for all large n . 14ow according to (2.51) and (2.55) P { Y N ′ y ,p xy ( n ) ≥ k } ≤ (1 + γp xy ( n ) ) N ′ y P { Z N ′ y pxy ( n )1 − pxy ( n ) ≥ k } (2.57) ≤ (1 + γp xy ( n ) ) n P { Z µ q ( y ) c ′ ψ ( x ) ψ ( y ) ≥ k } . Hence, if conditionally on B n at each (of at most n ) step of the exploration algorithm whichreveals τ n ( k ), we replace the Bin ( N ′ y , p xy ( n )) variable with the P o ( µ q ( y ) c ′ ψ ( x ) ψ ( y )) one, wearrive at the following bound using branching process B c ′ ,q ( k ) and bound (2.54): P (cid:8) | τ n ( k ) | > n / | B n (cid:9) ≤ γ (cid:18) c ( A log n ) n (cid:19) ! n P n X c ′ ,q ( k ) > n / o . (2.58)This implies statement (2.50) of the Proposition. ✷ Substituting (2.50) into (2.49) we derive for any q > c ′ > c that P n C (cid:16) G V ( n, κ ) (cid:17) > n / o ≤ b ne b (log n ) X k ∈ S µ q ( k ) P n X c ′ ,q ( k ) > n / o + o (1) (2.59)as n → ∞ , where b is some positive constant. By the Markov’s inequality P (cid:8) X c ′ ,q ( k ) > n / (cid:9) ≤ z − n / E z X c ′ ,q ( k ) (2.60)for all z ≥
1. This bound together with (2.59) yield P n C (cid:16) G V ( n, κ ) (cid:17) > n / o ≤ b ne b (log n ) z − n / X k ∈ S µ q ( k ) E z X c ′ ,q ( k ) + o (1) . (2.61)We are left to show that for some z > X k ∈ S µ q ( k ) E z X c ′ ,q ( k ) < ∞ . (2.62)Note that µ q and ψ satisfy the conditions of Lemma 2.1 for any 0 ≤ q < a/
2, and moreover c ′ < c cr ( q ). Hence, by Lemma 2.1 there exists z > X k ∈ S ψ ( k ) µ q ( k ) E z X c ′ ,q ( k )0 < ∞ , which clearly implies (2.62), and the statement of Lemma 2.2 follows by (2.61) where we set z = z > ✷ B ′ n := B n ∩ (cid:16) C (cid:16) G V ( n, κ ) (cid:17) ≤ n / (cid:17) . According to assumption (1.14) and Lemma 2.2 we have P {B ′ n } = 1 − o (1)as n → ∞ . This allows us to derive similar to (2.49) for any ω P n C (cid:16) G V ( n, κ ) (cid:17) > ω o ≤ P (cid:26) max ≤ i ≤ n | τ n ( x i ) | > ω | B ′ n (cid:27) + o (1) (2.63) ≤ n X k ∈ S ( µ ( k ) + εµ q ( k )) P {| τ n ( k ) | > ω | B ′ n } + o (1)as n → ∞ . Repeating the same argument which led to (2.58), we get the following boundusing the introduced branching process: P {| τ n ( k ) | > ω | B ′ n } ≤ γ (cid:18) c ( A log n ) n (cid:19) ! b n √ n P n X c ′ ,q ( k ) > ω o as n → ∞ , where we take into account that we can perform at most √ n steps of exploration(the maximal possible number of macro-vertices in any connected component conditioned on B ′ n ). Notice also that we can choose here c ′ and q arbitrarily close to c and 0, correspondingly,and so that condition (2.56) is fulfilled. The last bound implies P {| τ n ( k ) | > ω | B ′ n } ≤ (1 + o (1)) P n X c ′ ,q ( k ) > ω o (2.64)as n → ∞ . Substituting (2.64) into (2.63) we derive P n C (cid:16) G V ( n, κ ) (cid:17) > ω o ≤ bn X k ∈ S µ q ( k ) P n X c ′ ,q ( k ) > ω o + o (1) (2.65)as n → ∞ , where b is some positive constant. Then similar to (2.61) we derive from (2.65) P n C (cid:16) G V ( n, κ ) (cid:17) > ω o ≤ bnz − ω X k ∈ S µ q ( k ) E z X c ′ ,q ( k ) + o (1) . (2.66)16ince ψ and probability µ q satisfy the conditions of Lemma 2.1, for all values c ′ and q forwhich (2.56) holds, we have by this lemma r q ( c ′ ) := sup { z ≥ X x ∈ S ψ ( x ) E z X c ′ ,q ( x ) µ q ( x ) < ∞} > , and therefore for all 1 < z < r q ( c ′ ) X k ∈ S µ q ( k ) E z X c ′ ,q ( k ) < ∞ . (2.67)Note that r q ( c ′ ) is a continuous function of q and c ′ : this can be explicitly derived fromformula (2.43) where X is replaced by a random variable X q with a probability function µ q .Moreover, r ( c ) = r ( c ). Hence, for any δ > δ ′ > q, c ′ ) closeto (0 , c ) so that (2.67) holds with z = r q ( c ′ ) − δ ′ > , (2.68)and also (cid:18) r ( c ) + δ (cid:19) log( r q ( c ′ ) − δ ′ ) > . (2.69)Now setting ω = (cid:16) r ( c ) + δ (cid:17) log n and z = r q ( c ′ ) − δ ′ in (2.66) we derive with a help of(2.67) P (cid:26) C (cid:16) G V ( n, κ ) (cid:17) > (cid:18) r ( c ) + δ (cid:19) log n (cid:27) ≤ b nz − ω + o (1)= b n exp {− log( r q ( c ′ ) − δ ′ ) (cid:18) r ( c ) + δ (cid:19) log n } + o (1) (2.70)where b is some finite positive constant. This together with (2.69) clearly implies statement(2.44). ✷ Here we show that for any δ > n →∞ P (cid:26) C (cid:16) G V ( n, κ ) (cid:17) < (cid:18) r ( c ) − δ (cid:19) log n (cid:27) = 0 . (2.71)17iven a random graph G V ( n, κ ) let T be the number of its connected components, anddenote W i , i = 1 , . . . , T , the corresponding sets of the vertices in these components, orderedarbitrarily. Let us fix ε > n ≥ A ( n ) = (cid:26) max ≤ i ≤ T | W i | ≤ (cid:18) r ( c ) + ε (cid:19) log n (cid:27) ∩ B n with B n defined in (2.48). According to (2.44) and the assumption (1.14) P {A ( n ) } → n → ∞ . Let P A ( n ) ( · ) = P {· | A ( n ) } denote the conditional probability. Denote further ω = (cid:18) r ( c ) − δ (cid:19) log n, ω = (cid:18) r ( c ) + ε (cid:19) log n, (2.73) N = N ( n ) = nω . (2.74)Clearly, P A ( n ) { T ≥ [ N ] + 1 } = 1 . (2.75)We shall reveal recursively [ N ] + 1 connected components in the graph G V ( n, κ ) in thefollowing way. Let V be a random vertex uniformly distributed on V . Set L = τ n ( V ) tobe the set of the vertices in the connected component containing vertex V .Further for any U ⊂ V let τ Un ( v ) denote a set of vertices of the tree constructed in thesame way as τ n ( v ) but on the set of vertices V \ U instead of V . In particular, with thisnotation τ ∅ n ( v ) = τ n ( v ).Given constructed components L , . . . , L k for 1 ≤ k ≤ [ N ], let V k +1 be a vertex uniformlydistributed on V \ ∪ ki =1 L i , and set L k +1 = τ ∪ ki =1 L i n ( V k +1 ). Then according to (2.75) and (2.72)we have P (cid:26) C (cid:16) G V ( n, κ ) (cid:17) < (cid:18) r ( c ) − δ (cid:19) log n (cid:27) ≤ P A ( n ) (cid:26) max ≤ i ≤ [ N ]+1 | L i | < ω (cid:27) + o (1) (2.76)as n → ∞ .Let x ∈ S be such that ψ ( x ) = min x ∈ S ψ ( x ). Then a vertex of type x has among alldifferent types x ∈ S the smallest probabilities of the incident edges, which are cψ ( x ) ψ ( y ) /n , y ∈ S . This implies that for any U ⊂ V the size of τ Un ( x ) is stochastically dominated by18 τ Un ( x ) | for any x ∈ S . Notice also that if U ⊂ U ′ then | τ U ′ n ( x ) | is stochastically dominatedby | τ Un ( x ) | for any x ∈ S . This allows us to derive the following bound P A ( n ) (cid:26) max ≤ i ≤ [ N ]+1 | L i | < ω (cid:27) ≤ (cid:18) max U ⊂ V : | U |≤ Nω P A ( n ) (cid:8) | τ Un ( x ) | < ω (cid:9)(cid:19) N . (2.77)To approximate the distribution of | τ Un ( x ) | we introduce another branching process whichwill be stochastically dominated by B κ . First define for any value D ∈ S another auxiliaryprobability measure ˆ µ D ˆ µ D ( y ) = (cid:26) M − D µ ( y ) , if y ≤ D, , otherwise , (2.78)where M D := P y ≤ D µ ( y ) is a normalizing constant. Then for any positive c and D let ˆ B c,D be a process defined similar to B κ , but with the distribution of the offspring P o ( cψ ( x ) ψ ( y )ˆ µ D ( y )) (2.79)instead of P o ( cψ ( x ) ψ ( y ) µ ( y )). Notice, that ˆ B c, ∞ is defined exactly as B κ . Let ˆ X c,D ( x )denote the total number of the particles (including the initial one) produced by the branchingprocess ˆ B c,D starting with a single particle of type x . Lemma 2.3.
For any c ′ < c there exists finite D such that P A ( n ) (cid:8) | τ Un ( x ) | < ω (cid:9) ≤ (cid:18) b log nn (cid:19) nω P n ˆ X c ′ ,D ( x ) < ω o (2.80) for all large n uniformly in U ⊂ V with | U | ≤ N ω = (cid:16) r ( c ) + ε (cid:17) − n/ log n , where b = b ( c ) is some positive constant independent of c ′ and D . Proof.
At each step of the exploration algorithm which defines τ Un ( x ), the number of thetype y offspring of a particle of type x has a binomial distribution Bin ( N ′ y , p xy ( n )) where N ′ y is the number of remaining vertices of type y . We shall first find a lower bound for N ′ y .According to assumption (1.1) for any D ∈ S and ε > n ( D, ε ) such that N y := { x i ∈ V : x i = y } ≥ (1 − ε ) µ ( y ) n (2.81)for all y < D and n ≥ n ( D, ε ). Note that conditionally on A ( n ) the number of vertices in τ Un ( x ) is at most ω , and by deleting an arbitrary set U with | U | ≤ N ω from V , we maydelete at most N ω vertices of type y . Hence, at any step of the exploration algorithm which19efines τ Un ( x ), the number N ′ y of the remaining vertices of type y , is bounded from below asfollows N ′ y ≥ N y − ω − N ω , and thus according to (2.81) N ′ y ≥ n (1 − ε ) µ ( y ) − ω − N ω for all y < D and n ≥ n ( D, ε ). Taking into account definitions (2.74) and (2.73) we derivefrom here that for any ε > D > n ( D, ε ) such that N ′ y ≥ (1 − ε ) µ ( y ) n for all y ≤ D and n ≥ n ( D, ε ). This implies that conditionally on A ( n ) at any step of theexploration algorithm we have N ′ y p xy ( n )1 − p xy ( n ) ≥ µ ( y )(1 − ε ) cψ ( x ) ψ ( y ) (2.82)for any fixed ε , D , and all n ≥ n ( D, ε ) and y ≤ D . Now with a help of (2.78) we rewrite(2.82) as follows: N ′ y p xy ( n )1 − p xy ( n ) ≥ ˆ µ D ( y ) M D (1 − ε ) cψ ( x ) ψ ( y ) =: ˆ µ D ( y ) c ′ ψ ( x ) ψ ( y ) (2.83)for any fixed ε , D , and all n ≥ n ( D, ε ) and x, y ∈ S , where c ′ = M D (1 − ε ) c. Recall that lim D →∞ M D ↑
1. Therefore choosing appropriately constants D and ε we canmake c ′ arbitrarily close to c . Using again relation (2.51) between the Poisson and thebinomial distributions, and taking into account (2.83), we derive for all k P { Y N ′ y ,p xy ( n ) ≤ k } ≤ (1 + γp xy ( n )) N ′ y P { Z N ′ y pxy ( n )1 − pxy ( n ) ≤ k } (2.84) ≤ (1 + γp xy ( n )) n P { Z ˆ µ D ( y ) c ′ ψ ( x ) ψ ( y ) ≤ k } . This implies that if conditionally on A n , at each of at most ω steps of the explorationalgorithm which reveals τ Un ( x ), we replace the Bin ( N ′ y , p xy ( n )) variable with the P o (ˆ µ D ( y ) c ′ ψ ( x ) ψ ( y ))20ne, we arrive at the following bound using the branching process ˆ B c,D and bound (2.54)): P A ( n ) (cid:8) | τ Un ( x ) | < ω (cid:9) ≤ (cid:18) γc A log nn (cid:19) nω P n ˆ X c ′ ,D ( x ) < ω o (2.85)which holds for all large n. This implies the statement of Lemma 2.3. ✷ Lemma 2.3 together with (2.77) implies that for all large n P A ( n ) (cid:26) max ≤ i ≤ [ N ]+1 | L i | < ω (cid:27) ≤ (cid:18)(cid:18) b log nn (cid:19) nω P n ˆ X c ′ ,D ( x ) < ω o(cid:19) N . Substituting this into (2.76) we derive P n C (cid:16) G V ( n, κ ) (cid:17) < ω o ≤ (cid:18)(cid:18) b log nn (cid:19) nω P n ˆ X c ′ ,D ( x ) < ω o(cid:19) N + o (1)= (cid:18) b log nn (cid:19) n /ω P n ˆ X c ′ ,D ( x ) < ω o n/ω + o (1) ≤ e b log n (cid:16) − P n ˆ X c ′ ,D ( x ) ≥ ω o (cid:17) n/ω + o (1) (2.86)as n → ∞ , where b = b ( c ) is a positive constant which depends only on c .Since ˆ µ D and ψ obviously satisfy the conditions of Lemma 2.1 we get by (2.34) and (2.32)that if c ′ < ˆ c cr ( D ) := X S ψ ( x )ˆ µ D ( x ) ! − , then ˆ r ( c ′ , D ) := sup { z ≥ E z ˆ X c ′ ,D ( x ) < ∞} > , and ˆ r ( c ′ , D ) can be derived from (1.21) where X is replaced by a random variable on S witha distribution ˆ µ D . It is clear that lim D →∞ ˆ c cr ( D ) = c cr , ˆ r ( c, D ) is continuous in c , and lim D →∞ ˆ r ( c, D ) = r ( c ) . Hence we can find for any given δ > D and c ′ < c sufficiently close to c ,such that ˆ r ( c ′ , D ) < r ( c ) + δ / . r ( c ′ , D ) that for some positive constant A = A ( δ ) < ∞ and any positive ω P n ˆ X c ′ ,D ( x ) > ω o ≥ A (ˆ r ( c ′ , D ) + δ / − ω ≥ A ( r ( c ) + δ ) − ω . (2.87)This allows us to derive from (2.86) that for any δ > δ > A P n C (cid:16) G V ( n, κ ) (cid:17) < ω o (2.88) ≤ e b log n (cid:16) − A ( r ( c ) + δ ) − ( r ( c ) − δ ) log n (cid:17) α n/ log n + o (1)where α = (cid:16) r ( c ) + ε (cid:17) − . Now for any δ > δ so that γ := (cid:18) r ( c ) − δ (cid:19) log ( r ( c ) + δ ) < . Then (2.88) becomes P (cid:26) C (cid:16) G V ( n, κ ) (cid:17) < (cid:18) r ( c ) − δ (cid:19) log n (cid:27) ≤ e b log n (cid:18) − An γ (cid:19) α n/ log n + o (1) , (2.89)where the right-hand side goes to zero when n → ∞ . This completes the proof of (2.71),which together with (2.44) yields the assertion of Theorem 1.1. ✷ The proof of Theorem 1.3 almost exactly repeats the proof of Theorem 1.1. The onlydifference is that a random variable | τ n ( x ) | used in the proof of Theorem 1.1 should bereplaced by Ψ n ( x ) := X v ∈ τ n ( x ) ψ ( v ) , while X c,q and ˆ X c,D should be replaced by Φ c,q and ˆΦ c,D which denote the activity of thetotal progeny of the branching processes B c,q and ˆ B c,D , correspondingly (see definition of Φand (1.22)). Then due to the results (2.35) and (2.33) on α from Lemma 2.1 the proof ofTheorem 1.3 follows exactly the same lines as the proof of Theorem 1.1. ✷ Acknowledgment
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