Large deviation principle for the empirical degree measure of preferential attachment random graphs

We consider preferential attachment random graphs which may be obtained as follows: It starts with a single node. If a new node appears, it is linked by an edge to one or more existing node(s) with a probability proportional to function of their degree. For a class of linear preferential attachment random graphs we find a large deviation principle (LDP) for the empirical degree measure. In the course of the prove this LDP we establish an LDP for the empirical degree and pair distribution see Theorem 2.3, of the fitness preferential attachment model of random graphs.


Introduction
Preferential attachment (P.A) random graph models have become extremely popular in the last two decades since they were first studied by (Barabasi and Albert ,1999). Example (van der Hofstad ,2013), (Newman, 2003) and (Newman et. al, 2006) provide good overviews.
The P.A model of random graphs are graphs in which nodes are added sequentially and attach to exactly one randomly chosen existing node and the chance a new node connects to an existing node is proportional to its degree.
The model is typically generalized to allow for vertices to have m > 1 initial edges by collapsing m vertices in the one initial edge case into a single vertex (possibly causing loops). The most studied feature of these objects is the distribution of the degrees of the nodes; that is, the proportion of nodes that have degree k as the graph grows large. See, example (Collevecchio et. al, 2013), (Krapivsky et. al, 2000), (Rudas et. al, 2007) for results on more general attachment rules.
Few large deviation results for P.A model have so far been found. In paper ( Choi et. al, 2011), P.A schemes where the selection mechanism is possibly time-dependent are considered, and an in infinite dimensional large deviation principle for the sample path evolution of the empirical degree distribution is found by Dupuis-Ellis type methods. (Dereich and Moerters, 2009) studied a dynamic model of random networks, where new vertices are connected to old ones with a probability proportional to a sub-linear function of their degree. For this model of random networks, they obtained a strong limit law for the empirical degree distribution. Results on the temporal evolution of the degrees of individual vertices via large and moderate deviation principles were also found. (Bryc et. al, 2009) found the large deviation principle and related results for a class of Markov chains associated to the 'leaves'in P.A model of random graphs using both analytic and Dupuis-Ellis-type path arguments. Recently, (Doku-Amponsah et. al, 2014) proved a large deviation upper bound for fitness preferential attachment random network.
In this paper, we find a large deviation principle for the empirical degree distribution of preferential attachment random network in the linear regime. See, Theorem 2.1. In the course of the proof of Theorem 2.1, we find a large deviation principle for the empirical degree and pair measure of the fitness preferential attachment random networks, see Theorem 2.2 and a joint LDP for the empirical degree and pair measure, and the sample path empirical degree distribution of the fitness preferential attachment random networks, see Theorem 2.3. The main technique in our proof is exponential change of measure, see example (Doku-Amponsah et. al, 2014) and the method of mixtures, see (Biggins, 2004).

Main Results
2.1 LDP for the preferential attachment model of random graphs Let f : 0, 1, 2, ... → [0, ∞] be a weight function. We define a preferential attachment random graph as follows: It starts with single vertex serving as root. If a new vertex n is introduced, it connects to vertices v n ∈ { 1, . . . , n − 1 } independently with probability proportional to f (N (v n )), where N (m) is the in-degree of vertex m.
We write N = 0, 1, 2, ... . In this paper, we shall restrict ourself to functions of the form We define empirical degree measure measure L on N by We denote by M(N ) the space of probability measures on N , equipped with the topology generated by total variation metric π −π := 1 2 ∞ k=0 π(k) −π(k) . Theorem 2.1. Suppose X is P.A random graph with linear weight function f : Then, as n → ∞, the empirical degree measure L, satisfies a large deviation principle in M(N ) with good rate function 2.2 Large-deviations for fitness P.A random network . To establish Theorem 2.1 we pass to a more general random preferential random graph, the fitness or coloured preferential random graph. We write N = N ∪ {0}. Given a weight function f m/n : N × X → [0, ∞], m = 1, 2, 3, ...n and a probability law µ on finite alphabet X , we define coloured (fitness) P.A random network with n vertices as follows: • Assign vertex m = 1 (the root of the network) colour X(m) according to µ : X → [0, 1].
• If a new vertex m is introduced, it gets colour X(m) independently according µ, • it connects to vertices v m ∈ { 1, . . . , m − 1 } independently with probability proportional to where A(m) = X(v m ), X(m) and N (m) is the in-degree of vertex m. • Repeat the previous three steps until we have n vertices.
We consider (N (v m ), A(m)) : m = 1, 2, 3, ..., n . . . under the joint law of colour and tree. Denote by X a typed tree and by X(i) colour of vertex i. We write X * = X × X . In this paper, we shall restrict ourself to functions of the form where γ : (0, 1] × X * → (0, ∞], β : (0, 1] × X * → [0, ∞]. We assume γ(t, a) + β(t, a) := c t , for all (t, a) ∈ (0, 1] × X . (2.1) Let N (m) (i) be the degree of vertex i at time m and observe that at time n, the law of the fitness P.A graph is given by For every X, we define empirical degree and pair measure measure M X on N × X * by We write ℓ m (a) = j m ∈ {1, 2, 3, ..., m−1 : x(j m ) = a 1 , x(m) = a 2 and for every m = 2, 3, 4, ..., n−1 we define a probability measure on N × X * by and notice, . We denote by M(X ) the space of probability measures on X equipped with the weak topology and M(N × X * ) the space of probability measures on N × X * , equipped with the topology generated by total variation metric. π −π := 1 2 (k,a)∈N ×X * π(k, a) −π(k, a) .

4LARGE DEVIATION PRINCIPLE FOR THE EMPIRICAL DEGREE MEASURE OF FITNESS PREFERENTIAL ATTACHMENT RANDOM
Theorem 2.2. Suppose X is coloured P.A random graph with colour law µ : X → (0, 1] and linear weight functions Then, as n → ∞, the pair of empirical measures M X , where ω 2,1 is the X − marginal of the probability measure ω 2 and Our next theorem which generalizes Theorem 2.1 is a special case of Theorem 2.2 above.
Theorem 2.3. Suppose X is coloured P.A random graph with colour law µ : X → (0, 1] and linear weight function f : Then, as n → ∞, M X satisfies a large deviation principle in M(N × X * ) with good rate function Observe that J(ω) = 0 if and only if ω(k, a) = cω 2 (a) f (k, a) 1l − k j=0 ω(k | a) , and hence solving recursively for ω(· | a) we get . (2.4) Here we remark that conditions (2.1) and (2.3) are necessary for π f (· |a) to be a probability measure on N . See (Dereich and Morters, 2009, p. 13). Note, if f (k, a) = w(k) then (2.4) concise with the asymptotic degree distribution of random trees and general branching processes found in (Rudas et. al, 2008).

Proof of Results
3.1 Dynamics of the path empirical degree distribution. Denote by D([0, 1], R) the space of right continuous left limited(cadlag) paths from [0, 1] to R. We define the sample path space and endow it with the topology of uniform convergence associated with the norm for the time derivative of the measure ν t and we associate with each path ν ∈ D M the relaxed measure We call ν ∈ D M absolutely continuous if for each k ∈ N, there existsν(k|a) such that For each absolutely continuous path ν , we define ν ν (·|a),ν(·, ·|a)almost everywhere by By ν ν ≪ ν we mean ν is absolutely continuous. We write Note that the measure L X [nt]/n , for t ∈ [0, 1) is deterministic and its distribution is degenerate at some ν [nt]/n , for t ∈ [0, 1) converging to ν t , t ∈ [0, 1). gt ⊗ ν(a) = log eg t (·, a) ft(·, a) , ν t (·|a) =: Ug t ⊗ ν(a, t). We useg to define a new fitness P.A random graph with n vertices as follows:

Exponential
• At time m = assign the root m of the network fit X(m) according to the lawμ given bỹ µ(a 1 ) = eh (a 1 )−U (h) µ(a 1 ).
• For any other time m new node m which appear gets fit X(m) according to the fit lawμ. It connects to node v m , independently with probability proportional tõ • Repeat the previous three steps until we have n vertices.
We denote by Pf ,n the law of the new fitness P.A graph and observe that it is absolute continuous with respect to P f,n , as for fitness graph X we have that where id is the identity function from [0, 1] to [0, 1]. The following Lemma will be used to establish the upper bound in a variational formulation.

LARGE DEVIATION PRINCIPLE FOR THE EMPIRICAL DEGREE MEASURE OF FITNESS PREFERENTIAL ATTACHMENT RANDOM
As N ≤ k(l, δ) is pre-compact, Γ δ is compact in the weak topology by prokohov criterion. Moreover Now letting K θ be the closure of ∩ 1≥δ>0 Γ δ,θ and taking limit as n approaches ∞ we have (3.3) which ends the proof the Lemma.
3.3 Proof of Theorem 2.2. We derive the upper bound in a variational formulation. To do this, we denote by C 1 the space of all functions on X and by C 2 the space of all bounded continuous functions on N × X * . We define on the space of probability measures M(N × X ) the functionK given bŷ
We show that the functionK ν (ω) in Lemma 3.2 may be replaced by the good rate function Lemma 3.3. For every ν ∈ D M we have thatK ν (ω) ≥ K ν (ω). Moveover, the function K ν is good rate function and lower semi-continuous on M(N × X ).

Lower bound
B ω be open neighbourhood of ω such that for allω, ∈ B ω we have that We useP f,n the law of the coloured preferential attachment graph obtained by transforming P f,n using g t,ω . We observe that colour law in the transformed measure is ω 2,1 and the linear weight function is k=0 k 2 ν t (k|a) − 1 and that thereforeγ(t, a) +β(t, a) = 1. We use (3.2) to obtain (3.10) where we have used c t > 1 in the last inequality.
Therefore we have that where we have used (2.2) in the last inequality.
3.5 Proof of Theorem 2.1 By Mixing To use the technique of mixing LDP results developed in (Biggins, 2004), we check the main criteria needed for the validity of (See, Biggins, 2004, Theorem 5(a)) in the following Lemma. We write Θ n := D Mn(N ×X ) , Θ := D M(N ×X ) , and define P f,n (ν 1 ) := P M X = ν 1 L X [nt] n (·, a) = ν [nt] n (·, a), t ∈ [0, 1) and a ∈ X P n ν [nt] n , t ∈ [0, 1) := P L X n Then, the joint distribution of M X and L X is obtained by the mixture of P f,n and P n as follows: dP f,n (ν, ν 1 ) := dP n (ν)dP f,n (ν 1 ).
Lemma 3.5. The family of distributions (i) (P f,n , n ∈ N) (ii) (P f,n , n ∈ N) are exponentially tight.
Proof. (i) As this family distributions obey a large deviation upper bound with a good rate function K ν (ω), the family (P f,n , n ∈ N) is exponentially tight. See, e.g. (Dembo and Zeitouni, 1998, Exercise 4.1.10(c)).
(ii) By (i) for every θ 2 we can find K θ 2 , compact subset of D M(N ×X ) such that, we have lim sup Also by Lemma 3.1, for every θ 1 we can find K θ 1 , compact subset of M(N × X ) such that, we have lim sup n→∞ 1 n log P f,n (K c θ 1 ) ≤ −θ 1 .
Taking limit n → ∞ followed by δ ↓ 0 of above inequality, yields lim sup n→∞ 1 n logP f,n (Γ c θ ) ≤ −θ which proves the second part of the Lemma. Now, as J(ν 1 , ν) is lower semi-continuous by the continuity of the relative entropies, and by Lemma 3.5 the families of distributions (i) (P f,n , n ∈ N) (ii) (P f,n , n ∈ N) are exponentially tight, we have that the latter obeys a large deviation principle with good rate function give by J(ν 1 ). (See, Biggins, 2004, Theorem 5(a)).
3.6 Proof of Theorem 2.3 We note that in case of this theorem γ t = γ, β t = β, and hence c t = c for all t ∈ (0, 1]. Therefore, Theorem 2.2 and the contraction principle, (see Dembo and Zeitouni, 1998, Theorem 4.2.1) imply the large deviation principle for M X in the space M(N × X ) with good rate function for all (k, a) ∈ N × X and for all t ∈ [0, 1]. This ends the proof this Theorem.

Proof of Theorem 2.1
In the case of an preferential attachment graph, the function c = γ(a) + β(a) degenerates to a constant c = γ + β and M X = L ∈ M(N ). Theorem 2.3 and the contraction principle imply a large deviation principle for L with good rate function where (γ+β) f ⊗l(k) = (γ+β) f (k)l (k) andl(k) = 1l − k j=0 ℓ(k).