Geometric preferential attachment in non-uniform metric spaces

We investigate the degree sequences of geometric preferential attachment graphs in general compact metric spaces. We show that, under certain conditions on the attractiveness function, the behaviour of the degree sequence is similar to that of the preferential attachment with multiplicative fitness models investigated by Borgs et al. When the metric space is finite, the degree distribution at each point of the space converges to a degree distribution which is an asymptotic power law whose index depends on the chosen point. For infinite metric spaces, we can show that for vertices in a Borel subset of $S$ of positive measure the degree distribution converges to a distribution whose tail is close to that of a power law whose index again depends on the set.


Introduction
In [6] and [7], Flaxman, Frieze and Vera introduced a model for a growing graph driven by geometric preferential attachment. In this model, which is a variant of the Barabási-Albert preferential attachment model introduced in [1] and analysed in [3,4], vertices are given a random location in a metric space S and the probability that a new vertex is connected to an already existing vertex u depends on the distance between them in space as well as on the degree of u. The aim is to combine the ideas of the Barabási-Albert model with some of those found in spatial graph models, for example random geometric graphs, [10], online models such as the FKP model, [2], and its special case the online nearest neighbour graph, [11], which is also a degenerate case of geometric preferential attachment.
In [8], under a rather strong condition on the probability measure determining the locations of the vertices and the strength of the effect of distance on the connection probabilities, it was shown that the limiting proportion of vertices with degree d was the same as that found for the Barabási-Albert model in [4].
In this paper we consider one of the questions which was left open in [8], namely what it is possible to say if we weaken the assumptions on the probability measure µ determining the locations of the vertices. In [8] it was required that for any fixed r, µ(B r (x)) is constant as a function of x, where B r (x) is the open ball (in the underlying metric on S) of radius r centred on x, for example the case of Haar measure on a compact group with an invariant metric. We will show that if this assumption is weakened then (still assuming certain conditions on the the strength of the effect of distance on the connection probabilities and still assuming that the space is compact) the behaviour of the degree distribution of the model is similar to that found for preferential attachment with multiplicative fitness, as investigated by Borgs et al in [5]. In that paper each vertex has a random fitness, and the probability of a new vertex connecting to an existing vertex is proportional to the product of its degree and its fitness.
In addition we generalise the model of [8] so that the attractiveness of a vertex at location x to one at location y is not necessarily the same as the attractiveness of a vertex at location y to one at location x. We do this by replacing the attractiveness F (ρ(x, y)) (where ρ is the metric on S) by a function of two variables α(x, y). This also allows the preferential attachment with fitness model of [5] to be seen as a special case of our model, by allowing α(x, y) to only depend on x.

Our model and results
We assume S is a compact metric space with metric ρ and probability measure µ; the locations of the added vertices will be assumed to be independent random variables (X n ) n∈N with law µ. We denote the Borel σ-algebra of S by B(S).
Let α : S × S → R + be an attractiveness function; we will usually assume that α is continuous. The interpretation of α(x, y) here is that it is the attractiveness of a vertex at x to a new vertex at y. Note that the situation in [8], where the attractiveness was defined as F (ρ(x, y)) with F being a function from R + → R + , can be treated using the formulation in this paper by letting α(x, y) = F (ρ(x, y)). Also note that the preferential attachment with fitness model of [5] can be seen as a special case of this model with S being the set of possible fitnesses, the "location" of a vertex being its fitness, and α(x, y) = x for all x, y ∈ S.
Let m ∈ N be the number of vertices that each new vertex will be connected to when it is added to the graph, which as in other papers, such as [1,3,8], will be a parameter of the model.
In [8] it was assumed that the metric space S and measure µ satisfied the condition that, for any fixed r, µ(B r (x)) is constant as a function of x, where B r (x) is the open ρ-ball of radius r centred on x, a rather strict "uniformity" condition. The aim here is to discuss what happens when this assumption does not hold.
To start the process, we let G 0 be a connected graph with n 0 vertices and e 0 edges, and we give each vertex v ∈ V (G 0 ) a location X v ∈ S. Then, to form G n+1 from G n , we add a new vertex v n+1 to the graph whose location X n+1 = X vn+1 is a random variable on S with law µ independent of X 1 , X 2 , . . . , X n and the structure of G n . Let V (n+1) ι , 1 ≤ ι ≤ m be the random variables representing the m vertices chosen to be neighbours of the new vertex v n+1 at time n + 1. Conditional on X n+1 and F n , where F n is the σ-algebra generated by the graphs G 0 , G 1 , . . . , G n and the location in space of their vertices, we let the V (n+1) ι be chosen independently such that for v ∈ V (G n ) the probability that V where deg G (v) is the degree of the vertex v in the graph G and Note that we allow that V for some ι 1 = ι 2 , in which case multiple edges will form, but we do not allow loops.
Following [8], we define a sequence of (random) measures δ n on S by, for A ∈ B(S), so that δ n (A) is the total degree in A, normalised to give a probability measure on S. Then Thus the probability that V (n+1) i = v given G n and X n+1 = x can be rewritten as , and so the probability that V Given the uniformity and symmetry assumptions in [8] δ n converges weakly to µ, and furthermore S α(x, y) dµ(x) does not depend on x. This can be used to show that the integral in (1) tends to 1, so for large n the probability of choosing v k is close to what it would be in standard preferential attachment.
Our aim is to show that if the uniformity assumption does not hold, then under certain conditions δ n converges weakly to some measure ν on S. If this holds then for large n then P(V We have the following theorem. Theorem 1. Assume that α is a continuous function from S × S to R + , that log α is Lipschitz in both components, and that there exists α 0 > 0 such that α(x, y) ≥ α 0 for all x, y ∈ S. Then we have the following.
1. There exists a probability measure ν on S such that δ n converges weakly to ν.
2. Define, for u ∈ S, Note that, unlike in [8], we do not allow α(x, y) → ∞ as ρ(x, y) → 0 here. In the case where S is a finite metric space, we do not require the conditions on α; in that case we can simply assume that α is a function from R + ∪ {0} to R + ∪ {0}.
is approximately proportional to a constant φ(X v ) times the degree of v is reminiscent of the preferential attachment with fitness model of Borgs et al in [5]. This suggests that in our model the quantity φ(X v ), which depends on the location of the vertex v, might play a similar role to the fitness in the model of [5]. Indeed, in [5] it is shown that vertices of different fitnesses have degree distributions following approximately power law distributions where the index of the power law depends on the fitness, and we will show similar results here, starting with the case where S is a finite metric space.
Theorem 2. Assume that the metric space S consists of a finite number of points z 1 , z 2 , . . . , z N , and that α(x, y) is a function from S × S to R + ∪ {0}. Let µ i = µ({z i }), and let φ i = φ(z i ) where φ is as defined in Theorem 1. Then if we let p (n) d,i be the proportion of vertices of G n which are located at z i and have degree d, we have , almost surely, as n → ∞.
In a similar way to in [5] this gives an asymptotic (in d) power law for vertices at the same location. As in [5] the tail index of the power law depends on φ i , and here is 2φ −1 i . Indeed we can think of φ i as giving a "geometrical fitness" for a point at z i , thought of as a multiplicative fitness in the sense of [5].
We can also obtain a result on the degree sequence in the case where S is infinite.
Then, almost surely, for any φ > φ A , and lim sup , This shows that for vertices in A the limiting tail behaviour of the degree sequence is between those of power laws with tail indices 2ψ −1 A and 2φ −1 A .
To prove the above theorems, we will start off by considering, in section 3, the case where S is a finite metric space, where stochastic approximation techniques can be used to show the convergence of the measures (which in the finite case are points in a simplex). In section 4, we will then use a coupling between geometric preferential attachment on S and a process closely related to the finite space case to show that convergence of measures also applies in the infinite case, and deduce Theorem 3. The arguments are to some extent based on those in [5] but use more general stochastic approximation techniques. Before that, we will give some examples where the measure ν can be found explicitly.

Uniform measure
In [8], it was assumed that for any fixed r, µ(B r (x)) is constant as a function of x, where B r (x) is the open ball (in the underlying metric on S) of radius r centred on x. This includes for example the case of Haar measure on a compact group with an invariant metric. It was also assumed that α(x, y) = F (ρ(x, y)) for some function F . Under these assumptions, the results in [8] imply that the measure ν in Theorem 1 is equal to µ. As a result the function φ defined in the statement of Theorem 1 is 1 everywhere on S, and all subsets of S with positive measure under µ have the same limiting degree distribution for their vertices.

Preferential attachment with fitness
As mentioned above, the preferential attachment with fitness model of [5] can be considered as a special case of our model by letting the set S be a subset of R + , the location of a vertex being equal to its fitness, and taking α(x, y) = x, so that the attractiveness of a vertex does not depend on the location of the new vertex but simply on its own location, that is its fitness. (Note that for Theorem 1 to apply to this model as stated, we require that the set of fitnesses be bounded away from zero and to be contained within a compact subset of R + .) This model is analysed in detail in [5], considering finite, discrete countable and continuous fitness distributions separately. For example in Theorems 6 and 7 of [5] it is assumed that the fitness distribution is defined by a probability density function g(x) on an interval [0, h], and that g(x) is non-zero on (0, h).
It is also assumed that m = 1. Under these assumptions they show that, if M n,[a,b] the number of edge (this is described in [5] as the "fit get richer" phase) then λ0−x . In our notation, this shows that the measure ν satisfies ν( In what is described in [5] as the "innovation pays off" phase, where there is no solution to (2), the results given in Theorem 7 of [5] show that the measure ν has an atom at h. More generally, it is possible that in our setting there may be subsets of S for which the measure ν is positive but µ is not, indicating that a proportion of vertices tending to zero have a positive proportion of the edge ends, the innovation pays off phase of [5] being the simplest example where this happens.

A two-point metric space
We let S = {0, 1}, α(0, 0) = α(1, 1) = 1 and α(0, 1) = α(1, 0) = a > 0. We define the measure µ by µ({0}) = p and µ({1}) = 1 − p, with 0 < p < 1. This is then a special case of the framework in section 3, with N = 2. In this setting the simplex can be represented as [0, 1], and the Lyapunov function which we use in section 3 can be written as Let y 0 be the unique root in (0, 1) of which gives the location of the minimum of V (y) in [0, 1]; then Theorem 1 applies with the measure ν being defined by ν({0}) = y 0 and ν({1}) = 1 − y 0 . We can also calculate We note that if y = p then (3) only holds if either a = 1 (which would be equivalent to standard preferential attachment) or p = 1/2 (n which case the uniformity assumption of [8] would hold) so if neither of these two conditions hold the measures ν and µ are different.

The finite case
This section will prove Theorem 1 in the finite case and Theorem 2. We assume the metric space S consists of a finite number of points z 1 , z 2 , . . . , z N . Let µ i = µ({z i }), and let a i,j = α(z i , z j ).
Let Y Then we can write and so the probability that V (n+1) ι is at z i , conditional on the new vertex being at z j , is and so where g is a map from the N -simplex to itself given by the i co-ordinate being where G(y) = g(y) − y and so its components are given by

Proposition 4.
There exists ν in the N -simplex such that as n → ∞ we have y (n) → ν, almost surely.
Proof. For y in the interior of the N -simplex, let y k a k,j .
Then (using 1 = N j=1 y j ), G i (y) = −y i ∂ ∂y i V (y), and as y i > 0 this means that V is a Lyapunov function for G. By concavity of the logarithm V is a convex function and it tends to infinity near the boundary of the N -simplex, so it has a unique minimum, at a point which we will call ν, in the interior of the N -simplex.

Proposition 4 corresponds to Proposition 2 in Section 3 of [5].
Proposition 5. If vertex v is at location z i then Proof. By Proposition 4, To show that φ i = 2 − µi νi , we note that the probability that V (n+1) ι is a vertex at location z i is then and, because each edge has one end at a new vertex (location chosen with law µ) and one end at a vertex V (n) ι for some n and ι, also implies and from (5) we get φ i = 2 − µi νi .

Propositions 3 and 4 prove Theorem 1 in the finite case.
Let p (n) d,i be the proportion of the vertices of G n which are located at z i and have degree d. We will use the above to show an asymptotic power law for p (n) d,i . We will need to use the following lemma based on Lemma 2.6 and Corollary 2.7 of [9]. Lemma 6. For n ∈ N, let A n and B n be random variables taking non-negative values, ξ n , R n random variables taking real values, and k a positive constant, such that B n+1 − B n = 1 n (A n − kB n + ξ n ) + R n+1 and 1. A n → a as n → ∞, almost surely;
Then B n → a k as n → ∞, almost surely.
Proof. Fix δ > 0 and ǫ > 0. We note that if |A n − a| < ǫ and B n < a k − δ+ǫ k then A n − kB n > δ. The proof of Lemma 2.6 of [9] and the fact that |A n − a| < ǫ if n is large enough now shows that (B n ) n∈N cannot visit [0, a k − 2 δ+ǫ k ] infinitely often. Similarly if |A n − a| < ǫ and B n > a k + δ+ǫ k then A n − kB n < −δ and so (B n ) n∈N cannot visit [ a k + 2 δ+ǫ k , ∞) infinitely often. Hence, B n → a k , almost surely.
We can now prove Theorem 2.
Proof of Theorem 2. We start by showing that, when m > 1, the probability of a multiple edge being formed at time n from a vertex of degree d is O(n −1 ). Conditional on F n and X n+1 = j, the probability that a vertex u is connected to the new vertex at least twice is bounded above by and proposition 4 ensures that the expectation here is bounded.
Using Proposition 5, for each d, (where δ m,d = 1 if d = m and is zero otherwise) and so almost surely, as n → ∞. When d > m (7) becomes and repeatedly using Lemma 6, with B n = p almost surely, as n → ∞.

The infinite case 4.1 Coupling
We partition S into a finite set S = {S i , i = 1, 2, . . . , N S } with each S i ∈ B(S). Later on we will have each S i having diameter at most some small value.
We now construct a modified graph process, which will be similar to a finite space geometric preferential attachment, as in section 3, on {0, 1, 2, . . . , N S }. The extra point 0 will be used to allow the construction of a coupling, similar to that in Appendix C of [5], with the geometric preferential attachment on S.
We will construct a sequence of graphs (G (S) n ) n∈N whose vertices have locations in {0, 1, 2, . . . , N S } as follows. We start with a graph G We now show that the geometric preferential attachment process (G n ) n∈N on S can be coupled to the above process. In the geometric preferential attachment process, let Y and G 0 be the same graph, with the location X Then, we claim that the coupling can be done so that Y Given that X n+1 ∈ S j , that is that the new vertex in G n+1 is in S j , which occurs with probability µ j , let X For the denominator, define, for , which is non-negative by our assumption. Then the total degree of G n and G which is the probability that each new edge in G n+1 connects to a vertex in S i . Hence, for 1 ≤ i ≤ N S , the increase in the total degree at i from G Hence the coupling ensures that Y  where g (S) is a map from Y S := {y : y ∈ R NS , y i ≥ 0∀i, NS k=1 y k ≤ 1} to itself given by the i co-ordinate being

Analysis of the coupled process
where G (S) (y) = g (S) (y) − y and so its components are given by (Note that y ∈ R + such that as n → ∞ we have almost surely.
Proof. For y ∈ Y S , let y k a k,j .
and as y i > 0 this means that V is a Lyapunov function for G. Again V is a convex function and it tends to infinity as y i → 0, so it has a unique minimum, at a point which we will call ν (S) ∈ Y S . Proposition 2.18 of [9] now gives y (n,S) → ν (S) a.s. as n → ∞.
This shows that for vertex v at location i, (1), Let t = inf 1≤i,j≤NS b i,j h = inf x,y∈S α(x, y) sup x,y∈S α(x, y) .
The conditions of Theorem 1 ensure that t > 0.

Approximating S
We use the coupling in the previous section to complete the proof of Theorem 1.
Proposition 9. Let A ⊆ S be a Borel set. Then there existsν(A) such that as n → ∞ δ n (A) →ν(A), almost surely.
Proof. Given ǫ > 0 we can construct a partition S = {S 1 , S 2 , . . . , S NS } of S where each set S i , 1 ≤ i ≤ N S has diameter at most ǫ and such that A is the union of sets in S. Then assuming log α is a Lipschitz function (in both components) with Lipschitz constant K, we have for 1 ≤ i, j ≤ N S that 0 ≤ log a i,j − log b i,j ≤ 2Kǫ and so γ S ≥ e −2Kǫ . The analysis in section 4.2 shows that δ n (A) = 1 2(mn + e 0 )

Proof of Theorem 3
Let A ∈ B(S) be a Borel set with µ(A) > 0. Let p