The configuration model

MixPo

αW_i+αX

j6=i

W_j n−1

,MixPo(α(W_i+E(W_i))

≤E

α n−1

X

j6=i

W_j−αE(W_i)

.

ApplyingE|Z−E(Z)| ≤p

Var(Z) toZ = _n−1^α P

j6=i

W_j reveals that the right-hand side is smaller than or equal to

v u utVar

α n−1

X

j6=i

W_j

= α

n−1

p(n−1)Var(W_i) = α

√n−1

pVar(W_i)

and this expression converges to zero asn→ ∞. Since convergence in total variation distance implies

weak convergence, the desired result follows.

The following corollary is an immediate consequence of Theorem 5.3.13 5.3.14 Corollary

Let (W_i)i∈N be a sequence of positive independent and identically distributed random variables with E(W₁²)<∞. Then in the modified generalized random graph G⁽ⁿ⁾ with independent and identically distributed weightsW1, . . . , Wn andnnodes, the distribution of the degree of a node picked uniformly at random among allnnodes converges weakly to the MixPo(α(W₁+E(W₁))) distribution asn→ ∞ for alli∈ {1, . . . , n}, and we have the same convergence rate as in Theorem 5.3.13.

5.3.15 Remark

Note that we obtain an Erd˝os-R´enyi graph if we set Wi = 1 for all i and α = 1/2 in the modified generalized random graph with independent and identically distributed weights.

5.3.16 Remark

The weights in the modified generalized random graph model correspond to the social indices (Si)i∈N

in the Britton-Lindholm model. Recall that in the latter each node creates edges at rateαS_i and the second node is picked uniformly among all living nodes. Thus we would expect that the probability that two nodesiandjwith social indicesS_iandS_j are connected is in average approximately proportional to Si +Sj, which corresponds to the probability pij(n) in the modified generalized random graph model.

We obtain a in some sense similar asymptotic degree distribution in both models. The reason for the additional factor (1−e^−(β+µ)A)/(β+µ) in the random variable determining the mixing distribution of the Britton-Lindholm model is that this model is a (time-continuous) dynamic model in contrast to the modified generalized random graph model.

5.3.2 The configuration model

In this Section, which is based on Chapter 7 of [vdH16], we consider the case where an arbitrary distribution on N0 is given and we introduce a model that produces random graphs with node set

{1, . . . , n}, possibly containing multiple edges and loops, whose asymptotic degree distribution, i.e.

the distribution of the degree of a node picked uniformly at random, is this given distribution. If we condition on the absence of multiple edges and loops, we obtain a uniform distribution on all graphs with node set{1, . . . , n} that do not have any multiple edges and loops. Thus this model can be used in order to find out if the network is rather “purely” random or if it contains additional structure.

The configuration model for a given deterministic degree sequence 5.3.17 Definition and Remark (cf. Section 7.2 of [vdH16])

Let n∈ N0 and d = (di)i∈{1,...,n} be a deterministic sequence of positive integers such that the sum Pn

i=1d_i is even. We would like to produce a graph with node set {1, . . . , n} such that node i has degree d_i for all i ∈ {1, . . . , n}. If we only consider graphs without multiple edges and loops, this is not always possible (see Section 7.2 of [vdH16]). Therefore, we allow our graphs to have such edges.

Imagine that nodeiis equipped with d_i half-edges for any i∈ {1, . . . , n}such that we havePn i=1d_i in total. Now we pick two of those half-edges uniformly at random among all half-edges and combine them, i.e. if one of the randomly picked half-edge belongs to nodeiand the other one to nodej, we add the edge (i, j). Then we pick two half-edges uniformly at random among the Pn

i=1d_i−2 remaining half-edges, add the corresponding edge and continue this procedure until no half-edge is left. We call the resulting graphconfiguration model with degree sequence d.

Note that the first half-edges are not directly picked uniformly at random but in an arbitrary order in the definition of the configuration model in [vdH16]. However, as pointed out in [vdH16], this leads to the same random graph.

Obviously, the configuration model has the desired degree sequence. Moreover, we obtain from the following example that we can obtain any feasible distribution as asymptotic degree distribution.

5.3.18 Example (see Section 7.2 of [vdH16])

Let F be the cumulative distribution function of an arbitrary distribution on N0, and let n∈N. Note that a degree sequence d is, apart from the node labels, determined by the sequence (n_k)k∈N0

with n_k=Pn

i=11{di=k}, i.e. n_k denotes the number of nodes with degree k, for all k ∈ N0. Thus we can always find a corresponding configuration model if the sequence (n_k)k∈N0 is given. Let this sequence now be given bynk=dnF(k)e − dnF(k−1)e for all k∈N0, where d

·

e denotes the ceiling function, and let D_Jn denote the degree of a node picked uniformly at random among {1, . . . , n} in the configuration model with degree sequence d, where d satisfiesn_k=Pn

i=11{d_i=k} for all k∈ N0. Then

P(D_Jn ≤x) = 1 n

n

X

j=1

1{d_j≤x}

converges toF(x) for allx ≥ 0 as n→ ∞, i.e. the degree D_Jn converges weakly to the distribution with cumulative distribution functionF.

The configuration model with independent and identically distributed degrees

Let (Di)i∈{1,...,n} be a sequence ofN-valued independent and identically distributed random variables.

Since the sum of the degreesPn

i=1d_ihas to be even in Definition and Remark 5.3.17, the configuration

model with random degrees (Di)i∈{1,...,n} cannot be defined in the most natural way such that, given Di = di for alli∈ {1, . . . , n}, the random graph is just given by the configuration model with degrees d = (d_i)i∈{1,...,n} as defined above. Therefore, we add an additional half-edge to node nif the sum of the degrees would not be even otherwise. Formally, we let

D˜_n=











Dn if

n

P

i=1

Di is even, Dn+ 1 if

n

P

i=1

Di is odd

and define the model as described above but substitute (Di)i∈{1,...,n}by the sequenceD1, . . . , Dn−1,D˜n. For convenience, we denote the latter sequence by ( ˜Di)i∈{1,...,n}. By this procedure, we obtain a random graph where the degree ˜D_Jn of a node picked uniformly at random among{1, . . . , n}converges weakly to the distributionL(D₁) as n→ ∞(cf. Section 7.6 of [vdH16]).

The erased configuration model

As mentioned above, multiple edges and loops are not desired in many applications. Therefore, we erase all such edges in the configuration model now if they exist in the sense that we merge all multiple edges into single edges and ignore loops. This amounts in ignoring the additional edges of a multiple edge, which is the same we did when we considered the Britton-Lindholm model without loops in the introduction. We call the resulting model theerased configuration model. The following theorem states that the distribution of the degree of a node picked uniformly at random still converges weakly toL(D₁) asn→ ∞in this model if the degrees are deterministic and a couple of weak conditions are fulfilled. See Theorem 7.10 in [vdH16] for a similar result. The analogous theorem for independent and identically distributed degrees can be found in [BDML06] and is stated in Theorem 5.3.20 below.

5.3.19 Theorem LetD^(er)_J

n denote the degree of a node picked uniformly at random in the erased configuration model with deterministic degrees. Assume that the degree sequence d = (di)i∈{1,...,n} satisfies the following regularity conditions (cf. Condition 7.7 in [vdH16]).

(i) The distribution of the degree d_Jn of a node picked uniformly at random among {1, . . . , n}

converges weakly to some asymptotic distribution with cumulative distribution function F. (ii) LetDhave the cumulative distribution functionF from (i). Then we haveE(dJn)→E(D)<∞

asn→ ∞.

Then we obtain

L(D_J^(er)

n )→ L(D)^w asn→ ∞.

P roof: By Remark 3.1.10, it is sufficient to show that the total variation distancedT V(L(D_J^(er)

n ),L(D)) converges to zero asn→ ∞, where Dis defined as in (ii).

By the triangle inequality, we have d_{T V}(L(D_J^(er)

n ),L(D))≤d_{T V}(L(D_J^(er)

n ),L(d_Jn)) +d_{T V}(L(d_Jn),L(D)).

Since we know from above that L(d_Jn) converges weakly to L(D) as n → ∞, we obtain that d_{T V}(L(d_Jn),L(D)) converges to zero as n → ∞ by Remark 3.1.10. It remains to show that the total variation distanced_{T V}(L(D^(er)_J

n ),L(d_Jn)) converges to zero asn→ ∞as well. In order to do so, we note thatdT V(L(D_J^(er)

n ),L(d_Jn)) is smaller than or equal toP(D^(er)_Jn 6= d_Jn) by Theorem 3.1.8. For any nodei∈ {1, . . . , n}, letM_i⁽ⁿ⁾ denote the number of edges that are multiple edges or loops and are incident toi. Then we obtain by conditioning on J_n

P(D^(er)_Jn 6= d_Jn) = 1 In order to show that the right-hand side converges to zero, we use the idea of the last part of the proof of Proposition 7.11 in [vdH16]. Note that we know from [vdH16] that maxi∈{1,...,n}di = o(n) asn→ ∞ under the conditions (i) and (ii) of the theorem. Thus we can choose a sequence (an)n∈N

such that a_n → ∞ and a_nmaxi∈{1,...,n}d_i = o(n) as n→ ∞. For this sequence, we obtain that the right-hand side of (5.6) is smaller than or equal to

1

The first summand of the right-hand side converges to zero as n → ∞ by (7.3.30) in [vdH16]. The second summand is equal to the percentage of nodes with degree higher thanan. Since an → ∞ as n→ ∞ and (i) is equivalent to

the proof of Proposition 7.11 in [vdH16].

5.3.20 Theorem (cf. Theorem 2.1 in [BDML06]) LetD^(er)_J

n denote the degree of a node picked uniformly at random in the erased configuration model with independent and identically distributed degrees with finite expectation. Then we have

L(D^(er)_J

n )→ L(D^w ₁) asn→ ∞.

Further properties of the configuration model

A lot of further results for the configuration model and related models are given in [vdH16]. We only state the ones most interesting to us here.

The asymptotic probability that the random graph from the ordinary configuration model does not have any multiple edges or loops is considered in Theorem 7.11 and Theorem 7.19 in [vdH16].

This asymptotic probability is positive under some weak conditions. Thus instead of applying the erased configuration model, a random graph without multiple edges or loops can also be obtained by repeating the algorithm described in the definition of the configuration model until it produces such

graph. Formally, we condition on the graph having no multiple edges or loops. The corresponding model is called repeated configuration model (see also [BDML06]). For any deterministic degree se-quence d, the repeated configuration model with degree sese-quence d produces a random graph that is uniformly distributed on the set of all graphs without multiple edges or loops and degree sequence d (see Proposition 7.13 in [vdH16]).

Im Dokument Convergence Rates in Dynamic Network Models (Seite 71-75)