With an appropriate Wigner-noise, the noisy graph sequence (GAn) of Proposition 17 be-comes a generalized graph sequence on the model graph H, defined in Definition 21 and characterized from the point of view of discrepancy and spectra. Then we will establish so-called generalized quasirandom properties of expanding graph sequences, which are closely related to the properties of the generalized random graphs, and state implications between them, irrespective of the stochastic model.
3.2.1 Generalized random and quasirandom graphs
For the generalied random graphs, defined in Definition 21, the following can be proved (with subspace perturbation theorems and large deviations), see [Bol13], Chapter 2, and [Bol-El16].
Proposition 18 Let Gn(P,Pk) be a generalized random graph on n vertices with vertex-classes Pk= (C1, . . . , Ck)of sizes n1, . . . nk andk×k symmetric probability matrixP. Let k be a fixed positive integer andn→ ∞ in such a way that nnu ≥c (u= 1, . . . , k)with some constant0< c≤k1 (called balancing condition). Then the following hold almost surely for the adjacency matrix An= (a(n)ij )and the normalized modularity matrixMD,n of Gn(P,Pk).
1. An has k structural eigenvalues that are Θ(n)in absolute value, while the remaining eigenvalues are O(√
n). Further, the k-varianceSk,n2 of thek-dimensional vertex rep-resentatives, based on the eigenvectors corresponding to the structural eigenvalues of An (see (1.10)), is O(1n).
2. There exists a positive constant 0 < δ < 1 independent of n (it only depends on k) such that MD,n has exactly k−1 structural eigenvalues of absolute value greater than δ, while all the other eigenvalues are less than n−τ in absolute value, for every 0 < τ < 12. Further, the weighted k-variance S˜k,n2 of the (k−1)-dimensional vertex representatives, based on the transformed eigenvectors corresponding to the structural eigenvalues of MD,n (see (1.14)), isO(n−2τ), for every0< τ <12.
3. There exists a constant 0 < θ < 1 independent of n (it only depends on k) such that disc1(Gn(P,Pk))> θ, . . . , disck−1(Gn(P,Pk))> θ, and the k-way discrepancy disck(Gn(P,Pk);C1, . . . , Ck)is O(n−τ), for every0< τ < 12.
4. For every1≤u≤v≤kandi∈Cu: X
j∈Cv
a(n)ij =puvnv+o(n).
For every1≤u≤v≤kandi, j∈Cu: X
t∈Cv
a(n)it a(n)jt =p2uvnv+o(n).
Proof. Property 1 follows from Theorems 13 and 14, while Property 2 from Theorems 15 and 16. Property 3 is the consequence of Theorems 26 and 27.
The proof of Property 4 is as follows. Consider the generalized random graph sequence Gn(P,Pk), the subgraphs and the bipartite subgraphs of which have the following expected degrees. We will drop the index n, and use the notation A = (aij) for the entries of its adjacency matrix. As for the Cu, Cv pair (1 ≤ u ≤ v ≤ k), for any i ∈ Cu, the average degree ofiwith regard toCv is
E(X
j∈Cv
aij) =nvpuv,
each vertex in Cu has the same expected number of neighbors inCv. Observe that for i ∈ Cu, the sum P
j∈Cvaij has binomial distribution with the above expectation and variancenvpuv(1−puv). Therefore, by Lemma 2, the within- and between-cluster average degrees are highly concentrated on their expectations as n→ ∞under the balancing conditions nnu ≥c(u= 1, . . . , k)for the cluster sizes. Indeed, for any0< ε <1:
P(| 1 nv
X
j∈Cv
aij−puv|> ε) =P(| X
j∈Cv
aij−nvpuv|> nvε)≤e−
n2 v ε2 2(nv puv(1−puv)+ε/3)
that tends to 0 even with the choiceε=n−τ,0< τ < 12. Therefore, it holds almost surely that
| X
j∈Cv
aij−nvpuv| ≤nvn−τ =nv
nn1−τ=o(n).
This finishes the proof of the first part of 4.
As for every1 ≤u ≤v ≤k, the number of common neighbors in Cv of any i, j ∈ Cu
(i6=j)pair has binomial distribution with expectationnvp2uv and variancenvp2uv(1−p2uv), with the same calculations as above, we obtain that
| X
t∈Cv
aitajt−p2uvnv|=o(n)
holds almost surely. This finishes the proof of the second part of 4.
Recall that Proposition 17 implies thatGn(P,Pk)→WH almost surely when n→ ∞, under the strict balancing condition nni → ri (i = 1, . . . , k). In Lovász–Sós [Lov-Sos] the following definition of a generalized quasirandom graph sequence was given.
Definition 31 Given a model graphH onkvertices with vertex-weightsr1, . . . , rk and edge-weights puv = pvu, 1 ≤u≤v ≤k (entries ofP), the sequence (Gn) is H-quasirandom if Gn→WH asn→ ∞in term of the homomorphism densities.
The authors of [Lov-Sos] also proved that the vertex setV of a generalized quasirandom graph Gn can be partitioned into classesC1, . . . , Ckin such a way that |Cnu|→ru(u= 1, . . . , k)as n→ ∞, and the subgraph ofGn induced byCu is the general term of a quasirandom graph sequence with edge-density tending to puu (u= 1, . . . , k), whereas the bipartite subgraph between Cu and Cv is the general term of a quasirandom bipartite graph sequence with edge-density tending topuv (u6=v)asn→ ∞.
Because of the limit relation in the definition of the generalized quasirandom graphs, the properties, discussed in Proposition 18, are – with some modification – valid for them.
Actually, the authors in [Borgsetal1] proved that for any k, the k largest absolute value normalized adjacency eigenvalues of a convergent graph sequence converge (to the corre-sponding eigenvalues of the limiting graphon). In Section 3.1.1, we proved the same for the normalized modularity spectra of convergent graph sequences, see Theorem 29. As for the multiway discrepancies and spectra, we can use Theorems 26 and 27.
However, the order√
nand n−τ of the non-structural eigenvalues in the adjacency and normalized modularity spectrum, respectively, is not necessarily valid for the generalized quasirandom graphs; instead,o(n)ando(1)can be stated for their order. Indeed, in the case of a generalized random graph we can separate a Wigner-noise, the corresponding graphon to which tends to zero very quickly. The slower separation in the spectrum is supported by simulations and the following construction.
Vera T. Sós suggested the following construction of a generalized quasirandom graph with givenk,P, and vertex-weightsr1, . . . , rk of the model graphH. Consider the instance when there are k clusters C1, . . . , Ck of the vertices of sizes n1, . . . , nk such that nnu = ru
(u= 1, . . . , k). Let us choose the independent irrational numbersαuv(1≤u≤v≤k). Then the subgraph on the vertex-setCu is constructed as follows: i, j∈Cu,i < j are connected if and only if
{(i−j)2αuu}< puu (u= 1, . . . , k),
where {.} denotes the fractional part of a real number. The bipartite subgraph betweenCu
andCv is constructed as follows: i∈Cu andj∈Cv are connected if and only if {(i−j)2αuv}< puv (1≤u < v≤k).
In the k = 1 case, Bollobás and Erdős [Bo-Erd] recommended this construction, and Pinch [Pinch] in terms of the codegrees proved that it indeed produces a quasirandom graph. The analytical number theoretical considerations of [Bo, Kup-Nied] and particularly of [Pinch] imply that, for any1≤u≤v≤k, the sequence
yt:= ({(t−i)2αuv},{(t−j)2αuv})
iswell-distributed symmetrically in[0,1]2, uniformly ini−j. This means that the sequences (yt+h)are uniformly distributed symmetrically in[0,1]2 forh∈Z. With h=Pv−1
ℓ=1nℓ and the considerations of [Pinch], we get that
{t∈Cv : {(t−i)2αuv}< puv and {(t−j)2αuv}< puv}
=p2uvnv+o(nv) =p2uvnv+o(n)
for any i, j ∈ Cu (i 6=j)pair, when n→ ∞ and nnu →ru (u= 1, . . . , k). It is important that the role ofi, jis symmetric here: both are inCu and connected to anyt∈Cv with the
same rule. Possibly, it suffices to assume the weaker balancing condition to guarantee that n1, . . . , nktend to infinity at the same rate: n→ ∞in such a way thatnnu ≥c(u= 1, . . . , k), with some constant0< c≤ 1k. This ensures thato(n1) =· · ·=o(nk) =o(n).
For more examples of quasirandom graphs in the k= 1 case see [Bo, Thom87]. For an illustration of generalized random and quasirandom graphs see Figures 3.1, 3.2, 3.3, where we used the probability matrix
P =
0.7 0.1 0.15 0.2 0.25 0.1 0.75 0.3 0.35 0.4 0.15 0.3 0.8 0.45 0.5 0.2 0.35 0.45 0.85 0.55 0.25 0.4 0.5 0.55 0.9
.
Figure 3.1: Generalized random graph generated with k = 5, cluster sizes 60,80,100,120,140and prob-ability matrix P. The first non-trivial eigenvalues ofMD
are0.304,0.214,0.17,0.153,
−0.097,−0.094,−0.093,
−0.092,−0.091, . . .,
with a gap after the 4th one.
Figure 3.2: Generalized quasirandom graph con-structed with k = 5, cluster sizes 60,80,100,120,140 and probability matrix P. The first non-trivial eigenvalues of MD are 0.318,0.207,0.154,0.115,
−0.100,−0.099,−0.091,
−0.090,0.084, . . .,
exhibiting decreasing eigen-values up to the 4th one.
Figure 3.3: The former gener-alized quasirandom graph af-ter appropriately permuting the vertices within the blocks (made by Ahmed Elbanna).
3.2.2 Generalized quasirandom properties
Some properties similar to those of Proposition 18 are now formulated for expanding deter-ministic graph sequences.
Conjecture 1 Consider the sequence of graphs Gn with vertex-set Vn, adjacency matrix An = (a(n)ij ), and normalized modularity matrix MD,n. Let k be a fixed positive integer and |Vn|=n→ ∞ in such a way that there are no dominant vertices. Then the following properties are equivalent:
P0. There exists a vertex- and edge-weighted graph H on k vertices such that Gn → WH
asn→ ∞in terms of the homomorphism densities.
PI. An has k structural eigenvalues λ1,n, . . . , λk,n such that the normalized eigenvalues converge: n1|λi,n| → qi as n → ∞ (i = 1, . . . , k) with some positive reals q1, . . . , qk, and the remaining eigenvalues are o(n) in absolute value. The k-variance Sk,n2 of the k-dimensional vertex representatives, based on the eigenvectors corresponding to the structural eigenvalues ofAn, iso(1).
PII. There exists a constant 0< δ <1 (independent ofn) such thatMD,nhas k−1 struc-tural eigenvalues that are greater than δin absolute value, while the remaining eigen-values areo(1). Further, the weightedk-varianceS˜k,n2 of the(k−1)-dimensional vertex representatives, based on the transformed eigenvectors corresponding to the structural eigenvalues of MD,n, iso(1).
PIII. There are vertex-classes Pk = (C1, . . . , Ck)and a constant 0< θ <1 (independent of n) such that md1(Gn)> θ, . . . ,mdk−1(Gn)> θ, andmdk(Gn;C1, . . . , Ck) =o(1).
PIV. There are vertex-classesPk = (C1, . . . , Ck) and a k×k symmetric probability matrix P = (puv)(independent of n), such that every vertex of Cu has asymptotically nvpuv
neighbors in Cv for any 1 ≤ u ≤ v ≤ k pair. Further, for the codegrees (number of common neighbors) the following holds: every two different vertices i, j ∈Cu have asymptotically p2uvnv common neighbors in Cv for any 1 ≤ u ≤ v ≤ k pair. More exactly, for every1≤u≤v≤k andi, j∈Cu
X
t∈Cv
a(n)it =puvnv+o(n)
and X
t∈Cv
a(n)it a(n)jt =p2uvnv+o(n) hold, where nv=|Cv|,v= 1, . . . , k.
We will not prove the implications here, but have some remarks about the way how some implications follow from former results of others and theorems proved in the present dis-sertation; in particular, from Theorems 13, 14, 15, 16, 18, 26, and 27 of Chapter 2, and Theorems 29 and 30 of Chapter 3.
P0 is equivalent to PIV, due to the results of [Chu-G-W, Lov-Sos, Sim-Sos, Thom87, Thom89]. We can use that by [Lov-Sos], the vertex set of the generalized quasirandom graph Gn can be partitioned into classes C1, . . . , Ck in such a way that |C|Vu|| → ru (u = 1, . . . , k)and the subgraph ofGn induced byCu is the general term of a quasirandom graph sequence with edge-density tending to puu (u= 1, . . . , k), whereas the bipartite subgraph between Cu and Cv is the general term of a quasirandom bipartite graph sequence with edge-density tending to puv (u6=v)as n→ ∞. The converse is trivial. Then, with some modification, theorems of Thomason [Thom87, Thom89] about (p, α)-jumbled graphs are applicable for the subgraphs and bipartite subgraphs, wherepis somepuv andαis related to the k-way discrepancy. Lovász [Lov08] also discusses that quasirandom graphs (as our subgraphs) are asymptotically regular, while, in view of Thomason [Thom89], the bipartite quasirandom graphs (as our bipartite subgraphs) are asymptotically biregular. According to Chung–Graham–Wilson [Chu-G-W], these properties are weaker than the other properties of quasirandomness, therefore asymptotic behavior of the codegrees should be characterized too, see [Bo-Erd, Lov-Sos, Thom87, Thom89].
As for the equivalence between P0 and PI, we can use that the convergence of a graph sequence implies the convergence of its normalized spectrum, see [Borgsetal1]. By the con-siderations of the proof of Proposition 5, we can relate the eigenvalues of the limiting graphon
WH to the eigenvalues of ak×k matrix. Since the other eigenvalues of WH are 0, it fol-lows that |λi,n| = o(n) (i > k). As the spectral subspace corresponding to λ1,n, . . . , λk,n
also converges to that of the step-vectors, Sk,n2 = o(n) follows, see Theorem 14. In the backward direction, the convergence of the spectra usually does not imply the convergence of the graph or graphon sequence, but in the case of the quasirandom graphs it does as noted in [Borgsetal1]. Here we use both the separation in the spectrum and the convergence of the spectral subspaces, i.e., of the k-variances. By Theorem 18 and 23, we are able to find a blown-up matrix Bn of rankk and an error-matrixEn with kEnk=o(n) such that An =Bn+En. It is important that, provided Sk,n2 is ‘small’ enough, the so constructed Bn can have positive entries (see the considerations after the proof of Theorem 25), so that it can be the blown-up matrix of ak×k probability matrix.
In the equivalence between PI and PII we use that there are no dominant vertices, and the ideas of the proof of Theorem 15 and 16 extend to this case.
For the PII–PIII equivalence we plan to use the back and forth statements of Theorems 26 and 27. This equivalence suggest that low discrepancy clusters and cluster pairs can be obtained by spectral clustering tools, and so, justify the discrepancy minimizing spectral clustering.
Note that in thek= 1 case, the P0, PI, PIV properties are in accord with some of the properties of Chung–Graham–Wilson [Chu-G-W] and Simonovts–Sós [Sim-Sos], whereas PII, PIII rather harmonize with the properties of Chung–Graham [Chu-G] that do not contain a statement about the leading eigenvalue. As in the case of k= 1there is only one leading eigenvalue, this does not make too much difference, but in the k > 1 case the statements should deal with their asymptotic behavior too, see [Bol15] for some details.
Summarizing, we believe that the P0–>PIV–>PIII–>PII–>PI–>P0 implications can be proved and so, they close the circle. In view of the above, PIV–>PIII is the only missing chain. However, we hope that the results of [Thom87, Thom89] can be adopted for the subgraphs and bipartite subgraphs to estimate the number of common neighbors by means of discrepancy. In fact, our discrepancy is a bit different of α and our k×k probability matrixP is a bit different ofp(k= 1 case) of the notion of(p, α)-jumbledness.