Lower threshold ground state energy and testability of minimal balanced cut density
aa r X i v : . [ m a t h . P R ] J u l Lower threshold ground state energy and testability ofminimal balanced cut density
Andr´as Kr´amli ∗ Roland Mark´o † Abstract
Lov´asz and his coauthors in [4] defined the notion of microcanonical ground stateenergy ˆ E a ( G, J ) – borrowed from the statistical physics – for weighted graphs G , where a ∈ Pd q is a probability distribution on { , ..., q } and J is a symmetric q × q matrixwith real entries. We define a new version of the ground state energy ˆ E c ( G, J ) =inf a ∈ A c ˆ E a ( G, J ), called lower threshold ground state energy, where A c = { a ∈ Pd q : a i ≥ c, i = 1 , . . . , q } . Both types of energies can be extended for graphons W , thelimit objects of convergent sequences of simple graphs. The main result of the paperis Theorem 3.2 stating that if 0 ≤ c < c ≤
1, then the convergence of the sequences( ˆ E c /q ( G n , J )) for each J ∈ Sym q implies convergence of the sequences ( ˆ E c /q ( G n , J ))for each J ∈ Sym q . As a byproduct one can derive in a natural way the testability ofminimum balanced multiway cut densities – one of the fundamental problems of clusteranalysis – proved in [2]. The main goal of this paper is to introduce and to reveal the properties of an intermediateobject between the microcanonical ground state energy (MGSE) and ground state energy(GSE) of weighted graphs defined in [4]. For this purpose we need to define the fundamentalnotions used in [3],[4], and [5] and to cite the main results therein necessary for us, these willbe given below. The main contribution of the paper is that we give a convergence hierarchywith respect to the aforementioned intermediate objects that are Hamiltonians subject tocertain conditions. This can be regarded as a refined version of Theorem 2.9. (ii) from [4]combined with the equivalence assertion of Theorem 2.8. (v) from the same paper. In short,these state that MGSE convergence implies GSE convergence. Counterexamples are providedindicating that the implication is strict. We also reprove with the aid of the establishedhierarchy one of the main results of [2]. Our motivation comes from cluster analysis, where ∗ Bolyai Institute, University of Szeged. The Project is supported by the Hungarian Scientific ResearchFund OTKA 105645. E-mail: [email protected] † Hausdorff Center for Mathematics, University of Bonn. Supported in part by a Hausdorff scholarship.E-mail: [email protected] G (graphs without loops and multiple edges)and weighted graphs. We denote the node and edge sets of G by V ( G ) and E ( G ), respectively.Usually we denote by α i = α i ( G ) > i and β ij = β ij ( G ) ∈ R the weight associated with the edge ij . We set α G = P i α i ( G ).Definitions 1.1. – 1.9. (except 1.7.) are borrowed from [3] and [4]. In order to facilitatethe reading, at every definition we indicate its exact place of occurrence. Definition 1.1. ([3] Definition 3.1.) Let W denote the space of bounded symmetric mea-surable functions W : [0 , → R , that is W ( x, y ) = W ( y, x ) . Assume that the functions W ∈ W take their values in an interval I , usually I = [0 , . We can think of the interval [0 , as the set of nodes of graph with a node set that has cardinality continuum, and of thevalues W ( x, y ) as the weight of the edge xy . We call the functions in W I graphons. Definition 1.2. ([4] Definition 2.4.) Let G and G ′ be two weighted graphs with node set V and V ′ , respectively. For i ∈ V and u ∈ V ′ set µ i = α i ( G ) /α G and µ ′ u = α u ( G ′ ) /α G ′ . Thenwe define the set of fractional overlays χ ( G, G ′ ) as the set of probability distributions X on V × V ′ (or couplings of µ and µ ′ ) such that X u ∈ V ′ X iu = µ i for all i ∈ V, and X i ∈ V X iu = µ ′ u for all u ∈ V ′ , and set (1) δ (cid:3) ( G, G ′ ) = min X ∈ χ ( G,G ′ ) max S,T ⊂ V × V ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( i,u ) ∈ S ( j,v ) ∈ T X iu X jv ( β ij ( G ) − β uv ( G ′ )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Definition 1.3. ([3] formula (3.3)) The distance defined in Definition 1.2 can be extendedto graphons in terms of the cut norm k W k (cid:3) = sup S,T ⊂ [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z S × T W ( x, y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2) = sup f,g : [0 , → [0 , (cid:12)(cid:12)(cid:12)(cid:12)Z W ( x, y ) f ( x ) g ( y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12) , where the suprema go over measurable subsets and functions, respectively. Definition 1.4. ([4] formula (3.3)) The cut distance of two graphons U and W is definedas (3) δ (cid:3) ( U, W ) = inf φ k U − W φ k (cid:3) , here the infimum goes over all measure preserving permutations of [0 , , W φ ( x, y ) = W ( φ ( x ) , φ ( y )) . Now we define three versions of the ground state energies (GSE) borrowed from thestatistical physics that are the objects investigated in this paper (for the mathematicaltreatment of statistical physics, see, e.g. Sinai’s book [6]). They are defined in terms of afinite set of states [ q ] = { , . . . , q } , and a symmetric q × q matrix J with entries in R , the setof these matrices is denoted by Sym q . A spin configuration on a simple or weighted graph G is given by a map φ : V ( G ) → [ q ]. Definition 1.5. ([4] formulae (2.8) and (2.11))The energy density of a spin configuration of G with respect to J is given by (4) E φ ( G, J ) = − | V ( G ) | X uv ∈ E ( G ) J φ ( u ) φ ( v ) . The ground state energy (GSE) of G with respect to J is ˆ E ( G, J ) = min φ : V ( G ) → [ q ] E φ ( G, J ) . For a graphon W and a ρ = ( ρ , . . . , ρ q ) with ρ i : [0 , → [0 , being measurable and satisfying P i ρ ( x ) = 1 for each x ∈ [0 , called q -fractional partition the energy is defined as E ρ ( W, J ) = − q X i,j =1 J ij Z [0 , ρ i ( x ) ρ j ( y ) W ( x, y )d x d y, and the GSE is E ( W, J ) = min ρ E ρ ( W, J ) . Let Pd q be the set of all probability distributions on [ q ].Imposing some restrictions on the set where the minimum is taken in the above definitionwe can define another version of energies that are important in graph limit theory. Definition 1.6. ([4] formula (2.14)) Microcanonical ground state energy (MGSE) of G withrespect to J ∈ Sym q and a probability distribution a = ( a , . . . , a q ) ∈ Pd q is defined using theset (5) Ω a ( G ) = (cid:8) φ : V ( G ) → [ q ] : (cid:12)(cid:12) | φ − ( { i } ) | − a i | V ( G ) | (cid:12)(cid:12) ≤ for all i ∈ [ q ] (cid:9) , and is the quantity (6) ˆ E a ( G, J ) = min φ ∈ Ω a ( G ) E φ ( G, J ) . et ω a = n ρ : R ρ i ( x )d x = a i for all i ∈ [ q ] o be a subset of q -fractional partitions. Then theMGSE of a graphon W is defined as E a ( W, J ) = min ρ ∈ ω a E ρ ( W, J ) . Next we introduce the central object of our current investigation.
Definition 1.7.
Let G be a weighted graph, q ≥ , J ∈ Sym q , and ≤ c ≤ /q . We definethe set A c = { a ∈ Pd q : a i ≥ c, i = 1 , . . . , q } , and with its help the lower threshold groundstate energy (LTGSE): (7) ˆ E c ( G, J ) = inf a ∈ A c ˆ E a ( G, J ) . In a similar manner we introduce the LTGSEs for a graphon W for q ≥ , J ∈ Sym q , and ≤ c ≤ /q lower threshold: (8) E c ( W, J ) = inf a ∈ A c E a ( W, J ) . We remind the reader of the definition of testability of simple graph parameters. Beforedoing it, we should define the randomization procedure for graphs, used here.
Definition 1.8. ([3] Introduction of Section 2.5.3.) For a graph G and a positive integer k let G ( k, G ) denote the random induced subgraph G [ S ] where S is chosen uniformly from allsubsets of V ( G ) of cardinality k . Definition 1.9. ([3] Definition 2.11) A real function f defined on the set of simple graphsis a testable simple graph parameter, if for every ε > there exists a k = k ( ε ) ∈ N such thatfor every simple graph G on at least k vertices P ( | f ( G ) − f ( G ( k, G )) | > ε ) < ε. This paper is organized as follows. In the second section we prove yet another equiva-lent condition to left-convergence of a graph sequence relying on a subclass of MGSE, thereasoning will be instrumental for the proof of our main result in the subsequent section.In the third section we study the convergence of LTGSEs, see (8) for their definition. Wewill consider c : N → [0 ,
1] threshold functions with the property that c ( q ) q is constant as afunction of q . For this case we will prove that if 0 ≤ c ( q ) < c ( q ) ≤ /q (for all q ), thenthe convergence of ( E c ( q ) ( W n , J )) n ≥ for all q ≥ J ∈ Sym q implies the convergence of( E c ( q ) ( W n , J )) n ≥ for all q ≥ J ∈ Sym q .In the fourth section we provide some examples of graphs and graphons which supportthe fact, that the implication of convergence in the third section is strict in the sense thatconvergence of LTGSE sequences with smaller threshold do not imply convergence of LTGSEsequences with larger threshold in general. We also present a one-parameter family of block-diagonal graphons whose elements can be distinguished by LTGSEs for any threshold c > Microcanonical convergence
We start by showing that for each discrete probability distribution with rational probabilitiesthere exists a uniform probability distribution, such that the microcanonical ground stateenergies (MGSE) of it can be expressed as MGSEs corresponding to the uniform distribution.
Lemma 2.1.
Let q ≥ , and a ∈ Pd q be such that a = ( k q ′ , k − k q ′ , . . . , k q − k q − q ′ ) , where q ′ is apositive integer, k ≤ k ≤ · · · ≤ k q = q ′ are non-negative integers. Then for all J ∈ Sym q there exists a J ′ ∈ Sym q ′ , such that for b = (1 /q ′ , . . . , /q ′ ) ∈ Pd q ′ and every graphon W itholds that E a ( W, J ) = E b ( W, J ′ ) . Proof.
Set k = 0. If the i ’th component of a is 0, then erase this component from a , andalso erase the i ’th row and column of J . This transformation clearly will have no effect onthe value of the GSE. Let us define the q ′ × q ′ matrix J ′ by blowing up rows and columnsof J in the following way. For each u, v ∈ [ q ′ ] let J ′ uv = J ij , where k i − < u ≤ k i and k j − < v ≤ k j . The matrix J ′ defined this way is clearly symmetric.Now we will show that for every q -fractional partition with distribution a there exists a q ′ -fractional partition ρ ′ with distribution b , and vice versa, such that E ρ ( W, J ) = E ρ ′ ( W, J ′ ).On one hand, for 1 ≤ u ≤ q ′ let ρ ′ u = ρ i k i − k i − , where k i − < u ≤ k i . Then E ρ ′ ( W, J ′ ) = − q ′ X u,v =1 J ′ u,v Z [0 , ρ ′ u ( x ) ρ ′ v ( y ) W ( x, y )d x d y = − q X i,j =1 J i,j k i X l = k i − +1 k j X h = k j − +1 Z [0 , ρ ′ l ( x ) ρ ′ h ( y ) W ( x, y )d x d y = E ρ ( W, J ) . On the other hand, for 1 ≤ i ≤ q let ρ i := P kil = k i − +1 ρ ′ l . Then E ρ ( W, J ) = − q X i,j =1 J i,j Z [0 , ρ i ( x ) ρ j ( y ) W ( x, y )d x d y = − q X i,j =1 J i,j k i X l = k i − +1 k j X h = k j − +1 Z [0 , ρ ′ l ( x ) ρ ′ h ( y ) W ( x, y )d x d y = − q ′ X u,v =1 J ′ u,v Z [0 , ρ ′ u ( x ) ρ ′ v ( y ) W ( x, y )d x d y = E ρ ′ ( W, J ′ ) . So we conclude that E a ( W, J ) = inf ρ ∈ ω a E ρ ( W, J ) = inf ρ ′ ∈ ω b E ρ ′ ( W, J ′ ) = E b ( W, J ′ ) . W , J are close, whenever theircorresponding probability distribution parameters are close to each other. Lemma 2.2.
Let q ≥ , J ∈ Sym q , and W be an arbitrary graphon. Then for a , b ∈ Pd q we have that |E a ( W, J ) − E b ( W, J ) | < k a − b k k W k ∞ k J k ∞ . Proof.
Let ρ ∈ ω a , let us construct according to this a ρ ′ ∈ ω b the following way. First letus line up those i ’s, for which b i ≥ a i , for simplicity index them by integers from 1 to k . Let ρ ′ be such, that ρ ( x ) ≤ ρ ′ ( x ) ≤ x ∈ [0 ,
1] and R ρ ′ ( x )d x = b . It is clear thatsuch a function exists. We define ρ ′ in similar fashion: let ρ ( x ) ≤ ρ ′ ( x ) ≤ − ρ ′ ( x ) forall x ∈ [0 ,
1] and R ρ ′ ( x )d x = b , the existence is again clear. We define subsequently ρ ′ i for i ’s obeying b i ≥ a i by taking care that ρ i ( x ) ≤ ρ ′ i ( x ) ≤ − [ P i − j =1 ρ ′ j ( x )] holds at eachstep. In the other case, when b i < a i , we reverse the inequality we wish to be satisfied by thefunctions ρ i and ρ ′ i , and define ρ ′ i accordingly. For the constructed ρ ′ i either ρ ′ i ( x ) ≤ ρ i ( x ) forall x ∈ [0 , ρ ′ i ( x ) ≥ ρ i ( x ) for all x ∈ [0 , P i ρ i ( x ) = 1. Hence k ρ − ρ ′ k = q X i =1 1 Z | ρ i ( x ) − ρ ′ i ( x ) | d x = q X i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z ρ i ( x ) − ρ ′ i ( x )d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = q X i =1 | a i − b i | = k a − b k . Now we give an upper bound on the deviation of MGSEs. |E ρ ( W, J ) − E ρ ′ ( W, J ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q X i,j =1 J i,j Z [0 , ( ρ i ( x ) ρ j ( y ) − ρ ′ i ( x ) ρ ′ j ( y )) W ( x, y )d x d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k W k ∞ k J k ∞ q X i,j =1 Z [0 , (cid:12)(cid:12) ρ i ( x ) ρ j ( y ) − ρ i ( x ) ρ ′ j ( y ) (cid:12)(cid:12) + (cid:12)(cid:12) ρ i ( x ) ρ ′ j ( y ) − ρ ′ i ( x ) ρ ′ j ( y ) (cid:12)(cid:12) d x d y ≤ k W k ∞ k J k ∞ q X i,j =1 a i k ρ j − ρ ′ j k + b j k ρ i − ρ ′ i k = 2 k a − b k k W k ∞ k J k ∞ . The second inequality follows by Fubini’s theorem. From the definition of MGSE the state-ment of the lemma follows.With the aid of the two previous lemmas we are able to prove the main assertion of thesection. In the statement of the following theorem the LTGSE expression E /q ( W, J ) (which6s equal to E b ( W, J ), with b = (1 /q, . . . , /q )) appears, the notion will further be generalizedin what follows later on. Theorem 2.1.
Let I be a bounded interval, and ( W n ) n ≥ a sequence of graphons from W I .If for all q ≥ and J ∈ Sym q the sequences ( E /q ( W n , J )) n ≥ converge, then for all q ≥ , a ∈ Pd q and J ∈ Sym q the sequences ( E a ( W n , J )) n ≥ converge.Proof. Let q ≥ a ∈ Pd q and J ∈ Sym q be arbitrary and fixed. We will prove that wheneverthe conditions of the theorem are satisfied, then ( E a ( W n , J )) n ≥ is Cauchy convergent. Fixan arbitrary ε >
0. Let q ′ be such that 4 qq ′ k I k ∞ k J k ∞ < ε , and let b ∈ Pd q be such that b i = [ a i /q ′ ] ( i = 1 , . . . , q − b q = 1 − P q − i =1 b i (where [ x ] is the lower integer part x ). Then k a − b k = q X i =1 | a i − b i | ≤ q − q ′ < qq ′ . b is a q ′ -rational distribution, so by Lemma 2.1 there exists J ′ ∈ Sym q ′ , such that for all n ≥ E b ( W n , J ) = E /q ′ ( W n , J ′ ) . It follows from the conditions of the theorem that there exists n ∈ N such that for all m, n ≥ n it is true that (cid:12)(cid:12) E /q ′ ( W n , J ′ ) − E /q ′ ( W m , J ′ ) (cid:12)(cid:12) < ε . Applying Lemma 2.2 to all m, n ≥ n we get that |E a ( W n , J ) − E a ( W m , J ) | ≤ |E a ( W n , J ) − E b ( W n , J ) | + |E b ( W n , J ) − E b ( W m , J ) | + |E b ( W m , J ) − E a ( W m , J ) |≤ k a − b k k I k ∞ k J k ∞ + (cid:12)(cid:12)(cid:12) E /q ′ ( W n , J ′ ) − E /q ′ ( W m , J ′ ) (cid:12)(cid:12)(cid:12) + 2 k a − b k k I k ∞ k J k ∞ ≤ ε ε ε ε. We remark that Theorem 2.1 also appears in [4] as Corollary 7.4, but its proof follows adifferent line of thought in the present paper.
In various cases of testing, for certain cuts of graphs neither the notion of ground stateenergies, nor the notion of microcanonical ground state energies are satisfactory. For examplewhen investigating clusteredness of a graph in a certain sense these notions become useless,because the partition for which energies attain the minimal value are trivial partitions. Onthe other hand, in many applications one only asks for a lower bound on the size of these7lasses to keep a grade of freedom of the ground state case and at the same time achievea certain balance with respect to the sizes of classes. This setting can be regarded asan intermediate energy notion that manages to get rid of values corresponding to trivialpartitions. Recall Definition 1.7 of the lower threshold ground state energies.The next theorem will deliver an upper bound on the difference of the MGSEs of G and W G for fixed a an J , W G is the graphon constructed form the adjacency matrix of G in thenatural way. A straightforward consequence of this will be the analogous statement for theLTGSEs. Theorem 3.1. [4] Let G be a weighted graph, q ≥ , a ∈ Pd q and J ∈ Sym q . Then (cid:12)(cid:12)(cid:12) ˆ E a ( G, J ) − E a ( W G , J ) (cid:12)(cid:12)(cid:12) ≤ q α max ( G ) α G β max ( G ) k J k ∞ . Since the upper bound in the theorem for a given q is not dependent on a , it is easilypossible to apply it to the LTGSEs. Corollary 3.1.
Let G be a weighted graph, q ≥ , ≤ c ≤ /q and J ∈ Sym q . Then (cid:12)(cid:12)(cid:12) ˆ E c ( G, J ) − E c ( W G , J ) (cid:12)(cid:12)(cid:12) ≤ q α max ( G ) α G β max ( G ) k J k ∞ . Based on the preceding facts we are able to perform analysis on the LTGSEs the sameway as the authors of [4] did in the case of MGSE.
Corollary 3.2.
Let G n be a sequence of weighted graphs with uniformly bounded edge weights.Then if α max ( G n ) α Gn → ( n → ∞ ), then for all q ≥ , ≤ c ≤ /q and J ∈ Sym q the sequences ( ˆ E c ( G n , J )) n ≥ converge if, and only if ( E c ( W G n , J )) n ≥ converge, and then lim n →∞ ˆ E c ( G n , J ) = lim n →∞ E c ( W G n , J ) . Recall the definition of testability, Definition 1.9. It was shown in [3], among presentingother characterizations, that the testability of a graph parameter f is equivalent to theexistence of a δ (cid:3) -continuous extension ˆ f of f to the space W I , where extension here meansthat f ( G n ) − ˆ f ( W G n ) → | V ( G n ) | → ∞ (see [3], Theorem 6.1, the equivalenceof (a) and (d)). Using this we are able to present yet another consequence of Theorem 3.1,that was verified earlier using a different approach in [2] (see also [1], Chapter 4). Corollary 3.3.
For all q ≥ , ≤ c ≤ /q and J ∈ Sym q the simple graph parameter f ( G ) = ˆ E c ( G, J ) is testable. Choosing J appropriately, f ( G ) can be regarded as a type ofbalanced multiway minimal cut in [2].Proof. Let q ≥
1, 0 ≤ c ≤ /q and J ∈ Sym q be fixed, and we define ˆ f ( W ) = E c ( W, J ). Itfollows from Corollary 3.1 that f ( G n ) − ˆ f ( W G n ) → | V ( G n ) | → ∞ . It remains to8how that ˆ f is δ (cid:3) -continuous. To elaborate on this issue, let U, W ∈ W I and φ be a measure-preserving permutation of [0 ,
1] such that δ (cid:3) ( U, W ) = k U − W φ k (cid:3) , and let ρ = ( ρ , . . . , ρ q )be an arbitrary fractional partition. Then (cid:12)(cid:12) E ρ ( U, J ) − E ρ ( W φ , J ) (cid:12)(cid:12) ≤ q X i,j =1 | J ij | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z [0 , ( U − W φ )( x, y ) ρ i ( x ) ρ j ( y )d x d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ q k J k ∞ k U − W φ k (cid:3) = q k J k ∞ δ (cid:3) ( U, W ) . (9)This implies our claim, as E a ( W, J ) = E a ( W φ , J ) for any a ∈ Pd q and φ measure preservingpermutation, and the fact that the right-hand side of (9) does not depend on a , and that bydefinition E c ( W, J ) = inf a ∈ A c E a ( W, J ).In order to analyze the convergence relationship of LTGSEs with different thresholdsfor a given graph sequence it is sensible to consider c as a function of q . We restrict ourattention to lower threshold functions c with c ( q ) q being constant, which means that in thecase of graphons the total size of the thresholds stays the same relative to the size of theinterval [0 ,
1] (in the case of graphs relative to the cardinality of the vertex set). The mainstatement of the current section informally asserts that the convergence of LTGSEs withlarger lower threshold imply convergence of all LTGSEs with smaller ones. By the resultsof the previous section we know that in the case of c ( q ) = 1 /q the convergence of theseLTGSEs is equivalent convergence of the MGSEs for all probability distributions, and bythis, according to [4], to left convergence of graphs. Moreover, in the case of c ( q ) = 0 it isequivalent to the convergence of the unrestricted GSEs, that property is known to be strictlyweaker than left convergence. For technical purposes we introduce general LTGSEs and willrefer to the previously presented notion in all that follows as homogeneous
LTGSEs.
Definition 3.1.
Let q ≥ , x = ( x , . . . , x q ) , x , . . . x q ≥ and P qi =1 x i ≤ , and let A x = { a ∈ Pd q : a i ≥ x i , i = 1 , . . . , q } . For a graphon W and J ∈ Sym q we call thefollowing expression the lower threshold ground state energy corresponding to x : E x ( W, J ) = inf a ∈ A x E a ( W, J ) . The definition of ˆ E x ( G, J ) for graphs is analogous. Similarly to Lemma 2.1, the convergence of homogeneous LTGSEs is equivalent to theconvergence of certain general LTGSEs.
Lemma 3.1.
Let I be a bounded interval, ( W n ) n ≥ a sequence of graphons in W I . Let c bea lower threshold function, so that c ( q ) q = h for all q ≥ and for some ≤ h ≤ . If for all q ≥ and J ∈ Sym q the sequences ( E c ( q ) ( W n , J )) n ≥ converge, then for all q ≥ , all (10) x = ( x , . . . , x q ) x , . . . , x q ≥ q X i =1 x i = h, and J ∈ Sym q , the sequences ( E x ( W n , J )) n ≥ also converge. roof. Fix an arbitrary graphon W from W I , q ≥ J ∈ Sym q , and an arbitrary vector x that satisfies condition (10). Select for each of these vectors x a positive vector x ′ thatobeys the condition (10), and that has components which are integer multiples of c ( q ′ ) ( q ′ will be chosen later), so that k x − x ′ k ≤ qc ( q ′ ) = 2 h qq ′ . The sets A x and A x ′ have Hausdorff distance in the L -norm at most k x − x ′ k , in particularfor every a ∈ A x there exists a b ∈ A x ′ , such that k a − b k ≤ k x − x ′ k , and vice versa.Let ε > a ∈ A x be such that E x ( W, J ) + ε > E a ( W, J ) holds. Then byapplying Lemma 2.2 we have that E x ′ ( W, J ) − E x ( W, J ) < E x ′ ( W, J ) − E a ( W, J ) + ε ≤ E b ( W, J ) − E a ( W, J ) + ε ≤ k a − b k k W k ∞ k J k ∞ + ε ≤ k x − x ′ k k W k ∞ k J k ∞ + ε. The lower bound of the difference can be handled similarly, and therefore by the arbitrarychoice of ε it holds that (cid:12)(cid:12)(cid:12) E x ′ ( W, J ) − E x ( W, J ) (cid:12)(cid:12)(cid:12) ≤ k x − x ′ k k W k ∞ k J k ∞ ≤ h qq ′ k I k ∞ k J k ∞ . With completely analogous line of thought to the proof of Lemma 2.1, one can show thatthere exists a J ′ ∈ Sym q ′ such that E x ′ ( W, J ) = E c ( q ′ ) ( W, J ′ ). Finally, choose q ′ small enoughin order to satisfy 4 h qq ′ k I k ∞ k J k ∞ < ε , and n > m, n ≥ n the relation (cid:12)(cid:12)(cid:12) E c ( q ′ ) ( W n , J ′ ) − E c ( q ′ ) ( W m , J ′ ) (cid:12)(cid:12)(cid:12) < ε m, n ≥ n : |E x ( W n , J ) − E x ( W m , J ) | < (cid:12)(cid:12)(cid:12) E x ( W n , J ) − E x ′ ( W n , J ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) E c ( q ′ ) ( W n , J ′ ) − E c ( q ′ ) ( W m , J ′ ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) E x ′ ( W m , J ) − E x ( W m , J ) (cid:12)(cid:12)(cid:12) < ε ε ε ε. We did not only prove the statement of the lemma, but we also showed that the convergenceis uniform in the sense that n does not depend on x for fixed q and J .With the aid of the former lemma we can now prove that if all homogeneous LTGSEs withlarge thresholds converge, then all homogeneous LTGSEs with smaller ones also converge.10 heorem 3.2. Let I be a bounded interval, ( W n ) n ≥ a sequence of graphons in W I . Let c , c be two lower threshold functions, so that c ( q ) q = h < h = c ( q ) q for all q ≥ forsome ≤ h , h ≤ . If for every q ≥ and J ∈ Sym q the sequences ( E c ( q ) ( W n , J )) n ≥ converge, then for every q ≥ and J ∈ Sym q the sequences ( E c ( q ) ( W n , J )) n ≥ also converge.Proof. From Lemma 3.1 it follows that if the conditions of the theorem are satisfied then forevery q ≥
1, every(11) x = ( x , . . . , x q ) x , . . . x q ≥ q X i =1 x i = h , and J ∈ Sym q the sequences ( E x ( W n , J )) converge, for fixed q and J uniformly in x .Fix q . Our aim is to find for all a ∈ A c ( q ) an x , so that the condition (11) is satisfied, a ∈ A x and A x ⊆ A c ( q ) , where c ( q ) ≤ x i ≤ a i for i = 1 , . . . , q . As h < h ≤
1, there exists suchan x for all a ∈ A c ( q ) , let us denote it by x a , for convenience set ( x a ) i = h q + a i − h − h ( h − h ).According to this correspondence we have A c ( q ) = S a ∈ A c q ) A x a . So for an arbitrary graphon W and J ∈ Sym q we have E c ( q ) ( W, J ) = inf a ∈ A c q ) E x a ( W, J ) . We fix ε > J ∈ Sym q , and apply Lemma 3.1 for the case that the conditions of the theoremare satisfied. Then there exists a n ∈ N , so that for all n, m > n , for all x which satisfies(11), and implies |E x ( W n , J ) − E x ( W m , J ) | < ε. Let ε ′ > b ∈ A c ( q ) such that E c ( q ) ( W m , J ) + ε ′ > E x b ( W m , J ). Then E c ( q ) ( W n , J ) − E c ( q ) ( W m , J ) < E c ( q ) ( W n , J ) − E x b ( W m , J ) + ε ′ ≤ E x b ( W n , J ) − E x b ( W m , J ) + ε ′ < ε + ε ′ . The lower bound of E c ( q ) ( W n , J ) − E c ( q ) ( W m , J ) can be established completely similarly andas ε ′ was arbitrary, it follows that (cid:12)(cid:12) E c ( q ) ( W n , J ) − E c ( q ) ( W m , J ) (cid:12)(cid:12) < ε, which verifies the statement of the theorem.A direct consequence is the version of Theorem 3.2 for weighted graphs. Corollary 3.4.
Let G n be a sequence of weighted graphs with uniformly bounded edge weights,and α max ( G n ) α Gn → ( n → ∞ ). Let c and c be two lower threshold functions, so that c ( q ) q = h < h = c ( q ) q for all q ≥ for some ≤ h , h ≤ . If for every q ≥ and J ∈ Sym q the sequences ( ˆ E c ( q ) ( G n , J )) n ≥ converge, then for every q ≥ and J ∈ Sym q the sequences ( ˆ E c ( q ) ( G n , J )) n ≥ also converge. E c ↑ ( G, J ),is determined by a formula similar to (7) with the set A c replaced by A c , that is the set ofprobability distributions whose components are at most c , the general variant of the UTGSEis defined in the same manner. The equivalence corresponding to the one stated in Lemma3.1 between the general and the homogeneous version’s convergence follows by the sameblow-up trick as there, here for c ( q ) q = h ≥
1. The counterpart of Theorem 3.2 also holdstrue in the following form for 1 ≤ c ( q ) q ≤ c ( q ) q ≤ q : If for every q ≥ J ∈ Sym q thesequences ( E c ( q ) ↑ ( W n , J )) n ≥ converge, then for every q ≥ J ∈ Sym q the sequences( E c ( q ) ↑ ( W n , J )) n ≥ also converge. This conclusion comes not unexpected, it says, as in theLTGSE case, that less restriction on the set A c weakens the convergence property of a graphsequence. In this section we provide an example of a graphon family whose elements can be distin-guished for a larger c ( q ) lower threshold function for some pair of q ≥ J ∈ Sym q by looking at E c ( q ) ( W, J ), but whose LTGSEs are identical for some smaller c ( q ) lowerthreshold function for all q ≥ J ∈ Sym q . Based on this it is possible to constructa sequence of graphs, whose c ( q )-LTGES’s converge for every q ≥ J ∈ Sym q , butnot the c ( q )-LTGSEs through the same randomized method presented in [4] to show anon-convergent graph sequence with convergent ground state energies.In the second part of the section we demonstrate that there exist a family of graphons,where elements can be distinguished from each other by looking only at their LTGSEs for anarbitrary small, but positive c ( q ) lower threshold function, but whose corresponding GSEswithout any threshold are identical. Example 4.1.
An example which can be treated relatively easily are block-diagonal graphonswhich are defined for the parameters 0 ≤ α ≤
1, 0 ≤ β , β as W ( x, y ) = β , if 0 ≤ x, y ≤ αβ , if α < x, y ≤
10 , else.In the case of c ( q ) q = h , 1 − α ≥ h , β = 0, for arbitrary q ≥ J ∈ Sym q wehave E ( W, J ) = E c ( q ) ( W, J ). Choosing β = α , we get a one parameter family of graphonswhich have identical c ( q )-LTGSEs parametrized by α with 0 < α ≤ − h . This means that E c ( q ) ( W ( α ) , J ) = E ( I , J ), where I stands for the constant 1 graphon.12or every α > − h there are q ≥ J ∈ Sym q , so that the former equality does nothold anymore. Let J q ∈ Sym q be the q × q matrix, whose diagonal entries are 0, all otherentries being − q -partition mincut problem). Then E ( I , J q ) = 0 for all q ≥ q large enough E c ( q ) ( W ( α ) , J q ) >
0, we leave the details to the reader.With the aid of the previous example it is possible to construct a sequence of graphs whichverify that in Theorem 3.2 the implication of the convergence property of the sequence isstrictly one-way. This example is degenerate in the sense that the graphs consist of a quasi-random part and a sub-dense part with the bipartite graph spanned between the two partsalso being sub-dense.
Example 4.2.
Let us consider block-diagonal graphons with 0 < α < β , β >
0. Itwas shown in [4] that if we restrict our attention to a subfamily of block-diagonal graphons,where α β + (1 − α ) β is constant, then in these subfamilies the corresponding GSEs areidentical. Let c ( q ) be an arbitrarily small positive threshold function. Next we will showthat the c ( q )-LTGSEs determine the parameters of the block-diagonal graphon at least fora one-parameter family (up to graphon equivalence, since ( α, β , β ) belongs to the sameequivalence class as (1 − α, β , β )). The constant δ ij is 1, when i = j , and 0 otherwise.The value of the expression α β + (1 − α ) β is determined by the MAXCUT problemby E ( W, J ) with q = 2 and J ij = 1 − δ ij .In the second step let q be as large so that c ( q ) < min( α, − α ) holds, and let J be the q × q matrix with entries J ij = − δ i δ j . In this case E ( W, J ) = 0, but simple calculus gives −E c ( q ) ( W, J ) = − β β β + β c ( q ) . Hence β + β is determined by the LTGSEs.The extraction of a third dependency of the parameters from c ( q )-LTGSEs requires littlemore effort, we will only sketch details here. First consider α ’s with min( α, − α ) ≥ c (2).Let for q = 2 and k ≥ J k = (cid:18) − k − k (cid:19) . For every α with min( α, − α ) ≥ c (2) we havelim k →∞ −E c (2) ( W, J k ) = α β + (1 − α ) β + max( α β , (1 − α ) β ) . Now apply the notion of the general lower threshold: for q = 2 let c ( n ) = 2 c (2) /n and c ( n ) = 2 c (2)( n − /n two threshold functions , let us first consider the threshold x n = ( c ( n ) , c ( n )). If α ≥ c ( n ) or 1 − α ≥ c ( n ), then analogously to the case of thehomogeneous lower thresholdslim k →∞ −E x n ( W, J k ) = 2 α β + (1 − α ) β or α β + 2(1 − α ) β . If for example α < c ( n ), then it is easy to see that the LTGSE is going to infinity,because for some k , for all k > k : −E x n ( W, J k ) < − k ( c ( n ) − α ) c ( n ) min { β , β } + 2( α β + (1 − α ) β ) .
13o for fixed n then lim k →∞ −E x n ( W, J k ) = −∞ . To actually be able to extract the expression α β +(1 − α ) β +max( α β , (1 − α ) β ), we onlyhave to consider the lower threshold obtained by swapping the bounds, x ′ n = ( c ( n ) , c ( n )).Then, if α ≥ c ( n ) or 1 − α ≥ c ( n ), we havemax { lim k →∞ −E x n ( W, J k ) , lim k →∞ −E x ′ n ( W, J k ) } = α β + (1 − α ) β + max { α β , (1 − α ) β } , otherwise max { lim k →∞ −E x n ( W, J k ) , lim k →∞ −E x ′ n ( W, J k ) } = −∞ For every α there is a minimal n so that one of the conditions α ≥ c ( n ) and 1 − α ≥ c ( n )is satisfied, and for n < n the LTGSEs corresponding to x n and x ′ n tend to infinity when k goes to infinity. Therefore the expression α β + (1 − α ) β + max( α β , (1 − α ) β ) isdetermined by c ( q )-LTGSEs.Consider the one-parameter block-diagonal graphon family analyzed in [4], that is W ( α ) = W ( α, α , − α ), where 0 < α <
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