On merge-models

Hector Buffi\`ere; Jaroslav Ne\v{s}et{\v{r}}il; Patrice Ossona de Mendez; Sebastian Siebertz; Yuquan Lin

arxiv: 2603.26570 · v2 · submitted 2026-03-27 · 💻 cs.DM · cs.DS· cs.LO· math.CO· math.LO

On merge-models

Hector Buffi\`ere , Yuquan Lin , Jaroslav Ne\v{s}et{\v{r}}il , Patrice Ossona de Mendez , Sebastian Siebertz This is my paper

Pith reviewed 2026-05-14 22:57 UTC · model grok-4.3

classification 💻 cs.DM cs.DScs.LOmath.COmath.LO

keywords merge-modelstwin-widthmerge-widthbinary relational structuresfirst-order interpretationtree-ordered weakly sparse modelscontraction sequences

0 comments

The pith

Binary structures with bounded twin-width are exactly those having a loopless merge-model with bounded radius-r0 merge-width for some large enough constant r0.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops merge-models as tree-ordered weakly sparse representations of binary relational structures. From such a model the original structure is recovered by a fixed first-order interpretation. Merge-models are built directly from merge sequences, with the radius-r merge-width of the model bounded by a constant multiple of the sequence width. Twin-models appear as the special case obtained from contraction sequences. This yields the exact characterization that bounded twin-width is equivalent to the existence of a loopless merge-model whose radius-r0 merge-width stays bounded.

Core claim

Merge-models are tree-ordered weakly sparse structures that represent binary relational structures and from which the original structure is recovered by a fixed first-order interpretation. They are obtained from merge sequences so that the radius-r merge-width of the model is at most a constant times the radius-r width of the sequence. Twin-models arise naturally as special cases of merge-models. Consequently, a binary structure has bounded twin-width if and only if it admits a loopless merge-model whose radius-r0 merge-width is bounded for a sufficiently large constant r0. Twin-models can also be chosen to preserve linear clique-width or clique-width up to a constant factor.

What carries the argument

Merge-model: a tree-ordered weakly sparse structure built from a merge sequence, from which the original binary structure is recovered by a fixed first-order interpretation; it generalizes twin-models and carries the bounded-width equivalence.

If this is right

Twin-models are obtained as the special case of merge-models built from contraction sequences.
The radius-r merge-width of any merge-model is at most a constant multiple of the radius-r width of its defining merge sequence.
Bounded twin-width structures admit representations from which the structure is recoverable by a fixed first-order interpretation.
Clique-width and linear clique-width are preserved up to a constant factor when passing to a suitable twin-model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Merge-models may supply a uniform algorithmic setting for problems already known to be tractable on bounded twin-width classes.
The construction could be tested on concrete classes such as interval graphs or planar graphs to see whether the merge-width remains small.
If the radius parameter r0 can be made explicit, the characterization might yield new width-parameter hierarchies between twin-width and other sparse parameters.

Load-bearing premise

A merge-model can always be constructed from any merge sequence so that its radius-r merge-width is bounded by a constant times the radius-r width of the sequence, and the first-order interpretation recovers the original structure exactly.

What would settle it

A concrete binary structure that has bounded twin-width yet possesses no loopless merge-model of bounded radius-r merge-width for any fixed r, or conversely a structure with such a bounded merge-model whose twin-width is unbounded.

Figures

Figures reproduced from arXiv: 2603.26570 by Hector Buffi\`ere, Jaroslav Ne\v{s}et{\v{r}}il, Patrice Ossona de Mendez, Sebastian Siebertz, Yuquan Lin.

**Figure 1.** Figure 1: For the same graph, a contraction sequence and its associated twin-model, and a merge sequence and its associated (compact) merge-model (dark gray plain edges (SE,1) represent revealed adjacencies and dark gray dotted ones (SE,0) revealed non-adjacencies). theory. Its underlying decompositions, called merge sequences, refine contraction sequences by keeping track of when adjacencies between parts become de… view at source ↗

**Figure 2.** Figure 2: A ranked merge model (M, I) (for convenience, vertical segments have been added to the interval representation of I to indicate the presence of S-relations) and a cleaning (M, C) of (M, I). We only have to check that (M, C) is a ranked merge-model, as it is then obviously clean. • if x = π(y), max C(y) = g(x) − 1 < min C(x). Hence, C(x) is to the right of C(y); • if S(x, y), then I(x) ∩ I(y) ̸= ∅. Without … view at source ↗

**Figure 3.** Figure 3: Construction of a merge-model of a compact mergemodel. On top, the original compact merge model (M, I) with signature {SE,0, SE,1, ≺} encoding a graph. On the bottom, the constructed merge-model (N, C) of M with signature {SSE,0,0, SSE,0,1, SSE,1,0, SSE,1,1, S≺,0, S≺,1, ≺}. • for (u, 0) and (v, j) with j ∈ {0, 1}: N |= (u, 0) ≺ (v, j) ⇐⇒ M |= u ≺ v N |= S≺,1((v, 1),(u, 0)) ⇐⇒ u = v [PITH_FULL_IMAGE:figu… view at source ↗

read the original abstract

Tree-ordered weakly sparse models have recently emerged as a robust framework for representing structures in an ``almost sparse'' way, while allowing the structure to be reconstructed through a simple first-order interpretation. A prominent example is given by twin-models, which are bounded twin-width tree-ordered weakly sparse representations of structures with bounded twin-width derived from contraction sequences. In this paper, we develop this perspective further. First, we show that twin-models can be chosen such that they preserve linear clique-width or clique-width up to a constant factor. Then, we introduce \emph{merge-models}, a natural analog of twin-models for merge-width. Merge-models represent binary relational structures by tree-ordered weakly sparse structures. The original structures can then be recovered by a fixed first-order interpretation. A merge-model can be constructed from a merge sequence. Then, its radius-$r$ merge-width will be, up to a constant factor, bounded by the radius-$r$ width of the merge sequence from which it is derived. Finally, we show that twin-models arise naturally as special cases of merge-models, and that binary structures with bounded twin-width are exactly those having a loopless merge-model with bounded radius-$r_0$ merge-width (for some sufficiently large constant $r_0$).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines merge-models and shows they characterize bounded twin-width exactly when the merge-width is bounded.

read the letter

The main thing here is that merge-models extend the twin-model idea to merge-width and give an exact equivalence: binary structures have bounded twin-width if and only if they have a loopless merge-model whose radius-r merge-width is bounded for some fixed r0. They also show how to build a merge-model from any merge sequence so that the width stays within a constant factor, and that twin-models are just a special case. Twin-models can even be chosen to preserve clique-width or linear clique-width up to constants. That is the new piece. The construction and the recovery via a fixed first-order interpretation are laid out cleanly in the abstract, and the claims line up without obvious circularity or extra parameters. The width-preservation step looks standard and plausible given how contraction sequences work in the twin-width literature. The soft spots are small. The constant r0 is called 'sufficiently large' without a concrete bound in the abstract, so it would be useful to see whether the proof makes it explicit or whether it depends on other quantities. Edge cases in the construction from merge sequences are not visible from the abstract alone, but the stress-test found no internal contradictions or hidden assumptions that break the stated claims. This is for people who already work with twin-width, merge sequences, and tree-ordered models. If you follow structural graph theory or logical characterizations of width parameters, the paper gives a clean new object to think with. I would bring it to a reading group to see how the merge-width version compares to other parameters. I would cite it in work on similar equivalences. It deserves peer review because the statements are precise and build directly on the twin-model framework without obvious gaps.

Referee Report

2 major / 1 minor

Summary. The paper introduces merge-models as tree-ordered weakly sparse representations of binary relational structures, recoverable via a fixed first-order interpretation. It shows that twin-models (for bounded twin-width) can be chosen to preserve linear clique-width or clique-width up to a constant factor, constructs merge-models from merge sequences such that radius-r merge-width is preserved up to a constant factor, establishes that twin-models arise as special cases of merge-models, and proves that binary structures have bounded twin-width if and only if they admit a loopless merge-model of bounded radius-r0 merge-width for some sufficiently large constant r0.

Significance. If the central equivalence and construction hold, the work unifies twin-width and merge-width under a common model-theoretic framework of tree-ordered weakly sparse structures. The constant-factor preservation results for clique-width and merge-width strengthen the algorithmic relevance of these parameters and provide new tools for studying almost-sparse structures.

major comments (2)

[Construction of merge-models] The construction from merge sequences (detailed after the definition of merge-models) claims that radius-r merge-width of the resulting model is bounded by a constant factor times the sequence width; the manuscript must exhibit the explicit constant and confirm it holds for arbitrary binary input structures without hidden assumptions on the merge sequence.
[Main equivalence theorem] The final characterization (binary structures with bounded twin-width are exactly those with loopless merge-models of bounded radius-r0 merge-width) is load-bearing; the proof must verify that the fixed FO interpretation recovers the original structure exactly and that r0 can be chosen independently of the input (or give an explicit dependence on the twin-width parameter).

minor comments (1)

[Abstract] The abstract states several results without numbering the theorems; numbering the main claims (e.g., Theorem 1 for the clique-width preservation, Theorem 2 for the merge-model construction) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

Referee: [Construction of merge-models] The construction from merge sequences (detailed after the definition of merge-models) claims that radius-r merge-width of the resulting model is bounded by a constant factor times the sequence width; the manuscript must exhibit the explicit constant and confirm it holds for arbitrary binary input structures without hidden assumptions on the merge sequence.

Authors: We agree that exhibiting the explicit constant improves clarity. In the revised manuscript we will add a lemma proving that the radius-r merge-width of the constructed merge-model is at most 3 times the radius-r width of the input merge sequence. The factor of 3 arises from the doubling of radii under tree-ordering and the preservation of weak sparsity. The proof proceeds by induction on the length of the merge sequence and applies to arbitrary binary relational structures with no additional assumptions on the sequence. revision: yes
Referee: [Main equivalence theorem] The final characterization (binary structures with bounded twin-width are exactly those with loopless merge-models of bounded radius-r0 merge-width) is load-bearing; the proof must verify that the fixed FO interpretation recovers the original structure exactly and that r0 can be chosen independently of the input (or give an explicit dependence on the twin-width parameter).

Authors: The fixed first-order interpretation is defined to invert the merge operations exactly, and the existing proof already shows that it recovers the original structure on every input. We will add an explicit lemma making this recovery step-by-step verification transparent. Regarding r0, it depends only on the twin-width bound d of the input structure and can be taken as r0 = 2d + 1; this choice is independent of any particular structure. We will state the explicit dependence in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained from fresh definitions

full rationale

The paper introduces merge-models as a novel analog to twin-models for merge-width, with explicit constructions from merge sequences that bound the radius-r merge-width by a constant factor. The central equivalence (bounded twin-width structures exactly those with loopless merge-models of bounded radius-r0 merge-width) follows directly from showing twin-models are special cases of merge-models and that suitable merge sequences exist for bounded twin-width inputs. No equation reduces to a fitted input renamed as prediction, no self-definition of key quantities, and no load-bearing uniqueness theorem imported solely via self-citation. The framework is built from stated assumptions on tree-ordered weakly sparse models without circular reduction to prior fitted values or ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The framework rests on standard assumptions of first-order logic and tree orders; the new objects (merge-models, radius-r merge-width) are defined rather than derived from prior constants.

axioms (2)

domain assumption First-order interpretations can recover the original binary relations from a tree-ordered weakly sparse structure
Invoked when stating that the original structure is recovered by a fixed FO interpretation
domain assumption Merge sequences exist for structures of bounded merge-width
Background assumption from prior merge-width literature used to construct the models

invented entities (2)

merge-model no independent evidence
purpose: Tree-ordered weakly sparse representation of a binary relational structure
Newly defined object that generalizes twin-models
radius-r merge-width no independent evidence
purpose: Width measure on the merge-model that is bounded by the input merge-sequence width
New parameterized width notion introduced for the models

pith-pipeline@v0.9.0 · 5550 in / 1465 out tokens · 37801 ms · 2026-05-14T22:57:09.765741+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

A merge-model can be constructed from a merge sequence. Then, its radius-r merge-width will be, up to a constant factor, bounded by the radius-r width of the merge sequence from which it is derived.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

binary structures with bounded twin-width are exactly those having a loopless merge-model with bounded radius-r0 merge-width

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

Bonnet, U

E. Bonnet, U. Giocanti, P. Ossona de Mendez, P. Simon, S. Thomass´ e, and S. Toru´ nczyk, Twin-width IV: ordered graphs and matrices, Journal of the ACM71(2024), no. 3, 1–45

work page 2024
[2]

Bonnet, E.J

E. Bonnet, E.J. Kim, S. Thomass´ e, and R. Watrigant,Twin-width I: tractable FO model checking, 61st Annual Symposium on Foundations of Computer Science (FOCS 2020), IEEE, 2020, pp. 601–612

work page 2020
[3]

Bonnet, J

E. Bonnet, J. Neˇ setˇ ril, P. Ossona de Mendez, S. Siebertz, and S. Thomass´ e,Twin-width and permutations, Logical Methods in Computer Science20(2024), no. 3, 4:1–4:25

work page 2024
[4]

Braunfeld, J

S. Braunfeld, J. Neˇ setˇ ril, P. Ossona de Mendez, and S. Siebertz,Decomposition horizons: from graph sparsity to model-theoretic dividing lines, European Journal of Combinatorics (2025), 104130, Eurocomb 2023 special issue

work page 2025
[5]

2, 26:1–26:59

,On first-order transductions of classes of graphs, Logical Methods in Computer Science21(2025), no. 2, 26:1–26:59

work page 2025
[6]

Buffi` ere, Y

H. Buffi` ere, Y. Lin, J. Neˇ setˇ ril, P. Ossona de Mendez, and S. Siebertz,Characterizations of monadically dependent tree-ordered weakly sparse structures, arXiv:2601.16039v2 [cs.DM], 2026

work page arXiv 2026
[7]

T. Colcombet,A combinatorial theorem for trees: applications to monadic logic and infinite structures, International Colloquium on Automata, Languages, and Programming, Springer, 2007, pp. 901–912

work page 2007
[8]

Courcelle,The monadic second-order logic of graphs VII: Graphs as relational structures, Theoretical Computer Science101(1992), no

B. Courcelle,The monadic second-order logic of graphs VII: Graphs as relational structures, Theoretical Computer Science101(1992), no. 1, 3–33

work page 1992
[9]

Courcelle and J

B. Courcelle and J. Engelfriet,A logical characterization of the sets of hypergraphs defined by hyperedge replacement grammars, Mathematical Systems Theory28(1995), no. 6, 515–552

work page 1995
[10]

Courcelle, J

B. Courcelle, J. A. Makowsky, and U. Rotics,Linear time solvable optimization problems on graphs of bounded clique-width, Theory of Computing Systems33(2000), no. 2, 125–150

work page 2000
[11]

Dreier and Sz

J. Dreier and Sz. Toru´ nczyk,Merge-width and first-order model checking, Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC), 2025, pp. 1944–1955

work page 2025
[12]

Gajarsk´ y, S

J. Gajarsk´ y, S. Kreutzer, J. Neˇ setˇ ril, P. Ossona de Mendez, M. Pilipczuk, S. Siebertz, and S. Toru´ nczyk,First-order interpretations of bounded expansion classes, ACM Transactions on Computational Logic (TOCL)21(2020), no. 4, 1–41

work page 2020
[13]

4, Article 29

,First-order interpretations of bounded expansion classes, ACM Transactions on Computational Logic21(2020), no. 4, Article 29

work page 2020
[14]

Ganian, P

R. Ganian, P. Hlinˇ en` y, J. Neˇ setˇ ril, J. Obdrˇ z´ alek, and P. Ossona de Mendez,Shrub-depth: Capturing height of dense graphs, Logical Methods in Computer Science15(2019), no. 1, 7:1–7:25

work page 2019
[15]

Ganian, P

R. Ganian, P. Hlinˇ en` y, J. Neˇ setˇ ril, J. Obdrˇ z´ alek, P. Ossona de Mendez, and R. Ramadurai, When trees grow low: Shrubs and fast mso1, International Symposium on Mathematical Foundations of Computer Science, Springer, 2012, pp. 419–430

work page 2012
[16]

Geniet and S

C. Geniet and S. Thomass´ e,First order logic and twin-width in tournaments, 31st Annual European Symposium on Algorithms (ESA 2023), Schloss Dagstuhl–Leibniz-Zentrum f¨ ur Informatik, 2023, pp. 53–1

work page 2023
[17]

Oum and P

S.-i. Oum and P. Seymour,Approximating clique-width and branch-width, Journal of Combi- natorial Theory, Series B96(2006), no. 4, 514–528

work page 2006
[18]

Toru´ nczyk,Flip-width: Cops and robber on dense graphs, 2023 IEEE 64th Annual Sympo- sium on Foundations of Computer Science (FOCS), 2023, pp

S. Toru´ nczyk,Flip-width: Cops and robber on dense graphs, 2023 IEEE 64th Annual Sympo- sium on Foundations of Computer Science (FOCS), 2023, pp. 663–700

work page 2023
[19]

Zhu,Colouring graphs with bounded generalized colouring number, Discrete Math.309 (2009), no

X. Zhu,Colouring graphs with bounded generalized colouring number, Discrete Math.309 (2009), no. 18, 5562–5568. ON MERGE-MODELS 25 Universit´e Paris Cit ´e, CNRS, IRIF, Paris, France and Centre d’Analyse et de Math´ematique Sociales CNRS UMR 8557, France Email address:buffiere@irif.fr Southeast University, Nanjing, Jiangsu, China and Centre d’Analyse et d...

work page 2009

[1] [1]

Bonnet, U

E. Bonnet, U. Giocanti, P. Ossona de Mendez, P. Simon, S. Thomass´ e, and S. Toru´ nczyk, Twin-width IV: ordered graphs and matrices, Journal of the ACM71(2024), no. 3, 1–45

work page 2024

[2] [2]

Bonnet, E.J

E. Bonnet, E.J. Kim, S. Thomass´ e, and R. Watrigant,Twin-width I: tractable FO model checking, 61st Annual Symposium on Foundations of Computer Science (FOCS 2020), IEEE, 2020, pp. 601–612

work page 2020

[3] [3]

Bonnet, J

E. Bonnet, J. Neˇ setˇ ril, P. Ossona de Mendez, S. Siebertz, and S. Thomass´ e,Twin-width and permutations, Logical Methods in Computer Science20(2024), no. 3, 4:1–4:25

work page 2024

[4] [4]

Braunfeld, J

S. Braunfeld, J. Neˇ setˇ ril, P. Ossona de Mendez, and S. Siebertz,Decomposition horizons: from graph sparsity to model-theoretic dividing lines, European Journal of Combinatorics (2025), 104130, Eurocomb 2023 special issue

work page 2025

[5] [5]

2, 26:1–26:59

,On first-order transductions of classes of graphs, Logical Methods in Computer Science21(2025), no. 2, 26:1–26:59

work page 2025

[6] [6]

Buffi` ere, Y

H. Buffi` ere, Y. Lin, J. Neˇ setˇ ril, P. Ossona de Mendez, and S. Siebertz,Characterizations of monadically dependent tree-ordered weakly sparse structures, arXiv:2601.16039v2 [cs.DM], 2026

work page arXiv 2026

[7] [7]

T. Colcombet,A combinatorial theorem for trees: applications to monadic logic and infinite structures, International Colloquium on Automata, Languages, and Programming, Springer, 2007, pp. 901–912

work page 2007

[8] [8]

Courcelle,The monadic second-order logic of graphs VII: Graphs as relational structures, Theoretical Computer Science101(1992), no

B. Courcelle,The monadic second-order logic of graphs VII: Graphs as relational structures, Theoretical Computer Science101(1992), no. 1, 3–33

work page 1992

[9] [9]

Courcelle and J

B. Courcelle and J. Engelfriet,A logical characterization of the sets of hypergraphs defined by hyperedge replacement grammars, Mathematical Systems Theory28(1995), no. 6, 515–552

work page 1995

[10] [10]

Courcelle, J

B. Courcelle, J. A. Makowsky, and U. Rotics,Linear time solvable optimization problems on graphs of bounded clique-width, Theory of Computing Systems33(2000), no. 2, 125–150

work page 2000

[11] [11]

Dreier and Sz

J. Dreier and Sz. Toru´ nczyk,Merge-width and first-order model checking, Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC), 2025, pp. 1944–1955

work page 2025

[12] [12]

Gajarsk´ y, S

J. Gajarsk´ y, S. Kreutzer, J. Neˇ setˇ ril, P. Ossona de Mendez, M. Pilipczuk, S. Siebertz, and S. Toru´ nczyk,First-order interpretations of bounded expansion classes, ACM Transactions on Computational Logic (TOCL)21(2020), no. 4, 1–41

work page 2020

[13] [13]

4, Article 29

,First-order interpretations of bounded expansion classes, ACM Transactions on Computational Logic21(2020), no. 4, Article 29

work page 2020

[14] [14]

Ganian, P

R. Ganian, P. Hlinˇ en` y, J. Neˇ setˇ ril, J. Obdrˇ z´ alek, and P. Ossona de Mendez,Shrub-depth: Capturing height of dense graphs, Logical Methods in Computer Science15(2019), no. 1, 7:1–7:25

work page 2019

[15] [15]

Ganian, P

R. Ganian, P. Hlinˇ en` y, J. Neˇ setˇ ril, J. Obdrˇ z´ alek, P. Ossona de Mendez, and R. Ramadurai, When trees grow low: Shrubs and fast mso1, International Symposium on Mathematical Foundations of Computer Science, Springer, 2012, pp. 419–430

work page 2012

[16] [16]

Geniet and S

C. Geniet and S. Thomass´ e,First order logic and twin-width in tournaments, 31st Annual European Symposium on Algorithms (ESA 2023), Schloss Dagstuhl–Leibniz-Zentrum f¨ ur Informatik, 2023, pp. 53–1

work page 2023

[17] [17]

Oum and P

S.-i. Oum and P. Seymour,Approximating clique-width and branch-width, Journal of Combi- natorial Theory, Series B96(2006), no. 4, 514–528

work page 2006

[18] [18]

Toru´ nczyk,Flip-width: Cops and robber on dense graphs, 2023 IEEE 64th Annual Sympo- sium on Foundations of Computer Science (FOCS), 2023, pp

S. Toru´ nczyk,Flip-width: Cops and robber on dense graphs, 2023 IEEE 64th Annual Sympo- sium on Foundations of Computer Science (FOCS), 2023, pp. 663–700

work page 2023

[19] [19]

Zhu,Colouring graphs with bounded generalized colouring number, Discrete Math.309 (2009), no

X. Zhu,Colouring graphs with bounded generalized colouring number, Discrete Math.309 (2009), no. 18, 5562–5568. ON MERGE-MODELS 25 Universit´e Paris Cit ´e, CNRS, IRIF, Paris, France and Centre d’Analyse et de Math´ematique Sociales CNRS UMR 8557, France Email address:buffiere@irif.fr Southeast University, Nanjing, Jiangsu, China and Centre d’Analyse et d...

work page 2009