. Clearly, we have N 1 (z) ? b?N k?2 (x) N 1 (b) By definition, N 1 (z) = K(O k?3 (z) and b?N k?2 (x) N 1 (b) = K(O k?2 (x)). We then obtain K(O k?3 (z)) ? K(O k?2 (x)), which gives, pp.3-5

?. Since-z, Again, since z ? W k?2 (x), we have O k?3 (x) ? O k?3 (z) ? O k?2 (x) Since we already proved that N k?1 (x) = W k?1 (x), O k?4 (x)), and then, pp.3-3

N. Since, |N k?2 (x)| > 1, necessarily O k?3 (x) O k?2 (x) Similarly, the fact that N k?1 (x) = W k?1 (x) and |N k?1 (x)| > 1 implies that O k?2 (x) O k?1 (x) At last, from Remark 3, we have K(O k?1 (x)) = N 1 (x), and since |N 1 (x)| ? 2, it follows that |K(O k?1 (x))| ? 2. Combined with the fact that |O 1 (x)| ? 2

H. Proof-of and . .. Sur, From recursion hypothesis H sur (k ? 1), for any P ? O such that O k?3 P , there exists y P ? V k?1 such that S(y P ) = (O 1 , We denote by Y the set = (O 1 , . . . , O k?3 ) and O k?2 ? V * k and is therefore an element of V k . In order to conclude the proof of H sur (k), let us now show that the element x of V k has the desired characterising sequence (O 1 , . . . , O k?1 ). First, from H N (k), which we already proved, we know that O k?3 ), from the definition of x = Y y?Y N (y). Second, from Remark 3, we have K(O k?1 (x)) = N 1 (x) and we already know that |N 1 (x)| ? 2 (see beginning of the proof of H sur (k)), k?1 ) ? O k?1 such that O 1 . . . O k?1 O k?3 (y)) for any y ? N k?1 (x), which gives (O 1 (x), . . . , O k?3 (x)) = (O 1 This gives |K(O k?1 (x))| ? 2, and as we have |O 1 (x)| ? 2 and O 1 (x) ? . . . ? O k?2 (x) ? O k?1 (x), we obtain that O k?2 (x), O k?1 (x) ? O. Now, from H N (k), we know that the couple (O k?2 (x), O k?1 (x)) is such that N k?1 (x) = {y ? V k?1 | O k?2

R. Albert and A. L. Barabási, Statistical mechanics of complex networks, Reviews of Modern Physics, vol.74, issue.1, p.47, 2002.
DOI : 10.1103/RevModPhys.74.47

G. Alexe, S. Alexe, Y. Crama, S. Foldes, P. L. Hammer et al., Consensus algorithms for the generation of all maximal bicliques, Discrete Applied Mathematics, vol.145, issue.1, pp.11-21, 2004.
DOI : 10.1016/j.dam.2003.09.004

A. L. Barabási and Z. Toroczkai, Ccnr network databases, 2007.

A. Barrat, M. Barthélemy, and A. Vespignani, Dynamical processes on complex networks, 2010.
DOI : 10.1017/CBO9780511791383

E. A. Bender and E. Canfield, The asymptotic number of labeled graphs with given degree sequences, Journal of Combinatorial Theory, Series A, vol.24, issue.3, pp.296-307, 1978.
DOI : 10.1016/0097-3165(78)90059-6

C. F. Bornstein and J. L. Szwarcfiter, On clique convergent graphs, Graphs and Combinatorics, vol.103, issue.3, pp.213-220, 1995.
DOI : 10.1007/BF01793007

M. R. Cerioli, Clique Graphs and Edge-clique graphs, Electronic Notes in Discrete Mathematics, vol.13, pp.34-37, 2003.
DOI : 10.1016/S1571-0653(04)00432-9

G. Chartrand, S. F. Kapoor, T. A. Mckee, and F. Saba, Edge-clique graphs, Graphs and Combinatorics, vol.10, issue.3, pp.253-264, 1991.
DOI : 10.1007/BF01787631

T. Evans, Clique graphs and overlapping communities, Journal of Statistical Mechanics: Theory and Experiment, vol.2010, issue.12, p.12037, 2010.
DOI : 10.1088/1742-5468/2010/12/P12037

A. Gély, L. Nourine, and B. Sadi, Enumeration aspects of maximal cliques and bicliques, Discrete Applied Mathematics, vol.157, issue.7, pp.1447-1459, 2009.
DOI : 10.1016/j.dam.2008.10.010

M. Groshaus and L. P. Montero, The number of convergent graphs under the biclique operator with no twin vertices is finite, Electronic Notes in Discrete Mathematics, vol.35, pp.241-246, 2009.
DOI : 10.1016/j.endm.2009.11.040

M. Groshaus and L. P. Montero, On the Iterated Biclique Operator, Journal of Graph Theory, vol.33, issue.1-3, pp.181-190, 2013.
DOI : 10.1002/jgt.21666

J. L. Guillaume and M. Latapy, Bipartite structure of all complex networks, Information Processing Letters, vol.90, issue.5, pp.215-221, 2004.
DOI : 10.1016/j.ipl.2004.03.007

URL : https://hal.archives-ouvertes.fr/hal-00016855

J. L. Guillaume and M. Latapy, Bipartite graphs as models of complex networks, Physica A: Statistical Mechanics and its Applications, vol.371, issue.2, pp.795-813, 2006.
DOI : 10.1016/j.physa.2006.04.047

F. Larrión, C. P. De-mello, A. Morgana, V. Neumann-lara, and M. A. Pizaña, The clique operator on cographs and serial graphs, Discrete Mathematics, vol.282, issue.1-3, pp.183-191, 2004.
DOI : 10.1016/j.disc.2003.10.023

F. Larrión, V. Neumann-lara, and M. A. Pizaña, Clique divergent clockwork graphs and partial orders, Discrete Applied Mathematics, vol.141, issue.1-3, pp.195-207, 2004.
DOI : 10.1016/S0166-218X(03)00378-0

F. Larrión, M. A. Pizaña, and R. Villarroel-flores, Contractibility and the clique graph operator, Discrete Mathematics, vol.308, issue.16, pp.3461-3469, 2008.
DOI : 10.1016/j.disc.2007.07.004

M. Latapy, T. H. Phan, C. Crespelle, and T. Q. Nguyen, Termination of Multipartite Graph Series Arising from Complex Network Modelling, 4th International Conference on Combinatorial Optimization and Applications -COCOA'10, pp.1-10, 2010.
DOI : 10.1007/978-3-642-17458-2_1

URL : https://hal.archives-ouvertes.fr/hal-00644239

M. C. Lin, F. J. Soulignac, and J. L. Szwarcfiter, The clique operator on circular-arc graphs, Discrete Applied Mathematics, vol.158, issue.12, pp.1259-1267, 2010.
DOI : 10.1016/j.dam.2009.01.019

M. Molloy and B. Reed, A critical point for random graphs with a given degree sequence. Random Structures and Algorithms, 1995.

V. Neumann-lara, On clique-divergent graphs, Probì emes Combin. Théorie Graphes (Colloques internationaux du C, pp.313-315, 1978.

T. H. Phan and C. Crespelle, Termination of the iterated strong-factor operator on multipartite graphs, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01241925

E. Prisner, Convergence of iterated clique graphs, Discrete Mathematics, vol.103, issue.2, pp.199-207, 1992.
DOI : 10.1016/0012-365X(92)90270-P

J. L. Szwarcfiter, A Survey on Clique Graphs, Recent Advances in Algorithms and Combinatorics, pp.109-136, 2003.
DOI : 10.1007/0-387-22444-0_5

E. Tomita, A. Tanaka, and H. Takahashi, The worst-case time complexity for generating all maximal cliques and computational experiments, Theoretical Computer Science, vol.363, issue.1, pp.28-42, 2006.
DOI : 10.1016/j.tcs.2006.06.015

T. Uno, T. Asai, Y. Uchida, and H. Arimura, Lcm program (linear time closed itemset miner) Http

T. Uno, T. Asai, Y. Uchida, and H. Arimura, An Efficient Algorithm for Enumerating Closed Patterns in Transaction Databases, 7th International Conference, pp.16-31, 2004.
DOI : 10.1007/978-3-540-30214-8_2

L. Wan, B. Wu, N. Du, Q. Ye, and P. Chen, A New Algorithm for Enumerating All Maximal Cliques in Complex Network, Advanced Data Mining and Applications, Second International Conference -ADMA'06, pp.606-617, 2006.
DOI : 10.1007/11811305_67

D. Watts and S. Strogatz, Collective dynamics of small-world networks, Nature, vol.393, issue.6684, pp.440-442, 1998.
DOI : 10.1038/30918