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arxiv: 2603.09468 · v2 · pith:Q7N75QGBnew · submitted 2026-03-10 · 🪐 quant-ph

Multi-tasking through quantum annealing

Jargalsaikhan Artag , Koki Awaya , Takumi Kanezashi , Daisuke Tsukayama , Moe Shimada , Jun-ichi Shirakashi This is my paper

Pith reviewed 2026-05-21 12:13 UTC · model grok-4.3

classification 🪐 quant-ph
keywords multi-tasking quantum annealingquantum annealingcombinatorial optimizationminimum vertex cover problemgraph partitioning problemtime-to-solutioneigenspectrum analysisparallel embedding
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The pith

Multi-tasking quantum annealing embeds multiple problems into distinct hardware regions to solve them concurrently with reduced time-to-solution.

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

This paper proposes multi-tasking quantum annealing (MTQA) as a way to process several optimization problems simultaneously on quantum hardware by placing them in separate spatial regions. This method is tested on the minimum vertex cover problem and graph partitioning, showing that it matches the solution quality of running problems one at a time or using classical simulated annealing. The key benefit claimed is a reduction in time-to-solution metrics through better use of available qubits. Theoretical support comes from eigenspectrum analysis indicating that the parallel setup maintains the necessary quantum coherence without added complexity. Such an approach could make quantum annealing more practical for handling multiple tasks at once in applications involving up to 100 nodes.

Core claim

Multi-tasking quantum annealing (MTQA) enables the parallel processing of multiple optimization problems by embedding them into spatially distinct regions on quantum hardware. Evaluated on minimum vertex cover and graph partitioning problems, MTQA achieves solution quality comparable to single-problem quantum annealing and classical simulated annealing while reducing time-to-solution. Eigenspectrum analysis supports that this parallel embedding preserves quantum coherence and does not increase computational complexity, optimizing hardware utilization for concurrent tasks.

What carries the argument

Parallel embedding of multiple problems into spatially distinct regions on the quantum annealer, allowing concurrent optimization by utilizing idle qubits.

If this is right

  • MTQA optimizes quantum resource utilization by concurrently utilizing idle qubits.
  • Solution quality is comparable to single-problem quantum annealing and classical simulated annealing for NP-hard problems.
  • Time-to-solution metrics are notably reduced.
  • The method performs for problems up to 100 nodes.
  • Parallel embedding preserves quantum coherence without increasing computational complexity according to eigenspectrum analysis.

Where Pith is reading between the lines

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

  • This parallel embedding method could apply to other quantum optimization platforms that allow spatial separation of problems.
  • Future implementations might combine MTQA with classical post-processing to handle even larger problem sets.
  • Testing on hardware with more qubits could reveal scalability limits not addressed in the current experiments.

Load-bearing premise

Embedding multiple problems into spatially distinct regions on the hardware preserves quantum coherence and does not increase computational complexity.

What would settle it

An observation that eigenspectrum analysis shows increased complexity or that solution quality and TTS worsen in parallel embeddings compared to sequential ones would falsify the claim.

Figures

Figures reproduced from arXiv: 2603.09468 by Daisuke Tsukayama, Jargalsaikhan Artag, Jun-ichi Shirakashi, Koki Awaya, Moe Shimada, Takumi Kanezashi.

Figure 1
Figure 1. Figure 1: FIG. 1. Parallel embedding in D-Wave Advantage 6.4 hard [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Chain length variations as a function of graph size. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Ground-state probability (GSP) analysis comparing [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Time-to-solution (TTS) comparison across different [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. GPP solution energy distribution across different graph sizes ( [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. D-Wave default annealing schedule showing the evo [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Eigenspectrum analysis of parallel problem solving in [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Quantum mechanical analysis comparing MTQA, and PQA on embedded Hamiltonians. (a) Energy gap evolution [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
read the original abstract

Quantum annealing approximately solves combinatorial optimization problems by leveraging the principles of adiabatic quantum systems. In this approach, the system's Hamiltonian evolves from an initial general state to a problem-specific state. This study introduces multi-tasking quantum annealing (MTQA), a method that enables the parallel processing of multiple optimization problems by embedding them into spatially distinct regions on quantum hardware. MTQA is evaluated using two NP-hard problems: the minimum vertex cover problem (MVCP) and the graph partitioning problem (GPP). This parallel approach optimizes quantum resource utilization by concurrently utilizing idle qubits. The findings demonstrate that MTQA achieves a solution quality comparable to single-problem quantum annealing and classical simulated annealing (SA), while notably reducing the time-to-solution (TTS) metrics. Eigenspectrum analysis further theoretically supports the hypothesis that parallel embedding preserves quantum coherence and does not increase computational complexity by efficiently utilizing available quantum hardware (e.g., qubits and couplers). MTQA enables efficient multitasking in quantum annealing, optimizing hardware utilization and improving throughput for concurrent tasks and demonstrating performance for problems up to 100 nodes in real-world applications.

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.

Referee Report

2 major / 2 minor

Summary. The paper introduces multi-tasking quantum annealing (MTQA), a technique to embed and solve multiple combinatorial optimization problems, such as the minimum vertex cover problem (MVCP) and the graph partitioning problem (GPP), simultaneously in spatially distinct regions of quantum annealing hardware. It reports that this parallel approach yields solution quality comparable to single-problem quantum annealing and classical simulated annealing, with reduced time-to-solution (TTS), and uses eigenspectrum analysis to argue that parallel embedding preserves quantum coherence without increasing computational complexity. The method is tested on problems with up to 100 nodes.

Significance. If substantiated with quantitative details, MTQA could improve hardware utilization in quantum annealers by enabling concurrent solves on idle qubits and couplers, increasing throughput for optimization tasks without requiring larger devices. The evaluation on real hardware for problems up to 100 nodes provides a practical starting point for such resource-efficient approaches.

major comments (2)
  1. [Abstract] Abstract: The claims that MTQA achieves 'solution quality comparable to single-problem quantum annealing and classical simulated annealing' while 'notably reducing the time-to-solution (TTS) metrics' are presented without any quantitative values, error bars, specific baseline comparisons, or details on post-embedding interference. This leaves the central experimental result under-supported.
  2. [Eigenspectrum analysis] Eigenspectrum analysis: The manuscript states that eigenspectrum analysis of the combined Hamiltonian supports preservation of quantum coherence and unchanged computational complexity under parallel embedding. However, ideal-Hamiltonian eigenspectra do not automatically capture hardware noise, residual couplings between spatially distinct regions, or the effect of the joint annealing schedule on the instantaneous gap; without addressing these, the attribution of TTS reduction to preserved parallelism remains open.
minor comments (2)
  1. [Abstract] Abstract: The statement 'optimizing quantum resource utilization by concurrently utilizing idle qubits' would benefit from a concrete illustration of qubit and coupler counts before versus after multitasking.
  2. [Abstract] Abstract: The hardware platform and embedding procedure used for the MVCP and GPP experiments should be specified to allow reproducibility assessment.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address each major comment point by point below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claims that MTQA achieves 'solution quality comparable to single-problem quantum annealing and classical simulated annealing' while 'notably reducing the time-to-solution (TTS) metrics' are presented without any quantitative values, error bars, specific baseline comparisons, or details on post-embedding interference. This leaves the central experimental result under-supported.

    Authors: We agree that the abstract would be strengthened by including quantitative support. In the revised manuscript, we will incorporate specific values such as mean approximation ratios (with standard deviations) for solution quality and percentage reductions in TTS, along with explicit comparisons to single-problem QA and SA baselines. We will also add a brief statement that post-embedding interference was controlled via spatial separation of regions and confirmed through hardware experiments. revision: yes

  2. Referee: [Eigenspectrum analysis] Eigenspectrum analysis: The manuscript states that eigenspectrum analysis of the combined Hamiltonian supports preservation of quantum coherence and unchanged computational complexity under parallel embedding. However, ideal-Hamiltonian eigenspectra do not automatically capture hardware noise, residual couplings between spatially distinct regions, or the effect of the joint annealing schedule on the instantaneous gap; without addressing these, the attribution of TTS reduction to preserved parallelism remains open.

    Authors: The eigenspectrum analysis on the ideal combined Hamiltonian is intended to show that the ground-state structure and gap scaling are preserved relative to individual problems, indicating no inherent increase in complexity from parallelism. We acknowledge that this ideal analysis does not capture hardware noise, residual couplings, or schedule effects. In the revision, we will add a dedicated discussion paragraph addressing these limitations, noting that the observed TTS reductions are validated empirically on real hardware for instances up to 100 nodes, while the spectral analysis provides theoretical support for coherence preservation. revision: partial

Circularity Check

0 steps flagged

No circularity: claims rest on independent experimental evaluation and internal eigenspectrum calculation

full rationale

The paper presents MTQA as a method for embedding multiple problems into distinct hardware regions and evaluates it experimentally on MVCP and GPP instances up to 100 nodes, comparing solution quality and TTS directly against single-problem QA and classical SA. The eigenspectrum analysis is performed on the combined Hamiltonian constructed in the paper itself and is offered as supporting evidence for coherence preservation rather than as a derivation that reduces to a fitted parameter or prior self-citation. No load-bearing step matches any of the enumerated circular patterns; the central performance claims are grounded in measured outcomes rather than by-construction equivalence to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard assumptions of quantum annealing for combinatorial optimization and introduces MTQA as a new embedding strategy without new physical entities or fitted parameters visible in the abstract.

axioms (1)
  • domain assumption Quantum annealing approximately solves combinatorial optimization problems via adiabatic evolution of the Hamiltonian.
    Invoked in the opening description of the approach; standard in the field.

pith-pipeline@v0.9.0 · 5734 in / 1116 out tokens · 37763 ms · 2026-05-21T12:13:48.925155+00:00 · methodology

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Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages · 5 internal anchors

  1. [1]

    Quantum Computation by Adiabatic Evolution

    E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, Quantum computation by adiabatic evolution (2000), arXiv:quant-ph/0001106 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  2. [2]

    Morita and H

    S. Morita and H. Nishimori, J. Math. Phys.49, 125210 (2008)

    work page 2008

  3. [3]

    Albash and D

    T. Albash and D. A. Lidar, Rev. Mod. Phys.90, 015002 (2018)

    work page 2018

  4. [4]

    Kadowaki and H

    T. Kadowaki and H. Nishimori, Phys. Rev. E58, 5355 (1998)

    work page 1998

  5. [5]

    Hauke, H

    P. Hauke, H. G. Katzgraber, W. Lechner, H. Nishimori, and W. D. Oliver, Rep. Prog. Phys.83, 054401 (2020)

    work page 2020

  6. [6]

    Das and B

    A. Das and B. K. Chakrabarti, eds.,Quantum Anneal- ing and Related Optimization Methods,Vol.679(Springer Science & Business Media, 2005)

    work page 2005

  7. [7]

    M. W. Johnson, M. H. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. J. Berkley, J. Jo- hansson, P. Bunyk,et al., Nature473, 194 (2011)

    work page 2011

  8. [8]

    Boixo, T

    S. Boixo, T. F. Rønnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, Nat. Phys.10, 218 (2014)

    work page 2014

  9. [9]

    Chandarana, N

    P. Chandarana, N. N. Hegade, K. Paul, F. Albarrán- Arriagada, E. Solano, A. del Campo, and X. Chen, Phys. Rev. Res.4, 013141 (2022)

    work page 2022

  10. [10]

    Pelofske, G

    E. Pelofske, G. Hahn, and H. N. Djidjev, Sci. Rep.12, 4499 (2022)

    work page 2022

  11. [11]

    Artag, M

    J. Artag, M. Shimada, and J.-i. Shirakashi, inIEEE Int. Conf. Quantum Comput. Eng., Vol. 02 (IEEE, 2023) pp. 328–329

    work page 2023

  12. [12]

    Artag, M

    J. Artag, M. Shimada, and J.-i. Shirakashi, inIEEE Int. Conf. Quantum Comput. Eng., Vol. 02 (IEEE, 2024) pp. 506–507

    work page 2024

  13. [13]

    Pelofske, G

    E. Pelofske, G. Hahn, and H. N. Djidjev, Quantum Inf. Process.22, 219 (2023)

    work page 2023

  14. [14]

    Pelofske, A

    E. Pelofske, A. Bärtschi, and S. Eidenbenz, npj Quantum Inf.10, 30 (2024)

    work page 2024

  15. [15]

    Huang, Y

    T. Huang, Y. Zhu, R. S. M. Goh, and T. Luo, Array17, 100282 (2023)

    work page 2023

  16. [16]

    Blake, R

    G. Blake, R. G. Dreslinski, and T. Mudge, IEEE Signal Process. Mag.26, 26 (2009)

    work page 2009

  17. [17]

    J. L. Hennessy and D. A. Patterson,Computer Architec- ture: A Quantitative Approach(Elsevier, 2011)

    work page 2011

  18. [18]

    R. M. Karp, in50 Years of Integer Programming 1958- 2008: From the Early Years to the State-of-the-Art, edited by M. Jünger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, and L. A. Wolsey (Springer Berlin Heidelberg, 2010) pp. 219–241

    work page 1958

  19. [19]

    Lucas, Front

    A. Lucas, Front. Phys.2, 5 (2014)

    work page 2014

  20. [20]

    Ushijima-Mwesigwa, C

    H. Ushijima-Mwesigwa, C. F. A. Negre, and S. M. Mniszewski, inProc. Second Int. Workshop Post Moores Era Supercomput.(Association for Computing Machin- ery, New York, NY, USA, 2017) pp. 22–29

    work page 2017

  21. [21]

    Fu and P

    Y. Fu and P. W. Anderson, J. Phys. A: Math. Gen.19, 1605 (1986)

    work page 1986

  22. [22]

    Mezard and A

    M. Mezard and A. Montanari,Information, Physics, and Computation(Oxford University Press, 2009)

    work page 2009

  23. [23]

    Nickel, C

    S. Nickel, C. Steinhardt, H. Schlenker, and W. Burkart, Decision Optimization with IBM ILOG CPLEX Opti- mization Studio(Springer Berlin, Heidelberg, 2022)

    work page 2022

  24. [24]

    R. E. Bixby, Trans. Res. B41, 159 (2007)

    work page 2007

  25. [25]

    D-Wave Systems Inc., Technical description of the QPU, D-Wave System Documentation (2024)

    work page 2024

  26. [26]

    Boothby, P

    K. Boothby, P. Bunyk, J. Raymond, and A. Roy, Next-generation topology of D-wave quantum processors (2020), arXiv:2003.00133 [quant-ph]

    work page arXiv 2020

  27. [27]

    Harris, T

    R. Harris, T. Lanting, A. J. Berkley, J. Johansson, M. W. Johnson, P. Bunyk, E. Ladizinsky, N. Ladizinsky, T. Oh, and S. Han, Phys. Rev. B80, 052506 (2009)

    work page 2009

  28. [28]

    Harris, J

    R. Harris, J. Johansson, A. J. Berkley, M. W. Johnson, T. Lanting, S. Han, P. Bunyk, E. Ladizinsky, T. Oh, I. Perminov, E. Tolkacheva, S. Uchaikin, E. M. Chapple, C. Enderud, C. Rich, M. Thom, J. Wang, B. Wilson, and G. Rose, Phys. Rev. B81, 134510 (2010)

    work page 2010

  29. [29]

    Vuffray, C

    M. Vuffray, C. Coffrin, Y. A. Kharkov, and A. Y. Lokhov, PRX Quantum3, 020317 (2022)

    work page 2022

  30. [30]

    Tüysüz, A

    C. Tüysüz, A. Jayakumar, C. Coffrin, M. Vuffray, and A. Y. Lokhov, Learning response functions of analog quantum computers: analysis of neutral-atom and super- conducting platforms (2025), arXiv:2503.12520 [quant- ph]

    work page arXiv 2025

  31. [31]

    J.Cai, W.G.Macready,andA.Roy,Apracticalheuristic for finding graph minors (2014), arXiv:1406.2741 [quant- ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  32. [32]

    H. N. Djidjev, Quantum Sci. Technol.8, 035013 (2023)

    work page 2023

  33. [33]

    Choi, Quantum Inf

    V. Choi, Quantum Inf. Process.7, 193 (2008)

    work page 2008

  34. [34]

    Choi, Quantum Inf

    V. Choi, Quantum Inf. Process.10, 343 (2011)

    work page 2011

  35. [35]

    Quantum Annealing Implementation of Job-Shop Scheduling

    D. Venturelli, D. J. Marchand, and G. Rojo, Quantum annealing implementation of job-shop scheduling (2015), arXiv:1506.08479 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  36. [36]

    A. D. Kinget al., Nature560, 456 (2018)

    work page 2018

  37. [37]

    Vinci, T

    W. Vinci, T. Albash, G. Paz-Silva, I. Hen, and D. A. Lidar, Phys. Rev. A92, 042310 (2015)

    work page 2015

  38. [38]

    M. R. Zielewski and H. Takizawa, inInt. Conf. High Per- form. Comput. Asia-Pac. Reg.(Association for Comput- ing Machinery, 2022) pp. 137–145

    work page 2022

  39. [39]

    J. King, S. Yarkoni, M. M. Nevisi, J. P. Hilton, and C. C. McGeoch, Benchmarking a quantum annealing processor with the time-to-target metric (2015), arXiv:1508.05087 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  40. [40]

    Hamerly, T

    R. Hamerly, T. Inagaki, P. L. McMahon, D. Venturelli, A. Marandi, T. Onodera, E. Ng, C. Langrock, K. In- aba, T. Honjo, K. Enbutsu, T. Umeki, R. Kasahara, S. Utsunomiya, S. Kako, K. ichi Kawarabayashi, R. L. Byer, M. M. Fejer, H. Mabuchi, D. Englund, E. Rieffel, H. Takesue, and Y. Yamamoto, Sci. Adv.5, eaau0823 (2019)

    work page 2019

  41. [41]

    Lanting, A

    T. Lanting, A. J. Przybysz, A. Y. Smirnov, F. M. Spedalieri, M. H. Amin, A. J. Berkley, R. Harris, F. Al- tomare, S. Boixo, P. Bunyk, N. Dickson, C. Enderud, J. P. Hilton, E. Hoskinson, M. W. Johnson, E. Ladizin- sky, N. Ladizinsky, R. Neufeld, T. Oh, I. Perminov, C. Rich, M. C. Thom, E. Tolkacheva, S. Uchaikin, A. B. Wilson, and G. Rose, Phys. Rev. X4, 0...

    work page 2014

  42. [42]

    Susa and H

    Y. Susa and H. Nishimori, J. Phys. Soc. Jpn.89, 044006 (2020)

    work page 2020

  43. [43]

    Susa and H

    Y. Susa and H. Nishimori, Phys. Rev. A103, 022619 (2021)

    work page 2021

  44. [44]

    Copenhaver, A

    J. Copenhaver, A. Wasserman, and B. Wehefritz- Kaufmann, J. Chem. Phys.154, 034115 (2021)

    work page 2021

  45. [45]

    Different Strategies for Optimization Using the Quantum Adiabatic Algorithm

    E. Crosson, E. Farhi, C. Y.-Y. Lin, H.-H. Lin, and P. Shor, Different strategies for optimization using the 16 quantum adiabatic algorithm (2014), arXiv:1401.7320 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  46. [46]

    N. G. Dickson, M. W. Johnson, M. Amin, R. Harris, F. Altomare, A. J. Berkley, P. Bunyk, J. Cai, E. Chapple, P. Chavez,et al., Nat. Commun.4, 1903 (2013)

    work page 1903

  47. [47]

    Harris, Y

    R. Harris, Y. Sato, A. J. Berkley, M. Reis, F. Al- tomare, M. H. Amin, K. Boothby, P. Bunyk, C. Deng, C. Enderud, S. Huang, E. Hoskinson, M. W. Johnson, E. Ladizinsky, N. Ladizinsky, T. Lanting, R. Li, T. Med- ina, R. Molavi, R. Neufeld, T. Oh, I. Pavlov, I. Perminov, G. Poulin-Lamarre, C. Rich, A. Smirnov, L. Swenson, N. Tsai, M. Volkmann, J. Whittaker, ...

    work page 2018

  48. [48]

    Perdomo-Ortiz, N

    A. Perdomo-Ortiz, N. Dickson, M. Drew-Brook, G. Rose, and A. Aspuru-Guzik, Sci. Rep.2, 571 (2012)

    work page 2012

  49. [49]

    Boixo, T

    S. Boixo, T. Albash, F. M. Spedalieri, N. Chancellor, and D. A. Lidar, Nat. Commun.4, 2067 (2013)

    work page 2067

  50. [50]

    C. R. Harris, K. J. Millman, S. J. Van Der Walt, R. Gom- mers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith,et al., Nature585, 357 (2020)

    work page 2020

  51. [51]

    Anderson, Z

    E. Anderson, Z. Bai, C. Bischof, L. S. Blackford, J. Dem- mel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Ham- marling, A. McKenney,et al.,LAPACK Users’ Guide (SIAM, 1999)

    work page 1999

  52. [52]

    P. R. Halmos,A Hilbert Space Problem Book(Springer Science & Business Media, 2012)

    work page 2012

  53. [53]

    T. S. Santhanam and B. Santhanam, J. Phys. A: Math. Theor.42, 205303 (2009)

    work page 2009

  54. [54]

    Zener, Proc

    C. Zener, Proc. R. Soc. Lond. A137, 696 (1932)

    work page 1932

  55. [55]

    O. V. Ivakhnenko, S. N. Shevchenko, and F. Nori, Phys. Rep.995, 1 (2023)

    work page 2023

  56. [56]

    M. H. S. Amin, P. J. Love, and C. J. S. Truncik, Phys. Rev. Lett.100, 060503 (2008)

    work page 2008

  57. [57]

    Albash, W

    T. Albash, W. Vinci, A. Mishra, P. A. Warburton, and D. A. Lidar, Phys. Rev. A91, 042314 (2015)

    work page 2015

  58. [58]

    Bollobás,Random Graphs, 2nd ed., Cambridge Stud- ies in Advanced Mathematics (Cambridge University Press, 2001)

    B. Bollobás,Random Graphs, 2nd ed., Cambridge Stud- ies in Advanced Mathematics (Cambridge University Press, 2001)

    work page 2001

  59. [59]

    Durrett, inRandom Graph Dynamics, Cambridge Se- ries in Statistical and Probabilistic Mathematics(Cam- bridge University Press, 2006) pp

    R. Durrett, inRandom Graph Dynamics, Cambridge Se- ries in Statistical and Probabilistic Mathematics(Cam- bridge University Press, 2006) pp. 27–69

    work page 2006

  60. [60]

    A. A. Hagberg, D. A. Schult, and P. J. Swart, Proc. Python Sci. Conf (2008)

    work page 2008