Abstract
Stochastic population models are widely used to model phenomena in different areas such as chemical kinetics or collective animal behaviour. Quantitative analysis of stochastic population models easily becomes challenging, due to the combinatorial propagation of dependencies across the population. The complexity becomes especially prominent when model’s parameters are not known and available measurements are limited. In this paper, we illustrate this challenge in a concrete scenario: we assume a simple communication scheme among identical individuals, inspired by how social honeybees emit the alarm pheromone to protect the colony in case of danger. Together, n individuals induce a population Markov chain with n parameters. In addition, we assume to be able to experimentally observe the states only after the steady-state is reached. In order to obtain the parameters of the individual’s behaviour, by utilising the data measurements for population, we combine two existing techniques. First, we use the tools for parameter synthesis for Markov chains with respect to temporal logic properties, and then we employ CEGAR-like reasoning to find the viable parameter space up to desired coverage. We report the performance on a number of synthetic data sets.
TP’s research is supported by the Ministry of Science, Research and the Arts of the state of Baden-Württemberg, and the DFG Centre of Excellence 2117 ‘Centre for the Advanced Study of Collective Behaviour’ (ID: 422037984), MH’s research is supported by Young Scholar Fund (YSF), project no. \(P83943018 FP 430\_/18\). MN’s research is supported by the Mentorship grant from the Zukunftskolleg. DŠ’s research is supported by the Czech Grant Agency grant no. GA18-00178S.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In our case, ‘help’ does not involve interaction between agents, - it is simultaneously broadcasted from an agent to all the others.
- 2.
In general, the reachability probabilities for a pMC can be expressed by rational functions; In our case study, polynomials will suffice because the underlying transition system is acyclic.
- 3.
If the coverage is not set below 50%.
References
Alistarh, D., Gelashvili, R., Vojnović, M.: Fast and exact majority in population protocols. In: Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, pp. 47–56. ACM (2015)
Aspnes, J., Ruppert, E.: An introduction to population protocols. In: Garbinato, B., Miranda, H., Rodrigues, L. (eds.) Middleware for Network Eccentric and Mobile Applications, pp. 97–120. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-540-89707-1_5
Backenköhler, M., Bortolussi, L., Wolf, V.: Generalized method of moments for stochastic reaction networks in equilibrium. In: Bartocci, E., Lio, P., Paoletti, N. (eds.) CMSB 2016. LNCS, vol. 9859, pp. 15–29. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-45177-0_2
Backenkohler, M., Bortolussi, L., Wolf, V.: Moment-based parameter estimation for stochastic reaction networks in equilibrium. IEEE/ACM Trans. Comput. Biol. Bioinf. 15(4), 1180–1192 (2018)
Bartocci, E., Bortolussi, L., Nenzi, L., Sanguinetti, G.: System design of stochastic models using robustness of temporal properties. Theor. Comput. Sci. 587, 3–25 (2015)
Bortolussi, L., Cardelli, L., Kwiatkowska, M., Laurenti, L.: Approximation of probabilistic reachability for chemical reaction networks using the linear noise approximation. In: Agha, G., Van Houdt, B. (eds.) QEST 2016. LNCS, vol. 9826, pp. 72–88. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-43425-4_5
Bortolussi, L., Hillston, J., Latella, D., Massink, M.: Continuous approximation of collective system behaviour: a tutorial. Perform. Eval. 70(5), 317–349 (2013)
Bortolussi, L., Sanguinetti, G.: Learning and designing stochastic processes from logical constraints. In: Joshi, K., Siegle, M., Stoelinga, M., D’Argenio, P.R. (eds.) QEST 2013. LNCS, vol. 8054, pp. 89–105. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40196-1_7
Bortolussi, L., Silvetti, S.: Bayesian statistical parameter synthesis for linear temporal properties of stochastic models. In: Beyer, D., Huisman, M. (eds.) TACAS 2018. LNCS, vol. 10806, pp. 396–413. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-89963-3_23
Brim, L., Češka, M., Dražan, S., Šafránek, D.: Exploring parameter space of stochastic biochemical systems using quantitative model checking. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 107–123. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39799-8_7
Češka, M., Dannenberg, F., Paoletti, N., Kwiatkowska, M., Brim, L.: Precise parameter synthesis for stochastic biochemical systems. Acta Informatica 54(6), 589–623 (2017)
Daca, P., Henzinger, T.A., Křetínský, J., Petrov, T.: Faster statistical model checking for unbounded temporal properties. ACM Trans. Comput. Log. (TOCL) 18(2), 12 (2017)
Daws, C.: Symbolic and parametric model checking of discrete-time Markov Chains. In: Liu, Z., Araki, K. (eds.) ICTAC 2004. LNCS, vol. 3407, pp. 280–294. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-31862-0_21
de Moura, L., Bjørner, N.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78800-3_24
Dehnert, C., et al.: PROPhESY: a PRObabilistic ParamEter SYnthesis tool. In: Kroening, D., Păsăreanu, C.S. (eds.) CAV 2015. LNCS, vol. 9206, pp. 214–231. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21690-4_13
Dehnert, C., Junges, S., Katoen, J.-P., Volk, M.: A Storm is coming: a modern probabilistic model checker. In: Majumdar, R., Kunčak, V. (eds.) CAV 2017. LNCS, vol. 10427, pp. 592–600. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63390-9_31
Dorigo, M., Birattari, M., Blum, C., Clerc, M., Stützle, T., Winfield, A.: Ant Colony Optimization and Swarm Intelligence, vol. 5217. Springer, Heidelberg (2008)
Kluyver, T., et al.: Jupyter notebooks - a publishing format for reproducible computational workflows. In: Positioning and Power in Academic Publishing: Players, Agents and Agendas, pp. 87–90. IOS Press (2016)
Giacobbe, M., Guet, C.C., Gupta, A., Henzinger, T.A., Paixão, T., Petrov, T.: Model checking the evolution of gene regulatory networks. Acta Informatica 54(8), 765–787 (2017)
Giardina, I.: Collective behavior in animal groups: theoretical models and empirical studies. HFSP J. 2(4), 205–219 (2008)
Hansen, L.P.: Large sample properties of generalized method of moments estimators. Econometrica 50, 1029–1054 (1982)
Hillston, J.: Challenges for quantitative analysis of collective adaptive systems. In: Abadi, M., Lluch Lafuente, A. (eds.) TGC 2013. LNCS, vol. 8358, pp. 14–21. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-05119-2_2
Jansen, N., et al.: Accelerating parametric probabilistic verification. In: Norman, G., Sanders, W. (eds.) QEST 2014. LNCS, vol. 8657, pp. 404–420. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-10696-0_31
Katoen, J.-P.: The probabilistic model checking landscape. In: Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, pp. 31–45. ACM (2016)
Kwiatkowska, M., Norman, G., Parker, D.: PRISM 4.0: verification of probabilistic real-time systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 585–591. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22110-1_47
Loreti, M., Hillston, J.: Modelling and analysis of collective adaptive systems with CARMA and its tools. In: Bernardo, M., De Nicola, R., Hillston, J. (eds.) SFM 2016. LNCS, vol. 9700, pp. 83–119. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-34096-8_4
Mai, M., et al.: Monitoring pre-seismic activity changes in a domestic animal collective in central Italy. In: EGU General Assembly Conference Abstracts, vol. 20, p. 19348 (2018)
Nouvian, M., Reinhard, J., Giurfa, M.: The defensive response of the honeybee Apis mellifera. J. Exp. Biol. 219(22), 3505–3517 (2016)
Daca, P., Henzinger, T.A., Křetínský, J., Petrov, T.: Faster statistical model checking for unbounded temporal properties. In: Chechik, M., Raskin, J.-F. (eds.) TACAS 2016. LNCS, vol. 9636, pp. 112–129. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49674-9_7
Polgreen, E., Wijesuriya, V.B., Haesaert, S., Abate, A.: Data-efficient Bayesian verification of parametric Markov Chains. In: Agha, G., Van Houdt, B. (eds.) QEST 2016. LNCS, vol. 9826, pp. 35–51. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-43425-4_3
Quatmann, T., Dehnert, C., Jansen, N., Junges, S., Katoen, J.-P.: Parameter synthesis for Markov models: faster than ever. In: Artho, C., Legay, A., Peled, D. (eds.) ATVA 2016. LNCS, vol. 9938, pp. 50–67. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-46520-3_4
Schnoerr, D., Sanguinetti, G., Grima, R.: Approximation and inference methods for stochastic biochemical Kinetics–a tutorial review. J. Phys. A: Math. Theor. 50(9), 093001 (2017)
Shorter, J.R., Rueppell, O.: A review on self-destructive defense behaviors in social insects. Insectes Soc. 59(1), 1–10 (2012)
Sokolova, A., de Vink, E.P.: Probabilistic automata: system types, parallel composition and comparison. In: Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P., Siegle, M. (eds.) Validation of Stochastic Systems. LNCS, vol. 2925, pp. 1–43. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24611-4_1
Stoelinga, M.: An introduction to probabilistic automata. Bull. EATCS 78(176–198), 2 (2002)
Česka, M., Šafránek, D., Dražan, S., Brim, L.: Robustness analysis of stochastic biochemical systems. PLoS ONE 9(4), 1–23 (2014)
Wu, S.-H., Smolka, S.A., Stark, E.W.: Composition and behaviors of probabilistic I/O automata. Theor. Comput. Sci. 176(1–2), 1–38 (1997)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
A Performance Comparison
A Performance Comparison
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Hajnal, M., Nouvian, M., Šafránek, D., Petrov, T. (2019). Data-Informed Parameter Synthesis for Population Markov Chains. In: Češka, M., Paoletti, N. (eds) Hybrid Systems Biology. HSB 2019. Lecture Notes in Computer Science(), vol 11705. Springer, Cham. https://doi.org/10.1007/978-3-030-28042-0_10
Download citation
DOI: https://doi.org/10.1007/978-3-030-28042-0_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-28041-3
Online ISBN: 978-3-030-28042-0
eBook Packages: Computer ScienceComputer Science (R0)