BDD-Based Algorithm for SCC Decomposition of Edge-Coloured Graphs

Edge-coloured directed graphs provide an essential structure for modelling and computing complex problems arising in many scientific disciplines. The size of edge-coloured graphs appearing in practice can be enormous in the number of both vertices and colours. An important fundamental problem that needs to be solved over edge-coloured graphs is detecting strongly connected components. The problem becomes challenging for large graphs with a large number of colours. In this paper, we describe a novel symbolic algorithm that computes all the monochromatic strongly connected components of an edge-coloured graph. In the worst case, the algorithm performs O(p ·n · logn) symbolic steps, where p is the number of colours and n is the number of vertices. We evaluate the algorithm using an experimental implementation based on Binary Decision Diagrams (BDDs) and large (up to 2) coloured graphs produced by models appearing in systems biology.

[1]  D. Irons,et al.  Logical analysis of the budding yeast cell cycle. , 2009, Journal of theoretical biology.

[2]  E. Furlong,et al.  Logical modelling of Drosophila signalling pathways. , 2013, Molecular bioSystems.

[3]  Gregor Gößler,et al.  Efficient parameter search for qualitative models of regulatory networks using symbolic model checking , 2010, Bioinform..

[4]  Zhe Zhu,et al.  Efficient decomposition of strongly connected components on GPUs , 2014, J. Syst. Archit..

[5]  Alex Madrahimov,et al.  The Cell Collective: Toward an open and collaborative approach to systems biology , 2012, BMC Systems Biology.

[6]  Agustino Martínez-Antonio,et al.  Modeling Asymmetric Cell Division in Caulobacter crescentus Using a Boolean Logic Approach. , 2017, Results and problems in cell differentiation.

[7]  Steffen Klamt,et al.  A methodology for the structural and functional analysis of signaling and regulatory networks , 2006, BMC Bioinformatics.

[8]  Simona Orzan,et al.  On Distributed Verification and Verified Distribution , 2004 .

[9]  Jun Pang,et al.  Taming Asynchrony for Attractor Detection in Large Boolean Networks , 2017, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[10]  Alfons Laarman,et al.  Multi-core on-the-fly SCC decomposition , 2016, PPoPP.

[11]  Mária Ercsey-Ravasz,et al.  Principles of dynamical modularity in biological regulatory networks , 2016, Scientific Reports.

[12]  John H. Reif,et al.  Depth-First Search is Inherently Sequential , 1985, Inf. Process. Lett..

[13]  M. Scheffer,et al.  Trajectories of the Earth System in the Anthropocene , 2018, Proceedings of the National Academy of Sciences.

[14]  Gianfranco Ciardo,et al.  The saturation algorithm for symbolic state-space exploration , 2006, International Journal on Software Tools for Technology Transfer.

[15]  Saieed Akbari,et al.  Multicolored trees in complete graphs , 2007, J. Graph Theory.

[16]  Dietmar Dorninger,et al.  Hamiltonian circuits determining the order of chromosomes , 1994, Discret. Appl. Math..

[17]  Andrew Thomason,et al.  Complete graphs with no rainbow path , 2007, J. Graph Theory.

[18]  Lubos Brim,et al.  Symbolic Coloured SCC Decomposition , 2021, TACAS.

[19]  Thomas A. Henzinger,et al.  Model checking the evolution of gene regulatory networks , 2016, Acta Informatica.

[20]  Lubos Brim,et al.  Computing Strongly Connected Components in Parallel on CUDA , 2011, 2011 IEEE International Parallel & Distributed Processing Symposium.

[21]  Aurélien Naldi,et al.  Logical modelling of gene regulatory networks with GINsim. , 2012, Methods in molecular biology.

[22]  A. E. Matouk,et al.  Complex dynamics in susceptible-infected models for COVID-19 with multi-drug resistance , 2020, Chaos, Solitons & Fractals.

[23]  Bin Jiang,et al.  I/O-and CPU-Optimal Recognition of Strongly Connected Components , 1993, Inf. Process. Lett..

[24]  Sivasankaran Rajamanickam,et al.  BFS and Coloring-Based Parallel Algorithms for Strongly Connected Components and Related Problems , 2014, 2014 IEEE 28th International Parallel and Distributed Processing Symposium.

[25]  Rachid Saad,et al.  Paths and trails in edge-colored graphs , 2008, Theor. Comput. Sci..

[26]  Emmanuel Barillot,et al.  Mathematical Modelling of Molecular Pathways Enabling Tumour Cell Invasion and Migration , 2015, PLoS Comput. Biol..

[27]  Dragan Bosnacki,et al.  GPU-Based Graph Decomposition into Strongly Connected and Maximal End Components , 2014, CAV.

[28]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[29]  Lubos Brim,et al.  Model Checking of Biological Systems , 2013, SFM.

[30]  Carla Piazza,et al.  Symbolic Graphs: Linear Solutions to Connectivity Related Problems , 2007, Algorithmica.

[31]  Gregory Gutin,et al.  Alternating cycles and paths in edge-coloured multigraphs: A survey , 1994, Discret. Math..

[32]  Gianfranco Ciardo,et al.  Symbolic computation of strongly connected components and fair cycles using saturation , 2011, Innovations in Systems and Software Engineering.

[33]  M. Sharir,et al.  A strong-connectivity algorithm and its applications in data flow analysis. , 2018 .

[34]  E. Aurell,et al.  Dynamics inside the cancer cell attractor reveal cell heterogeneity, limits of stability, and escape , 2016, Proceedings of the National Academy of Sciences.

[35]  G. Palm,et al.  Predicting Variabilities in Cardiac Gene Expression with a Boolean Network Incorporating Uncertainty , 2015, PloS one.

[36]  Xueliang Li,et al.  Monochromatic and Heterochromatic Subgraphs in Edge-Colored Graphs - A Survey , 2008, Graphs Comb..

[37]  Lubos Brim,et al.  Detecting Attractors in Biological Models with Uncertain Parameters , 2017, CMSB.

[38]  S. Kauffman.  Metabolic stability and epigenesis in randomly constructed genetic nets. , 1969, Journal of theoretical biology.

[39]  Jean-Michel Couvreur,et al.  Hierarchical Decision Diagrams to Exploit Model Structure , 2005, FORTE.

[40]  Jun Pang,et al.  A new decomposition-based method for detecting attractors in synchronous Boolean networks , 2019, Sci. Comput. Program..

[41]  Jaco van de Pol,et al.  Distributed Algorithms for SCC Decomposition , 2011, J. Log. Comput..

[42]  Julian D. Schwab,et al.  Concepts in Boolean network modeling: What do they all mean? , 2020, Computational and structural biotechnology journal.

[43]  Fabio Somenzi,et al.  An Algorithm for Strongly Connected Component Analysis in n log n Symbolic Steps , 2000, Formal Methods Syst. Des..

[44]  Adrien Richard,et al.  Graph-Based Modeling of Biological Regulatory Networks: Introduction of Singular States , 2004, CMSB.

[45]  Kwang-Hyun Cho,et al.  An efficient algorithm for identifying primary phenotype attractors of a large-scale Boolean network , 2016, BMC Systems Biology.

[46]  Lawrence Rauchwerger,et al.  Identifying Strongly Connected Components in Parallel , 2000, PPSC.

[47]  N. Alon,et al.  Properly colored Hamilton cycles in edge-colored complete graphs , 1997, Random Struct. Algorithms.

[48]  Ioannis Xenarios,et al.  A method for the generation of standardized qualitative dynamical systems of regulatory networks , 2005, Theoretical Biology and Medical Modelling.

[49]  R. Thomas,et al.  Boolean formalization of genetic control circuits. , 1973, Journal of theoretical biology.

[50]  Kunle Olukotun,et al.  On fast parallel detection of strongly connected components (SCC) in small-world graphs , 2013, 2013 SC - International Conference for High Performance Computing, Networking, Storage and Analysis (SC).

[51]  Lubos Brim,et al.  Formal Analysis of Qualitative Long-Term Behaviour in Parametrised Boolean Networks , 2019, ICFEM.

[52]  Randal E. Bryant,et al.  Graph-Based Algorithms for Boolean Function Manipulation , 1986, IEEE Transactions on Computers.

[53]  David Šafránek,et al.  AEON: Attractor Bifurcation Analysis of Parametrised Boolean Networks , 2020, CAV.

[54]  Peter A. Beerel,et al.  Implicit enumeration of strongly connected components and anapplication to formal verification , 2000, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[55]  Joshua E. S. Socolar,et al.  Global control of cell-cycle transcription by coupled CDK and network oscillators , 2008, Nature.

[56]  Zoltán Király,et al.  Monochromatic components in edge-colored complete uniform hypergraphs , 2011, Eur. J. Comb..

[57]  Krishnendu Chatterjee,et al.  Lower Bounds for Symbolic Computation on Graphs: Strongly Connected Components, Liveness, Safety, and Diameter , 2017, SODA.

[58]  Edmund M. Clarke,et al.  Symbolic Model Checking: 10^20 States and Beyond , 1990, Inf. Comput..

[59]  Carla Piazza,et al.  Computing strongly connected components in a linear number of symbolic steps , 2003, SODA '03.