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The hourglass effect in hierarchical dependency networks



Many hierarchically modular systems are structured in a way that resembles an hourglass. This “hourglass effect” means that the system generates many outputs from many inputs through a relatively small number of intermediate modules that are critical for the operation of the entire system, referred to as the waist of the hourglass. We investigate the hourglass effect in general, not necessarily layered, hierarchical dependency networks. Our analysis focuses on the number of source-to-target dependency paths that traverse each vertex, and it identifies the core of a dependency network as the smallest set of vertices that collectively cover almost all dependency paths. We then examine if a given network exhibits the hourglass property or not, comparing its core size with a “flat” (i.e., non-hierarchical) network that preserves the source dependencies of each target in the original network. As a possible explanation for the hourglass effect, we propose the Reuse Preference model that captures the bias of new modules to reuse intermediate modules of similar complexity instead of connecting directly to sources or low complexity modules. We have applied the proposed framework in a diverse set of dependency networks from technological, natural, and information systems, showing that all these networks exhibit the general hourglass property but to a varying degree and with different waist characteristics.

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Akhshabi, S., & Dovrolis, C. (2011). The evolution of layered protocol stacks leads to an hourglass-shaped architecture. In ACM SIGCOMM Computer Communication Review, vol. 41, ACM, pp. 206217.
Akhshabi, S., Sarda, S., Dovrolis, C., & Yi, S. (2014). An explanatory evo-devo model for the developmental hourglass. f1000research, 3 (156). doi: 10.12688/f1000research.4583.2.
Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., & Walter, P. (2002). Molecular biology of the cell (4th ed.). New York: Garland Science.
Baldwin, C. Y., & Clark, K. B. (2000). Design rules: The power of modularity. Vol. 1. Cambridge, MA: MIT Press.
Barabási, A.-L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286 (5439), 509512.
Beutler, B. (2004). Inferences, questions and possibilities in toll-like receptor signalling. Nature, 430 (6996), 257263.
Bhardwaj, N., Yan, K.-K., & Gerstein, M. B. (2010). Analysis of diverse regulatory networks in a hierarchical context shows consistent tendencies for collaboration in the middle levels. Proceedings of the National Academy of Sciences, 107 (15), 68416846.
Bhattacharya, P., Iliofotou, M., Neamtiu, I., & Faloutsos, M. (2012). Graph-based analysis and prediction for software evolution. In Proceedings of the 2012 International Conference on Software Engineering, IEEE Press, pp. 419429.
Borgatti, S. P., & Everett, M. G. (2000). Models of core/periphery structures. Social Networks, 21 (4), 375395.
Broder, A., Kumar, R., Maghoul, F., Raghavan, P., Rajagopalan, S., Stata, R., . . . Wiener, J. (2000). Graph structure in the web. Computer Networks, 33 (1), 309320.
Callebaut, W. & Rasskin-Gutman, D. (2005). Modularity: Understanding the Development and Evolution of Natural Complex Systems. Cambridge, MA: MIT Press.
Capocci, A., Servedio, V., Colaiori, F., Buriol, L. S., Donato, D., Leonardi, S., & Caldarelli, G. (2006). Preferential attachment in the growth of social networks: The Internet encyclopedia Wikipedia. Physical Review E, 74 (3), 036116.
Casci, T. (2011). Development: Hourglass theory gets molecular approval. Nature Reviews Genetics, 12 (2), 7676.
Clune, J., Mouret, J.-B., & Lipson, H. (2013). The evolutionary origins of modularity. Proceedings of the Royal Society of London B: Biological Sciences, 280 (1755), 20122863.
Corominas-Murtra, B., Goñi, J., Solé, R., & Rodríguez-Caso, C. (2013). On the origins of hierarchy in complex networks. Proceedings of the National Academy of Sciences, 110 (33), 1331613321.
Csermely, P., London, A., Wu, L.-Y., & Uzzi, B. (2013). Structure and dynamics of core/periphery networks. Journal of Complex Networks, 1 (2), 93123.
Csete, M., & Doyle, J. (2004). Bow ties, metabolism and disease. TRENDS in Biotechnology, 22 (9), 446450.
Domazet-Lošo, T., & Tautz, D. (2010). A phylogenetically based transcriptome age index mirrors ontogenetic divergence patterns. Nature, 468 (7325), 815818.
Easley, D., & Kleinberg, J. (2010). Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge: Cambridge University Press.
Fortuna, M. A., Bonachela, J. A., & Levin, S. A. (2011). Evolution of a modular software network. Proceedings of the National Academy of Sciences, 108 (50), 1998519989.
Fowler, J. H., & Jeon, S. (2008). The authority of supreme court precedent. Social Networks, 30 (1), 1630.
Fowler, J. H., Johnson, T. R., Spriggs, J. F., Jeon, S., & Wahlbeck, P. J. (2007). Network analysis and the law: Measuring the legal importance of precedents at the us supreme court. Political Analysis, 15 (3), 324346.
Friedlander, T., Mayo, A. E., Tlusty, T., & Alon, U. (2015). Evolution of bow-tie architectures in biology. PLoS Computational Biology, 11 (3), e1004055.
Gorman, M. (2015). Codeviz: A callgraph visualiser. Available at:
Gousios, G. (2015). java-callgraph: Java call graph utilities. Available at:
Hinton, G. E., & Salakhutdinov, R. R. (2006). Reducing the dimensionality of data with neural networks. Science, 313 (5786), 504507.
Holme, P. (2005). Core-periphery organization of complex networks. Physical Review E, 72 (4), 046111.
Huang, C.-C., & Kusiak, A. (1998). Modularity in design of products and systems. IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans, 28 (1), 6677.
Ishakian, V., Erdös, D., Terzi, E., & Bestavros, A. (2012). A framework for the evaluation and management of network centrality. In Sdm, SIAM, pp. 427–438.
Jain, S., & Krishna, S. (2002). Large extinctions in an evolutionary model: The role of innovation and keystone species. Proceedings of the National Academy of Sciences, 99 (4), 20552060.
Kanehisa, M., & Goto, S. (2000). Kegg: Kyoto encyclopedia of genes and genomes. Nucleic Acids Research, 28 (1), 2730.
Kanehisa, M., Goto, S., Sato, Y., Kawashima, M., Furumichi, M., & Tanabe, M. (2014). Data, information, knowledge and principle: Back to metabolism in kegg. Nucleic Acids Research, 42 (D1), D199D205.
Kashtan, N., & Alon, U. (2005). Spontaneous evolution of modularity and network motifs. Proceedings of the National Academy of Sciences of the United States of America, 102 (39), 1377313778.
Kashtan, N., Noor, E., & Alon, U. (2007). Varying environments can speed up evolution. Proceedings of the National Academy of Sciences, 104 (34), 1371113716.
Kirschner, M., & Gerhart, J. (1998). Evolvability. Proceedings of the National Academy of Sciences, 95 (15), 84208427.
Kirsten, H., & Hogeweg, P. (2011). Evolution of networks for body plan patterning; interplay of modularity, robustness and evolvability. PLoS Computational Biology, 7 (10), e1002208.
Kitano, H. (2004). Biological robustness. Nature Reviews Genetics, 5 (11), 826837.
Kleinberg, J. M., Kumar, R., Raghavan, P., Rajagopalan, S., & Tomkins, A. S. (1999). The web as a graph: Measurements, models, and methods. In International Computing and Combinatorics Conference. Lecture Notes in Computer Science book series, vol. 1627. Springer, pp. 117.
Krapivsky, P. L., & Redner, S. (2005). Network growth by copying. Physical Review E, 71 (3), 036118.
Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tompkins, A., & Upfal, E. (2000). The web as a graph. In Proceedings of the 19th ACM sigmod-sigact-sigart Symposium on Principles of Database Systems, ACM.
Legal Information Institute, Cornell University Law School. (2015). Historic Supreme Court Decisions. Available at: Accessed: 2015-10-30.
Lorenz, D. M., Jeng, A., & Deem, M. W. (2011). The emergence of modularity in biological systems. Physics of Life Reviews, 8 (2), 129160.
Ma, H.-W., & Zeng, A.-P. (2003). The connectivity structure, giant strong component and centrality of metabolic networks. Bioinformatics, 19 (11), 14231430.
Mengistu, H., Huizinga, J., Mouret, J. B., & Clune, J. (2016). The evolutionary origins of hierarchy. PLoS Computational Biology, 12 (6), e1004829.
Meunier, D., Lambiotte, R., & Bullmore, E. T. (2010). Modular and hierarchically modular organization of brain networks. Frontiers in Neuroscience, 4, 200. doi: 10.3389/fnins.2010.00200.
Mihm, J., Loch, C. H., Wilkinson, D., & Huberman, B. A. (2010). Hierarchical structure and search in complex organizations. Management Science, 56 (5), 831848.
Myers, C. R. (2003). Software systems as complex networks: Structure, function, and evolvability of software collaboration graphs. Physical Review E, 68 (4), 046116.
Nemhauser, G. L., Wolsey, L. A., & Fisher, M. L. (1978). An analysis of approximations for maximizing submodular set functions. Mathematical Programming, 14 (1), 265294.
Newman, M. (2010). Networks: An Introduction. Oxford, UK: Oxford University Press.
Newman, M. E. J., Forrest, S., & Balthrop, J. (2002). Email networks and the spread of computer viruses. Physical Review E, 66 (3), 035101.
Oda, K., & Kitano, H. (2006). A comprehensive map of the toll-like receptor signaling network. Molecular Systems Biology, 2, 2006.0015. doi: 10.1038/msb4100057.
Palsson, B. O. (2015). Systems Biology. Cambridge: Cambridge University Press.
Parnas, D. L., Clements, P. C., & Weiss, D. M. (1984). The modular structure of complex systems. In Proceedings of the 7th International Conference on Software Engineering, IEEE Press, pp. 408417.
Quint, M., Drost, H.-G., Gabel, A., Ullrich, K. K., Bönn, M., & Grosse, I. (2012). A transcriptomic hourglass in plant embryogenesis. Nature, 490 (7418), 98101.
Quiroga, R. Q., Reddy, L., Kreiman, G., Koch, C., & Fried, I. (2005). Invariant visual representation by single neurons in the human brain. Nature, 435 (7045), 11021107.
Ravasz, E., & Barabási, A.-L. (2003). Hierarchical organization in complex networks. Physical Review E, 67 (2), 026112.
Ravasz, E., Somera, A. L., Mongru, D. A., Oltvai, Z. N., & Barabási, A.-L. (2002). Hierarchical organization of modularity in metabolic networks. Science, 297 (5586), 15511555.
Ravindra, K. A., Magnanti, T. L., & Orlin, J. B. (1993). Network flows: Theory, algorithms, and applications. Englewood Cliffs, NJ: Prentice Hall.
Rexford, J., & Dovrolis, C. (2010). Future internet architecture: clean-slate versus evolutionary research. Communications of the ACM, 53 (9), 3640.
Riesenhuber, M., & Poggio, T. (1999). Hierarchical models of object recognition in cortex. Nature Neuroscience, 2 (11), 10191025.
Rombach, M. P., Porter, M. A., Fowler, J. H., & Mucha, P. J. (2014). Core-periphery structure in networks. SIAM Journal on Applied Mathematics, 74 (1), 167190.
Saito, H., Toyoda, M., Kitsuregawa, M., & Aihara, K. (2007). A large-scale study of link spam detection by graph algorithms. In Proceedings of the 3rd International Workshop on Adversarial Information Retrieval on the Web, ACM, pp. 4548.
Sales-Pardo, M., Guimera, R., Moreira, A. A., & Amaral, L. A. N. (2007). Extracting the hierarchical organization of complex systems. Proceedings of the National Academy of Sciences, 104 (39), 1522415229.
Schilling, M. A. (2000). Toward a general modular systems theory and its application to interfirm product modularity. Academy of Management Review, 25 (2), 312334.
Simon, H. A. (1991). The architecture of complexity. In Facets of Systems Science. New York: Springer US, pp. 457476.
Siyari, P., Dilkina, B., & Dovrolis, C. (2016). Lexis: An optimization framework for discovering the hierarchical structure of sequential data. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. KDD '16, New York, NY, USA: ACM, pp. 11851194.
Smolke, C. (2009). The metabolic pathway engineering handbook: Tools and applications, Vol. 2. Boca Raton, FL: CRC Press.
Stelling, J., Sauer, U., Szallasi, Z., Doyle, F., & Doyle, J. (2004). Robustness of cellular functions. Cell, 118 (6), 675685.
Supper, J., Spangenberg, L., Planatscher, H., Dräger, A., Schröder, A., & Zell, A. (2009). Bowtiebuilder: Modeling signal transduction pathways. BMC Systems Biology, 3 (1), 1.
Swaminathan, J. M., Smith, S. F., & Sadeh, N. M. (1998). Modeling supply chain dynamics: A multiagent approach. Decision Sciences, 29 (3), 607632.
Tanaka, R., Csete, M., & Doyle, J. (2005). Highly optimised global organisation of metabolic networks. IEE Proceedings-Systems Biology, 152 (4), 179184.
Tarjan, R. (1972). Depth-first search and linear graph algorithms. SIAM Journal on Computing, 1 (2), 146160.
Valverde, S., & Solé, R. V. (2007). Self-organization versus hierarchy in open-source social networks. Physical Review E, 76 (4), 046118.
Vitali, S., Glattfelder, J. B., & Battiston, S. (2011). The network of global corporate control. PloS One, 6 (10), e25995.
Wagner, G. P., Pavlicev, M., & Cheverud, J. M. (2007). The road to modularity. Nature Reviews Genetics, 8 (12), 921931.
Yan, K.-K., Fang, G., Bhardwaj, N., Alexander, R. P., & Gerstein, M. (2010). Comparing genomes to computer operating systems in terms of the topology and evolution of their regulatory control networks. Proceedings of the National Academy of Sciences, 107 (20), 91869191.
Yu, H., & Gerstein, M. (2006). Genomic analysis of the hierarchical structure of regulatory networks. Proceedings of the National Academy of Sciences, 103 (40), 1472414731.
Yu, H., Kim, P. M., Sprecher, E., Trifonov, V., & Gerstein, M. (2007). The importance of bottlenecks in protein networks: Correlation with gene essentiality and expression dynamics. PLoS Computational Biology, 3 (4), e59.
Zhao, J., Yu, H., Luo, Jian-Hua, Cao, Zhi-Wei, & Li, Yi-Xue. (2006). Hierarchical modularity of nested bow-ties in metabolic networks. BMC Bioinformatics, 7 (1), 386.
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