Abstract
Graph isomorphism is a problem for which there is no known polynomial-time solution. The more general problem of computing graph similarity metrics, graph edit distance or maximum common subgraph, is NP-hard. Nevertheless, assessing (dis)similarity between two or more networks is a key task in many areas, such as image recognition, biology, chemistry, computer and social networks. In this article, we offer a statistical answer to the following questions: (a) “Are networks \(G_1\) and \(G_2\) similar?”, (b) “How different are the networks \(G_1\) and \(G_2\)?” and (c) “Is \(G_3\) more similar to \(G_1\) or \(G_2\)?”. Our comparisons begin with the transformation of each graph into an all-pairs distance matrix. Our node-node distance, Jaccard distance, has been shown to offer an accurate reflection of the graph’s connectivity structure. We then model these distances as probability distributions. Finally, we use well-established statistical tools to gauge the (dis)similarities in terms of probability distribution (dis)similarity. This comparison procedure aims to detect (dis)similarities in connectivity structure and community structure in particular, not in easily observable graph characteristics, such as degrees, edge counts or density. We validate our hypothesis that graphs can be meaningfully summarized and compared via their node-node distance distributions, using several synthetic and real-world graphs. Empirical results demonstrate its validity and the accuracy of our comparison technique.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Akara-pipattana, P., Chotibut, T., Evnin, O.: Resistance distance distribution in large sparse random graphs (2021). arXiv:2107.12561
Bai, Y., Dingand S. Bian, H., Chen, T., Sun, Y., Wang, W.: SimGNN: A Neural Network Approach to Fast Graph Similarity Computation (2018). arXiv:1808.05689
Bunke, H.: Graph matching: Theoretical foundations, algorithms, and applications. Proc. Vision Interf. 21 (2000)
Camby, E., Caporossi, G.: The extended Jaccard distance in complex networks. Les Cahiers du GERAD G-2017-77 (2017)
Chebotarev, P., Shamis, E.: The Matrix-Forest Theorem and Measuring Relations in Small Social Groups. arXiv Mathematics e-prints math/0602070 (2006)
Coupette, C., Vreeken, J.: Graph similarity description: how are these graphs similar? In: Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 185–195. KDD ’21, Association for Computing Machinery, New York, NY, USA (2021). https://doi.org/10.1145/3447548.3467257
Du, Z., Yang, Y., Gao, C., Huang, L., Huang, Q., Bai, Y.: The temporal network of mobile phone users in Changchun Municipality, Northeast China. Sci. Data 5, 180228 (2018)
Fouss, F., Francoisse, K., Yen, L., Pirotte, A., Saerens, M.: An experimental investigation of kernels on graphs for collaborative recommendation and semisupervised classification. Neural Netw. 31, 53–72 (2012). https://www.sciencedirect.com/science/article/pii/S0893608012000822
Grohe, M., Rattan, G., Woeginger, G.: Graph Similarity and Approximate Isomorphism (2018). arXiv:1802.08509
Hagberg, A., Schult, D., Swart, P.: Exploring network structure, dynamics, and function using network X. In: Varoquaux, G., Vaught, T., Millman, J. (eds.), Proceedings of the 7th Python in Science Conference, pp. 11–15. Pasadena, CA USA (2008)
Han, J.: Autonomous systems graphs (2016). https://doi.org/10.7910/DVN/XLGMJR
Huang, S., Hitti, Y., Rabusseau, G., Rabbany, R.: Laplacian Change Point Detection for Dynamic Graphs (2020). arXiv:2007.01229
Jaccard, P.: Étude de la distribution florale dans une portion des Alpes et du Jura. Bulletin de la Société Vaudoise des Sciences Naturelles 37, 547–579 (1901)
Tang, J., Leontiadis, I., Scellato, S., Nicosia, V., Mascolo, C., M. Musolesi, M., Latora, V.: Applications of temporal graph metrics to real-world networks. In: Temporal Networks, p. 135 (2013)
Koutra, D., Parikh, A., Ramdas, A., Xiang, J.: Algorithms for graph similarity and subgraph matching (2011). http://www.cs.cmu.edu/jingx/docs/DBreport.pdf. Accessed on 01 Dec 2015
von Luxburg, U., Radl, A., Hein, M.: Getting lost in space: large sample analysis of the resistance distance. In: Lafferty, J.D., Williams, C.K.I., Shawe-Taylor, J., Zemel, R.S., Culotta, A. (eds.), Advances in Neural Information Processing Systems 23, pp. 2622–2630. Curran Associates, Inc. (2010). http://papers.nips.cc/paper/3891-getting-lost-in-space-large-sample-analysis-of-the-resistance-distance.pdf
von Luxburg, U., Radl, A., Hein, M.: Hitting and commute times in large random neighborhood graphs. J. Mach. Learn. Res. 15(52), 1751–1798 (2014). http://jmlr.org/papers/v15/vonluxburg14a.html
Maduako, I., Wachowicz, M., Hanson, T.: STVG: an evolutionary graph framework for analyzing fast-evolving networks. J. Big Data 6 (2019)
Miasnikof, P., Shestopaloff, A.Y., Pitsoulis, L., Ponomarenko, A.: An empirical comparison of connectivity-based distances on a graph and their computational scalability. J. Complex Netw. 10(1) (2022). https://doi.org/10.1093/comnet/cnac003
Miasnikof, P., Shestopaloff, A.Y., Pitsoulis, L., Ponomarenko, A., Lawryshyn, Y.: Distances on a graph. In: Benito, R.M., Cherifi, C., Cherifi, H., Moro, E., Rocha, L.M., Sales-Pardo, M. (eds.) Complex Networks & Their Applications IX, pp. 189–199. Springer International Publishing, Cham (2021)
Perozzi, B., Al-Rfou, R., Skiena, S.: DeepWalk: Online Learning of Social Representations (2014). arXiv:1403.6652
Ponomarenko, A., Pitsoulis, L., Shamshetdinov, M.: Overlapping community detection in networks based on link partitioning and partitioning around medoids. PLOS One 16(8), 1–43 (2021). https://doi.org/10.1371/journal.pone.0255717
Schieber, T., Carpi, L., Diaz-Guilera, A., Pardalos, P., Masoller, C., Ravetti, M.: Quantification of network structural dissimilarities. Nat. Commun. 8, 13928 (2017)
Shrivastava, N., Majumder, A., Rastogi, R.: In: 2008 IEEE 24th International Conference on Data Engineering, pp. 486–495 (2008)
Tang, J., Mascolo, C., Musolesi, M., Latora, V.: Exploiting temporal complex network metrics in mobile malware containment. In: 2011 IEEE International Symposium on a World of Wireless, Mobile and Multimedia Networks, pp. 1–9 (2011)
Wang, Z., Zhan, X.X., Liu, C., Zhang, Z.K.: Quantification of network structural dissimilarities based on network embedding. iScience 104446 (2022). https://www.sciencedirect.com/science/article/pii/S2589004222007179
Yan, H., Zhang, Q., Mao, D., Lu, Z., Guo, D., Chen, S.: Anomaly detection of network streams via dense subgraph discovery. In: 2021 International Conference on Computer Communications and Networks (ICCCN), pp. 1–9 (2021)
Ying, X., Wu, X., Barbará, D.: Spectrum based fraud detection in social networks. In: 2011 IEEE 27th International Conference on Data Engineering, pp. 912–923 (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Miasnikof, P., Shestopaloff, A.Y., Bravo, C., Lawryshyn, Y. (2023). Statistical Network Similarity. In: Cherifi, H., Mantegna, R.N., Rocha, L.M., Cherifi, C., Micciche, S. (eds) Complex Networks and Their Applications XI. COMPLEX NETWORKS 2016 2022. Studies in Computational Intelligence, vol 1078. Springer, Cham. https://doi.org/10.1007/978-3-031-21131-7_25
Download citation
DOI: https://doi.org/10.1007/978-3-031-21131-7_25
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-21130-0
Online ISBN: 978-3-031-21131-7
eBook Packages: EngineeringEngineering (R0)