Succinct Summarisation of Large Networks via Small Synthetic Representative Graphs
Talk by Jérôme Kunegis at BeNet'17 in Gent. In this talk, Jérôme Kunegis presented ongoing work, see the abstract below. Abstract: We describe a method for visually summarising the structure of large network datasets that works by drawing smaller graphs generated to have similar structural properties to the input graphs. Visualising complex networks is crucial to understand and make sense of networked data and the relationships it represents. Due to the large size of many networks, visualisation is extremely difficult; the simple method of \emph{drawing} large networks like those of Facebook or Twitter leads to graphics that convey little or no information. While modern graph layout algorithms can scale computationally to large networks, their output tends to a common \emph{hairball} look, which makes it difficult to even distinguish different graphs from each other. Graph sampling and graph coarsening techniques partially address these limitations but they are only able to preserve a subset of the properties of the original graphs. In this paper we take the problem of visualising large graphs from a novel perspective: we leave the original graph's nodes and edges behind, and instead summarise its properties such as the clustering coefficient and bipartivity by generating a completely new graph whose structural properties match that of the original graph. To verify the utility of this approach as compared to other graph visualisation algorithms, we perform an experimental evaluation in which we repeatedly asked experimental subjects (professionals in graph mining and related areas) to determine which of two given graphs has a given structural property and then assess which visualisation algorithm helped in identifying the correct answer. Our summarisation approach SynGraphy compares favourably to other techniques on a variety of networks.
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Succinct Summarisation of Large Networks via Small Synthetic Representative Graphs
- 1. Jérôme KUNEGIS Pawan KUMAR SŪN Jùn Giuseppe PIRRÒ Anna SAMOILENKO 6 December 2017, BeNet'17 Succinct Summarisation of Large Networks via Small Synthetic Representative Graphs
- 2. Who Can Answer This? G = ({1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}, {{1,2}, {1,5}, {1,8}, {2,3}, {2,10}, {3,4}, {3,12}, {4,5}, {4,14}, {5,6}, {6,7}, {6,15}, {7,8}, {7,17}, {8,9}, {9,10}, {9,18}, {10,11}, {11,12}, {11,19}, {12,13}, {13,14}, {13,20}, {14,15}, {15,16}, {16,17}, {16,20}, {17,18}, {18,19}, {19,20}}) What structure does this graph have? What are its symmetries? Find central nodes! Find clusters!
- 3. Graph Drawing Using eigenvectors of the Laplacian matrix Dodecahedron!
- 4. Drawing Real-World Networks Twitter user–user following, Fruchterman–Reingold algorithm (1991, Software: Pract and Exp 21(11):1129–1164) http://konect.cc/networks/ego-twitter/
- 5. Drawing Real-World Networks Gnutella connection network, „Delaunay“ graph drawing method http://konect.cc/networks/p2p-Gnutella05/
- 7. Graph Visualisation, Not Graph Drawing Idea: ● We don't need to show the exact nodes and edges for visualisation ● Only want to show global graph properties ⇒ Summarise the graph
- 8. Input: G = (V, E) (large graph) Global graph properties / measures / statistics Scaled down properties / measures / statistics Small, representative graph G' (n = 100) Output: Drawing of G' (d) Fruchterman–Reingold (c) Graph generator (b) Scaling algorithm [2 variants] (a) Measure properties
- 9. (a) Measure Properties Subgraph counts: Number of . . . ● Nodes [a.k.a. 0-paths, 0-stars, 1-cliques, 1-cycles] (n) ● Edges [a.k.a. 1-paths, 1-stars, 2-cliques] (m) ● Wedges [a.k.a. 2-stars, cherries] (s) ● Claws [a.k.a. 3-stars] (z) ● Crosses [a.k.a. 4-stars] (x) ● Triangles [a.k.a. 3-cliques, 3-cycles] (t) ● Squares [a.k.a. 4-cycles] (q)
- 10. (b) Scaling of Statistics Number of nodes is always set to n' = 100 Other statistics need to be scaled: (SI) By variable change (NO) Empirically
- 11. (SI) Scaling by Change to Size- Independent Variables The average degree d is a size-independent proxy for the edge count m.
- 12. (SI) Other Statistics Assuming an unchanging degree distribution
- 13. (SI) Other Statistics Assuming a constant clustering coefficient c = 3t / s Assuming a constant 4-clustering coefficient y = 4q / P₃ Definition: q = Number of squares ; P = Number of paths of length three₃
- 14. (NO) Normal Empirical Scaling
- 15. (NO) Normal Empirical Scaling Let x be the vector of all log-statistics. Fit f over a collection of known real-world networks (http://KONECT.cc). α n' G G'α' f n(for α being any statistic except n)
- 17. (d) Graph Drawing Use the Fruchterman–Reingold algorithm, which works well for small graphs
- 18. Example Reactome network: http://konect.cc/networks/reactome/ Baselines: FR Fruchterman–Reingold ; LA Laplacian ; SU Uniform vertex sampling ; SN Node sampling Our methods: NO Normal empirical ; SI Change to size-indepedent variables
- 19. User Experiments Website is now down
- 21. Success by Statistic Legend: d = Avg. degree ; s = Wedge count ; c = Clustering coeff. ; δ = Diameter ; z = Claw count ρ = Degree assortativity ; q = Square count ; y = 4-Clustering coefficient More at http://konect.cc/statistics/
- 22. Gallery Questions? Jérôme KUNEGIS et al., Univ. of Namur
- 23. Appendix : All Experimental Results
- 25. Appendix : Varying n' Dataset: Enron email network http://konect.cc/networks/enron/