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.

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/

6. The Hairball Effect

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)

16. (c) Graph Generator
Additionally:
Vectorisation

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

20. Success by Method

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

24. Appendix : Runtime(MainlyComputingStatistics)

25. Appendix : Varying n'
Dataset: Enron email network http://konect.cc/networks/enron/