Visualizing Correlation Networks

Transcription

1 Visualizing Correlation Networks Jürgen Lerner University of Konstanz Workshop on Networks in Finance Barcelona, 4 May 2012

2 From correlation matrices to network visualization. correlation distance coordinates graphical representation (r ij ) i,j=1,...,n (d ij ) i,j=1,...,n (p (1) i, p (2) i ) i=1,...,n BMW.logret DAI.logret Time Time Typically not realizable in 2-dim. Optimal 2-dim approximation of (d ij ). Main topic: from pairwise distances to coordinates. 1. General framework: network vis. by stress minimization. 2. Preserving the mental map in dynamic network vis. 3. Wrap-up, open issues, discussion,...

3 From correlation matrices to network visualization. correlation distance coordinates graphical representation (r ij ) i,j=1,...,n (d ij ) i,j=1,...,n (p (1) i, p (2) i ) i=1,...,n BMW.logret DAI.logret Time Time Typically not realizable in 2-dim. Optimal 2-dim approximation of (d ij ). Main topic: from pairwise distances to coordinates. 1. General framework: network vis. by stress minimization. 2. Preserving the mental map in dynamic network vis. 3. Wrap-up, open issues, discussion,...

4 Reducing (d ij ) to a minimum spanning tree (MST). Connected, acyclic subgraph with minimum sum of distances. source: screenshot from Some issues arise distances of most pairs are ignored; arbitrary cyclic ordering of subtrees around a node; potentially unstable in dynamic scenarios.

5 MST from two consecutive intervals in June 21st November 07th June 22nd November 08th source: screenshots from

6 MST from two consecutive intervals in June 21st November 07th June 22nd November 08th data source: layout: v isone

7 What causes this change in the MST? Cycle is broken up at different edges. (ALV,BAS) (BAS,SIE) (SIE,DPW) (DPW,ALV) 0621/ / Dyad with minimum correlation on this cycle changed this is reflected in the MST.

8 Network visualization by stress minimization.

9 Network visualization by stress minimization. Find coordinates P = (p i ) i=1,...,n minimizing stress(p) = i<j w ij ( pi p j d ij ) 2. d ij optimal distance between node i and j; w ij weighting the deviation from d ij ; often w ij = 1/d 2 ij ; Note: MST drawings fit into this framework. Established in multidimensional scaling and graph drawing [Kruskal and Seery, 1980]; [Kamada and Kawai, 1988]. Efficient iterative optimization through stress majorization [Gansner et al., 2004]. Empirically good results when initializing with classical scaling [Brandes and Pich, 2009].

10 Network visualization by stress minimization. Find coordinates P = (p i ) i=1,...,n minimizing stress(p) = i<j w ij ( pi p j d ij ) 2. d ij optimal distance between node i and j; w ij weighting the deviation from d ij ; often w ij = 1/d 2 ij ; Note: MST drawings fit into this framework. Established in multidimensional scaling and graph drawing [Kruskal and Seery, 1980]; [Kamada and Kawai, 1988]. Efficient iterative optimization through stress majorization [Gansner et al., 2004]. Empirically good results when initializing with classical scaling [Brandes and Pich, 2009].

11 Network visualization by stress minimization. Find coordinates P = (p i ) i=1,...,n minimizing stress(p) = i<j w ij ( pi p j d ij ) 2. d ij optimal distance between node i and j; w ij weighting the deviation from d ij ; often w ij = 1/d 2 ij ; Note: MST drawings fit into this framework. Established in multidimensional scaling and graph drawing [Kruskal and Seery, 1980]; [Kamada and Kawai, 1988]. Efficient iterative optimization through stress majorization [Gansner et al., 2004]. Empirically good results when initializing with classical scaling [Brandes and Pich, 2009].

12 Correlations among DAX stocks in coordinates obtained by stress minimization data source: layout: v isone

13 Correlations among DAX stocks in after dividing prices by daily sum of prices data source: layout: v isone

14 Correlations among DAX stocks in all nodes are pairwise connected do not draw the edges data source: layout: v isone

15 Correlations among DAX stocks in or, at least, reduce label overlap data source: layout: v isone

16 Correlation networks from consecutive time intervals. recall: MST visualization June 21st November 07th June 22nd November 08th Huge structural change from one day to the next.

17 Stress minimization of the same intervals. June 21st November 07th June 22nd November 08th Almost no change from one day to the next.

18 Preserving the mental map in dynamic network visualization.

19 Tradeoff between individual quality and stability. So far, individual layouts have been computed separately. June 21st November 07th June 22nd November 08th Try to add stability over time while not loosing too much individual quality.

20 Dynamic layout techniques. [Brandes and Mader, 2012] Aggregation Anchoring Linking Positions do not change over time. Nodes anchored at reference positions. Nodes linked to copies of themselves. Another possibility: start standard layout algorithm at previous positions empirically leads to inferior results.

21 Dynamic layout techniques. [Brandes and Mader, 2012] Aggregation Anchoring Linking Positions do not change over time. Nodes anchored at reference positions. Nodes linked to copies of themselves. Another possibility: start standard layout algorithm at previous positions empirically leads to inferior results.

22 Can be implemented by modifying the stress function. Aggregation stress ( P) ( = ω ij δij p i p j ) 2 i<j V Anchoring stress A ( ) α P (t) = (1 α) stress ( P (t)) + α p (t) 2 i p i i V Linking stress L ( α P (1),..., P (T )) = T t=1 (1 α) stress ( P (t)) + α i V T ζ(t, t ) p (t) i p (t ) i t =1 t t 2

23 Dynamic layout (linking) of MSTs. June 21st November 07th June 22nd November 08th Changes and stable substructures are more easily recognizable.

24 Wrap-up, open issues, discussion,...

25 Discussion points. Network visualization by stress minimization stress(p) = i<j w ij ( pi p j d ij ) 2. Claim: methodological framework is general enough. However, some choices have to be made. How to define the optimal distance d ij to represent pairwise correlations r ij? How to choose the weights w ij (importance of edges)? Enforce layout stability to preserve the mental map? Use of additional graphical variables (shape, color, size, link width,... ) to enhance readability. Experimentation could provide answers; compare [Brandes and Pich, 2009] and [Brandes and Mader, 2012].

26 Some thoughts on defining (d ij ) and (w ij ). Network visualization by stress minimization stress(p) = i<j w ij ( pi p j d ij ) 2. The optimal distance d ij and weight (importance) w ij of the dyad (i, j) can be defined independently of other dyads,... or based on some structural arguments, e. g., weight edges of the MST more heavily; consider only the k strongest correlations of each node; define distances and weights to enhance clustering: structural equivalence: put nodes with similar correlations to others closer together; identify cross-cluster edges (e. g., by edge betweenness); other concepts from network analysis.

27 Software (for the images in these slides). Financial network analytics ( MST drawings of correlation networks visone ( Network visualization by classical scaling, stress minimization, dynamic layouts,...

28 Cited references I U. Brandes and M. Mader. A quantitative comparison of stress-minimization approaches for offline dynamic graph drawing. Proc. 19th Intl. Symp. Graphdrawing (GD 11), U. Brandes and C. Pich. An experimental study on distance-based graph drawing. Proc. 16th Intl. Symp. Graphdrawing (GD 08), E.R. Gansner, Y. Koren, and S. North. Graph drawing by stress majorization. Proc. 12th Intl. Symp. Graphdrawing (GD 04), T. Kamada and S. Kawai. A simple method for computing general positions in displaying three-dimensional objects. Computer Vision, Graphics, and Image Processing, 41, 1988.

29 Cited references II J. Kruskal and J. Seery. Designing network diagrams. Proc. 1st Conf. Social Graphics, 1980.