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Visual Analysis of Large Graphs Using ( X , Y )-clustering and Hybrid Visualizations

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### Visual Analysis of Large Graphs Using (X, Y)-clustering and Hybrid Visualizations

V. Batagelj, W. Didimo, G. Liotta,P. Palladino, M. Patrignani

(Univ. Ljubljana, Univ. Perugia, Univ. Roma Tre)

In Proc. IEEE Pacific Visualization 2010

Outline

- The problem of visualizing large graphs
- State of the art
- Our contribution
- Conclusions and open problems

The problem of visualizing large graphs

- Some major issues in the visualization of large graphs:
- Readability: optimization of aesthetic criteria
- Scalability: fast computation
- Visual complexity: interaction tools that allow users to limit the amount of information displayed on the screen
- overview of the graph
- details on demand
- user’s mental map preservation

State of the art

- Readability: there are many effective algorithms that are computationally fast for relatively small and sparse graphs (see the graph drawing book of Di Battista, Eades, Tamassia, Tollis , 1999)

State of the art

- Scalability: there are some fast graph drawing algorithms based on physical or algebraic models; the drawings have high visual complexity and do not allow detailed views (see the survey of Hacul and Jünger, 2007)

State of the art

- Visual complexity: draw the whole graph and then interact with it; ex. focus+context techniques, like fisheye view or hyperbolic layouts; conceived for tree-like graphs (see the survey of Herman, Melançon, Marshall, 2000)

State of the art

- Interactive approaches for visualizing and exploring large graphs:
- graph visualized incrementally or at different levels of details
- strong interaction between the user and the drawing

Interactive Approaches

- Bottom-up strategies: the graph is visualized a piece at a time
- topological window moving through canvas (Eades et al. ,1997)
- Limits: no overview, the user’s mental map preservation is difficult

Interactive Approaches

- Bottom-up strategies: the graph is visualized a piece at a time
- incremental enhancement of the drawing (ex. Carmignani et al., 2002)
- Limits: no overview, the user’s mental map preservation is difficult without readability degradation

Interactive Approaches

- Top-down approaches

Interactive Approaches

- Top-down approaches
- the graph is clustered (vertices are grouped together)

Interactive Approaches

- Top-down approaches
- the graph is clustered (vertices are grouped together)
- a simplified view is shown (overview)

Interactive Approaches

- Top-down approaches
- the graph is clustered (vertices are grouped together)
- a simplified view is shown (overview)
- the user interactively explores the clusters (detailed views)

Interactive Approaches

- Top-down strategies
- the graph is clustered (vertices are grouped together)
- a simplified view is shown
- the user interactively explores the clusters
- Limits
- someone/something has to define clustering rules
- existing clustering algorithms do not guarantee properties on the graph of clusters

Our contribution

- A top-down approach with these ingredients:
- a new clustering framework
- new clustering algorithm within the framework
- hybrid visualizations
- A system: VHyXY
- Some case studies

Basic Terminology: Clustering

- G=(V, E): graph with vertex set V and edge set E
- A cluster of G=(V, E) is a subset of V
- A clusteringC of G isa set of disjoint clusters of G

Basic Terminology: Clustering

- Thegraph of clusters H(G, C)is the graph obtained by collapsing each cluster of C into a single vertex and by replacing multiple edges with a single one

Basic Terminology: Clustering

- Thegraph of clusters H(G, C)is the graph obtained by collapsing each cluster of C into a single vertex and by replacing multiple edges with a single one

A new clustering framework

- Clustering algorithms usually detect groups of highly connected vertices without taking care of the graph of clusters
- We adopt a new framework for the design of automatic clustering algorithms that guarantee:
- desired properties for the clusters
- desired properties for the graph of clusters

The (X,Y)-clustering

- X and Y are two classes of graphs with certain properties
- G is called an (X,Y)-graph if there exists a clustering of G such that:
- each cluster induces a subgraphthat belongs to Y
- the graph of clusters belongs to X

(X,Y)-graph example

- Let X be the class of cycles and let Y be the class of K4

(X,Y)-graph example

- Let X be the class of cycles and let Y be the class of K4

(X,Y)-graph example

- Let X be the class of cycles and let Y be the class of K4

- The graph is a (cycle,K4)-graph

Interesting combinations

- Xis some class of sparse graphs:
- planar graphs, cycles, trees, paths, …
- Y is some class of highly connected graphs:
- cliques, subgraphs with high-degree vertices, …
- One can think of using different visual paradigms and algorithms for drawing the graph of clusters and the subgraph induced by each cluster (hybrid visualization)

Remark on (X,Y)-clustering

- (X, Y)-clustering was previously defined by Brandenburg (GD 1997), but his model requires that every vertex belongs to some cluster
- Our model does not have this requirement, which poses severe practical limitations

The (X,Y)-clustering problem

- Problem: Given a graph G and two desired classes X and Y, is G an (X,Y)-graph?
- This problem is NP-hard in general
- Theorem: Deciding whether G is a (planar, k-clique)-graph for desired k ≥ 5 is NP-hard
- This result motivates us to look for some relaxation of cliques

K-core components

- The subgraph induced by a cluster is ak-core component if it is a maximal connected subgraph such that every vertex has degree at least k

5-core component

4-core component

4-core component

(Planar, K-core component)-graphs

- We investigate (X,Y)-graphs G such that:
- X is the class of planar graphs
- Y is the class of k-core components of G
- In particular, for a given k > 0, one can ask whether G is a (planar, k-core component)-graph
- this decision problem can be solved in polynomial time
- we give a polynomial-time algorithm that finds the maximum k for which G is a (planar, k-core component)-graph, and that computes the corresponding clustering

Properties of (planar, k-core component)-graphs

The union of all k-core components of G is called the k-core of G (the k-core of G, if it exists, is unique)

Property. If Ghas the k-core Gk (for some k≥ 1), then Ghas the (k−1)-core G(k−1) and Gk ⊆ G(k−1)

Lemma. If G is a (planar, k-core component)-graph then it is a (planar, (k−1)-core component)-graph

Clustering Algorithm

- Theorem: Let G be a graph with n vertices and m edges. There exists an O((n+m)log n)-time algorithm that computes the maximum k for which G is a (planar, k-core component)-graph, and the corresponding clustering
- Steps of the algorithm:
- Compute core-numbers for the vertices
- Perform a binary search on core-numbers
- For each graph of clusters, test its planarity

Algorithm animation

- Compute the core number of each vertex, i.e., the maximum k for which there exists a k-core that contains the vertex

Algorithm animation

- Compute the core number of each vertex, i.e., the maximum k for which there exists a k-core that contains the vertex

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Hybrid Visualizations

- The (X, Y)-clustering technique can be used to design hybrid visualizations
- combination of different drawing conventions for different parts of the graph
- Example:
- node-link representation for sparse subgraphs
- matrix-based representation for dense subgraphs
- Highly readable drawings for the graph of clusters (which is always planar)

Matrix based representation

- Matrix-based representation
- vertices are rows and columns
- edges are cells
- The ordering of vertices in rows/columns may strongly affect the number of crossings in the drawing

Crossings minimization heuristic

vertex1

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Crossings minimization heuristic

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Crossings minimization heuristic

vertex20

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Crossings minimization heuristic

vertex20

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Remark about hybrid visualizations

- A hybrid visualization that combines node-link and matrix-based representations was previously used in the literature (Henry et al., 2007 - NodeTrix)
- Clusters are manually defined
- no automatic clustering
- no automatic ordering for rows-columns

The System VHyXY

- VHyXYintegrates the clustering algorithm and hybrid visualizations
- X-class chooser (e.g., planar, forest)
- Y-class chooser (e.g., k-core component)
- Filters on edge weights
- Specific drawing algorithms for each component

Case Study: Co-authorship networks

- DBLP: on-line database of publications in Computer Science
- VHyXYallows user to query DBLP on a specific topic
- It retrieves data about all papers on that topic (looking at the title of the papers)
- It builds a network where
- authors are vertices
- there is an edge between two authors if they share a paper (edge’s weight = number of papers)

Hybrid visualizations: a matrix and a circular in an orthogonal layout

- Larger network for “graph drawing”

Clustering algorithm performance

- Graph clustering
- Property of a graph: the higher the value the better can be the clustering
- Coverage
- How the computed clusters covers edges of the whole graph
- Performance
- Counts the number of “correctly interpreted pairs of nodes” in a graph
- Error
- 1-performance

0.94

0.999

[Brandes et al. “Engineering graph clustering: Models and experimental evaluation” ACM Journal of Experimental Algorithmics 2007]

Open problems

- Explore additional X-classes or Y-classes for which polynomial-time clustering algorithms exist
- X: forest, path, outerplanar, …
- Y: relaxations of cliques, …
- Extend our techniques to
- multi-level clustering (hierarchical clustering)
- overlapping clusters
- Experiment the system on a larger set of application domains
- biological networks, criminal networks, …

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