## Small World Graphs

### BibTeX

@MISC{Ham_smallworld,

author = {Frank Ham and Jarke J. Wijk},

title = {Small World Graphs},

year = {}

}

### OpenURL

### Abstract

Many real world graphs have small world characteristics, that is, they have a small diameter compared to the number of nodes and exhibit a local cluster structure. Examples are social networks, software structures, bibliographic references and biological neural nets. Their high connectivity makes both finding a pleasing layout and a suitable clustering hard. In this paper we present a method to create scalable, interactive visualizations of small world graphs, allowing the user to inspect local clusters while maintaining a global overview of the entire structure. The visualization method uses a combination of both semantical and geometrical distortions, while the layout is generated by a spring embedder algorithm using a recently developed force model. We use a cross referenced database of 500 artists as a running example.

### Citations

2163 |
Collective dynamics of small-world networks
- Watts, Strogatz
- 1998
(Show Context)
Citation Context ...h between nodes) compared to their number of nodes, but have a high degree of clustering compared to a random graph of the same size. The degree of clustering can be expressed by the clustering index =-=[35, 2]-=-. The small world property has been identified in many real world graphs such as social networks, neural networks, software systems, power grids, cross referenced knowledge bases and the in∗ e-mail: f... |

2137 | Data Mining: Concepts and Techniques
- Han, Kamber
- 2000
(Show Context)
Citation Context ...ction d(i, j) to determine the distance between two clusters. Widely used metrics in clustering literature define the inter cluster distance as a function of the distances between their (leaf)members =-=[14]-=-. Well known variants are minimum link, maximum link and average link distance. Minimum link defines the distance d(i, j) between clusters i and j as the distance between the two closest members of bo... |

914 |
The small world problem
- Milgram
- 1967
(Show Context)
Citation Context ...late? One way to answer these questions is to provide a picture of the network in which these global properties emerge visually. In this paper we will focus on small world graphs. First identified by =-=[20]-=- in social networks, small world graphs are graphs which have a small average path length (average shortest path between nodes) compared to their number of nodes, but have a high degree of clustering ... |

897 |
Generalized fisheye views
- Furnas
- 1986
(Show Context)
Citation Context ...a he is interested in, as well as a consistent global mental map of the entire structure. One often used way of embedding local detail in global context is the Fisheye View. Generalized Fisheye views =-=[12]-=- define a Degree of Interest for an information element based on its a priori importance and its distance to a focal point. Most applications of the fisheye technique fall into one of two categories. ... |

841 |
Finding and evaluating community structure in networks
- Newman, Girvan
- 2004
(Show Context)
Citation Context ...ed number of partitions, but in the context of information visualization it is generally not desirable (or even possible) to specify this number in advance. Other methods, most notably those based on =-=[22]-=- avoid this pitfall but suffer from O(N3 ) complexity. A second problem is that having a decent clustering does not absolve us from constructing a consistent visualization of this clustered graph afte... |

544 | A framework for dynamic graph drawing
- Cohen, Battista, et al.
- 1992
(Show Context)
Citation Context ...died intensively in the Graph Drawing [4] and Information Visualization [16] communities. In both fields, a popular method to lay out graphs is the force directed algorithm (FDA), introduced by Eades =-=[8]-=-. FDAs are widely used because they are simple to understand and implement, yet they can provide a pleasing layout for sparse graphs, showing global graph structure and graph automorphisms. A disadvan... |

473 | Graph Drawing by Force-Directed Placement
- Fruchterman, Reingold
- 1991
(Show Context)
Citation Context ... rPolyLog energy models. In these models the total potential energy is defined as P = ∑ (pij− x0) eij∈E r − ∑ ln(pij) i̸= j Note that 3-PolyLog has an energy function that is equal to the one used by =-=[11]-=-. The most interesting in this respect however is the 1-PolyLog model, that has associated force functions and f (x)=1 g(x)= 1 x Noack proves that by using the 1-PolyLog (which he dubs LinLog) model w... |

429 | Toolglass and magic lenses: The see-through interface
- Bier, Stone, et al.
- 1993
(Show Context)
Citation Context ...ferent zooming actions (one geometric and the other semantic) more understandable for the user (Fig 4c). The overall effect obtained when moving the focus bears a lot of resemblance to the Magic Lens =-=[5]-=-. Fig 8 shows a number of intermediate frames when moving the focus. The fact that only O(Log(N)) nodes are visible on average (depending on the choice of the DOA function) allows us to maintain inter... |

368 |
Fast algorithm for detecting community structure in networks,” Phys
- Newman
- 2004
(Show Context)
Citation Context ... graphs with small diameters (as their authors also note). The clustering step currently has an O(N 2 Log(N)) complexity and may also benefit from this space subdivision technique. Recent research by =-=[23]-=- suggests a promising O(N 2 ) algorithm to explicitly compute a dendrogram of clusters based on the graph structure. We intend to use this method to evaluate the quality of our layout further. Both st... |

367 | A focus+context technique based on hyperbolic geometry for visualizing large hierarchies
- Lamping
- 1995
(Show Context)
Citation Context ...ed, artists belonging to the Pop-Art movement in green. of nodes based on their geometric distance to the focus. This can be either done in the layout stage by employing a hyperbolic distance measure =-=[18, 21]-=- or as a successive projection step on an existing layout [28, 19, 6, 31]. The other are data suppression techniques that hide or abstract nodes based on the distance to a focus, creating an abstracte... |

349 |
Graph Drawing: Algorithms for the Visualization of Graphs
- Battista, Eades, et al.
- 1999
(Show Context)
Citation Context ...proach by first constructing a layout that reflects the cluster structure and basing the clustering on this layout. The visual presentation of graphs has been studied intensively in the Graph Drawing =-=[4]-=- and Information Visualization [16] communities. In both fields, a popular method to lay out graphs is the force directed algorithm (FDA), introduced by Eades [8]. FDAs are widely used because they ar... |

276 | A review and taxonomy of Distortion-Oriented Presentation Techniques
- Leung, Apperley
- 1994
(Show Context)
Citation Context ...ased on their geometric distance to the focus. This can be either done in the layout stage by employing a hyperbolic distance measure [18, 21] or as a successive projection step on an existing layout =-=[28, 19, 6, 31]-=-. The other are data suppression techniques that hide or abstract nodes based on the distance to a focus, creating an abstracted layout. Although conceptually closer to Furnas’ original idea, relative... |

255 | Visualizing the non-visual: spatial analysis and interaction with information from text documents
- Wise, Thomas, et al.
- 1995
(Show Context)
Citation Context ... (0.18), whereas [2] used graphs with very high clustering indices (mostly over 0.90). Other methods to obtain insight into large cluttered structures are Information Galaxies, Information Landscapes =-=[36]-=- and the related Graph Splatting method [33]. Galaxies and Information Landscapes are mostly used to visualize large document collections. Documents are represented as multidimensional points, cluster... |

239 |
Graph clustering by flow simulation
- Dongen
- 2000
(Show Context)
Citation Context ...as attracted much interest, from diverse areas such as data mining, VLSI design and finite element simulations. Prior literature is substantial and is spread out over many different application areas =-=[1, 27, 7, 17, 32]-=-. However, a hard problem with the use of any automatic clustering method is the fuzzyness of the problem itself. Although the concept of a graph cluster is easy to understand intuitively, there is no... |

232 | Graphical Fisheye Views of Graphs
- Sarkar, Brown
(Show Context)
Citation Context ...ased on their geometric distance to the focus. This can be either done in the layout stage by employing a hyperbolic distance measure [18, 21] or as a successive projection step on an existing layout =-=[28, 19, 6, 31]-=-. The other are data suppression techniques that hide or abstract nodes based on the distance to a focus, creating an abstracted layout. Although conceptually closer to Furnas’ original idea, relative... |

202 | Recent directions in netlist partitioning: a survey
- Alpert, Kahng
- 1995
(Show Context)
Citation Context ...as attracted much interest, from diverse areas such as data mining, VLSI design and finite element simulations. Prior literature is substantial and is spread out over many different application areas =-=[1, 27, 7, 17, 32]-=-. However, a hard problem with the use of any automatic clustering method is the fuzzyness of the problem itself. Although the concept of a graph cluster is easy to understand intuitively, there is no... |

155 |
A hierarchical O(N log N) force calculation algorithm
- BARNES, HUT
(Show Context)
Citation Context ...nds of nodes and more. The current layout method inherits the O(N 3 ) complexity of the standard force directed algorithm, but recent efficiency improvements for this, such as Barnes-Hut optimization =-=[3, 26]-=- can be used, such that a O(N 2 logN) complexity is achieved. Note that multigrid/multiscale methods such as [15], perform poorly on dense graphs with small diameters (as their authors also note). The... |

140 |
SemNet: Threedimensional graphic representations of large knowledge bases
- Fairchild, Poltrok, et al.
- 1988
(Show Context)
Citation Context ...tract nodes based on the distance to a focus, creating an abstracted layout. Although conceptually closer to Furnas’ original idea, relatively few researchers have pursued this path. Most notable are =-=[9, 26]-=-, while others, such as [29] also use abstraction coupled with a distortion function but require the user to manually expand and collapse clusters of interest. Distortion oriented techniques provide a... |

129 | Navigating hierarchically clustered networks through fisheye and full-zoom methods
- Schaffer, Greenberg, et al.
- 1996
(Show Context)
Citation Context ...ce to a focus, creating an abstracted layout. Although conceptually closer to Furnas’ original idea, relatively few researchers have pursued this path. Most notable are [9, 26], while others, such as =-=[29]-=- also use abstraction coupled with a distortion function but require the user to manually expand and collapse clusters of interest. Distortion oriented techniques provide a local geometric zoom that i... |

126 |
H3: Laying Out Large Directed Graphs in 3D Hyperbolic Space
- Munzner
- 1997
(Show Context)
Citation Context ...ed, artists belonging to the Pop-Art movement in green. of nodes based on their geometric distance to the focus. This can be either done in the layout stage by employing a hyperbolic distance measure =-=[18, 21]-=- or as a successive projection step on an existing layout [28, 19, 6, 31]. The other are data suppression techniques that hide or abstract nodes based on the distance to a focus, creating an abstracte... |

97 | A multilevel algorithm for force-directed graph drawing
- Walshaw
- 2000
(Show Context)
Citation Context ...tween every node pair and the total number of iterations is typically in the order of the number of nodes N, the total complexity is O(N 3 ).Mostof the research in this area focuses on improving this =-=[15, 26, 34, 13]-=-. Other research on FDA’s tries to create layouts that are more aesthetically pleasing by introducing different force models. A recent promising direction is the introduction of a force model that map... |

83 | A fast multi-scale method for drawing large graphs
- Harel, Koren
- 2000
(Show Context)
Citation Context ...tween every node pair and the total number of iterations is typically in the order of the number of nodes N, the total complexity is O(N 3 ).Mostof the research in this area focuses on improving this =-=[15, 26, 34, 13]-=-. Other research on FDA’s tries to create layouts that are more aesthetically pleasing by introducing different force models. A recent promising direction is the introduction of a force model that map... |

67 |
Multiscale visualization of small world networks
- Auber, Chiricota, et al.
- 2003
(Show Context)
Citation Context ...h between nodes) compared to their number of nodes, but have a high degree of clustering compared to a random graph of the same size. The degree of clustering can be expressed by the clustering index =-=[35, 2]-=-. The small world property has been identified in many real world graphs such as social networks, neural networks, software systems, power grids, cross referenced knowledge bases and the in∗ e-mail: f... |

45 |
FADE: Graph drawing, clustering, and visual abstraction
- Quigley, Eades
- 2001
(Show Context)
Citation Context ...tween every node pair and the total number of iterations is typically in the order of the number of nodes N, the total complexity is O(N 3 ).Mostof the research in this area focuses on improving this =-=[15, 26, 34, 13]-=-. Other research on FDA’s tries to create layouts that are more aesthetically pleasing by introducing different force models. A recent promising direction is the introduction of a force model that map... |

44 | An Energy Model for Visual Graph Clustering
- Noack
- 2003
(Show Context)
Citation Context .... Nodes are colored based on artistic movement. ily. Section 2 discusses related work. In section 3 we describe how the layout can be improved by employing a force model based on recent work by Noack =-=[24]-=-. In section 4 we show how we can reduce the number of screen items and still provide a detailed impression of the area we are interested in. Finally, in section 5 we draw conclusions and discuss futu... |

38 | A multidimensional approach to force-directed layouts of large graphs
- Gajer, Goodrich, et al.
(Show Context)
Citation Context |

35 | H-BLOB: A hierarchical visual clustering method using implicit surfaces
- TC, Brunella, et al.
- 2000
(Show Context)
Citation Context ... apart when we zoom in. By keeping the screen size of nodes constant while zooming in on the structure, we obtain progressively more details. In fact, this method can be seen as a fast alternative to =-=[30]-=-, with the added ability to continuously display the graph at any level of abstraction by zooming in or out. Figure 3 shows our sample graph at three zoom levels, nodes are colored based on birthdate.... |

24 |
Heiko Mehldau. A Fast Adaptive Layout Algorithm for Undirected Graphs
- Frick, Ludwig
- 1994
(Show Context)
Citation Context ... often produce disappointing results when it comes to dense, highly connected graphs in general and small world graphs in particular. Figure 1 shows a layout for out artists graph, generated with GEM =-=[10]-=-, a typical example of a force directed graph drawing package. Based on the semantics of the input data and the relatively high clustering index, we would like the visualization to clearly display any... |

22 | A.: Graph Clustering Using Multiway Ratio Cut
- Roxborough, Sen
- 1997
(Show Context)
Citation Context ...as attracted much interest, from diverse areas such as data mining, VLSI design and finite element simulations. Prior literature is substantial and is spread out over many different application areas =-=[1, 27, 7, 17, 32]-=-. However, a hard problem with the use of any automatic clustering method is the fuzzyness of the problem itself. Although the concept of a graph cluster is easy to understand intuitively, there is no... |

18 | Graph Folding: Extending Detail and Context Viewing into a Tool for Subgraph Comparisons
- Carpendale, Cowperthwaite, et al.
- 1996
(Show Context)
Citation Context ...ased on their geometric distance to the focus. This can be either done in the layout stage by employing a hyperbolic distance measure [18, 21] or as a successive projection step on an existing layout =-=[28, 19, 6, 31]-=-. The other are data suppression techniques that hide or abstract nodes based on the distance to a focus, creating an abstracted layout. Although conceptually closer to Furnas’ original idea, relative... |

16 | Customizing a Fisheye View Algorithm to Preserve the Mental Map
- Storey, Fracchia, et al.
- 1999
(Show Context)
Citation Context |

6 |
Graph visualization and navigation in information visualization
- HERMAN, G, et al.
- 2000
(Show Context)
Citation Context ...out that reflects the cluster structure and basing the clustering on this layout. The visual presentation of graphs has been studied intensively in the Graph Drawing [4] and Information Visualization =-=[16]-=- communities. In both fields, a popular method to lay out graphs is the force directed algorithm (FDA), introduced by Eades [8]. FDAs are widely used because they are simple to understand and implemen... |

4 |
Eigenvalues, flows and separators of graphs
- Chung, Yau
- 1997
(Show Context)
Citation Context |

4 |
Liere and Wim de Leeuw. Graphsplatting: Visualizing graphs as continuous fields
- van
(Show Context)
Citation Context ...igh clustering indices (mostly over 0.90). Other methods to obtain insight into large cluttered structures are Information Galaxies, Information Landscapes [36] and the related Graph Splatting method =-=[33]-=-. Galaxies and Information Landscapes are mostly used to visualize large document collections. Documents are represented as multidimensional points, clustered and subsequently projected onto a two dim... |

1 |
A fully interactive system for clustering and navigating large graphs
- Huang, Eades
- 1998
(Show Context)
Citation Context |