## Energy Models for Graph Clustering

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Citations: | 16 - 1 self |

### BibTeX

@MISC{Noack_energymodels,

author = {Andreas Noack},

title = {Energy Models for Graph Clustering},

year = {}

}

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### Abstract

The cluster structure of many real-world graphs is of great interest, as the clusters may correspond e.g. to communities in social networks or to cohesive modules in software systems. Layouts can naturally represent the cluster structure of graphs by grouping densely connected nodes and separating sparsely connected nodes. This article introduces two energy models whose minimum energy layouts represent the cluster structure, one based on repulsion between nodes (like most existing energy models) and one based on repulsion between edges. The latter model is not biased towards grouping nodes with high degrees, and is thus more appropriate for the many real-world graphs with right-skewed degree distributions. The two energy models are shown to be closely related to widely used quality criteria for graph clusterings – namely the density of the cut, Shi and Malik’s normalized cut, and Newman’s modularity – and to objective functions optimized by eigenvector-based graph drawing methods.

### Citations

2756 | Normalized cuts and image segmentation
- Shi, Malik
- 2000
(Show Context)
Citation Context ...pulsion LinLog and edgerepulsion LinLog, will be shown to group nodes according to two well-known clustering criteria, namely the density of the cut (e.g. [33, 38]) and Shi and Malik’s normalized cut =-=[45]-=-. The normalized cut and the edge-repulsion LinLog energy model are not biased towards grouping nodes with high degree, and are thus particularly appropriate for graphs with right-skewed degree distri... |

2303 | Algorithms for Clustering Data - Jain, Dubes - 1988 |

1635 | The structure and function of complex networks
- Newman
(Show Context)
Citation Context ...ular paper P. Eades and P. Healy January 2006 February 2007sNoack, Energy Models for Graph Clustering, JGAA, 11(2) 453–480 (2007) 454 1 Introduction Researchers from Herbert Simon [47] to Mark Newman =-=[39]-=- have observed that many real-world systems share a common structure: They are decomposable into subsystems with strong intra-subsystem interactions and relatively weak inter-subsystem interactions. T... |

1392 | Statistical mechanics of complex networks
- Barabsi, Albert
- 2002
(Show Context)
Citation Context ... LinLog energy model are not biased towards grouping nodes with high degree, and are thus particularly appropriate for graphs with right-skewed degree distributions, which are very common in practice =-=[48, 1, 39]-=-. The difference between conventional energy models, node-repulsion LinLog, and edge-repulsion LinLog can be illustrated with a model of the trade between ten North American and European countries 2 .... |

1043 |
Spectral Graph Theory
- Chung
- 1997
(Show Context)
Citation Context ...sed as additional coordinates in higher-dimensional layouts (see [26, 32] for details). More recently, solutions of the generalized Laplacian eigensystem Ly = µDy have received considerable attention =-=[10, 45, 32]-=-. We denote the generalized eigenvector corresponding to the second smallest generalized eigenvalue as the degree-normalized Fiedler vector. Theorem 6 (Chung [10, Chapter 1.2], similarly Koren [32, Se... |

828 |
Finding and evaluating community structure in networks
- Newman, Girvan
(Show Context)
Citation Context ...ts, and this is easier for a coupling measure that is nonnegative and takes the value 0 in the case of no coupling. 5 This version of the measure differs slightly from an earlier version published in =-=[41]-=-, where the second term is the squared fraction of edges that connect to nodes in Vi.sNoack, Energy Models for Graph Clustering, JGAA, 11(2) 453–480 (2007) 461 3 Energy Models for Graph Clustering As ... |

540 | A heuristic for graph drawing
- Eades
- 1984
(Show Context)
Citation Context ...ble the efficient computation and comprehensible presentation of the subsystem structure in many real-world systems. Most existing energy models and force models 1 for general undirected graphs (e.g. =-=[16, 30, 20, 13]-=-) have not been designed to find clusters, but to produce readable visualizations. They enforce small and uniform edge lengths, which often prevents the separation of nodes in different clusters. As a... |

511 | An Algorithm for Drawing General Undirected Graphs
- Kamada, Kawai
- 1989
(Show Context)
Citation Context ...ble the efficient computation and comprehensible presentation of the subsystem structure in many real-world systems. Most existing energy models and force models 1 for general undirected graphs (e.g. =-=[16, 30, 20, 13]-=-) have not been designed to find clusters, but to produce readable visualizations. They enforce small and uniform edge lengths, which often prevents the separation of nodes in different clusters. As a... |

497 | The architecture of complexity
- Simon
- 1962
(Show Context)
Citation Context ...ubmitted Revised Regular paper P. Eades and P. Healy January 2006 February 2007sNoack, Energy Models for Graph Clustering, JGAA, 11(2) 453–480 (2007) 454 1 Introduction Researchers from Herbert Simon =-=[47]-=- to Mark Newman [39] have observed that many real-world systems share a common structure: They are decomposable into subsystems with strong intra-subsystem interactions and relatively weak inter-subsy... |

468 | Graph drawing by force-directed placement
- Fruchterman, Reingold
- 1991
(Show Context)
Citation Context ...ble the efficient computation and comprehensible presentation of the subsystem structure in many real-world systems. Most existing energy models and force models 1 for general undirected graphs (e.g. =-=[16, 30, 20, 13]-=-) have not been designed to find clusters, but to produce readable visualizations. They enforce small and uniform edge lengths, which often prevents the separation of nodes in different clusters. As a... |

454 | The geometry of graphs and some of its algorithmic applications
- Linial, London, et al.
- 1995
(Show Context)
Citation Context ...tios, and thus to reveal the cluster structure. Earlier works have minimized similar distance ratios to find clusters. In particular, some approximation algorithms for graph clustering problems (e.g. =-=[36, 4]-=-) derive partitions from layouts that minimize the ratio arithmean(E,p) arithmean(V (2) . However, ,p) these layouts are not suitable for human viewers (e.g. many nodes are placed at the same position... |

402 |
Exploring complex networks
- Strogatz
- 2001
(Show Context)
Citation Context ... LinLog energy model are not biased towards grouping nodes with high degree, and are thus particularly appropriate for graphs with right-skewed degree distributions, which are very common in practice =-=[48, 1, 39]-=-. The difference between conventional energy models, node-repulsion LinLog, and edge-repulsion LinLog can be illustrated with a model of the trade between ten North American and European countries 2 .... |

299 |
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
- Leighton, Rao
- 1999
(Show Context)
Citation Context ...conductance are NP-hard [29, 46]. A recent O( � log(|V |))-approximation algorithm for both problems by Arora, Rao and Vazirani [3] improves the classic O(log(|V |))-approximation of Leighton and Rao =-=[33, 34]-=-. The expansion is related to the node-normalized cut by and thus expansion(V1, V2) = max(|V1|, |V2]) ncut(V1, V2) 1 2 (|V1| + |V2|) ncut(V1, V2) ≤ expansion(V1, V2) ≤ (|V1| + |V2|) ncut(V1, V2) . The... |

263 | On clusterings: good, bad and spectral
- Kannan, Vempala, et al.
(Show Context)
Citation Context ...wever, Shi and Malik’s normalized cut is biased towards small clusters when deg(V1) + deg(V2) is not fixed. Expansion and Conductance Two other well-known measures of coupling are the expansion (e.g. =-=[31]-=-) and the conductance (e.g. [31]) expansion(V1, V2) = cut(V1, V2) min(|V1|, |V2]) conductance(V1, V2) = cut(V1, V2) min(deg(V1), deg(V2)) . The terms isoperimetric number and Cheeger constant of a gra... |

247 |
Multiterminal Network Flows
- Gomory, Hu
- 1961
(Show Context)
Citation Context ... the coupling between two disjoint sets of nodes V1 and V2 of a graph (V, E) is their cut cut(V1, V2). There exist efficient algorithms for finding a bipartition of a given graph with the minimum cut =-=[23]-=-. However, the cut prefers bipartitions that consist of a very small and a very large subgraph, as the following calculation shows. Among the 1 2 (|V |2 −|V |) unordered pairs of nodes from V , there ... |

245 |
A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory
- Fiedler
- 1975
(Show Context)
Citation Context ...ith associated eigenvector (1, 1, ..., 1) T ), and all other eigenvalues are positive. The eigenvector corresponding to the second smallest eigenvalue is called the Fiedler vector. Theorem 5 (Fiedler =-=[17]-=-) The Fiedler vector of a graph (V, E) minimizes � {u,v}∈E (x(u) − x(v))2 � {u,v}∈V (2)(x(u) − x(v))2 over all vectors x ∈ IR |V | that are non-constant (i.e. have at least two different entries). Thi... |

241 |
An Approximate Max–Flow Min–Cut Theorem for Uniform Multicommodity Flow Problems with Application to Approximation Algorithm
- Leighton, Rao
- 1988
(Show Context)
Citation Context ...ew energy models in this work, called node-repulsion LinLog and edgerepulsion LinLog, will be shown to group nodes according to two well-known clustering criteria, namely the density of the cut (e.g. =-=[33, 38]-=-) and Shi and Malik’s normalized cut [45]. The normalized cut and the edge-repulsion LinLog energy model are not biased towards grouping nodes with high degree, and are thus particularly appropriate f... |

237 | Expander flows, geometric embeddings and graph partitioning
- Arora, Rao, et al.
- 2004
(Show Context)
Citation Context ... of finding the bipartition of a given graph with minimum node-normalized cut is NP-hard for edge-weighted graphs [38], but approximable within factor O( � log(|V |)) in deterministic polynomial time =-=[3]-=-. The node-normalized cut is still biased towards bipartitions with a very small and a very large subgraph if the number of edges is used as measure of subgraph size. Consider two bipartitions of the ... |

216 |
A hierarchical o(n log n) force-calculation algorithm
- Barnes, Hut
- 1986
(Show Context)
Citation Context ...ain contribution of this work are the LinLog energy models; algorithms for minimizing these energy models are already available. In our experiments we use the hierarchical algorithm of Barnes and Hut =-=[5]-=-, which was introduced to graph drawing by Tunkelang [50] and Quigley and Eades [44]. Its runtime is in O(e + n log n) per iteration, where e is the number of edges and n is the number of nodes. The o... |

201 | Recent directions in netlist partitioning: A survey
- Alpert, Kahng
- 1995
(Show Context)
Citation Context ...es |V1| and |V2|, the node-normalized 2|E| cut has the same expected value |V | 2−|V | . This measure is also known as the density of the cut or the ratio of the cut, and has been used in VLSI design =-=[2]-=- and software engineering [37]. The problem of finding the bipartition of a given graph with minimum node-normalized cut is NP-hard for edge-weighted graphs [38], but approximable within factor O( � l... |

190 | Drawing Graphs Nicely using Simulated Annealing
- Davidson, Harel
- 1996
(Show Context)
Citation Context |

169 |
An r-dimensional quadratic placement algorithm
- Hall
- 1970
(Show Context)
Citation Context ...e non-constant (i.e. have at least two different entries). This property justifies the use of the Fiedler vector not only as node coordinate vector in one-dimensional graph layouts (pioneered by Hall =-=[26]-=-), but also for deriving graph clusters e.g. by simple thresholding (pioneered by Donath and Hoffman [15]). The next eigenvectors of the Laplacian have similar properties and can be used as additional... |

144 |
Lower bounds for the partitioning of graphs
- Donath, Hoffman
- 1973
(Show Context)
Citation Context ... vector not only as node coordinate vector in one-dimensional graph layouts (pioneered by Hall [26]), but also for deriving graph clusters e.g. by simple thresholding (pioneered by Donath and Hoffman =-=[15]-=-). The next eigenvectors of the Laplacian have similar properties and can be used as additional coordinates in higher-dimensional layouts (see [26, 32] for details). More recently, solutions of the ge... |

134 | Using Automatic Clustering to Produce High-Level System Organizations of Source Code
- Mancoridis, Mitchell, et al.
- 1998
(Show Context)
Citation Context ...rmalized 2|E| cut has the same expected value |V | 2−|V | . This measure is also known as the density of the cut or the ratio of the cut, and has been used in VLSI design [2] and software engineering =-=[37]-=-. The problem of finding the bipartition of a given graph with minimum node-normalized cut is NP-hard for edge-weighted graphs [38], but approximable within factor O( � log(|V |)) in deterministic pol... |

129 |
Analysis of weighted networks
- Newman
- 2004
(Show Context)
Citation Context ... the expected expansion is only |E| |V |−1 |V | , while for |V1| = |V2| = 1 2 . The conductance has a similar bias when the total degree is used as measure of cluster size. Newman’s Modularity Newman =-=[40]-=- proposed a measure of coupling for k disjoint sets of nodes called modularity 5 : Q(V1, ..., Vk) = k� � 2 cut(Vi, Vi) deg(Vi) − |E| deg(V ) 2 � . i=1 The first term is the fraction of all edges that ... |

123 | An O(1og k) approximate min-cut max-flow theorem and approximation algorithm
- Aumann, Rabani
- 1994
(Show Context)
Citation Context ...tios, and thus to reveal the cluster structure. Earlier works have minimized similar distance ratios to find clusters. In particular, some approximation algorithms for graph clustering problems (e.g. =-=[36, 4]-=-) derive partitions from layouts that minimize the ratio arithmean(E,p) arithmean(V (2) . However, ,p) these layouts are not suitable for human viewers (e.g. many nodes are placed at the same position... |

112 | A fast adaptive layout algorithm for undirected graphs
- Frick, Ludwig, et al.
- 1994
(Show Context)
Citation Context ...ances of the connected components approach infinity in layouts with minimum LinLog energy. This can be avoided by adding a gravitational energy that attracts each node to the barycenter of the layout =-=[19]-=-. For a� weighted graph G = (V, E, w), a layout p of G with the barycenter v∈V b(p) := w(v)p(v) � v∈V w(v) , and a small constant g that determines the distances of the components, the gravitational e... |

96 | A Multilevel Algorithm for Force-Directed Graph Drawing
- Walshaw
- 1984
(Show Context)
Citation Context .... A Java implementation of the algorithm is freely available 3 . Efficient multi-scale algorithms for energy-based graph layout have been developed by Gajer et al. [21], Harel and Koren [27], Walshaw =-=[51]-=-, and Hachul and Jünger [24]. These algorithms rely on the assumption that nodes with a small graph-theoretic distance (e.g. adjacent nodes) also have a small Euclidean distance in the optimal layout.... |

82 | A fast multi-scale method for drawing large graph
- Koren, Harel
- 2002
(Show Context)
Citation Context ...to grow with n. A Java implementation of the algorithm is freely available 3 . Efficient multi-scale algorithms for energy-based graph layout have been developed by Gajer et al. [21], Harel and Koren =-=[27]-=-, Walshaw [51], and Hachul and Jünger [24]. These algorithms rely on the assumption that nodes with a small graph-theoretic distance (e.g. adjacent nodes) also have a small Euclidean distance in the o... |

72 | Graph Drawing by Stress Majorization - Gansner, Koren, et al. |

48 | An experimental comparison of force-directed and randomized graph drawing algorithms
- Brandenburg, Himsolt, et al.
- 1995
(Show Context)
Citation Context ...nections. Of course, these properties could also be checked empirically for example layouts, like previous studies of force and energy models have verified the conformance to aesthetic criteria (e.g. =-=[9, 25]-=-). However, such empirical validation is inferior to theoretical validation, because it is limited to a relatively small number of graphs, and because properties of the used minimization heuristics in... |

45 | Clustering Software Artifacts Based on Frequent Common Changes
- Beyer, Noack
- 2005
(Show Context)
Citation Context ...JGAA, 11(2) 453–480 (2007) 472 identifying entry points for the exploration of topics, or for finding semantically related terms even if they are not explicitly linked. Coupling of Software Artifacts =-=[7]-=- In large software systems, the individual software artifacts (e.g. files or classes) are hierarchically organized into subsystems. Artifacts that are frequently changed together should belong to the ... |

45 |
FADE: Graph drawing, clustering, and visual abstraction
- Quigley, Eades
- 2001
(Show Context)
Citation Context ...ng these energy models are already available. In our experiments we use the hierarchical algorithm of Barnes and Hut [5], which was introduced to graph drawing by Tunkelang [50] and Quigley and Eades =-=[44]-=-. Its runtime is in O(e + n log n) per iteration, where e is the number of edges and n is the number of nodes. The overall runtime grows somewhat faster because the number of iterations needed for con... |

43 | An energy model for visual graph clustering
- Noack
- 2004
(Show Context)
Citation Context ...in p. For this simplified situation it can be shown that the distance between V1 and V2 equals the inverse normalized cut between V1 and V2. A similar theorem for less restricted layouts is proved in =-=[42]-=-. Theorem 3 Let G = (V, E) be a connected graph, and let (V1, V2) be a bipartition of its set of nodes. Let P be the set of layouts of G that assign the same position to all nodes in V1, and the same ... |

38 | A multidimensional approach to force-directed layouts of large graphs - Gajer, Goodrich, et al. |

36 | 3D graph drawing with simulated annealing
- CRUZ, TWAROG
(Show Context)
Citation Context ...e between edge midpoints, and Davidson and Harel [13] and Bertault [6] introduce a repulsive force between edges and nodes, all to avoid edges that are very close or cross each other. Cruz and Twarog =-=[12]-=- suggest (without giving details) that for 3D layouts, the latter force can be replaced with a repulsive force between non-adjacent edges. Lin and Yen [35] use a repulsive force only between adjacent ... |

35 |
Drawing large graphs with a potential-field-based multilevel algorithm
- Hachul, Jnger
- 2004
(Show Context)
Citation Context ...he algorithm is freely available 3 . Efficient multi-scale algorithms for energy-based graph layout have been developed by Gajer et al. [21], Harel and Koren [27], Walshaw [51], and Hachul and Jünger =-=[24]-=-. These algorithms rely on the assumption that nodes with a small graph-theoretic distance (e.g. adjacent nodes) also have a small Euclidean distance in the optimal layout. This assumption is usually ... |

33 | Finding social groups: A meta-analysis of the southern women data
- Freeman
- 2002
(Show Context)
Citation Context ...ation (Figure 4) The graph represents the participation of 18 women in 14 informal social events. Each woman and each event is modeled by a node, and each participation is modeled by an edge. Freeman =-=[18]-=- performed a meta-analysis of 21 earlier studies that assigned the women to groups. Applying consensus analysis to combine the results of these studies, he obtained a decomposition into two groups, wi... |

26 |
A force-directed algorithm that preserves edge-crossing properties
- Bertault
(Show Context)
Citation Context ...c criteria, and do not enable interpretations with respect to the cluster structure. Coleman and Parker [11] propose a repulsive force between edge midpoints, and Davidson and Harel [13] and Bertault =-=[6]-=- introduce a repulsive force between edges and nodes, all to avoid edges that are very close or cross each other. Cruz and Twarog [12] suggest (without giving details) that for 3D layouts, the latter ... |

23 | Aesthetics-based graph layout for human consumption
- Coleman, Parker
- 1996
(Show Context)
Citation Context ... pairs of edges. Second, the forces are intended to improve the conformance to specific aesthetic criteria, and do not enable interpretations with respect to the cluster structure. Coleman and Parker =-=[11]-=- propose a repulsive force between edge midpoints, and Davidson and Harel [13] and Bertault [6] introduce a repulsive force between edges and nodes, all to avoid edges that are very close or cross eac... |

23 | An experimental comparison of fast algorithms for drawing general large graphs
- Hachul, Junger
(Show Context)
Citation Context ...nections. Of course, these properties could also be checked empirically for example layouts, like previous studies of force and energy models have verified the conformance to aesthetic criteria (e.g. =-=[9, 25]-=-). However, such empirical validation is inferior to theoretical validation, because it is limited to a relatively small number of graphs, and because properties of the used minimization heuristics in... |

20 |
The effect of graph layout on inference from social network data
- Blythe, McGrath, et al.
- 1995
(Show Context)
Citation Context ...ted with their cluster, and how clearly clusters are separated; and they facilitate the comprehension of the clusters, because viewers naturally interpret closely positioned nodes as strongly related =-=[8, 14]-=-. This section introduces two energy models that correspond to the two unbiased clustering criteria of the previous section, and demonstrates the internal validity of their minimum energy layouts. Spe... |

20 |
Graph drawing by the magnetic spring model
- Sugiyama, Misue
- 1995
(Show Context)
Citation Context ...of the graph. Examples of such properties include the density of subgraphs (in this work), the graph-theoretic distances of nodes (e.g. in [30]), or the direction of edges in directed graphs (e.g. in =-=[49]-=-). Interpretable layouts can be seen as simple models of a graph, which reflect some properties of the graph and abstract from others, and which have the additional benefit of being easily visualizabl... |

15 | Drawing graphs by eigenvectors: theory and practice
- Koren
- 2005
(Show Context)
Citation Context ...simple thresholding (pioneered by Donath and Hoffman [15]). The next eigenvectors of the Laplacian have similar properties and can be used as additional coordinates in higher-dimensional layouts (see =-=[26, 32]-=- for details). More recently, solutions of the generalized Laplacian eigensystem Ly = µDy have received considerable attention [10, 45, 32]. We denote the generalized eigenvector corresponding to the ... |

14 | A space of layout styles for hierarchical graph models of software systems
- Noack, Lewerentz
- 2005
(Show Context)
Citation Context ...of the clustering criteria in Section 2. This can be achieved by generalizing the gravitational energy (introduced in the previous paragraph) to attract each node to the barycenter of its subset (see =-=[43]-=- for details). 4 Real-World Examples This section discusses example layouts of the two LinLog energy models and, for comparison, of the widely used Fruchterman-Reingold force model [20]. It provides e... |

13 |
Human perception of laid-out graphs
- Dengler, Cowan
- 1998
(Show Context)
Citation Context ...ted with their cluster, and how clearly clusters are separated; and they facilitate the comprehension of the clusters, because viewers naturally interpret closely positioned nodes as strongly related =-=[8, 14]-=-. This section introduces two energy models that correspond to the two unbiased clustering criteria of the previous section, and demonstrates the internal validity of their minimum energy layouts. Spe... |

9 | On the NP-completeness of some graph cluster measures
- Sima, Schaeffer
- 2006
(Show Context)
Citation Context ... the number of nodes and the total degree of a (sub)graph can be considered as its area or volume. The problems of finding the bipartition of a graph with minimum expansion or conductance are NP-hard =-=[29, 46]-=-. A recent O( � log(|V |))-approximation algorithm for both problems by Arora, Rao and Vazirani [3] improves the classic O(log(|V |))-approximation of Leighton and Rao [33, 34]. The expansion is relat... |

6 | On the expansion of graphs of 0/1-polytopes
- Kaibel
(Show Context)
Citation Context ... the number of nodes and the total degree of a (sub)graph can be considered as its area or volume. The problems of finding the bipartition of a graph with minimum expansion or conductance are NP-hard =-=[29, 46]-=-. A recent O( � log(|V |))-approximation algorithm for both problems by Arora, Rao and Vazirani [3] improves the classic O(log(|V |))-approximation of Leighton and Rao [33, 34]. The expansion is relat... |

6 | A new force-directed graph drawing method based on edge-edge repulsion
- Lin, Yen
- 2005
(Show Context)
Citation Context ...y close or cross each other. Cruz and Twarog [12] suggest (without giving details) that for 3D layouts, the latter force can be replaced with a repulsive force between non-adjacent edges. Lin and Yen =-=[35]-=- use a repulsive force only between adjacent edges, mainly to improve angular resolution. Frick, Ludwig and Mehldau [19, Section 4.3] scale 1 the attractive force acting on each node v with a factor d... |

5 |
Jiggle: Java interactive graph layout environment
- Tunkelang
- 1998
(Show Context)
Citation Context ...ls; algorithms for minimizing these energy models are already available. In our experiments we use the hierarchical algorithm of Barnes and Hut [5], which was introduced to graph drawing by Tunkelang =-=[50]-=- and Quigley and Eades [44]. Its runtime is in O(e + n log n) per iteration, where e is the number of edges and n is the number of nodes. The overall runtime grows somewhat faster because the number o... |