Using Adjacency Matrices to Lay out Larger Small-World Networks
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Using adjacency matrices to lay out larger small-world networks GIBSON, Helen <http://orcid.org/0000-0002-5242-0950> and VICKERS, Paul Available from Sheffield Hallam University Research Archive (SHURA) at: http://shura.shu.ac.uk/11564/ This document is the author deposited version. You are advised to consult the publisher's version if you wish to cite from it. Published version GIBSON, Helen and VICKERS, Paul (2016). Using adjacency matrices to lay out larger small-world networks. Applied Soft Computing, 42, 80-92. Copyright and re-use policy See http://shura.shu.ac.uk/information.html Sheffield Hallam University Research Archive http://shura.shu.ac.uk Applied Soft Computing 42 (2016) 80–92 Contents lists available at ScienceDirect Applied Soft Computing j ournal homepage: www.elsevier.com/locate/asoc Using adjacency matrices to lay out larger small-world networks a,∗ b,1 Helen Gibson , Paul Vickers a CENTRIC, Sheffield Hallam University, Cantor Building, 153 Arundel Street, Sheffield S1 2NU, UK b Northumbria University, Department of Computer Science and Digital Technologies, Pandon Building, Camden Street, Newcastle-upon-Tyne NE2 1XE, UK a r t i c l e i n f o a b s t r a c t Article history: Many networks exhibit small-world properties. The structure of a small-world network is characterized Received 4 January 2015 by short average path lengths and high clustering coefficients. Few graph layout methods capture this Received in revised form structure well which limits their effectiveness and the utility of the visualization itself. Here we present 12 November 2015 an extension to our novel graphTPP layout method for laying out small-world networks using only their Accepted 20 January 2016 topological properties rather than their node attributes. The Watts–Strogatz model is used to generate Available online 29 January 2016 a variety of graphs with a small-world network structure. Community detection algorithms are used to generate six different clusterings of the data. These clusterings, the adjacency matrix and edgelist are Keywords: loaded into graphTPP and, through user interaction combined with linear projections of the adjacency Small world networks matrix, graphTPP is able to produce a layout which visually separates these clusters. These layouts are Adjacency matrix Node attributes compared to the layouts of two force-based techniques. graphTPP is able to clearly separate each of the Graph visualization communities into a spatially distinct area and the edge relationships between the clusters show the Targeted projection pursuit strength of their relationship. As a secondary contribution, an edge-grouping algorithm for graphTPP is demonstrated as a means to reduce visual clutter in the layout and reinforce the display of the strength of the relationship between two communities. © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). 1. Introduction layout methods while Gibson and Faith [8] put forward a method based on node-attribute data. Small-world networks are a commonly occurring graph struc- How clusters are represented in a graph is extremely impor- ture characterized by short average path lengths and high tant as the human perceptual system will naturally cause users to clustering coefficients [1]. This means that even when the net- assume that there is a relationship between nodes that are placed work is large there are very few steps between each pair of nodes. close together [9]. For graphs that are highly clustered adhering to Despite their prevalence very few methods are able to lay them this principle when laying out the graph is key for communicat- out such that their structure is communicated optimally [2]. Exam- ing the structure of the network. Generally, force-directed layouts ples of networks that display these characteristics in the real-world do not produce an accurate representation of small-world network include social networks [3], biological networks [4] and even geo- structures. This is because they try to optimize the layout to have physical ones [5]. uniform edge lengths but longer edge lengths are usually required The high clustering coefficient of small-world networks pro- to separate clusters [2]. vides an interesting problem for layout particularly as it has been Force-directed methods such as LinLog (linear-logarithmic) shown that users seek to lay out graphs such that their clustered [2,10] and OpenOrd (a successor to VxOrd) [11] do try to optimize structure is apparent [6]. Thus, ensuring that the clustered struc- for clustering based on topology while there are other methods ture of the graph is well represented in the layout seems crucial that use attributes or pre-computed clusterings. Muelder and Ma’s for enhancing users’ understanding of the graph in the context of treemap [12] and space-filling [13] approaches use a pre-computed its clusters and the relationships between them. van Ham and van clustering, while the group-in-the-box layout [14] can take any user Wijk [7] have proposed one of the few small-world network specific input or pre-computed clustering. graphTPP (graph targeted projection pursuit) [8] is a method, encapsulated in an interactive software program, that has pre- ∗ viously been used for the layout of small-world networks using Corresponding author. Tel.: +44 0114 225 6819; fax: +44 0114 225 6702. node-attributes. The graph is laid out, assisted by direct user inter- E-mail addresses: [email protected] (H. Gibson), action, according to the specified clustering resulting in a clear [email protected] (P. Vickers). 1 Tel.: +44 0191 243 7614.. visual separation of the clusters. Here we propose that rather http://dx.doi.org/10.1016/j.asoc.2016.01.036 1568-4946/© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). H. Gibson, P. Vickers / Applied Soft Computing 42 (2016) 80–92 81 than using node-attributes, the adjacency matrix of the graph can result in high degree nodes being placed close to the centre of the replace the multidimensional matrix of attributes and by signif- graph and this encourages the well-known hairball-style layout to icantly increasing the size of the graph (in terms of the number form (for an example in this paper see Fig. 9(c)). Further, many of nodes) demonstrates that graphTPP is scalable beyond the very force-based techniques also try to optimize the layout according to small examples used in the previous work. certain aesthetic criteria. One such criterion is uniformity of edge The paper proceeds as follows. Section 2 discusses related work length; however, long edge lengths are required to separate clus- on the layout of small-world networks. Section 3 introduces the ters and as such the force-style layouts do not well represent the Watts–Strogatz model that is used for computing the small-world clusters formed by the small-world network [2]. networks and the community detection algorithms used by the lay- Ten years have passed since Auber et al. [18] commented that out. Section 4 presents the graphTPP layouts of the small-world the “the structural properties of small-world networks have not networks for both one and two-dimensional Watts–Strogatz mod- yet been fully exploited from a visualization perspective” but to our els whilst comparing them to results obtained using the OpenOrd knowledge still very few techniques exist that deal specifically with and ForceAtlas layout algorithms. It also introduces an edge- layout of graphs exhibiting small-world properties. Auber et al.’s grouping technique for reducing visual clutter caused by the edges [18] multi-scale small-world layout is one solution. It requires in the graphTPP layout. Section 5 discusses the results and the lim- decomposing the graph in a hierarchical manner by rating the itations of this work while also recommending directions for future strength of each edge and removing the so-called ‘weaker’ ones. work. Section 6 then concludes the paper. This helps with exploring the individual clusters. This research shows that graphTPP outperforms OpenOrd [11] Other small-world network layouts include van Ham and van and ForceAtlas [15] as a method for laying out small-world Wijk’s [7] that uses an adapted version of Noack’s [2] force-model networks where the aim is to optimize the layout for the communi- to further separate the clusters while van Ham and Wattenberg ties detected through various community detection algorithms. The [19] create a sparse backbone of the graph by removing edges with main contribution then is the demonstration that graphTPP can be a low betweenness centrality, laying out the graph at this level and viable layout method even when there is no typical node-attribute then proceeding to add edges back in. Topolayout [20] also detects data available upon which to base the layout. small-world features while HiMap [21] has adapted Kamada and Kawai’s [22] algorithm to produce clustered layouts. Recently, Gibson and Faith [8] used a projection based technique 2. Related work to lay out small-world networks based on node-attributes aiming to produce a clustered layout. Here we extend that technique to a 2.1. Small-world networks much larger class of small-world networks and no longer rely on the node-attributes but instead use the topological structure of the Small-world networks are characterized by short average path graph itself. lengths (the shortest path between any pair of nodes) and high clustering coefficients (e.g., in social networks this would be the number of friends a person has who are also friends, i.e., they 2.2.