InfoVis.net>Magazine>message nº 202  Published 20100920 
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In the real world but specially in Internet, the relationship between individuals, objects and concepts are increasingly important. Just as an example, who is not connected today to one or more social networks?. These have become ubiquitous and constitute a vast truss of professional and leisure relations where we knit together our interests. Consequently, the networks have ended up as important sources of information. How can we visualise them in an effective way? From a mathematical standpoint a network can be considered as a graph (see also issue number 137). Each element of a graph (also called node) can be related to other elements of the same. Each relation is called arc, edge or also link. The traditional representation of a graph consists of a set of points representing the nodes, linked by lines that bind the nodes that hold a relationship. Nevertheless, when the number of nodes begins to become high (above about 20 nodes and 2030 links depending on the authors*), the problems of occlusion between links and even between nodes make understanding and interacting with the representation very difficult. An alternative representation that, despite the fact that it's relatively unknown, appears to be very useful is the matrix representation. For our purposes the matrix representation of a graph is an array of rows and columns where each row and column represent a node (for example a person) and in the intersections (cells) between them a 0 or a 1 (a coloured square or its absence) denotes that there exists a relation (or not) between the corresponding nodes. So what we are actually painting is a boolean matrix of connectivity, also called and adjacency matrix. Notice that this allows us to visualise one to one, one to many and many to one links in a very simple way. Obviously the matrix paradigm can be extended beyond the adjacency matrix by assigning a visual variable like the colour to each cell as a function of the value of a given numerical value like, for example, the traffic of a web link or the number of publications of which two nodes are coauthors. The matrix layout guarantees that there's no occlusion neither between links nor between nodes. On the other hand the study of the visual patterns that arise let you identify clusters and "communities" by permuting the ordering of rows and columns so that the most mutually linked nodes become closer. To be more systematic, the advantages and inconveniences of such a representation , according to the PhD Thesis by Nathalie Henry and the articles mentioned at the footer of this page, are the following: Advantages
Inconveniences
J. Bertin, who sadly passed away just a few weeks ago, had already introduced in his book Semiologie Graphique the reordenable matrix. Effectively, being able to properly permute the order of rows and columns allows you to understand better the structure of the links and the identification of features making interesting patterns emerge. In some ways what Bertin came to say is that the matrix provides us with a valid representation, but what makes it usable is choosing an appropriate ordering of rows and columns. Particularly you can detect
Nevertheless the permutation of rows and columns is not unique so it's possible to obtain different views of the communities that make up a social network. It's not easy, anyway, to choose which ordering is the right one in general. It's clear that different reordering produces different views and, hence, different approaches to the problem. Which of them is the best one is a key aspect of the utility you can give to it and most probably depends on the needs of the user, i.e. on the objective we are pursuing when analysing a social network. Matrix representations have been successfully used in the study of scientific publications to ascertain, in function of the citations and coauthorship: which researchers work together more often, which are the professional relations between them, the fields where they thrive, which are the main researchers etc. In the field of patent analysis, as we already saw in the issue number 167 , they have been extensively used to detect alliances between companies and communities of researchers working collaboratively between different institutions or within the same. But there are many other applications that can benefit from this type of representation allowing you to find social groups sharing purchase habits or groups of people connected by mobile telephony that have common interests. It isn't difficult to imagine the interest that a revealing visualisation can have when taking marketing decisions affecting considerably large social networks. Nevertheless the bigger the network the more complex it becomes to acquire insight and knowledge. Visualisation has a long road ahead in this field. On the readability of graphs using nodelink and matrix based representations: a controlled experiment and statistical analysis. Information Visualization (2005) 4, 114–135, Palgrave Mac Millan Dedicated to the memory of Jacques Bertin Links of this issue:

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