Basic features of the visualization's layout

The following list is a set of simple rules for a complete understanding of the visualization layout. They should provide the reader with a method to extract information on the original graph starting from the visual observation of the representation's properties and a way to compare different graphs. All points are illustrated with figures.

 

 

The main features of the layout's structure listed below are visible in Fig.2 where, for the sake of simplicity, we don't show any edge. The leftmost panel displays the case in which all $k$ -cores have a single component, while in the rightmost one an example of $k$ -core fragmentation is reported. Indeed, it is possible that, during the pruning procedure, the remaining nodes forming a $k$ -core do not belong to the same connected component. When such a fragmentation occurs, LaNet-vi computes the separed components of the core and displays all of them in a coherent way (see below).

• The visualization's layout is two-dimensional, composed of a series of concentric circular shells (see the five different shells in Fig.2).
• Each shell corresponds to a single shell index and all vertices in it are therefore drawn with the same color.
• A color scale allows to distinguish different values of shell index: in LaNet-vi's images, as in Fig.2, the violet is used for the minimum value of shell index $k_{min}$ , then nuances of blue, green and yellow compose a graduated scale for higher and higher values of shell index up to the maximum value $k_{max}$ that is colored in red.
• The diameter of each $k$ -shell depends on the shell index $k$ , and is proportionnal to $k_{max}-k$ , (In Fig.2, the position of each shell is schematized by a circle having the corresponding diameter). The presence of a trivial order relation in the values of shell index ensures that all shells are placed in a concentric arrangement. On the other hand, when a $k$ -core is fragmented in two or more components, the diameter of the different components depends also on the relative number of vertices belonging to each of them, i.e. the fraction between the number of vertices belonging to that component and the total number of vertices in that $k$ -shell. This is a very important information, providing a way to distinguish between multiple components at a given coreness value. Looking at the two central components for high coreness values in Fig.2 (right), we immediately realize that the bigger one contains a larger fraction of vertices.
• Finally, the size of each node is proportional to the original degree of that vertex; we use a logarithmic scale for the size of the drawn bullets.