A guide for using this applet
1. Generating a network
All these networks admit as many as five different layouts and two layout algorithms.
“Real colours” which represent opinions are assigned when pressing set opinions.
Note that degree statistics will only show after setting opinions (see set opinions).
Regular lattice
Simply select the side of the lattice and whether there exists links uniting the edges (wrap). Note that wrapping will usually be hard to visualise (see spring). Finally, setup-Lattice.
Érdös–Renyi
Choose the number of nodes and the number of links . This can be achieved by moving the sliders or by writing them down (then pressing enter) in the input monitors. The last step is setup Érdös–Renyi.
Bárabási-Albert
Setup–BA will create the initial core or seed, which will be determined by the degree (1 or 2) chosen. Go will add new nodes; look at the node counter to decide when to stop. Layout enables an algorithm that is useful for the visualisation of tree-like degree–one networks. For degree–two networks, we may prefer to change the visualisation as explained later (see set structure). Redo layout can be used for other networks as well. By selecting plot, the degree distribution of the network will be represented all along the construction process.
Watts-Strogatz
Setup WS will generate a circle of nodes, where each node is linked to its two or four closet neighbours, depending on the value of kini (2 or 4). Note that all links are straight lines, so for a high number of nodes long distance (two–step) links will be overshadowed by short links. Generating a few-nodes example –10 nodes should do– will make this evident. When we press rewire, one of the links will be erased and redirected to a different node -avoiding self and multiple connections- chosen at random. Rewire all will rewire each and every link with the probability we select in the slider. (For a network with 5 nodes – and kini 4–, links cannot be redirected since the network is connected and has all possible links already assigned).
2. Voter Dynamics
Black/White
This will change the colour of the background. Active nodes will only change their colour (to the complementary of the background) while voting or if pressing set structure.
Set opinions
This control distributes initial–condition opinions –represented by colours– with uniform probability. A slider in the voter control (diversity–of–opinions) will determine the number of opinions. Setting will automatically call set structure.
Set structure
Set structure will colour the links in agreement with the nodes. Links between nodes sharing colour will adopt this characteristic, while active links will be black (white) for a white (black) background. Set structure will preserve the link distribution.
- If default structure is selected, the structure of the network will remain as is currently displayed.
- If we choose random, the nodes will spread over the area at random.
- Circle creates a ring of nodes.
- Radial sets the nodes in a radial pattern with “node #0” in the centre of the image (this structure is often interesting).
- Tutte places nodes at the centre of mass of its neighbours, while nodes of degree four or less remain as anchors and are arranged in a circle.
- Remember we can also use degree–one Bárabási-Albert’s algorithm for every network by pressing redo layout.
- Spring is the second layout algorithm. As suggested by the name, links are treated as springs (characterised by the slide constant) between nodes that try to separate from each other to a certain distance (long) with a certain force (repulsion).
Highlight
This option can be selected at any time. It will turn the network grey, but if we move the mouse next of a node, this node as well as its neighbours and links will shine in full colour –note that highlight will only show true colours or opinions, so it will only work properly after distributing opinions. If dynamics are in progress, they will not be affected by this measure, but reprint must be set to Off to prevent colours from blinking annoyingly. Reprint should be selected when we deselect highlight.
Vote
Voting will proceed with the dynamics over whatever network is drawn. This process will finish when there are no active links left, or when vote is deselected.
Move
While voting is in process, the user may drag nodes with the mouse along the screen.
3. Things to try, notice and play
This applet can be used to explore the “Small-World” phenomenon7. Two monitors will show clustering and the average path length magnitudes, normalised in relation to their values with no rewiring. One can notice that the average path length decreases quickly as we rewire a few links, dropping to half its value or even less (this is more evident the bigger the network, which can result very slow as well), which is the so-called “Small-World” phenomenon present in most real-world networks. Opposed to this, the clustering coefficient is found to drop in a much more slow way. This procedure enables us to construct networks with small average path length and yet a noticeable clustering coefficient (compared to that of random networks), the purpose for which WS networks were created.
WS networks can also be useful to understand the voter model dynamics, this time being advisable to choose distance 1 and not doing rewiring at all, which will create a plain ring of nodes. Once opinions have been distributed (two opinions would be ideal in this case), segments of colour will form. In this situation, the bows of colour will be isolated, except for their limits. Therefore, they can only grow or diminish in one dimension. It can also help to understand the model to create a very small network, of around 10 nodes, and perform the voting process step by step in lieu of pressing the continuous button.
When the average degree is small (around 2 or less) it is likely that a variety of components will form in an ER network. Using layout can disentangle them. If we want to get rid of isolated nodes tutte will do so. Any other structure may bring them back.
BA is a dynamic process, which means that it can be run over anything previously created. This is a nice way to construct peculiar networks, which might reveal interesting characteristics.
Lattices (if not restructured) will show the formation of islands of opinion, which will compete amongst them. This is clearer in the packed form of lattices (not using spring).
Having so many different networks and options will permit comparison of the convergence times, the prominence of fluctuations, etc. It is suggested to compare networks with equal number of nodes. It is also recommended to compare the active links fluctuations of one network and very different amounts of nodes.
Create a lattice (17 x 17 would be fine) and then combine spring and layout to obtain beautiful patterns. Radial structure for lattices looks nice as well. ER networks with circular structure and all links allowed will also look beautiful, although less creative.