EXPLORABLES

This explorable illustrates a dynamic network model that was designed to capture the emergence of community structures, heterogeneities and clusters that are frequently observed in social networks. Clusters are characterized by a high probability that a person's *'friends are also friends'*. In this model not only the connectivity evolves but also the strength of links between nodes. The model was orginally proposed by Jussi Kumpula, Jukka-Pekka Onnela, Jari Saramäki, János Kertész and Kimmo Kaski.

* Press Play* and keep on reading....

## This is how it works

The *Jujujajáki Network* is a dynamic, **weighted network**. Existing links between nodes \(i\) and \(j\) have weights \(w_{ij}>0\) that quantify the connection strength.
The system starts with an initial configuration of \(N=100\) nodes and a few links that randomly connect some of the nodes. In the initial configuration all existing links have identical weight \(w_0\).

### The dynamics

Now we let the connectivity evolve: nodes can i) **establish new links**, ii) **cut their links** and iii) **reinforce existing links**. Estabilishing new links is either done by * exploring* the entire network or by

*of a node's neighbors as described in detail below. Here are the steps of the iteration:*

**leveraging the local connectivity**- A node \(i\) is chosen randomly.
- With probability \(P_0\) the node
itself (cuts all its links). We can also think of this step as**isolates***'the node is removed from the system and immediately reborn without any links'*. This way the number of nodes \(N\) is kept constant. - With probability \(P_\text{E}\) node \(i\) picks another random node \(j\) (that it isn't connected to already) anywhere in the network and establishes a link with initial weight \(w_{ij}=w_0\). This is the
.**exploration step** - Finally, with probability \(P_\text{L}\) the node seeks to establish a new link using one of its neighbors \(j\) as a promotor. This
**local search**step is a bit more complicated and is explained below.

### Local search and link reinforcement

Node \(i\) selects one of its neighbors \(j\) with a propensity proportional to their link weight \(w_{ij}\). Next, node \(j\) picks one of its neighbors \(k\neq i\) with propensity \(w_{jk}\). If \(i\) and \(k\) are not connected, a link between \(i\) and \(k\) is established with weight \(w_0\) forming an \((i,j,k)\)-triangle. We can interpret this rule as: 'Node \(j\) introduces two good friends \(i\) and \(k\).

Additionally, the involved links get reinforced by a little amound \(\delta\) so \(w_{ij}\rightarrow w_{ij}+\delta\) and \(w_{jk}\rightarrow w_{jk}+\delta\).

If a link between \(i\) and \(k\) already exists, it also gets reinforced by an increment \(\delta\).

### Parameters and properties

The key parameters of the system are the * isolation rate*, the

*, the*

**exploration probability***and the*

**local search probability***all of which you can*

**reinforcement increment***.*

**control with the respective sliders**In the display, apart from the layout, three node / link specific properties are illustrated. Nodes vary in size proportional to their degree (the number of links of a node). Nodes are colored (from white to black) proportional to their clustering coefficient. Links are colored from black to darkred according to their strength and vary in width accordingly.

### Local clustering

The local clustering coefficient is defined by the ratio of connections between a node's neighbors and the maximum number of connections they can have. So, if say a node has 4 neighbors, than the maximum number of links they can have is \((4\times 3) / 2 = 6\). A large clustering coefficient means your friends are also friends. For isolated nodes and nodes with only one neighbor, the clustering coefficient is zero.

## Try this

Initially, the network has only a few links altogether. So it is very unlikely that a node's neighbors are also connected. When you press play every now and then a black node appears because it is part of a spurious triangle.

* For the default parameters nodes do not execute local search.* All new links are established by exploration, so randomly connecting anywhere in the networks. Nothing much happens in terms of structure. The network exhibits no tendency for clustering.

* If you now increase the local search probability,* strong links will appear, as well as tightly knit groups of triangles. This structure will eventually come to a dynamic equilibrium, exhibiting structures observed in real networks.

By increasing the isolation rate, you can 'thin out' the network.

## Also....

As pointed out by Hiroki Sayama **Jujujajáki** written in Japanese is **呪呪邪邪鬼** which, according to google translate means: **Curse evil evil demon**.

## References

Jussi M. Kumpula, Jukka-Pekka Onnela, Jari Saramäki, and János Kertész, Kimmo Kaski, Model of community emergence in weighted social networks,

*Computer Physics Communications*,**180**, 517-522, (2009)Jussi M. Kumpula, Jukka-Pekka Onnela, Jari Saramäki, Kimmo Kaski, and János Kertész, Emergence of Communities in Weighted Networks,

*Phys. Rev. Lett.***99**, 228701 (2007)