# A Graph Visualization Tool

[Fabian Schubert, 2022]

You can use this application to explore the properties of two types of randomly generated graphs: The Erdős–Rényi model and the Barabasi-Albert model. You can vary the network size as well as the average number of edges of the nodes in the network.

The plot on the left is a visualization of the resulting graph structure. Nodes belonging to the same connected component have the same color. On the right, you see the size of the largest connected component relative to the network size as a function of the mean degree. Changing the network size or the mean degree will add more samples to the plot. Click on "clear plot" to remove these samples.

## Erdős–Rényi Model

In the Erdős–Rényi model, each potential edge in the graph is added with a probability $$p=\langle k \rangle / (2 N)$$, where $$k$$ is the average node degree and $$N$$ is the network size. If you vary $$k$$, you will observe a transition for $$k \approx 1$$in the plot on the right, where the relative size of the largest component increases. This transition becomes more "sharp" as you increase the network size. For $$N \rightarrow \infty$$, this becomes a second-order phase transition.

## Barabasi-Albert Model

In this model, edges and nodes are subsequently added. Each new node that is added to the network connects on average to $$\langle k \rangle / (2 N)$$ existing nodes. The probability of connecting to an existing node $$i$$ is weighted by its current degree $$k_i$$. This preferential attachment leads to a scale-free powerlaw distribution of edge degrees. Furthermore, the relative size of the largest connected component in the network becomes non-zero in the $$N \rightarrow \infty$$ for all $$\langle k \rangle > 0$$: Try varying $$\langle k \rangle$$ with the Barabasi-Albert model and observe the orange samples on the right plot.