Birth and Death Networks

In many growing network models, nodes are added to a network and attach their incoming links to existing nodes with a specified probability that typically depends on the existing nodes’ properties. For example, in the famous preferential attachment model, nodes with a high connectivity are more likely to attract incoming links which leads to the emergence of scaling in such networks. When incoming links attach preferentially to the local density of connections clusters emerge naturally.

Because these models are purely growing models, they do not approach an equilibrium. To describe an equilibrium situation for every node that is added on average a node must also die. Below is an interactive network birth death process. Nodes come in at a constant rate and each existing node can die with a small probability. When a node is born it links to existing nodes with 1-2 incoming links. When a node dies all its links die with it.

As the simulation proceeds nodes age (age is quantified by color from white to darkgreen). In this particular case new nodes attach with a higher probability to other “young” nodes and with a smaller probability to the “old” folks. When you watch the simulation sufficiently long you observe that in dynamic equilibrium (approx. 300 nodes) the old nodes will be “expelled” to the periphery of the network and will be isolated while an active core of young nodes emerges in the core of the network.