Evolutionary Architecture model (EvoArch)

This is a model built to explain the Internet protocol stack hourglass shape. It builds a DAG in discrete time steps $G_i = (V_i,E_i)$ over time $i \in \mathbb{N}$ to model the protocols in the OSI model.

• Define a set of layers $L$,
• Each vertex $v \in V_i$ gets mapped to a layer $l(v) \in L$,
• Each edge $(u,v) \in E_i$ has $l(v) = l(u) +1$ and describes the relationship protocol $v$ can use protocol $u$ as its lower level protocol.
• We define the set of parents of a node $S_i(v) = \{u \in V_i \vert (u,v) \in E_i\}$ to be substrates.
• The children of node $P_i(u) = \{v \in V_i \vert (u,v) \in E_i\}$ we define to be its Product.
• Each layer $l \in L$ has a generality probability associated to it, $s(l) \in [0,1]$. This determine the probability a higher level protocol will depend on a new protocol in this layer in the future. For each $v \in V_i$ where $l(v) > 1$ for each $u \in V_i \backslash V_{i+1}$ where $l(u) = l(v) +1$ then $(u,v) \in E_i$ with probability $s(l)$.
• Each node $v \in V$ has a value at time $i$, $v_i(v)$, this is defined by the product of that node i.e. $v_i(v) = f(P_i(v))$.
• There is a competitor threshold $c \in [0,1]$ that defines to competitors for nodes $$C_i(v) = \{u \in V_i \vert l(u) = l(v) \ \land \ \frac{\vert P(v) \cap P(u)\vert}{\vert P(u) \vert} > 1\}.$$
• Nodes die if their competitors have too much value. This also kills upstream nodes if they only depended on that node for its layer.
• Nodes also get added randomly to the graph as a percentage of the total size of the current network. Though this can very depending on the implementation

To update this model we do the following steps.

1. Introduce new nodes to the model.
2. Going from top to bottom carry out the following for each layer 1. Add new edges for new nodes. 2. Calculate new values for the nodes. 3. Examine nodes in decreasing value order and remove if needed.
3. Stop if we reach a predetermined number of nodes.