Temporal network classes¶
Undirected and unweighted temporal networks are composed of \(N\) nodes and up to \(m_{\mathrm{max}}=N(N+1)/2\) edges, where each edge \((i,j)\) can be described as a series of events where the edge is either switched on or switched off. One way of expressing that is to define the temporal adjacency matrix
In tacoma, we will interpret temporal networks as if they were recorded in an experiment. We expect that over the course of time \(t_0\leq t < t_\mathrm{max}\) in which we record activity, we will encounter \(N\) nodes from the node set \(V={0,1,\dots,N-1}\) (nodes posses an integer label).
The experiment begins at time \(t_0\), where the network consists of an edge set \(E_0 \subseteq \{i,j: V\times V, i<j\}\). Then, each time the network changes, we denote that time by an entry in a time vector \(t\). Each entry in the time vector corresponds to a network change event and thus to a change in the edge set. We call the total number of change events \(N_e\), such that the vector \(t\) has \(N_e\) entries. In between consecutive times, the network is constant. After the last recorded event, we kept the experiment running until the maximum time \(t_\mathrm{max}\) without observing any change and stopped recording at \(t_\mathrm{max}\).
There’s three data structures implemented in this package, all of which capture the situation described above in different ways and are useful in different situations.
Edge lists¶
The class _tacoma.edge_lists consists of a collection of complete edge lists,
each time the network changes, a complete edge list of the network after the change is saved.
It has the following attributes.
- \(N\) : The total number of nodes
- \(t\) : A vector of length \(N_e+1\). The 0-th entry contains the time of the beginning of the experiment \(t_0\)
- edges : A vector of length \(N_e+1\) where each entry contains an edge list, describing the network after the change which occured at the corresponding time in \(t\). The 0-th entry contains the edge list of the beginning of the experiment \(t_0\)
- \(t_\mathrm{max}\) : The time at which the experiment ended.
Additionally,
Edge changes¶
The class _tacoma.edge_changes consists of a collection of both edges being created
and edges being deleted.
It has the following attributes.
- \(N\) : The total number of nodes.
- \(t_0\) : The time of the beginning of the experimen.
- edges_initial : The edge list of the beginning of the experiment at \(t_0\).
- \(t\) : A vector of length \(N_e\), each time corresponding to a change in the network.
- \(t_\mathrm{max}\) : The time at which the experiment ended.
Additionally,