Master equation transition rates

We indicate in Table 2 the relationship between (k, m) and its neighbouring classes in configuration space. Classes that can be reached from (k, m) via ego, neighbour, and edge transitions are shown in the left column, with the corresponding transition type given in the right column. The rates at which nodes flow between classes are given in the middle column. The following notation is used to describe relative changes to the class (k, m). First, e_j is the jth unit vector, and determines the change to a degree vector as a result of a neighbour transition. Second, we define Δj±=−ej+ej±1, which determines the change to a degree vector as a result of an edge transition. When an edge increments or decrements state, adjacent nodes lose a j-type edge, and gain a j ± 1-type edge, all while preserving the underlying degree k. Transitions appear as directed edges in the lattice diagram illustrations of configuration space (see Supplementary Note ¹).

Class transition rates.

Classes neighbouring (k,m) in configuration space are shown in the leftmost column. Those marked with and without an asterisk flow into and out of (k,m), respectively. The rate at which they do so is given in the centre column. Transition types are shown in the rightmost column, and include ego, neighbour and edge transitions. Classes related to (k,m) by neighbour transitions differ only by e_j, the jth unit vector. Classes related by edge transitions differ by Δj±=−ej+ej±1, that is, by the loss of an edge of type j, and the gain of an edge of type j ± 1.

Ego transitions occur at rates F_k,m, and involve the flow of nodes from set S_k,m to I_k,m. As such, no change to the ego’s local neighbourhood (k, m) takes place, and the transition represents a type of self-edge, or loop, in the lattice representation of configuration space. The rates F_k,m are encoded in transmission functions such as those shown in Table 1. Flux measurements of these transitions, such as those in Supplementary Note ⁷, are expected to be exact. This is because node infection is directly determined by F_k,m. As nodes are infected at constant rates, we draw the waiting time to infection from an exponential distribution with mean 1/F_k,m. As such, flux measurements of ego transitions must agree with F_k,m, by construction. This makes them a useful benchmark for verifying one’s implementation. The rates F_k,m are contained in the matrix W_ego.

Neighbour transitions are based on the probability β_jdt that an uninfected neighbour of an uninfected node becomes infected over an interval dt. To calculate β_j we use a straightforward ensemble average over S. To obtain the expected fraction of neighbours undergoing transitions, we observe the number of nodes undergoing ego transitions at time t, and count the number of neighbour transitions produced as a result. That is, when an uninfected node in class (k, m) becomes infected, which occurs with probability F_k,mdt, it has k_j − m_j uninfected neighbours that observe this transition, or k_j − m_j nodes undergoing neighbour transitions. The number of such edges across the entire network is given by ∑_Sp_k(k_j − m_j)F_k,ms_k,m, where the sum is over all uninfected classes. We compare this with the total number of uninfected-uninfected edges, ∑_Sp_k(k_j − m_j)s_k,m, giving the neighbour transition rate

which has previously been used in master equation solutions of binary-state dynamics on static networks. The rates β_j are contained in the matrix W_neigh, weighted by the values k_j and m_j of the relevant classes (k, m), as detailed in Supplementary Note ¹.

Edge transitions occur at rates μ_j and ν_j, and give the probability of edges in state j transitioning to state j + 1 or j − 1, respectively, over an interval dt. Their value depends upon the temporal network model in question. In this work, edge transition rates are determined by renewal processes following interevent time distributions ψ(τ), with complementary cumulative distributions Ψ. If the state of an edge is determined by the number of events j having occurred in the preceding time window of duration η owing to a renewal process, edge transition rates are

and

with

giving the probability that a randomly selected edge is in state j. It is this quantity that provides the normalising constant for the rates μ_j and ν_j. Here, ψ^*j is the jth convolution power of ψ. A complete derivation of these quantities is given in Supplementary Note ¹. The Gaver–Stehfest algorithm is used to compute the inverse Laplace transforms, and an efficient numerical procedure reducing μ_j and ν_j to a matrix-vector product is developed in Supplementary Note ⁵. These expressions hold for j > 0, with Eqs. (²) and (³) in the main text giving the special case of j = 0 for E_j and μ_j, respectively. Regardless of the form of ψ, the mean edge state η/〈τ〉 is always conserved on a network-wide level. Applying Eqs. (¹¹) and (¹²) at the level of class transitions amounts to a mean field approximation, since flux measurements of Monte Carlo simulation show edge transition rates to deviate slightly from μ_j and ν_j at the class level (k, m), even if exact for the network as a whole, as shown in Supplementary Note ⁷.