Injection without space (variable slope)

 

Without space, clusters do not grow via neighbors, but via random selection of one of their members. That is, we pick a cluster, remove it from the system, and then give its members to the other clusters one by one. The probability that the agent choses a cluster $i$ is proportional to that cluster's size $s$_i. If we assume that time advances with each member which is given back, we obtain the rate equation

$d\; N(s)\; dt\; =\; -\; s\; N(s)\; +\; (s-1)\; N(s-1)\; -\; \epsilon \hspace{0.17em}\; N(s)\; -\; \epsilon \hspace{0.17em}\; p$_inj   N(s) + ε  p_inj   N(s+1) .

The first and second term on the RHS represent growth by addition of another member; the third term represents random deletion; the fourth and fifth term the decrease by one which happens if one of the members is converted to a start-up via injection. $\epsilon$ is the rate of cluster deletion; since we first give all members of a deleted cluster back to the population before we delete the next cluster, it is proportional to the inverse of the average cluster size, which is in turn proportional to the injection rate: $\epsilon 1/s\; p$_inj. This is similar to an urn process with additional deletion.

Via the typical approximations $s\; \hspace{0.17em}\; N(s)\; -\; (s-1)\; N(s-1)\; \approx d\; ds\; (s\; \hspace{0.17em}\; N(s))$ etc. we obtain, for the steady state, the differential equation $0\; =\; -\; N\; -\; s\; \hspace{0.17em}\; dN\; ds\; -\; \epsilon \hspace{0.17em}\; N\; +\; \epsilon \hspace{0.17em}\; p$_inj   dN ds . This leads to

$N(s)\; \propto (s\; -\; \epsilon p$_inj)^-(1+ε) s^-(1+ε) .

That is, the exponent depends on the injection rate, and in the limit of $p$_inj →0 it goes to $-1$.

Note that the approach in this section corresponds to measuring the cluster size distribution every time we give an agent back to the system, while in the simulations we measured the cluster size distribution only just before a cluster was picked for deletion. In how far this is important is an open question; preliminary simulation results indicate that it is important for the spatial case with injection.