Recruitment dynamics in adaptive social networks

We consider a social structure consisting of $M$ individuals represented by nodes in a network. An existing relation between any two individuals is represented by a link between the two nodes. New individuals join the society at a constant total rate $\mu$, and the individuals in the network can leave the network via death at rate $\delta$ per individual. When new individuals join the system, they arrive with $\sigma$ links, which are connected to $\sigma$ randomly selected nodes in the population. When individuals die, their links are removed.

Some members of the society, recruiters (also referred to as R-nodes), are carriers of an ideology that they try to spread to the individuals they come into contact with. The individuals that do not belong to the recruiting class are divided into two groups: those who are susceptible to the recruitment (S-nodes) and those who are non-susceptible (N-nodes). We assume that individuals can spontaneously change their state from non-susceptible to susceptible and vice versa (with rates $\lambda_{1}$ and $\lambda_{2}$ respectively). An individual in the recruiting class remains in that class until death Udwadia et al. (2006).

A susceptible individual joins the recruiting class at a rate that is proportional (with proportionality constant $\gamma$) to the number of contacts it has with the recruiting class. In order to improve their recruiting capabilities, the recruiting individuals can rewire their links with rate $w$, abandoning a connection to a nonsusceptible individual in favor of a connection to a susceptible one. As the links are rewired, the network topology changes based on the current states of its nodes. A schematic representation of the node dynamics is shown on Fig. 1.

We perform Monte Carlo simulations of the above system following the continuous time algorithm described in Gillespie (1976). We start with an Erdos-Renyi random network, which then evolves according to the rules of birth, death, and rewiring. The results presented in the rest of the paper are for the systems that have reached a steady state.

To describe the dynamics of this system, we construct a mean field model for nodes and links as in Gross et al. (2006). The time evolution of the nodes of each type is described by the following rate equations:

Here the functions $\mathcal{N}_{\text{N}}\equiv\mathcal{N}_{\text{N}}(t)$, $\mathcal{N}_{\text{S}}\equiv\mathcal{N}_{\text{S}}(t)$, $\mathcal{N}_{\text{R}}\equiv\mathcal{N}_{\text{R}}(t)$ represent the number of nodes of each type. The process of recruitment is captured by the $\gamma\mathcal{N}_{\text{RS}}$ term, where the recruitment is shown to take place at a rate proportional to the number of links between the recruiting class and the susceptible class, $\mathcal{N}_{\text{RS}}$.

In order to capture the rewiring process, we follow the evolution of the number of different types of links present in the network:

where the terms $\mathcal{N}_{\text{xy}}$ correspond to the number of links connecting nodes from classes x and y, e.g., $\mathcal{N}_{\text{\text{NN}}}$ corresponds to the number of NN links. The terms proportional to $\sigma\mu$ correspond to the influx of edges due to the birth of new nodes, where the probability for the new node to attach itself to a node from class X is proportional to the number of nodes in that class. The third order terms, $\mathcal{N}_{\text{NSR}}$, $\mathcal{N}_{\text{SSR}}$ and $\mathcal{N}_{\text{RSR}}$, describe the formation of triples of nodes with an S-node at the center, and at least one of the edges terminating at an R-node. These terms describe the rate at which NS-, SS- and RS-links become NR-, SR- and RR- links respectively, due to the interaction of the central S-node with its neighboring R-node. Note that our definition of RSR triples includes degenerate triples, i.e., RSR triples where both of the R-nodes correspond to a single R-node, by analogy with degenerate triangles.

The resulting system of equations is not closed, as it contains higher order terms. Following earlier works in epidemiology Keeling et al. (1997), we introduce the closure based on the assumption of homogeneous distribution of RS-links:

We thus obtain a system of nine mean field equations, which we analyze.

We first consider the bifurcation point of the recruitment model where the zero recruit (trivial) steady state becomes unstable, which we call the recruiting threshold. This is a transcritical bifurcation point, which is analogous to the epidemic onset in epidemic spreading models. It can be found analytically as a function of parameters for the mean field model, and certain asymptotic limits have a simple form.

We nondimensionalize Eqns. (1) and (2) to simplify the analysis. We introduce the dimensionless time variable $\tilde{t}\equiv\delta t$, where time is rescaled by the average node lifetime $\delta^{-1}$. Further, the node variables are rescaled by the expected population size at steady state, $\mu/\delta$, while the link variables are rescaled by the expected number of links at steady state, $(\mu/\delta)(\sigma/2)$. Let ${\bf x}\equiv[\mathcal{N}_{\text{N}},\mathcal{N}_{\text{S}},\mathcal{N}_{\text{R}},\mathcal{N}_{\text{NN}}2\sigma^{-1},\mathcal{N}_{\text{SN}}2\sigma^{-1},\mathcal{N}_{\text{SS}}2\sigma^{-1},\\ \mathcal{N}_{\text{RN}}2\sigma^{-1},\mathcal{N}_{\text{RS}}2\sigma^{-1},\mathcal{N}_{\text{RR}}2\sigma^{-1}]\delta\mu^{-1}$ denote the 9-dimensional vector of rescaled node and link state variables. Note that in the steady state the rescaled node variables sum to 1, and, therefore, in steady state they correspond to the probability for a node of a given type to exist in the system. Similarly, the rescaled link variables sum to 1, and correspond to the probability for an edge chosen at random to be of a given type, e.g., at steady state $x_{8}$ corresponds to the probability for a randomly chosen edge to be an RS link.

We introduce the following rescaled parameters:

We can now write the dimensionless equations of motion as

where $\bar{\Lambda}\equiv[\Lambda_{1},\Lambda_{2},W,\Gamma,\sigma]$ is a vector of all the system parameters. (Recall that $\sigma$ is a dimensionless integer.) The full system of dimensionless equations is given in Eqs. (11).

We can now find the trivial steady state solution, where the number of R nodes is zero. This restricts the state to be of the form ${\bf x}_{0}=[x_{0,1},x_{0,2},0,x_{0,4},x_{0,5},x_{0,6},0,0,0]$, where the number of links involving R nodes is zero as well. This guarantees that $F_{i}({\bf x}_{0},\bar{\Lambda})=0$ for $i=3,7,8,9$. Since this subset of equations represents an invariant manifold, we concentrate on solving the equations for the rest of the 5 components, $x_{0,1}$, $x_{0,2}$, $x_{0,4}$, $x_{0,5}$ and $x_{0,6}$.

The first observation is that the $x_{1}$ and $x_{2}$ equations are solvable when the number of recruiting nodes is zero, and they yield steady state values of

Now that we have the full trivial solution, we can examine its stability. In order to do this, we linearize the vector field about ${\bf x_{0}}$ and examine where it has a one dimensional null space. That is, at the bifurcation point, there is only one real eigenvalue passing through zero. This is equivalent to examining where the determinant of the Jacobian vanishes; i.e., we compute those parameters where

The following relation describes the location of the bifurcation:

We next examine the limits of Eq. (8) when either the recruitment rate $\Gamma$ or rewiring rate $W$ is large. The minimum amount of recruitment required to maintain nonzero values of recruited population, when $W$ approaches infinity, can be found by setting the least common denominator of Eq. (8) to zero, which yields a simple expression for $\Gamma$:

If the recruitment rate is below this value, additional rewiring is not sufficient to enable spreading of the recruiting class.

Similarly, we find the smallest amount of rewiring required for existence of a nontrivial steady state solution, conditioned on our ability to control $\Gamma$, with all the other parameters fixed:

For lower rewiring rates, the recruiting class cannot spread even if the recruiting rate is large. The value of this asymptote is greatest, meaning rewiring is most necessary, when $\Lambda_{1}$ approaches zero. In this case few non-susceptible nodes become susceptible to recruiting, and the necessary rewiring value approaches

We next compare the mean field predictions for the spread of recruiters with simulations of the full stochastic network system. Thus, we compare the average size of the recruited portion of the population in statistical steady state to the solution of the mean field equations at steady state, which we solve exactly in the Appendix A. We assume that the parameters $\Lambda_{1}$ and $\Lambda_{2}$ (rates for gaining and losing susceptibility) and $\sigma$ (determines average degree) depend on details of the society. On the other hand, we assume that the parameters $\Gamma$ and $W$ can be controlled by the recruiters, i.e., the recruiters may choose to be more or less aggressive in their recruitment, as well as in how quickly they rewire their links to susceptible members of society. Therefore, we investigate the recruiting effectiveness for a given choice of $\Lambda_{1}$, $\Lambda_{2}$ and $\sigma$, while varying $\Gamma$ and $W$.

We distinguish two parameter regimes, corresponding to two qualitatively different behaviors of the system: small values of $\Lambda_{1}$ ($\Lambda_{1}\ll 1$) and large values of $\Lambda_{1}$ ($\Lambda_{1}\gg 1$). The small $\Lambda_{1}$ regime corresponds to the case where nodes become susceptible to recruiting only rarely, on average much less than once in their lifetime (where the average lifetime of a node is $\delta^{-1}$ in the original time units and $1$ in the dimensionless $\tilde{t}$ units). The large $\Lambda_{1}$ regime corresponds to the case where, on average, nodes become susceptible to recruitment at least once in their lifetime.

In Figs. 2 and 3 we show the results of simulating the system in these two regimes. The density plots in Figs. 2 and 3 show the fraction of recruited nodes as a function of recruiting rate $\Gamma$ and rewiring rate $W$ in networks with $\Lambda_{1}\gg 1$ and $\Lambda_{1}\ll 1$ respectively. The black curves in the two figures represent the location of the recruiting threshold as derived from the mean field equations and given by Eq. (8). Here mean field allows us to accurately predict the onset of the stable nontrivial solution. In Figs. 2 and 3 we compare the simulation results to the mean field predictions. Even though there is some discrepancy between the direct simulations and the mean field near the bifurcation, the recruiting threshold and the asymptotic behavior for large $\Gamma$ show excellent agreement with the simulations.

We observe in simulations that the steady state trivial solution undergoes a forward transcritical bifurcation in the recruiting rate $\Gamma$ for all values of $W$ for which a nontrivial solution exists. This is in contrast with epidemiological models where the purpose of the rewiring is avoidance of the nodes spreading infection Shaw and Schwartz (2008); Gross et al. (2006), which can undergo a backward transcritical bifurcation and exhibit bistability. Note that, unlike those models, the purpose of the rewiring in the recruitment model is attraction of the susceptible population by the recruiters.

We would like to draw the reader’s attention to an important difference in the system behavior in the two regimes. In the regime where $\Lambda_{1}\gg 1$, the nontrivial solution exists for all values of $W$ as long as $\Gamma$ is large enough. On the other hand, in the regime where $\Lambda_{1}\ll 1$, the nontrivial solution may fail to exist for any value of $\Gamma$ unless rewiring is aggressive enough. In Fig. 3 there is a range of rewiring rates $W$ for which only trivial solutions exist. The horizontal dashed line, given by Eq. (10), indicates the mean field level of rewiring that is sufficient for the nontrivial solution to exist for rapid recruiting (large $\Gamma$). Also, we can see that the rewiring level that is necessary for the emergence of the nontrivial solution can be very close to the predicted sufficient level of Eq. (10), for small values of $\Lambda_{1}$.

Another important difference between the two regimes of $\Lambda_{1}$ values is seen when we compare two systems with different values of $\Lambda_{1}$ and $\Lambda_{2}$ while the ratio $\Lambda_{1}/(1+\Lambda_{2})$ is kept fixed. Note that in the absence of recruitment, such systems have identical fractions of susceptible nodes. In other words, in the presence of recruitment, the pool of individuals available for recruitment would appear to be the same. However, as we can see from Fig. 4, there is a significant difference in the size of the recruited population as well as the recruiting threshold. The difference appears to be caused by the difference in the dynamics in the two regimes. Thus, for the large values of $\Lambda_{1}$ and $\Lambda_{2}$ a node can become susceptible several times during its life, while as the value of $\Lambda_{1}$ decreases, only some nodes will ever become susceptible, with low likelihood of doing so more than once.

We now investigate the structure of the portion of the network (later referred to as the R-subnetwork) consisting only of R-nodes and links between them. In particular, we are interested in the mean degree of its nodes when the system reaches a steady state. The mean field approach, in addition to predicting the fraction of recruited nodes (which corresponds to the size of the R-subnetwork), also provides information about topology of the subnetwork in the form of mean degree of the nodes. The mean degree of a node in the subnetwork serves as a low order description of how well the nodes in the network are connected, which may be important for communication within the established subnetwork. The density plot in Fig. 5 shows the dependence of the mean degree on $W$ and $\Gamma$. The nonmonotonic behavior as a function of $\Gamma$ for fixed $W$ is predicted by the mean field model. The solid line corresponds to the analytical prediction of the maximum’s location as given by Eq. (32) derived in Appendix B. Note that the analytic description of the maximum’s location would allow the recruiting class to optimize connectivity within the subnetwork, if that happens to be an important goal for that class. The mean field approximation fails to capture the nonmonotonic behavior of the mean degree in the limit where $\Lambda_{1}\ll 1$, and we leave this issue to a future study.