Emergent centrality in rank-based supplanting process

We propose a stochastic process of interacting many agents, which is inspired by rank-based supplanting dynamics commonly observed in a group of Japanese macaques. In order to characterize the breaking of permutation symmetry with respect to agents' rank in the stochastic process, we introduce a rank-dependent quantity, overlap centrality, which quantifies how often a given agent overlaps with the other agents. We give a sufficient condition in a wide class of the models such that overlap centrality shows perfect correlation in terms of the agents' rank in the zero-supplanting limit. We also discuss a singularity of the correlation in the case of interaction induced by a Potts energy.

Figure 1

Schematic illustration of a transition step from a configuration described by (a) to a configuration described by (c) in the model. From (a) to (b), agent 2 hops to the right site, and from (b) to (c), agent 3 among two possibly supplanted agents, which are 3 and 5, is supplanted by agent 2 and hops to the left site. Four arrows in (b) mean that, in the above transition, two agents 3 and 5 in $S(\mathit{x},2,+)$ could be supplanted, and the direction of supplanting process could be either to the left or to the right.

Figure 2

Probabilities of the configurations determined by stationary distribution for $\beta =2$ (circles), $\beta =0$ (rectangles), and $\beta =-1$ (diamonds); $p=0$ (blue or the left at each column) and $p=10$ (red or the right); $N=L=3$. The state of (0,0,0) means that three agents are located at the same site. The state of (0,1,2) means that each of three agents is located at the different position, respectively. The other three states mean that two agents are located at the same site, and another agent is located at one of the different sites.

Figure 3

The $\beta$ dependence of the normalized expectation value of the energy $M=\frac{1}{N^2}{\sum }_x{\sum }_{i,j}\delta (x_i,x_j)\langle \mathit{x}\left|P\right(\beta ,p)\rangle$ for various values of $p$ with $L=6,N=6$.

Figure 4

Heatmap of neighbor matrix $\mathcal{R}$ determined by the stationary distribution with the diagonal components left out. The color corresponds to $r_{ij}$ for pair of two agents $(i,j)$. Parameters: (top) $\beta =-1,p=1,L=6,N=6$; (bottom) $\beta =1,p=1,L=6,N=6$.

Figure 5

The overlap centrality $O_i$ determined by the stationary distribution obtained by the exact diagonalization as a function of agent $i$ for $L=N=6$. The solid lines are linear regressions for $O_i$ and $i$. The parameters are set as follows; $p=0.1$ (left) and $p=10.0$ (right); $\beta =1$ (top), $\beta =0$ (center), and $\beta =-1$ (bottom), respectively. For negative $\beta , \varphi$ is close to 1 regardless of the value of $p$.

Figure 6

Correlation coefficient $\varphi$ as a function of $\beta$ for $L=N=6$.

Figure 7

The overlap centrality $O_j$ computed by the exact diagonalization for $L=N=6$ is expressed by $O_j\simeq a\left(p\right)j+b\left(p\right)$, where coefficient $a\left(p\right)$ and $b\left(p\right)$ are determined by linear regression. These two figures plot the regression coefficient $a\left(p\right)$ vs $p/(1+p)$ for $p=0.001,0.002,\cdots ,0.01$; $\beta =-1$ (left) and $\beta =1$ (right). The correlation coefficient between $a\left(p\right)$ and $p/(1+p)$ is $0.9999998...$ for $\beta =-1$ and $-0.99997...$ for $\beta =1$. Note that $p$ dependence of the first term in (85) corresponds to that the correlation coefficient is equal to 1 for $\beta =-1$ and $-1$ for $\beta =1$, respectively.

Figure 8

${\mathcal{B}}_2-{\mathcal{B}}_1$ as a function of $\beta$ for $L=N=3$. The line corresponds to the computation through its definition (81). The blue crosses correspond to the estimation through (85) where $O_i$ is calculated by the exact diagonalization. Practically, the estimated value is calculated as $\tilde{a}\left(\beta \right)$ where $a\left(p\right)\simeq \tilde{a}\left(\beta \right)\frac{p}{1+p}+\tilde{b}\left(\beta \right)$ through linear regression.

Figure 9

The overlap centrality (blue circles) $O_i$ and the normalized eigenvector centrality (black crosses) determined by the stationary distribution obtained by the exact diagonalization as a function of agent $i$ for $L=N=6$. The parameters are set as follows; $p=0.1$ (left) and $p=10.0$ (right); $\beta =1$ (top), $\beta =0$ (center), and $\beta =-1$ (bottom), respectively. The eigenvector centrality is normalized so that the two data points of $i=1$ and $i=2$ in the ranking axis are perfectly matched. The overlap centrality gets closer to the eigenvector centrality for $p=0.1$ compared to the case of $p=10.0$.

Figure 10

Probability tree of hopping by equilibrium dynamics.

Figure 11

Probability tree of supplanting process after the first hopping.