Mean-field approach to evolving spatial networks, with an application to osteocyte network formation

We consider evolving networks in which each node can have various associated properties (a state) in addition to those that arise from network structure. For example, each node can have a spatial location and a velocity, or it can have some more abstract internal property that describes something like a social trait. Edges between nodes are created and destroyed, and new nodes enter the system. We introduce a “local state degree distribution” (LSDD) as the degree distribution at a particular point in state space. We then make a mean-field assumption and thereby derive an integro-partial differential equation that is satisfied by the LSDD. We perform numerical experiments and find good agreement between solutions of the integro-differential equation and the LSDD from stochastic simulations of the full model. To illustrate our theory, we apply it to a simple model for osteocyte network formation within bones, with a view to understanding changes that may take place during cancer. Our results suggest that increased rates of differentiation lead to higher densities of osteocytes, but with a smaller number of dendrites. To help provide biological context, we also include an introduction to osteocytes, the formation of osteocyte networks, and the role of osteocytes in bone metastasis.

Figure 1

Diagrammatic illustration of our model of evolving spatial networks. In each panel, the square box represents the state space $\mathbb{S}$. In each time step of size $\mathrm{\Delta }t$ (where $0<\mathrm{\Delta }t\ll 1$), the following events can occur: (a) edge creation, (b) edge deletion, (c) node creation, and (d) evolution of node state.

Figure 2

Numerical illustration of the validity of the mean-field assumption (15). We average over 100 realizations of Algorithm lg1 using $n=125$ particles and a time step of $\mathrm{\Delta }t={10}^{-3}$. No new particles enter the system ($\mathcal{J}=0$), the edge-creation rate is $\mathcal{C}(k_i,k_j)=2$, and the edge-deletion rate is $\mathcal{D}(k_i,k_j)=k_i+k_j$. We show (a) the two-particle distribution $P_{k_1,k_2}^{\left(2\right)}\left(t\right)$, (b) the product $P_{k_1}\left(t\right)P_{k_2}\left(t\right)$ of the one-particle distributions, and (c) the difference $P_{k_1,k_2}^{\left(2\right)}\left(t\right)-P_{k_1}\left(t\right)P_{k_2}\left(t\right)$ at time $t=1/10$.

Figure 3

Illustration of the example scenario in Sec. 6b. Nodes enter the system in a strip on the left-hand side of the unit square and then diffuse and drift to the right according to Eq. (26). There are reflective boundary conditions at $x=0$ and $x=1$ and periodic boundary conditions at $y=0$ and $y=1$. We create edges between nearby nodes according to Eq. (27). Edges are deleted at the rate given by Eq. (28).

Figure 4

Comparison of (a) the mean of 200 Monte Carlo simulations using Algorithm lg1 binned into compartments of size $1/100$ along the $x$-axis and (b) $u_{k_1}(1/2,x_1)$ obtained from the numerical solution of Eq. (30). The parameter values are $\mathcal{J}=500$, $\mu =3/4$, $\sigma =1/4$, and $\epsilon =0.1$; and the time step in the stochastic (i.e., Monte Carlo) simulations is $\mathrm{\Delta }t={10}^{-4}$. We initialize the simulations with ${10}^3$ particles placed uniformly randomly in the domain $[0,1/10]\times [0,1]$. We show results at the final time $T_{\text{end}}=1/2$.

Figure 5

Comparison of degree distribution at time $T_{\text{end}}=1/2$ determined from Eq. (30) (red) and from a mean over multiple Monte Carlo simulations using Algorithm lg1 with a time step of $\mathrm{\Delta }t={10}^{-3}$ (blue). With $n$ nodes (constant in time, so $\mathcal{J}\equiv 0$), we average the simulations are over ${10}^6/n$ realizations, where $n=125$ (dotted curve), $n=500$ (dash-dotted curve), $n=2000$ (dash-dotted curve), and $n=8000$ (solid curve). (a) No edge deletion (i.e., $\mathcal{D}\equiv 0$), and the edge-creation rate is $500/n$ times that given in Eq. (27). (b) The edge-deletion rate is given by Eq. (28), and the edge-creation rate is $750/n$ times that given in Eq. (27). The other parameter values are as in Sec. 6b.

Figure 6

Analytical approximation to the degree distribution (solid blue curve) of the latent social-space model given in Ref. [33] versus the result from a kinetic formulation (red dash-dotted curve) using Eq. (19). We uniformly distribute nodes with a social parameter $h\in [0,125]$. We use the parameter values $n=1000$, $\alpha =3$, and $b=1/2$. The probability of connection and choice of $\mathcal{C}$ is given by Eq. (33). There is no edge deletion (i.e., $\mathcal{D}\equiv 0$), and the system is of constant size (i.e., $\mathcal{J}\equiv 0$). We solve the kinetic equation until a final time of $T_{\text{end}}=1$.

Figure 7

Scanning electron micrograph of osteocytes in bone. The sample was prepared by embedding the bone in resin, which was subsequently etched with perchloric acid. The image was created by removing the entire mineral in the sample, leaving a replica of the cells. Therefore, what is observed is the resin that filled the spaces in the bone and the spaces inside the cells. (This picture is copyrighted work and is available via Ref. [69] from Ref. [70].)

Figure 8

Diagrammatic illustration of the bone-formation process. Lighter shades of blue indicate more differentiated cells. The lighter shade of pink indicates the deposition front, and the darker shade of pink indicates the mineralization front. Panel (a) occurs earlier than panel (b). Dendritic osteocytes (light blue) have dendrites that extend towards the osteoblast layer (dark blue). The osteoblasts secrete bone matrix. Osteoblast cells marked with “A” are signaled by the osteocyte network to differentiate into osteocytes. Osteoblast cells marked with “B” do not differentiate and stay on the outer bone surface. Osteoblast cells marked with “C” arrive at the deposition front after differentiating from precursor osteoblasts (pre-osteoblasts). (This figure is inspired by a similar illustration in Ref. [21].)

Figure 9

(a) Traveling-wave profile of mineral density $m$ normalized by $C_m$. At the front of the wave, $m=r_{\text{exc}}^{\left(\text{Ob}\right)}/{\kappa }_{\text{form}}$ (dashed line), which we choose to be $0.5C_m$ in the figure. Behind the front, $m$ approaches $C_m$. (b) The solid curve is a traveling-wave profile of osteocyte density $f={\sum }_{k=0}^{\infty }{u}_k$, normalized by $b=D_{\text{burial}}/{\kappa }_{\text{form}}$. We illustrate the degree distribution of osteocytes as a function of position by showing ${\sum }_{k=0}^K{u}_k$ (dashed curves) for the case ${\lambda }_{\infty }=5$. The lower ($K=0$) curve illustrates the proportion of osteocytes with degree 0, and the difference between the $K=i$ and $K=i-1$ curves illustrates the proportion of osteocytes with degree $i$.