Principle of Locality I: Hacking the Continuum Mean-Field Technique in Network Modeling

Figure 1. Space-network created by Stephen Wolfram

Outline

• What does "Continuum Mean-Field" mean ?
• From Dynamics assumption to Stable Distribution
• The Preferential Attachment Model
• The Preferential Return Model
• The Spatial-constrained Attachment Model
• From Stable Distribution to Dynamics assumption

1. What does "Continuum Mean-Field" mean ?

1. "Mean-Field": many -> one
   The complex interaction between a large number of components can be simplified into a single averaged effect of all the other individuals on any given one.

2. "Continuum": discrete -> Continuous
   A discrete variables that is "smooth enough", i.e. increase by one unit in each period, such as time steps [1], spatial jumps [2-3], and population, can be viewed as a continuous geometric structure. Any measure on this structure, like degree [1], number of unique locations visited [2], and distance [3], can be described by differential equations.

2. From Dynamics assumption to Stable Distribution

2.1 The Preferential Attachment Model

Figure 2. An example network of the preferential attachment model

Figure 3. Time continuum

Geometric structure	Measure	Differential form
Time t	Degree k

The "preferential attachment" assumption can be translated as


In other words, we have

Eq. 2

or

Eq. 3

which gives

Eq. 4

Considering the boundary condition (the purple node in Figure 2) j=t, k=m leads to

=\frac{1}{2}ln(j)+c \rightarrow c=ln(m)-\frac{1}{2}lnj.)

Therefore, Eq. 4 reads

Eq. 6

or

Eq. 7

Now, here comes the most interesting step. Observe Figure 2 again, we find that all nodes on the right of the ith node (including the ith node itself) have a degree <= ki(t), whereas all nodes on the left have a degree > ki(t). It means that we can write the cumulative probability function as

![Eq. 8][eq8]
[eq8]:http://latex.codecogs.com/svg.latex?P(K\leq,k_j)=\frac{t-j}{t}=1-\frac{j}{t}=1-\frac{m^2}{k_j2}=F(k_j),

and thus naturally we have the probability density function as

![Eq. 9][eq9]
[eq9]:http://latex.codecogs.com/svg.latex?f(k_j)=\frac{dF(k_j)}{dk_j}=\frac{2m^2}{k_j3},

To summarize, PA predicts two scaling relationships, Eq. 7 and Eq. 9.

Now, let's summarize the steps:

1. Define the quantity of interest y (e.g., degree k) and the background variable x (e.g., time t);
2. Give the single-step assumption as differential equation dy/dx = f(x,y);
3. Solve this differential equation to obtain y = g(x);
4. Get the distribution P(Y<=y)=h(x)=h(g'(y))

This approach is very general and can be applied to any kind of social dynamics satisfying the "Continuum" and "Mean-Field" assumption.

2.2 The Preferential Return Model

Figure 4. An example network of the preferential return model

Figure 5. Interest continuum

The major (and very critical) difference between Preferential Return (PR) and Preferential Attachment (PA) is that, the number of nodes increase sub-linearly over time in PR. Therefore, in PR we have to choose either time or the interest space as the geometric background. Here we choose the latter and study the dynamics.

Besides this, PR use a linear function of degree (1+\lambda k) for calculating weights so that new nodes with zero degree may also be visited. We can call this variable votes (v=1+\lambda k) for it is vote and not degree that determine the probability directly.

As shown by Figure 3, nodes are sorted by their visiting orders. The jth node is firstly visited at tj. In other words, util time ti there are i nodes being visited, and each node is visited by kj times. We have

}{\sum_{j=1}^M{(1+\lambda k_j)}}=\frac{M-i}{M+\lambda \sum _{j=1}^Mk_j}=\frac{M-i}{M+\lambda t}.)

Re-arrange it to obtain


or


which gives

=-\frac{1}{\lambda}ln(M+\lambda t)+c.)

or

^{\frac{1}{\lambda}})

Considering the boundary condition when t=1, i=1,

=-\frac{1}{\lambda}ln(M+1)+c \rightarrow e^{{-c}=(M-1)(M+\lambda)}{\frac{1}{\lambda}}.)

Therefore Eq 14 reads

(\frac{M+\lambda}{M+\lambda t})^{\frac{1}{\lambda}})

or

Eq. 17

Eq. 17 predicts the waiting time before the first visit of the jth node.

Meanwhile, we derive the increase of degree over time as

}}=\frac{1+\lambda k_{j}}{M+\lambda t}.)

Re-arrange it to obtain


or


which gives

=ln(M+\lambda t)+c.)

Considering the boundary condition when t=tj, kj=1

=ln(M+\lambda t_j)+c \rightarrow c=ln(1+ \lambda)-ln(M+\lambda t_j).)

Therefore, Eq. 21 reads

=ln(M+\lambda t)+ln(1+ \lambda)-ln(M+\lambda t_j).)

or


Considering the condition tj=g(j) given by Eq. 17, we derive

^{\lambda}-\frac{1}{\lambda})

Considering the current time step ti=g(i) we also have

Eq. 26

When j=i, kj=ki=1.

Again, here comes the most interesting step. In Figure 3 we find that all nodes on the right of the ith node (including the ith node itself) have a degree <= ki, whereas all nodes on the left have a degree > ki. It means that we can write the cumulative probability function as

![Eq. 27][eq27]
[eq27]:http://latex.codecogs.com/svg.latex?P(K\leq,k_i)=\frac{M-j}{M}\approx\frac{M-j}{M-1}=[\frac{\lambda}{\lambda+1}\frac{\lambda+M}{\lambda{t}+M}(k+\frac{1}{\lambda})]^{\frac{1}{\lambda}}=F(k_i),

and thus naturally we have the probability density function as

![Eq. 28][eq28]
[eq28]:http://latex.codecogs.com/svg.latex?f(k_i)=\frac{dF(k_i)}{dk_i}=\frac{1}{\lambda}(\frac{\lambda}{\lambda+1}\frac{\lambda+M}{\lambda{t}+M})^{{\frac{1}{\lambda}}(k+\frac{1}{\lambda})}{\frac{1}{\lambda}-1},

To summarize, PR predicts three scaling relationships, Eq. 16, Eq. 26, and Eq. 28.

2.3 The Spatial-constrained Attachment Model

Figure 6. An example network of the space-constrained attachment model

Figure 7. Similarity continuum

}{dt}=\Delta R(t)=\int_0^{rxf(x)dx=\int_0}rx\frac{1}{L}dx=\frac{r^2}{2L}.)

which gives

Eq. 30

in which we can consider the boundary condition t=2, R(t)<=r to obtain that c<=r.

If the similarity space is D dimensional, then the volume of the space

\sim R(t)^D\sim t^D,)

As the radius increase linearly with time, the number of nodes at distance rho from the origin is approximately R(t) - rho, which is the time steps passed by after the time step rho:

\sim R(t)-\rho,)

Therefore, the total number of nodes in one dimensional space is

\sim\int_0^R(R-\rho)d\rho = \frac{R(t)^2}{2},)

If the similarity space is D dimensional, then

\sim\int_0^R(R-\rho)Dd\rho = \frac{R(t)^{D+1}}{D+1}\sim t^{D+1},)

Meanwhile, we know that for every "local area" of the space with distance r, n nodes will be fully connected to each other, generating n^2/2 undirected links, therefore the total number of edges

\sim\int_0^RN(t) d\rho \sim R(t)^{D+2} \sim t^{D+2},)

Putting these results together, we have the following scaling relationships:

\sim V(t)^{\frac{D+1}{D}},)

and

\sim N(t)^{\frac{D+2}{D+1}}.)

That is, the super-linear scaling between nodes and covered areas, and the super-linear scaling between edges and nodes.

3. From Stable Distribution to Dynamics assumption

References

[1] What Is Spacetime, Really? by Stephen Wolfram.
[2] Barabási, A. L., & Albert, R. (1999). Emergence of scaling in random networks. science, 286(5439), 509-512.
[3] Zhao, Y. M., Zeng, A., Yan, X. Y., Wang, W. X., & Lai, Y. C. (2015). Universal underpinning of human mobility in the real world and cyberspace. arXiv preprint arXiv:1512.04669.
[4] Zhang, J., Li, X., Wang, X., Wang, W. X., & Wu, L. (2015). Scaling behaviours in the growth of networked systems and their geometric origins.Scientific reports, 5.