Principle of Locality I: Hacking
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 ?
-
"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. -
"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 |
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
or 
Figure 6. An example network of the space-constrained attachment model
Figure 7. Similarity continuum
The expected increase of the radius R(t), which is obviously a function of time t, is }{dt}=\Delta R(t)=\int_0rxf(x)dx=\int_0rx\frac{1}{L}dx=\frac{r^2}{2L}.) which gives 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_0R(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 assumptionReferences[1] What Is Spacetime, Really? by Stephen Wolfram. 猜你喜欢
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