Finite Size Effects on barabasy Albert Model
?
0 likes?749 views
This document summarizes simulation results of the Barab¨¢si-Albert preferential attachment model of network growth. Key results include: 1) The degree distribution follows a power law with an exponential cutoff due to finite size effects. 2) Scaled finite size corrections appear as a Gaussian beyond a threshold point. 3) The maximum degree scales as a power law with exponent 0.5, and its distribution is well fit by the Gumbel distribution, influenced by finite size effects.
![Simulating the model: The Algorithm
Start with Initial clique of m+1 nodes.
For Each time step t (up to T):
Choose connections between new node and m existing nodes:
Generate m random integers {r1... rm} in [1,m(2t+m-1)].
Store node indices {i1... im}={Edges(r1)... Edges(rm)}.
Check if they are different. If not, pick a new edge.
Update Degree - add m for the new node and 1 for each of the m old
nodes.
Update Edges - add m cells containing the new node¡¯s index and one
cell for each of the old node¡¯s index.
Repeat for suf?cient number of times for statistical consistence.](https://image.slidesharecdn.com/presentation-111008071022-phpapp02/85/Finite-Size-Effects-on-barabasy-Albert-Model-5-320.jpg)
Finite Size Effects on barabasy Albert Model
- 1. Finite-size effects in the Barab¨¢si-Albert Model DAFNA MATHOV OLEGUER SAGARRA Complex Systems Master in computational Physics UB-UPC 2011.
- 2. Overview Introduction: The Barabasi-Albert (BA) Model Simulating The Model: The Algorithm Expected Results Results of the Simulation Concluding Remarks
- 3. The BA model I Scale-Free Network Algorithm Proposed by Bar¨¢basi and Albert (1999) The model is based on: Preferential Attachment Growth of the network
- 4. The BA model II Initial network of m0 nodes. At each time step we connect a new node to m existing nodes. The probability of these new connections is: Can be undirected, directed, weighted and so on...
- 5. Simulating the model: The Algorithm Start with Initial clique of m+1 nodes. For Each time step t (up to T): Choose connections between new node and m existing nodes: Generate m random integers {r1... rm} in [1,m(2t+m-1)]. Store node indices {i1... im}={Edges(r1)... Edges(rm)}. Check if they are different. If not, pick a new edge. Update Degree - add m for the new node and 1 for each of the m old nodes. Update Edges - add m cells containing the new node¡¯s index and one cell for each of the old node¡¯s index. Repeat for suf?cient number of times for statistical consistence.
- 6. Expected Results Degree distribution and CCDF Scaled Finite size corrections Maximum Degree Statistics
- 7. Results: CCDF and Probability Distribution Distributions do behave according to power laws, accompanied by an exponential truncation.
- 8. Results: Scaled Finite Size Corrections The Scaled Finite Size Corrections appear in the form of a Gaussian beyond the threshold point. Divergences start at k¡«N1/2
- 9. Results: Maximum Degree Statistics The Maximum Degree behaves as a power law, with exponent (¦Á=0.5) Gumbel distribution ?ts well to the numerical Maximum degree distribution Finite size effects dictates the form of the maximum degree
- 10. Concluding Remarks The degree distribution of the BA model is a power law with an exponential truncation. The Scaled Finite Size Corrections appear in the form of a Gaussian beyond the threshold point. The Maximum averaged degree scales as a power law with ¦Á=0.5 The maximum degree distribution is highly in?uenced by the ?nite size correction and seem to behave according to the Gumbel distribution.