From Local to Global Symmetry: Activation Dynamics in the Independent Cascade Model on Undirected Graphs
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911485084565504 |
|---|---|
| author | Liu, Peiyao |
| author_facet | Liu, Peiyao |
| contents | The independent cascade model is a widely used framework for simulating the spread of influence in social networks. In this model, activations propagate stochastically through the network, with each edge having a probability of transmitting activation. We study the independent cascade model on undirected graphs with symmetric influence probabilities ($p_{ij} = p_{ji}$ for all nodes $i$ and $j$). We focus on persistent activations, where activated nodes remain active indefinitely. Our main result is to demonstrate that this local symmetry in the graph structure induces a global symmetry in the activation dynamics. Specifically, the probability of node $j$ being activated within $n$ steps, starting with only node $i$ activated, equals the probability of node $i$ being activated within $n$ steps, starting with only node $j$ activated, for all $n$. We establish this result using a novel approach based on random matrices, offering a fresh perspective on the model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04483 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | From Local to Global Symmetry: Activation Dynamics in the Independent Cascade Model on Undirected Graphs Liu, Peiyao Probability The independent cascade model is a widely used framework for simulating the spread of influence in social networks. In this model, activations propagate stochastically through the network, with each edge having a probability of transmitting activation. We study the independent cascade model on undirected graphs with symmetric influence probabilities ($p_{ij} = p_{ji}$ for all nodes $i$ and $j$). We focus on persistent activations, where activated nodes remain active indefinitely. Our main result is to demonstrate that this local symmetry in the graph structure induces a global symmetry in the activation dynamics. Specifically, the probability of node $j$ being activated within $n$ steps, starting with only node $i$ activated, equals the probability of node $i$ being activated within $n$ steps, starting with only node $j$ activated, for all $n$. We establish this result using a novel approach based on random matrices, offering a fresh perspective on the model. |
| title | From Local to Global Symmetry: Activation Dynamics in the Independent Cascade Model on Undirected Graphs |
| topic | Probability |
| url | https://arxiv.org/abs/2409.04483 |