Cornell University

2020

presented by Albert M Orozco Camacho

**Opinion dynamics** has been previously studied within the mathematical social
sciences literature; yet not much attention has been put to current social network studies.

On the other hand, *perturbations that induce discord in social media* are of important interest for
current research in politics, sociology, economics, computer science, etc.

The authors present a formal study on the intuition that a (*maliscious*) actor
attacks the overall opinion consensus of a network.

These perturbations will difuse via an opinion dynamics model.

An adversarial agent targets a few individual opinions to difuse a sense of discord.

Identify the most vulverable structures within a network. Mitigate these attacks by insulating nodes.

**Diagonal Matrix**
$$D_{i,i} = \sum_{j \in V} w(i,j)$$

**Laplacian Matrix**
$$L := D - A$$

$$ L = \sum_{i=1}^n \lambda_i\mathbf{v}_i\mathbf{v}_i^T $$

Start with $\mathbf{z}^{(0)} = \mathbf{s}$.

For each node $i \in V$, update $$ \mathbf{z}^{(t+1)}(i) = \frac{\mathbf{s}(i) + \sum_{j \in V} w(i,j)\mathbf{z}^{(T)}(j)} {1 + \sum_{j \in V} w(i,j)} $$

Convergence will be reached and the limiting final opinion vector is given by $$\mathbf{z} = (I + L)^{-1}\mathbf{s}$$

$$ \max_{s \in \mathbb{R}^n: ||\mathbf{s}||_2 \leq R} \mathbf{s}^T (I + L)^{-1}f(L)(I + L)^{-1}\mathbf{s} $$

**CONCATENATION**
$$[h_v^{(1)},\ldots,h_v^{(k)}]$$

**MAX-POOLING**. Select the most informative layer for each feature coordinate.

**LSTM-ATTENTION**. Input $h_v^{(1)},\ldots,h_v^{(k)}$ into a bi-directional
LSTM to generate forward and backward features $f_v^{(l)}$ and $b_v^{(l)}$ for each
layer $l$; finally compute an attention score per each node by combining those
for each layer.

**CONCATENATION**
$$[h_v^{(1)},\ldots,h_v^{(k)}]$$

**MAX-POOLING**. Select the most informative layer for each feature coordinate.

**LSTM-ATTENTION**. Input $h_v^{(1)},\ldots,h_v^{(k)}$ into a bi-directional
LSTM to generate forward and backward features $f_v^{(l)}$ and $b_v^{(l)}$ for each
layer $l$; finally compute an attention score per each node by combining those
for each layer.

The influence score $I(x, y)$ for any $x, y \in V$ under a $k$-layer JK-Net with layer-wise max-pooling is equivalent in expectation to a mixture of $0,\ldots,k$-step random walk distributions on $\tilde{G}$ at $y$ starting at $x$, the coefficients of which depend on the values of the layer features $h_x^{(l)}$.

**Goal**: Provide a representation learning scheme that can generalize
better on diverse variety of network structure, than the one proposed for GCN's

**Problem**: Denser subgraphs may cause aggregation algorithms to converge
in expectation to biased random walks. ☹

**Solution**: JK-Nets aggregate and leverage information from more than
one hidden layers.😁

JK-Nets with the LSTM-attention aggregators outperform the non-adaptive models GraphSAGE, GAT and JK-Nets with concatenation aggregators.

https://github.com/ShinKyuY/Representation_Learning_on_Graphs_with_Jumping_Knowledge_Networks

Exploring other layer aggregators and studying the effect of the combination of various layer-wise and node-wise aggregators on different types of graph structures.

How can sequence modelling by itself impact the task of layer aggregation?

Are there *smarter* ways to keep track of node/community correlations within a network?

**Goal**: Provide a representation learning scheme that can generalize
better on diverse variety of network structure, than the one proposed for GCN's

**Problem**: Denser subgraphs may cause aggregation algorithms to converge
in expectation to biased random walks. ☹

**Solution**: JK-Nets aggregate and leverage information from more than
one hidden layers.😁

JK-Nets with the LSTM-attention aggregators outperform the non-adaptive models GraphSAGE, GAT and JK-Nets with concatenation aggregators.

https://github.com/ShinKyuY/Representation_Learning_on_Graphs_with_Jumping_Knowledge_Networks

Exploring other layer aggregators and studying the effect of the combination of various layer-wise and node-wise aggregators on different types of graph structures.

How can sequence modelling by itself impact the task of layer aggregation?

Are there *smarter* ways to keep track of node/community correlations within a network?