  In this paper, we propose a novel pooling layer for graph neural networks
based on maximizing the mutual information between the pooled graph and the
input graph. Since the maximum mutual information is difficult to compute, we
employ the Shannon capacity of a graph as an inductive bias to our pooling
method. More precisely, we show that the input graph to the pooling layer can
be viewed as a representation of a noisy communication channel. For such a
channel, sending the symbols belonging to an independent set of the graph
yields a reliable and error-free transmission of information. We show that
reaching the maximum mutual information is equivalent to finding a maximum
weight independent set of the graph where the weights convey entropy contents.
Through this communication theoretic standpoint, we provide a distinct
perspective for posing the problem of graph pooling as maximizing the
information transmission rate across a noisy communication channel, implemented
by a graph neural network. We evaluate our method, referred to as Maximum
Entropy Weighted Independent Set Pooling (MEWISPool), on graph classification
tasks and the combinatorial optimization problem of the maximum independent
set. Empirical results demonstrate that our method achieves the
state-of-the-art and competitive results on graph classification tasks and the
maximum independent set problem in several benchmark datasets.
