26 research outputs found
Weighted (Co)homology and Weighted Laplacian
In this paper, we generalize the combinatorial Laplace operator of Horak and
Jost by introducing the -weighted coboundary operator induced by a weight
function . Our weight function is a generalization of Dawson's
weighted boundary map. We show that our above-mentioned generalizations include
new cases that are not covered by previous literature. Our definition of
weighted Laplacian for weighted simplicial complexes is also applicable to
weighted/unweighted graphs and digraphs.Comment: 22 page
Max-Plus Synchronization in Decentralized Trading Systems
We introduce a decentralized mechanism for pricing and exchanging
alternatives constrained by transaction costs. We characterize the
time-invariant solutions of a heat equation involving a (weighted) Tarski
Laplacian operator, defined for max-plus matrix-weighted graphs, as approximate
equilibria of the trading system. We study algebraic properties of the solution
sets as well as convergence behavior of the dynamical system. We apply these
tools to the "economic problem" of allocating scarce resources among competing
uses. Our theory suggests differences in competitive equilibrium, bargaining,
or cost-benefit analysis, depending on the context, are largely due to
differences in the way that transaction costs are incorporated into the
decision-making process. We present numerical simulations of the
synchronization algorithm (RRAggU), demonstrating our theoretical findings
Cell Attention Networks
Since their introduction, graph attention networks achieved outstanding
results in graph representation learning tasks. However, these networks
consider only pairwise relationships among nodes and then they are not able to
fully exploit higher-order interactions present in many real world data-sets.
In this paper, we introduce Cell Attention Networks (CANs), a neural
architecture operating on data defined over the vertices of a graph,
representing the graph as the 1-skeleton of a cell complex introduced to
capture higher order interactions. In particular, we exploit the lower and
upper neighborhoods, as encoded in the cell complex, to design two independent
masked self-attention mechanisms, thus generalizing the conventional graph
attention strategy. The approach used in CANs is hierarchical and it
incorporates the following steps: i) a lifting algorithm that learns {\it edge
features} from {\it node features}; ii) a cell attention mechanism to find the
optimal combination of edge features over both lower and upper neighbors; iii)
a hierarchical {\it edge pooling} mechanism to extract a compact meaningful set
of features. The experimental results show that CAN is a low complexity
strategy that compares favorably with state of the art results on graph-based
learning tasks.Comment: Preprint, under revie
CIN++: Enhancing Topological Message Passing
Graph Neural Networks (GNNs) have demonstrated remarkable success in learning
from graph-structured data. However, they face significant limitations in
expressive power, struggling with long-range interactions and lacking a
principled approach to modeling higher-order structures and group interactions.
Cellular Isomorphism Networks (CINs) recently addressed most of these
challenges with a message passing scheme based on cell complexes. Despite their
advantages, CINs make use only of boundary and upper messages which do not
consider a direct interaction between the rings present in the underlying
complex. Accounting for these interactions might be crucial for learning
representations of many real-world complex phenomena such as the dynamics of
supramolecular assemblies, neural activity within the brain, and gene
regulation processes. In this work, we propose CIN++, an enhancement of the
topological message passing scheme introduced in CINs. Our message passing
scheme accounts for the aforementioned limitations by letting the cells to
receive also lower messages within each layer. By providing a more
comprehensive representation of higher-order and long-range interactions, our
enhanced topological message passing scheme achieves state-of-the-art results
on large-scale and long-range chemistry benchmarks.Comment: 21 pages, 9 figure