93,565 research outputs found
Size Generalization of Graph Neural Networks on Biological Data: Insights and Practices from the Spectral Perspective
We investigate size-induced distribution shifts in graphs and assess their
impact on the ability of graph neural networks (GNNs) to generalize to larger
graphs relative to the training data. Existing literature presents conflicting
conclusions on GNNs' size generalizability, primarily due to disparities in
application domains and underlying assumptions concerning size-induced
distribution shifts. Motivated by this, we take a data-driven approach: we
focus on real biological datasets and seek to characterize the types of
size-induced distribution shifts. Diverging from prior approaches, we adopt a
spectral perspective and identify that spectrum differences induced by size are
related to differences in subgraph patterns (e.g., average cycle lengths).
While previous studies have identified that the inability of GNNs in capturing
subgraph information negatively impacts their in-distribution generalization,
our findings further show that this decline is more pronounced when evaluating
on larger test graphs not encountered during training. Based on these spectral
insights, we introduce a simple yet effective model-agnostic strategy, which
makes GNNs aware of these important subgraph patterns to enhance their size
generalizability. Our empirical results reveal that our proposed
size-insensitive attention strategy substantially enhances graph classification
performance on large test graphs, which are 2-10 times larger than the training
graphs, resulting in an improvement in F1 scores by up to 8%.Comment: 21 pages, including appendi
Topology Discovery of Sparse Random Graphs With Few Participants
We consider the task of topology discovery of sparse random graphs using
end-to-end random measurements (e.g., delay) between a subset of nodes,
referred to as the participants. The rest of the nodes are hidden, and do not
provide any information for topology discovery. We consider topology discovery
under two routing models: (a) the participants exchange messages along the
shortest paths and obtain end-to-end measurements, and (b) additionally, the
participants exchange messages along the second shortest path. For scenario
(a), our proposed algorithm results in a sub-linear edit-distance guarantee
using a sub-linear number of uniformly selected participants. For scenario (b),
we obtain a much stronger result, and show that we can achieve consistent
reconstruction when a sub-linear number of uniformly selected nodes
participate. This implies that accurate discovery of sparse random graphs is
tractable using an extremely small number of participants. We finally obtain a
lower bound on the number of participants required by any algorithm to
reconstruct the original random graph up to a given edit distance. We also
demonstrate that while consistent discovery is tractable for sparse random
graphs using a small number of participants, in general, there are graphs which
cannot be discovered by any algorithm even with a significant number of
participants, and with the availability of end-to-end information along all the
paths between the participants.Comment: A shorter version appears in ACM SIGMETRICS 2011. This version is
scheduled to appear in J. on Random Structures and Algorithm
Topological Resonances in Scattering on Networks (Graphs)
We report on a hitherto unnoticed type of resonances occurring in scattering
from networks (quantum graphs) which are due to the complex connectivity of the
graph - its topology. We consider generic open graphs and show that any cycle
leads to narrow resonances which do not fit in any of the prominent paradigms
for narrow resonances (classical barriers, localization due to disorder,
chaotic scattering). We call these resonances `topological' to emphasize their
origin in the non-trivial connectivity. Topological resonances have a clear and
unique signature which is apparent in the statistics of the resonance
parameters (such as e.g., the width, the delay time or the wave-function
intensity in the graph). We discuss this phenomenon by providing analytical
arguments supported by numerical simulation, and identify the features of the
above distributions which depend on genuine topological quantities such as the
length of the shortest cycle (girth). These signatures cannot be explained
using any of the other paradigms for narrow resonances. Finally, we propose an
experimental setting where the topological resonances could be demonstrated,
and study the stability of the relevant distribution functions to moderate
dissipation
Distribution of shortest cycle lengths in random networks
We present analytical results for the distribution of shortest cycle lengths
(DSCL) in random networks. The approach is based on the relation between the
DSCL and the distribution of shortest path lengths (DSPL). We apply this
approach to configuration model networks, for which analytical results for the
DSPL were obtained before. We first calculate the fraction of nodes in the
network which reside on at least one cycle. Conditioning on being on a cycle,
we provide the DSCL over ensembles of configuration model networks with degree
distributions which follow a Poisson distribution (Erdos-R\'enyi network),
degenerate distribution (random regular graph) and a power-law distribution
(scale-free network). The mean and variance of the DSCL are calculated. The
analytical results are found to be in very good agreement with the results of
computer simulations.Comment: 44 pages, 11 figure
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