179 research outputs found
Distances and Isomorphism between Networks and the Stability of Network Invariants
We develop the theoretical foundations of a network distance that has
recently been applied to various subfields of topological data analysis, namely
persistent homology and hierarchical clustering. While this network distance
has previously appeared in the context of finite networks, we extend the
setting to that of compact networks. The main challenge in this new setting is
the lack of an easy notion of sampling from compact networks; we solve this
problem in the process of obtaining our results. The generality of our setting
means that we automatically establish results for exotic objects such as
directed metric spaces and Finsler manifolds. We identify readily computable
network invariants and establish their quantitative stability under this
network distance. We also discuss the computational complexity involved in
precisely computing this distance, and develop easily-computable lower bounds
by using the identified invariants. By constructing a wide range of explicit
examples, we show that these lower bounds are effective in distinguishing
between networks. Finally, we provide a simple algorithm that computes a lower
bound on the distance between two networks in polynomial time and illustrate
our metric and invariant constructions on a database of random networks and a
database of simulated hippocampal networks
The Importance of Forgetting: Limiting Memory Improves Recovery of Topological Characteristics from Neural Data
We develop of a line of work initiated by Curto and Itskov towards
understanding the amount of information contained in the spike trains of
hippocampal place cells via topology considerations. Previously, it was
established that simply knowing which groups of place cells fire together in an
animal's hippocampus is sufficient to extract the global topology of the
animal's physical environment. We model a system where collections of place
cells group and ungroup according to short-term plasticity rules. In
particular, we obtain the surprising result that in experiments with spurious
firing, the accuracy of the extracted topological information decreases with
the persistence (beyond a certain regime) of the cell groups. This suggests
that synaptic transience, or forgetting, is a mechanism by which the brain
counteracts the effects of spurious place cell activity
The Metric Space of Networks
We study the question of reconstructing a weighted, directed network up to
isomorphism from its motifs. In order to tackle this question we first relax
the usual (strong) notion of graph isomorphism to obtain a relaxation that we
call weak isomorphism. Then we identify a definition of distance on the space
of all networks that is compatible with weak isomorphism. This global approach
comes equipped with notions such as completeness, compactness, curves, and
geodesics, which we explore throughout this paper. Furthermore, it admits
global-to-local inference in the following sense: we prove that two networks
are weakly isomorphic if and only if all their motif sets are identical, thus
answering the network reconstruction question. Further exploiting the
additional structure imposed by our network distance, we prove that two
networks are weakly isomorphic if and only if certain essential associated
structures---the skeleta of the respective networks---are strongly isomorphic
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