4 research outputs found
Persistent Homology Lower Bounds on High Order Network Distances
High order networks are weighted hypergraphs col- lecting relationships
between elements of tuples, not necessarily pairs. Valid metric distances
between high order networks have been defined but they are difficult to compute
when the number of nodes is large. The goal here is to find tractable
approximations of these network distances. The paper does so by mapping high
order networks to filtrations of simplicial complexes and showing that the
distance between networks can be lower bounded by the difference between the
homological features of their respective filtrations. Practical implications
are explored by classifying weighted pairwise networks constructed from
different gener- ative processes and by comparing the coauthorship networks of
engineering and mathematics academic journals. The persistent homology methods
succeed in identifying different generative models, in discriminating
engineering and mathematics commu- nities, as well as in differentiating
engineering communities with different research interests
Metrics in the Space of High Order Networks
This paper presents methods to compare high order networks, defined as
weighted complete hypergraphs collecting relationship functions between
elements of tuples. They can be considered as generalizations of conventional
networks where only relationship functions between pairs are defined. Important
properties between relationships of tuples of different lengths are
established, particularly when relationships encode dissimilarities or
proximities between nodes. Two families of distances are then introduced in the
space of high order networks. The distances measure differences between
networks. We prove that they are valid metrics in the spaces of high order
dissimilarity and proximity networks modulo permutation isomorphisms. Practical
implications are explored by comparing the coauthorship networks of two popular
signal processing researchers. The metrics succeed in identifying their
respective collaboration patterns
Network Comparison: Embeddings and Interiors
This paper presents methods to compare networks where relationships between
pairs of nodes in a given network are defined. We define such network distance
by searching for the optimal method to embed one network into another network,
prove that such distance is a valid metric in the space of networks modulo
permutation isomorphisms, and examine its relationship with other network
metrics. The network distance defined can be approximated via multi-dimensional
scaling, however, the lack of structure in networks results in poor
approximations. To alleviate such problem, we consider methods to define the
interiors of networks. We show that comparing interiors induced from a pair of
networks yields the same result as the actual network distance between the
original networks. Practical implications are explored by showing the ability
to discriminate networks generated by different models
Persistence Homology of Networks: Methods and Applications
Information networks are becoming increasingly popular to capture complex
relationships across various disciplines, such as social networks, citation
networks, and biological networks. The primary challenge in this domain is
measuring similarity or distance between networks based on topology. However,
classical graph-theoretic measures are usually local and mainly based on
differences between either node or edge measurements or correlations without
considering the topology of networks such as the connected components or holes.
In recent years, mathematical tools and deep learning based methods have become
popular to extract the topological features of networks. Persistent homology
(PH) is a mathematical tool in computational topology that measures the
topological features of data that persist across multiple scales with
applications ranging from biological networks to social networks. In this
paper, we provide a conceptual review of key advancements in this area of using
PH on complex network science. We give a brief mathematical background on PH,
review different methods (i.e. filtrations) to define PH on networks and
highlight different algorithms and applications where PH is used in solving
network mining problems. In doing so, we develop a unified framework to
describe these recent approaches and emphasize major conceptual distinctions.
We conclude with directions for future work. We focus our review on recent
approaches that get significant attention in the mathematics and data mining
communities working on network data. We believe our summary of the analysis of
PH on networks will provide important insights to researchers in applied
network science.Comment: Submitted to Applied Network Science Special Issue on Machine
Learning with Graph