17 research outputs found
Formal Concept Lattice Representations and Algorithms for Hypergraphs
There is increasing focus on analyzing data represented as hypergraphs, which
are better able to express complex relationships amongst entities than are
graphs. Much of the critical information about hypergraph structure is
available only in the intersection relationships of the hyperedges, and so
forming the "intersection complex" of a hypergraph is quite valuable. This
identifies a valuable isomorphism between the intersection complex and the
"concept lattice" formed from taking the hypergraph's incidence matrix as a
"formal context": hypergraphs also generalize graphs in that their incidence
matrices are arbitrary Boolean matrices. This isomorphism allows connecting
discrete algorithms for lattices and hypergraphs, in particular s-walks or
s-paths on hypergraphs can be mapped to order theoretical operations on the
concept lattice. We give new algorithms for formal concept lattices and
hypergraph s-walks on concept lattices. We apply this to a large real-world
dataset and find deep lattices implying high interconnectivity and complex
geometry of hyperedges
Hypergraph Topological Features for Autoencoder-Based Intrusion Detection for Cybersecurity Data
In this position paper, we argue that when hypergraphs are used to capture
multi-way local relations of data, their resulting topological features
describe global behaviour. Consequently, these features capture complex
correlations that can then serve as high fidelity inputs to autoencoder-driven
anomaly detection pipelines. We propose two such potential pipelines for
cybersecurity data, one that uses an autoencoder directly to determine network
intrusions, and one that de-noises input data for a persistent homology system,
PHANTOM. We provide heuristic justification for the use of the methods
described therein for an intrusion detection pipeline for cyber data. We
conclude by showing a small example over synthetic cyber attack data
Size-Aware Hypergraph Motifs
Complex systems frequently exhibit multi-way, rather than pairwise,
interactions. These group interactions cannot be faithfully modeled as
collections of pairwise interactions using graphs, and instead require
hypergraphs. However, methods that analyze hypergraphs directly, rather than
via lossy graph reductions, remain limited. Hypergraph motif mining holds
promise in this regard, as motif patterns serve as building blocks for larger
group interactions which are inexpressible by graphs. Recent work has focused
on categorizing and counting hypergraph motifs based on the existence of nodes
in hyperedge intersection regions. Here, we argue that the relative sizes of
hyperedge intersections within motifs contain varied and valuable information.
We propose a suite of efficient algorithms for finding triplets of hyperedges
based on optimizing the sizes of these intersection patterns. This formulation
uncovers interesting local patterns of interaction, finding hyperedge triplets
that either (1) are the least correlated with each other, (2) have the highest
pairwise but not groupwise correlation, or (3) are the most correlated with
each other. We formalize this as a combinatorial optimization problem and
design efficient algorithms based on filtering hyperedges. Our experimental
evaluation shows that the resulting hyperedge triplets yield insightful
information on real-world hypergraphs. Our approach is also orders of magnitude
faster than a naive baseline implementation
Attributed Stream Hypergraphs: temporal modeling of node-attributed high-order interactions
Recent advances in network science have resulted in two distinct research
directions aimed at augmenting and enhancing representations for complex
networks. The first direction, that of high-order modeling, aims to focus on
connectivity between sets of nodes rather than pairs, whereas the second one,
that of feature-rich augmentation, incorporates into a network all those
elements that are driven by information which is external to the structure,
like node properties or the flow of time. This paper proposes a novel toolbox,
that of Attributed Stream Hypergraphs (ASHs), unifying both high-order and
feature-rich elements for representing, mining, and analyzing complex networks.
Applied to social network analysis, ASHs can characterize complex social
phenomena along topological, dynamic and attributive elements. Experiments on
real-world face-to-face and online social media interactions highlight that
ASHs can easily allow for the analyses, among others, of high-order groups'
homophily, nodes' homophily with respect to the hyperedges in which nodes
participate, and time-respecting paths between hyperedges.Comment: Submitted to "Applied Network Science
Stepping out of Flatland: Discovering Behavior Patterns as Topological Structures in Cyber Hypergraphs
Data breaches and ransomware attacks occur so often that they have become
part of our daily news cycle. This is due to a myriad of factors, including the
increasing number of internet-of-things devices, shift to remote work during
the pandemic, and advancement in adversarial techniques, which all contribute
to the increase in both the complexity of data captured and the challenge of
protecting our networks. At the same time, cyber research has made strides,
leveraging advances in machine learning and natural language processing to
focus on identifying sophisticated attacks that are known to evade conventional
measures. While successful, the shortcomings of these methods, particularly the
lack of interpretability, are inherent and difficult to overcome. Consequently,
there is an ever-increasing need to develop new tools for analyzing cyber data
to enable more effective attack detection. In this paper, we present a novel
framework based in the theory of hypergraphs and topology to understand data
from cyber networks through topological signatures, which are both flexible and
can be traced back to the log data. While our approach's mathematical grounding
requires some technical development, this pays off in interpretability, which
we will demonstrate with concrete examples in a large-scale cyber network
dataset. These examples are an introduction to the broader possibilities that
lie ahead; our goal is to demonstrate the value of applying methods from the
burgeoning fields of hypernetwork science and applied topology to understand
relationships among behaviors in cyber data.Comment: 18 pages, 11 figures. This paper is written for a general audienc
A Hypergraph-Based Machine Learning Ensemble Network Intrusion Detection System
Network intrusion detection systems (NIDS) to detect malicious attacks
continues to meet challenges. NIDS are vulnerable to auto-generated port scan
infiltration attempts and NIDS are often developed offline, resulting in a time
lag to prevent the spread of infiltration to other parts of a network. To
address these challenges, we use hypergraphs to capture evolving patterns of
port scan attacks via the set of internet protocol addresses and destination
ports, thereby deriving a set of hypergraph-based metrics to train a robust and
resilient ensemble machine learning (ML) NIDS that effectively monitors and
detects port scanning activities and adversarial intrusions while evolving
intelligently in real-time. Through the combination of (1) intrusion examples,
(2) NIDS update rules, (3) attack threshold choices to trigger NIDS retraining
requests, and (4) production environment with no prior knowledge of the nature
of network traffic 40 scenarios were auto-generated to evaluate the ML ensemble
NIDS comprising three tree-based models. Results show that under the model
settings of an Update-ALL-NIDS rule (namely, retrain and update all the three
models upon the same NIDS retraining request) the proposed ML ensemble NIDS
produced the best results with nearly 100% detection performance throughout the
simulation, exhibiting robustness in the complex dynamics of the simulated
cyber-security scenario.Comment: 12 pages, 10 figure
The magnitude homology of a hypergraph
The magnitude homology, introduced by R. Hepworth and S. Willerton, offers a
topological invariant that enables the study of graph properties. Hypergraphs,
being a generalization of graphs, serve as popular mathematical models for data
with higher-order structures. In this paper, we focus on describing the
topological characteristics of hypergraphs by considering their magnitude
homology. We begin by examining the distances between hyperedges in a
hypergraph and establish the magnitude homology of hypergraphs. Additionally,
we explore the relationship between the magnitude and the magnitude homology of
hypergraphs. Furthermore, we derive several functorial properties of the
magnitude homology for hypergraphs. Lastly, we present the K\"{u}nneth theorem
for the simple magnitude homology of hypergraphs
Hypergraph models of biological networks to identify genes critical to pathogenic viral response
BACKGROUND: Representing biological networks as graphs is a powerful approach to reveal underlying patterns, signatures, and critical components from high-throughput biomolecular data. However, graphs do not natively capture the multi-way relationships present among genes and proteins in biological systems. Hypergraphs are generalizations of graphs that naturally model multi-way relationships and have shown promise in modeling systems such as protein complexes and metabolic reactions. In this paper we seek to understand how hypergraphs can more faithfully identify, and potentially predict, important genes based on complex relationships inferred from genomic expression data sets.
RESULTS: We compiled a novel data set of transcriptional host response to pathogenic viral infections and formulated relationships between genes as a hypergraph where hyperedges represent significantly perturbed genes, and vertices represent individual biological samples with specific experimental conditions. We find that hypergraph betweenness centrality is a superior method for identification of genes important to viral response when compared with graph centrality.
CONCLUSIONS: Our results demonstrate the utility of using hypergraphs to represent complex biological systems and highlight central important responses in common to a variety of highly pathogenic viruses