6,063 research outputs found
Graph Similarity and Approximate Isomorphism
The graph similarity problem, also known as approximate graph isomorphism or graph matching problem, has been extensively studied in the machine learning community, but has not received much attention in the algorithms community: Given two graphs G,H of the same order n with adjacency matrices A_G,A_H, a well-studied measure of similarity is the Frobenius distance dist(G,H):=min_{pi}|A_G^{pi}-A_H|_F, where pi ranges over all permutations of the vertex set of G, where A_G^pi denotes the matrix obtained from A_G by permuting rows and columns according to pi, and where |M |_F is the Frobenius norm of a matrix M. The (weighted) graph similarity problem, denoted by GSim (WSim), is the problem of computing this distance for two graphs of same order. This problem is closely related to the notoriously hard quadratic assignment problem (QAP), which is known to be NP-hard even for severely restricted cases.
It is known that GSim (WSim) is NP-hard; we strengthen this hardness result by showing that the problem remains NP-hard even for the class of trees. Identifying the boundary of tractability for WSim is best done in the framework of linear algebra. We show that WSim is NP-hard as long as one of the matrices has unbounded rank or negative eigenvalues: hence, the realm of tractability is restricted to positive semi-definite matrices of bounded rank. Our main result is a polynomial time algorithm for the special case where the associated (weighted) adjacency graph for one of the matrices has a bounded number of twin equivalence classes. The key parameter underlying our algorithm is the clustering number of a graph; this parameter arises in context of the spectral graph drawing machinery
On Ranked Approximate Matching Of Large Attributed Graphs
Many emerging database applications entail sophisticated graph based query manipulation, predominantly evident in large-scale
scientific applications. To access the information embedded in
graphs, efficient graph matching tools and algorithms have become of prime importance. Although the prohibitively expensive time
complexity associated with exact sub-graph isomorphism techniques has limited its efficacy in the application domain, approximate yet efficient graph matching techniques have received much attention due to their pragmatic applicability. Since public domain databases are noisy and incomplete in nature, inexact graph matching techniques have proven to be more promising in terms of inferring knowledge from numerous structural data repositories.
Contemporary algorithms for approximate graph matching incur
substantial cost to generate candidates, and then test and rank them for possible match. Leading algorithms balance processing time and overall resource consumption cost by leveraging sophisticated data structures and graph properties to improve overall performance.
In this dissertation, we propose novel techniques for approximate graph matching based on two different techniques called TraM or
Top-k Graph Matching and Approximate Network Matching or AtoM respectively. While TraM off-loads a significant amount of its processing on to the database making the approach viable for large graphs, AtoM provides improved turn around time by means of graph
summarization prior to matching. The summarization process is aided by domain sensitive similarity matrices, which in turn helps improve the matching performance. The vector space embedding of the graphs and efficient filtration of the search space enables computation of approximate graph similarity at a throw-away cost. We combine domain similarity and topological similarity to obtain overall graph similarity and compare them with neighborhood biased segments of the data-graph for proper matches. We show that our approach can naturally support the emerging trend in graph pattern queries and discuss its suitability for large networks as it can be seamlessly transformed to adhere to map-reduce framework.
We have conducted thorough experiments on several synthetic and real data sets, and have demonstrated the effectiveness and efficiency of the proposed method
Gradual Weisfeiler-Leman: Slow and Steady Wins the Race
The classical Weisfeiler-Leman algorithm aka color refinement is fundamental
for graph learning and central for successful graph kernels and graph neural
networks. Originally developed for graph isomorphism testing, the algorithm
iteratively refines vertex colors. On many datasets, the stable coloring is
reached after a few iterations and the optimal number of iterations for machine
learning tasks is typically even lower. This suggests that the colors diverge
too fast, defining a similarity that is too coarse. We generalize the concept
of color refinement and propose a framework for gradual neighborhood
refinement, which allows a slower convergence to the stable coloring and thus
provides a more fine-grained refinement hierarchy and vertex similarity. We
assign new colors by clustering vertex neighborhoods, replacing the original
injective color assignment function. Our approach is used to derive new
variants of existing graph kernels and to approximate the graph edit distance
via optimal assignments regarding vertex similarity. We show that in both
tasks, our method outperforms the original color refinement with only moderate
increase in running time advancing the state of the art
Maximum common subgraph isomorphism algorithms for the matching of chemical structures
The maximum common subgraph (MCS) problem has become increasingly important in those aspects of chemoinformatics that involve the matching of 2D or 3D chemical structures. This paper provides a classification and a review of the many MCS algorithms, both exact and approximate, that have been described in the literature, and makes recommendations regarding their applicability to typical chemoinformatics tasks
Towards an Efficient Discovery of the Topological Representative Subgraphs
With the emergence of graph databases, the task of frequent subgraph
discovery has been extensively addressed. Although the proposed approaches in
the literature have made this task feasible, the number of discovered frequent
subgraphs is still very high to be efficiently used in any further exploration.
Feature selection for graph data is a way to reduce the high number of frequent
subgraphs based on exact or approximate structural similarity. However, current
structural similarity strategies are not efficient enough in many real-world
applications, besides, the combinatorial nature of graphs makes it
computationally very costly. In order to select a smaller yet structurally
irredundant set of subgraphs, we propose a novel approach that mines the top-k
topological representative subgraphs among the frequent ones. Our approach
allows detecting hidden structural similarities that existing approaches are
unable to detect such as the density or the diameter of the subgraph. In
addition, it can be easily extended using any user defined structural or
topological attributes depending on the sought properties. Empirical studies on
real and synthetic graph datasets show that our approach is fast and scalable
Models and Algorithms for Graph Watermarking
We introduce models and algorithmic foundations for graph watermarking. Our
frameworks include security definitions and proofs, as well as
characterizations when graph watermarking is algorithmically feasible, in spite
of the fact that the general problem is NP-complete by simple reductions from
the subgraph isomorphism or graph edit distance problems. In the digital
watermarking of many types of files, an implicit step in the recovery of a
watermark is the mapping of individual pieces of data, such as image pixels or
movie frames, from one object to another. In graphs, this step corresponds to
approximately matching vertices of one graph to another based on graph
invariants such as vertex degree. Our approach is based on characterizing the
feasibility of graph watermarking in terms of keygen, marking, and
identification functions defined over graph families with known distributions.
We demonstrate the strength of this approach with exemplary watermarking
schemes for two random graph models, the classic Erd\H{o}s-R\'{e}nyi model and
a random power-law graph model, both of which are used to model real-world
networks
A simple yet effective baseline for non-attributed graph classification
Graphs are complex objects that do not lend themselves easily to typical
learning tasks. Recently, a range of approaches based on graph kernels or graph
neural networks have been developed for graph classification and for
representation learning on graphs in general. As the developed methodologies
become more sophisticated, it is important to understand which components of
the increasingly complex methods are necessary or most effective.
As a first step, we develop a simple yet meaningful graph representation, and
explore its effectiveness in graph classification. We test our baseline
representation for the graph classification task on a range of graph datasets.
Interestingly, this simple representation achieves similar performance as the
state-of-the-art graph kernels and graph neural networks for non-attributed
graph classification. Its performance on classifying attributed graphs is
slightly weaker as it does not incorporate attributes. However, given its
simplicity and efficiency, we believe that it still serves as an effective
baseline for attributed graph classification. Our graph representation is
efficient (linear-time) to compute. We also provide a simple connection with
the graph neural networks.
Note that these observations are only for the task of graph classification
while existing methods are often designed for a broader scope including node
embedding and link prediction. The results are also likely biased due to the
limited amount of benchmark datasets available. Nevertheless, the good
performance of our simple baseline calls for the development of new, more
comprehensive benchmark datasets so as to better evaluate and analyze different
graph learning methods. Furthermore, given the computational efficiency of our
graph summary, we believe that it is a good candidate as a baseline method for
future graph classification (or even other graph learning) studies.Comment: 13 pages. Shorter version appears at 2019 ICLR Workshop:
Representation Learning on Graphs and Manifolds. arXiv admin note: text
overlap with arXiv:1810.00826 by other author
Structural Data Recognition with Graph Model Boosting
This paper presents a novel method for structural data recognition using a
large number of graph models. In general, prevalent methods for structural data
recognition have two shortcomings: 1) Only a single model is used to capture
structural variation. 2) Naive recognition methods are used, such as the
nearest neighbor method. In this paper, we propose strengthening the
recognition performance of these models as well as their ability to capture
structural variation. The proposed method constructs a large number of graph
models and trains decision trees using the models. This paper makes two main
contributions. The first is a novel graph model that can quickly perform
calculations, which allows us to construct several models in a feasible amount
of time. The second contribution is a novel approach to structural data
recognition: graph model boosting. Comprehensive structural variations can be
captured with a large number of graph models constructed in a boosting
framework, and a sophisticated classifier can be formed by aggregating the
decision trees. Consequently, we can carry out structural data recognition with
powerful recognition capability in the face of comprehensive structural
variation. The experiments shows that the proposed method achieves impressive
results and outperforms existing methods on datasets of IAM graph database
repository.Comment: 8 page
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