Unlabeled Graphs Research Articles

Structural Entropy of the Stochastic Block Models.

With the rapid expansion of graphs and networks and the growing magnitude of data from all areas of science, effective treatment and compression schemes of context-dependent data is extremely desirable. A particularly interesting direction is to compress the data while keeping the “structural information” only and ignoring the concrete labelings. Under this direction, Choi and Szpankowski introduced the structures (unlabeled graphs) which allowed them to compute the structural entropy of the Erdős–Rényi random graph model. Moreover, they also provided an asymptotically optimal compression algorithm that (asymptotically) achieves this entropy limit and runs in expectation in linear time. In this paper, we consider the stochastic block models with an arbitrary number of parts. Indeed, we define a partitioned structural entropy for stochastic block models, which generalizes the structural entropy for unlabeled graphs and encodes the partition information as well. We then compute the partitioned structural entropy of the stochastic block models, and provide a compression scheme that asymptotically achieves this entropy limit.

On the expected number of perfect matchings in cubic planar graphs

A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that a bridgeless cubic graph has exponentially many perfect matchings. It was solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the special case of cubic planar graphs. In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with $n$ vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically $c\gamma^n$, where $c>0$ and $\gamma \sim 1.14196$ is an explicit algebraic number. We also compute the expected number of perfect matchings in (non necessarily bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs. Our starting point is a correspondence between counting perfect matchings in rooted cubic planar maps and the partition function of the Ising model in rooted triangulations.

SGML: A Symmetric Graph Metric Learning Framework for Efficient Hyperspectral Image Classification

Recently, the semi-supervised graph convolutional network (SSGCN) has been verified effective for hyperspectral image (HSI) classification. However, constrained by the limited training data and spectral uncertainty, the classification performance is remained to be further improved. Moreover, attribute to the massive data, the SSGCN with complex computation is generally too time- and resource-consuming to be applicable in real-time needs. To conquer these issues, we propose an efficient symmetric graph metric learning (SGML) framework by incorporating metric learning into the SSGCN paradigm. Specifically, we first conduct multilevel pixel-to-superpixel projection (P-SP) on the HSI to investigate the multiscale spatial information, where the suitable superpixel numbers are adaptively determined. Then, to extract more expressive representations, we design a new structure denoted as GSvolution, comprising the graph convolution (G-Conv) and a novel self-channel-enhanced convolution (S-Conv), to propagate the labeled and unlabeled graph node information and simultaneously enhance the critical intranode channel features. Finally, the superpixel node features are reprojected to the pixel level (SP-P) so that the distilled multistream features can be integrated to obtain the final decision. Noticeably, this ingenious symmetric mechanism (P-SP and SP-P) can alleviate the spectral variability and facilitate the framework to be an efficient model. Furthermore, in the metric learning module, we propose an innovative metric loss function to enhance the discrimination of the embedding features, i.e., inter class far apart and intraclass close. In the experiments, we demonstrate that the classification capacity of the proposed SGML can surpass the comparators on three benchmark data sets.

Asymptotic Properties of Random Unlabelled Block-Weighted Graphs

We study the asymptotic shape of random unlabelled graphs subject to certain subcriticality conditions. The graphs are sampled with probability proportional to a product of Boltzmann weights assigned to their $2$-connected components. As their number of vertices tends to infinity, we show that they admit the Brownian tree as Gromov–Hausdorff–Prokhorov scaling limit, and converge in a strengthened Benjamini–Schramm sense toward an infinite random graph. We also consider models of random graphs that are allowed to be disconnected. Here a giant connected component emerges and the small fragments converge without any rescaling towards a finite random limit graph. Our main application of these general results treats subcritical classes of unlabelled graphs. We study the special case of unlabelled outerplanar graphs in depth and calculate its scaling constant.

Graph kernels based on linear patterns: Theoretical and experimental comparisons

Graph kernels are powerful tools to bridge the gap between machine learning and data encoded as graphs. Most graph kernels are based on the decomposition of graphs into a set of patterns. The similarity between two graphs is then deduced to the similarity between corresponding patterns. Kernels based on linear patterns constitute a good trade-off between accuracy and computational complexity. In this work, we propose a thorough investigation and comparison of graph kernels based on different linear patterns, namely walks and paths. First, all these kernels are explored in detail, including their mathematical foundations, structures of patterns and computational complexity. After that, experiments are performed on various benchmark datasets exhibiting different types of graphs, including labeled and unlabeled graphs, graphs with different numbers of vertices, graphs with different average vertex degrees, linear and non-linear graphs. Finally, for regression and classification tasks, accuracy and computational complexity of these kernels are compared and analyzed, in the light of baseline kernels based on non-linear patterns. Suggestions are proposed to choose kernels according to the types of graph datasets. This work leads to a clear comparison of strengths and weaknesses of these kernels. An open-source Python library containing an implementation of all discussed kernels is publicly available on GitHub to the community, thus allowing to promote and facilitate the use of graph kernels in machine learning problems.

Graph Matching between Bipartite and Unipartite Networks: to Collapse, or not to Collapse, that is the Question.

Graph matching consists of aligning the vertices of two unlabeled graphs in order to maximize the shared structure across networks; when the graphs are unipartite, this is commonly formulated as minimizing their edge disagreements. In this paper we address the common setting in which one of the graphs to match is a bipartite network and one is unipartite. Commonly, the bipartite networks are collapsed or projected into a unipartite graph, and graph matching proceeds as in the classical setting. This potentially leads to noisy edge estimates and loss of information. We formulate the graph matching problem between a bipartite and a unipartite graph using an undirected graphical model, and introduce methods to find the alignment with this model without collapsing. We theoretically demonstrate that our methodology is consistent, and provide non-asymptotic conditions that ensure exact recovery of the matching solution. In simulations and real data examples, we show how our methods can result in a more accurate matching than the naive approach of transforming the bipartite networks into unipartite, and we demonstrate the performance gains achieved by our method in simulated and real data networks, including a co-authorship-citation network pair, and brain structural and functional data.

Operator algebras for higher rank analysis and their application to factorial languages

We study strong compactly aligned product systems of ℤ N+ over a C*-algebra A. We provide a description of their Cuntz-Nica-Pimsner algebra in terms of tractable relations coming from ideals of A. This approach encompasses product systems where the left action is given by compacts, as well as a wide class of higher rank graphs (beyond row-finite).Moreover we analyze higher rank factorial languages and their C*-algebras. Many of the rank one results in the literature find here their higher rank analogues. In particular, we show that the Cuntz-Nica-Pimsner algebra of a higher rank sofic language coincides with the Cuntz-Krieger algebra of its unlabeled follower set higher rank graph. However, there are also differences. For example, the Cuntz-Nica-Pimsner can lie in-between the first quantization and its quotient by the compactly supported operators.

Contrastive Self-supervised Learning for Graph Classification

Graph classification is a widely studied problem and has broad applications. In many real-world problems, the number of labeled graphs available for training classification models is limited, which renders these models prone to overfitting. To address this problem, we propose two approaches based on contrastive self-supervised learning (CSSL) to alleviate overfitting. In the first approach, we use CSSL to pretrain graph encoders on widely-available unlabeled graphs without relying on human-provided labels, then finetune the pretrained encoders on labeled graphs. In the second approach, we develop a regularizer based on CSSL, and solve the supervised classification task and the unsupervised CSSL task simultaneously. To perform CSSL on graphs, given a collection of original graphs, we perform data augmentation to create augmented graphs out of the original graphs. An augmented graph is created by consecutively applying a sequence of graph alteration operations. A contrastive loss is defined to learn graph encoders by judging whether two augmented graphs are from the same original graph. Experiments on various graph classification datasets demonstrate the effectiveness of our proposed methods. The code is available at https://github.com/UCSD-AI4H/GraphSSL.

Minimal Deterministic Traversable Vertex Labelling of Infinite Square Grid Graph

Automata walking on graphs are a mathematical formalization of autonomous mobile agents with limited memory operating in discrete environments. Under this model a broad area of studies of the behaviour of automata in labyrinths arose and intensively developing last decades (a labyrinth is an embedded directed graph of special form). Research in this regard received a wide range of applications, for example, in the problems of image analysis and navigation of mobile robots. Automata operating in labyrinths can distinguish directions, that is, they have a compass. This paper deals with the problem of constructing square grid graph vertex labelling thanks to which a finite automaton without a compass can walk on graph in any arbitrary direction. The automaton looking over neighbourhood of the current vertex and may travel to some neighbouring vertex selected by its label. In this paper, we propose a minimal deterministic traversable vertex labelling that satisfies the required property. A labelling is said to be deterministic if all vertices in closed neighbourhood of every vertex have different labels. In previous works we have proved that minimal deterministic traversable vertex labelling of square grid graph uses labels of five different types. In this paper we prove that a collective of one automaton and three pebbles can construct this labelling on initially unlabelled infinite square grid graph. We consider pebbles as automata of the simplest form, whose positions are completely determined by the remaining automata of the collective.

Scalable generalized median graph estimation and its manifold use in bioinformatics, clustering, classification, and indexing

In this paper, we present GMG-BCU—a local search algorithm based on block coordinate update for estimating a generalized median graph for a given collection of labeled or unlabeled input graphs. Unlike all competitors, GMG-BCU is designed for both discrete and continuous label spaces and can be configured to run in linear time w.r.t. the size of the graph collection whenever median node and edge labels are computable in linear time. These properties make GMG-BCU usable for applications such as differential microbiome data analysis, graph classification, clustering, and indexing. We also prove theoretical properties of generalized median graphs, namely, that they exist under reasonable assumptions which are met in almost all application scenarios, that they are in general non-unique, that they are NP-hard to compute and APX-hard to approximate, and that no polynomial α-approximation exists for any α unless the graph isomorphism problem is in P. Extensive experiments on six different datasets show that our heuristic GMG-BCU always outperforms the state of the art in terms of runtime or quality (on most datasets, both w.r.t. runtime and quality), that it is the only available heuristic which can cope with collections containing several thousands of graphs, and that it shows very promising potential when used for the aforementioned applications. GMG-BCU is freely available on GitHub: https://github.com/dbblumenthal/gedlib/.

Harmonic analysis of symmetric random graphs

This note attempts to understand graph limits as defined by Lovasz and Szegedy (2006)} in terms of harmonic analysis on semigroups. This is done by representing probability distributions of random exchangeable graphs as mixtures of characters on the semigroup of unlabeled graphs with node-disjoint union, thereby providing an alternative derivation of de Finetti's theorem for random exchangeable graphs.

A ranking based multi-view learning method for positive and unlabeled graph classification

A ranking based multi-view learning method for positive and unlabeled graph classification

Establish the expected number of induced motifs on unlabeled graphs through analytical models

Complex networks are usually characterized by the presence of small and recurrent patterns of interactions between nodes, called network motifs. These small modules can help to elucidate the structure and the functioning of complex systems. Assessing the statistical significance of a pattern as a motif in a network G is a time consuming task which entails the computation of the expected number of occurrences of the pattern in an ensemble of random graphs preserving some features of G, such as the degree distribution. Recently, few models have been devised to analytically compute expectations of the number of non-induced occurrences of a motif. Less attention has been payed to the harder analysis of induced motifs. Here, we illustrate an analytical model to derive the mean number of occurrences of an induced motif in an unlabeled network with respect to a random graph model. A comprehensive experimental analysis shows the effectiveness of our approach for the computation of the expected number of induced motifs up to 10 nodes. Finally, the proposed method is helpful when running subgraph counting algorithms to get the number of occurrences of a topology become unfeasible.

Classes of graphs with low complexity: The case of classes with bounded linear rankwidth

Classes with bounded rankwidth are MSO-transductions of trees and classes with bounded linear rankwidth are MSO-transductions of paths – a result that shows a strong link between the properties of these graph classes considered from the point of view of structural graph theory and from the point of view of finite model theory. We take both views on classes with bounded linear rankwidth and prove structural and model theoretic properties of these classes. The structural results we obtain are the following. (1) The number of unlabeled graphs of order n with linear rank-width at most r is at most [(2r+1)(r+1)!2r23r+1]n (2) Graphs with linear rankwidth at most r are linearly χ-bounded. Actually, they have bounded c-chromatic number, meaning that they can be colored with f(r) colors, each color inducing a cograph. (3) To the contrary, based on a Ramsey-like argument, we prove for every proper hereditary family ℱ of graphs (like cographs) that there is a class with bounded rankwidth that does not have the property that graphs in it can be colored by a bounded number of colors, each inducing a subgraph in ℱ.From the model theoretical side we obtain the following results: (1) A direct short proof that graphs with linear rankwidth at most r are first-order transductions of linear orders. This result could also be derived from Colcombet’s theorem on first-order transduction of linear orders and the equivalence of linear rankwidth with linear cliquewidth. (2) For a class C with bounded linear rankwidth the following conditions are equivalent: (a) C is stable, (b) C excludes some half-graph as a semi-induced subgraph, (c) C is a first-order transduction of a class with bounded pathwidth. These results open the perspective to study classes admitting low linear rankwidth covers.

A categorial approach to reaction systems: First steps

In the literature, one encounters the intensely studied classical set-based reaction systems and the more recently introduced generalization to graph-based reaction systems where the considered graphs are directed, simple, and edge-labeled. In this paper, we propose a categorical approach to reaction systems so that a wider spectrum of data structures becomes available on which reaction systems can be based including various types of graphs and of graph-like structures like unlabeled graphs, vertex-labeled graphs, bipartite graphs, and a variety of types of hypergraphs. But also algebraic structures like monoids fit into the framework.

Graph Classification Using Back Propagation Learning Algorithms

Due to the propagation of graph data, there has been a sharp focus on developing effective methods for classifying the graph object. As most of the proposed graph classification techniques though effective are constrained by high computational overhead, there is a consistent effort to improve upon the existing classification algorithms in terms of higher accuracy and less computational time. In this paper, an attempt has been made to classify graphs by extracting various features and selecting the important features using feature selection algorithms. Since all the extracted graph-based features need not be equally important, only the most important features are selected by using back propagation learning algorithm. The results of the proposed study of feature-based approach using back propagation learning algorithm lead to higher classification accuracy with faster computational time in comparison to other graph kernels. It also appears to be more effective for large unlabeled graphs.

Succinct Encodings for Families of Interval Graphs

We consider the problem of designing succinct data structures for interval graphs with n vertices while supporting degree, adjacency, neighborhood and shortest path queries in optimal time. Towards showing succinctness, we first show that at least $$n\log _2{n} - 2n\log _2\log _2 n - O(n)$$ bits are necessary to represent any unlabeled interval graph G with n vertices, answering an open problem of Yang and Pippenger (Proc Am Math Soc Ser B 4(1):1–3, 2017). This is augmented by a data structure of size $$n\log _2{n} +O(n)$$ bits while supporting not only the above queries optimally but also capable of executing various combinatorial algorithms (like proper coloring, maximum independent set etc.) on interval graphs efficiently. Finally, we extend our ideas to other variants of interval graphs, for example, proper/unit interval graphs, k-improper interval graphs, and circular-arc graphs, and design succinct data structures for these graph classes as well along with supporting queries on them efficiently.

On Non-Eliminative Structuralism. Unlabeled Graphs as a Case Study, Part A†

Abstract This is Part A of an article that defends non-eliminative structuralism about mathematics by means of a concrete case study: a theory of unlabeled graphs. Part A summarizes the general attractions of non-eliminative structuralism. Afterwards, it motivates an understanding of unlabeled graphs as structures sui generis and develops a corresponding axiomatic theory of unlabeled graphs. As the theory demonstrates, graph theory can be developed consistently without eliminating unlabeled graphs in favour of sets; and the usual structuralist criterion of identity can be applied successfully in graph-theoretic proofs. Part B will turn to the philosophical interpretation and assessment of the theory.

On Non-Eliminative Structuralism. Unlabeled Graphs as a Case Study, Part B†

AbstractThis is Part B of an article that defends non-eliminative structuralism about mathematics by means of a concrete case study: a theory of unlabeled graphs. Part A motivated an understanding of unlabeled graphs as structures sui generis and developed a corresponding axiomatic theory of unlabeled graphs. Part B turns to the philosophical interpretation and assessment of the theory: it points out how the theory avoids well-known problems concerning identity, objecthood, and reference that have been attributed to non-eliminative structuralism. The part concludes by explaining how the theory relates to set theory, and what remains to be accomplished for non-eliminative structuralists.

Right-angled Coxeter groups, universal graphs, and Eulerian polynomials

We define a class of representations for any right-angled Coxeter group R and its Hecke algebra H(R). The group R and the algebra H(R) act on any Bruhat interval of any Coxeter system (W,S), once given a suitable function from the set of Coxeter generators of R to the power set of S. The existence of such a function is related to the problem of universality of a graph G2 constructed from the unlabeled Coxeter graph G of (W,S). When G=Pn is a path with n vertices, we conjecture that G2 is n-universal; this property is equivalent to the existence of an action of R and H(R) on the Bruhat intervals of the symmetric group Sn+1, for all right-angled groups with n generators. We prove that (Pn)2 is n-universal for forests. Eulerian polynomials arise as characters of our representations, when the Coxeter graph of R is a path. We also give a formula for the toric h-polynomial of any lower Bruhat interval in a universal Coxeter group Un, using results on the Kazhdan–Lusztig basis of H(Un).