Graph Dissimilarity Research Articles

Filtered simplicial homology, graph dissimilarity and überhomology

We introduce a filtration on the simplicial homology of a finite simplicial complex X using bi-colourings of its vertices. This yields two dual homology theories closely related to discrete Morse matchings on X. We give an explicit expression for the associated graded object of these homologies when X is the matching complex of the Tait graph of a plane graph G, in terms of subgraphs determined by certain matchings on the dual of G. We then use one of these homologies, in the case where X is a graph, to define a conjecturally optimal dissimilarity pseudometric for graphs; we prove various results for this dissimilarity and provide several computations. We further show that, by organising the horizontal homologies of a simplicial complex in the poset of its colourings, we obtain a triply graded homology theory which we call überhomology. This latter homology is not a homotopy invariant, but nonetheless encodes both combinatorial and topological information on X. For example, we prove that if X is a subdivision, the überhomology vanishes in its lowest degree, while for an homology manifold it coincides with the fundamental class in its top degree. We compute the überhomology on several classes of examples and infinite families and prove some of its properties; namely that, in its extremal degrees, it is well-behaved under coning and taking suspension. We then focus on the case where X is a simple graph and prove a detection result. Finally, we define some singly graded homologies for graphs obtained by specialising the überhomology in certain bi-degrees, provide some computations and use computer-aided calculations to make some conjectures.

EmbAssi: embedding assignment costs for similarity search in large graph databases

The graph edit distance is an intuitive measure to quantify the dissimilarity of graphs, but its computation is mathsf {NP}-hard and challenging in practice. We introduce methods for answering nearest neighbor and range queries regarding this distance efficiently for large databases with up to millions of graphs. We build on the filter-verification paradigm, where lower and upper bounds are used to reduce the number of exact computations of the graph edit distance. Highly effective bounds for this involve solving a linear assignment problem for each graph in the database, which is prohibitive in massive datasets. Index-based approaches typically provide only weak bounds leading to high computational costs verification. In this work, we derive novel lower bounds for efficient filtering from restricted assignment problems, where the cost function is a tree metric. This special case allows embedding the costs of optimal assignments isometrically into ell _1 space, rendering efficient indexing possible. We propose several lower bounds of the graph edit distance obtained from tree metrics reflecting the edit costs, which are combined for effective filtering. Our method termed EmbAssi can be integrated into existing filter-verification pipelines as a fast and effective pre-filtering step. Empirically we show that for many real-world graphs our lower bounds are already close to the exact graph edit distance, while our index construction and search scales to very large databases.

An IOHMM-Based Framework to Investigate Drift in Effectiveness of IoT-Based Systems.

IoT-based systems, when interacting with the physical environment through actuators, are complex systems difficult to model. Formal verification techniques carried out at design-time being often ineffective in this context, these systems have to be quantitatively evaluated for effectiveness at run-time, i.e., for the extent to which they behave as expected. This evaluation is achieved by confronting a model of the effects they should legitimately produce in different contexts to those observed in the field. However, this quantitative evaluation is not informative on the drifts in effectiveness, it does not help designers investigate their possible causes, increasing the time needed to resolve them. To address this problem, and assuming that models of legitimate behavior can be described by means of Input-Output Hidden Markov Models (IOHMMs), a novel generic unsupervised clustering-based IOHMM structure and parameters learning algorithm is developed. This algorithm is first used to learn a model of legitimate behavior. Then, a model of the observed behavior is learned from observations gathered in the field. A second algorithm builds a dissimilarity graph that makes clear structural and parametric differences between both models, thus providing guidance to designers to help them investigate possible causes of drift in effectiveness. The approach is validated on a real world dataset collected in a smart home.

Ultrametric fitting by gradient descent * *This article is an updated version of: Chierchia G and Perret B 2019 Ultrametric fitting by gradient descent 33rd Conf. Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada vol 32 eds H Wallach, H Larochelle, A Beygelzimer, F d’Alché-Buc, E Fox and R Garnett pp 3181–92

We study the problem of fitting an ultrametric distance to a dissimilarity graph in the context of hierarchical cluster analysis. Standard hierarchical clustering methods are specified procedurally, rather than in terms of the cost function to be optimized. We aim to overcome this limitation by presenting a general optimization framework for ultrametric fitting. Our approach consists of modeling the latter as a constrained optimization problem over the continuous space of ultrametrics. So doing, we can leverage the simple, yet effective, idea of replacing the ultrametric constraint with a min–max operation injected directly into the cost function. The proposed reformulation leads to an unconstrained optimization problem that can be efficiently solved by gradient descent methods. The flexibility of our framework allows us to investigate several cost functions, following the classic paradigm of combining a data fidelity term with a regularization. While we provide no theoretical guarantee to find the global optimum, the numerical results obtained over a number of synthetic and real datasets demonstrate the good performance of our approach with respect to state-of-the-art agglomerative algorithms. This makes us believe that the proposed framework sheds new light on the way to design a new generation of hierarchical clustering methods. Our code is made publicly available at https://github.com/PerretB/ultrametric-fitting.

Thought Chart: tracking the thought with manifold learning during emotion regulation

The Nash embedding theorem demonstrates that any compact manifold can be isometrically embedded in a Euclidean space. Assuming the complex brain states form a high-dimensional manifold in a topological space, we propose a manifold learning framework, termed Thought Chart, to reconstruct and visualize the manifold in a low-dimensional space. Furthermore, it serves as a data-driven approach to discover the underlying dynamics when the brain is engaged in a series of emotion and cognitive regulation tasks. EEG-based temporal dynamic functional connectomes are created based on 20 psychiatrically healthy participants’ EEG recordings during resting state and an emotion regulation task. Graph dissimilarity space embedding was applied to all the dynamic EEG connectomes. In order to visualize the learned manifold in a lower dimensional space, local neighborhood information is reconstructed via k-nearest neighbor-based nonlinear dimensionality reduction (NDR) and epsilon distance-based NDR. We showed that two neighborhood constructing approaches of NDR embed the manifold in a two-dimensional space, which we named Thought Chart. In Thought Chart, different task conditions represent distinct trajectories. Properties such as the distribution or average length in the 2-D space may serve as useful parameters to explore the underlying cognitive load and emotion processing during the complex task. In sum, this framework is a novel data-driven approach to the learning and visualization of underlying neurophysiological dynamics of complex functional brain data.

Filters for graph-based keyword spotting in historical handwritten documents

The accessibility to handwritten historical documents is often constrained by the limited feasibility of automatic full transcriptions. Keyword Spotting (KWS), that allows to retrieve arbitrary query words from documents, has been proposed as alternative. In the present paper, we make use of graphs for representing word images. The actual keyword spotting is thus based on matching a query graph with all documents graphs. However, even with relative fast approximation algorithms the shear amount of matchings might limit the practical application of this approach. For this reason we present two novel filters with linear time complexity that allow to substantially reduce the number of graph matchings actually required. In particular, these filters estimate a graph dissimilarity between a query graph and all document graphs based on their node and edge distribution in a polar coordinate system. Eventually, all graphs from the document with distributions that differ to heavily from the query’s node/edge distribution are eliminated. In an experimental evaluation on four different historical documents, we show that about 90% of the matchings can be omitted, while the KWS accuracy is not negatively affected.

Dissimilarity of Graph Invariant Features from EEG Phase-space Analysis

Electroencephalogram (EEG) data has been used in a variety of linear and nonlinear time series analysis techniques for predicting epileptic seizures. We examine phase-space dissimilarity measures for forewarning of seizure events based on time-delay embedding and state space recreation of the underlying brain dynamics.Given novel states which form graph nodes and dynamical linkages between states which form graph edges, we use graph dissimilarity to detect dynamical phase shifts which indicate the onset of epileptic events. In this paper, we report on observed trends and characteristics of graphs based on event and nonevent data from human EEG observations, and extend previous work focused on node and link dissimilarity by analyzing other graph properties as well. Our analysis includes measured properties and dissimilarity features that influence forewarning prediction accuracy.

Some properties of a dissimilarity measure for labeled graphs

We investigate the problem of comparing different graphs on the same set of vertices. It is a problem arising when using different biological networks to elucidate cellular processes. We wish to see their similarity and difference via connectivity-aware graph dissimilarity for graphs with the same node set. We extend a previous result and present some results concerning the orders of magnitude of the dissimilarity as the graphs’ sizes grow to infinity. We find that removing an edge playing a very important role in graph connectivity, such as a bridge between two fully connected subgraphs, can have a dramatic effect on the dissimilarity compared to the removal of any “ordinary” edge.

Spectral classes of regular, random, and empirical graphs

We define a (pseudo-)distance between graphs based on the spectrum of the normalized Laplacian. Since this quantity can be computed easily, or at numerically estimated, it is suitable for comparing in particular large graphs. Numerical experiments demonstrate that the spectral distance provides a practically useful measure of graph dissimilarity. The asymptotic behavior of the Laplacian spectrum furthermore yields a tool for classifying families of graphs in such a way that the distance of two graphs from the same family is bounded by O(1/n) in terms of size n of their vertex sets.

Predicting the correctness of node assignments in bipartite graph matching

Due to their high representational power and flexibility, graphs have emerged as a widely used data structure for complex pattern representation. The advantages of graphs are, however, at the expense of a substantial increase of the computational complexity of the associated algorithms. For instance, exact graph similarity or distance can be computed in exponential time only. An algorithmic framework based on bipartite graph matching (allowing graph dissimilarity computation in cubic time) has been presented recently. However, this fast computation leads in general to some incorrect node assignments with respect to a globally optimal graph matching. In this paper we present a novel methodology for predicting which of the node assignments are actually incorrectly assigned by this approximation. To this end, a comprehensive set of features – which numerically characterize the assignments – is defined and extracted from the underlying graphs. These features are in turn used for training a statistical classifier, which can eventually be employed for predicting unseen assignments. The power and applicability of our novel approach is empirically verified on eight real world data sets.

Spatial analyses of wildlife contact networks.

Datasets from which wildlife contact networks of epidemiological importance can be inferred are becoming increasingly common. A largely unexplored facet of these data is finding evidence of spatial constraints on who has contact with whom, despite theoretical epidemiologists having long realized spatial constraints can play a critical role in infectious disease dynamics. A graph dissimilarity measure is proposed to quantify how close an observed contact network is to being purely spatial whereby its edges are completely determined by the spatial arrangement of its nodes. Statistical techniques are also used to fit a series of mechanistic models for contact rates between individuals to the binary edge data representing presence or absence of observed contact. These are the basis for a second measure that quantifies the extent to which contacts are being mediated by distance. We apply these methods to a set of 128 contact networks of field voles (Microtus agrestis) inferred from mark–recapture data collected over 7 years and from four sites. Large fluctuations in vole abundance allow us to demonstrate that the networks become increasingly similar to spatial proximity graphs as vole density increases. The average number of contacts, , was (i) positively correlated with vole density across the range of observed densities and (ii) for two of the four sites a saturating function of density. The implications for pathogen persistence in wildlife may be that persistence is relatively unaffected by fluctuations in host density because at low density is low but hosts move more freely, and at high density is high but transmission is hampered by local build-up of infected or recovered animals.

Leveraging graph dimensions in online graph search

Graphs have been widely used due to its expressive power to model complicated relationships. However, given a graph database D g = { g 1 , g 2 , · · ·, g n }, it is challenging to process graph queries since a basic graph query usually involves costly graph operations such as maximum common subgraph and graph edit distance computation, which are NP-hard. In this paper, we study a novel DS-preserved mapping which maps graphs in a graph database D g onto a multidimensional space M g under a structural dimension M using a mapping function φ(). The DS-preserved mapping preserves two things: distance and structure. By the distance-preserving, it means that any two graphs g i and g j in D g must map to two data objects φ( g i ) and φ( g j ) in M g , such that the distance, d (φ( g i ), φ( g j )), between φ( g i ) and φ( g j ) in M g approximates the graph dissimilarity δ( g i , g j ) in D g . By the structure-preserving, it further means that for a given unseen query graph q , the distance between q and any graph g i in D g needs to be preserved such that δ( q , g i ) ≈ d (φ( q ), φ( g i )). We discuss the rationality of using graph dimension M for online graph processing, and show how to identify a small set of subgraphs to form M efficiently. We propose an iterative algorithm DSPM to compute the graph dimension, and discuss its optimization techniques. We also give an approximate algorithm DSPMap in order to handle a large graph database. We conduct extensive performance studies on both real and synthetic datasets to evaluate the top- k similarity query which is to find top- k similar graphs from D g for a query graph, and show the effectiveness and efficiency of our approaches.

A quantum Jensen–Shannon graph kernel for unattributed graphs

In this paper, we use the quantum Jensen–Shannon divergence as a means of measuring the information theoretic dissimilarity of graphs and thus develop a novel graph kernel. In quantum mechanics, the quantum Jensen–Shannon divergence can be used to measure the dissimilarity of quantum systems specified in terms of their density matrices. We commence by computing the density matrix associated with a continuous-time quantum walk over each graph being compared. In particular, we adopt the closed form solution of the density matrix introduced in Rossi et al. (2013) [27,28] to reduce the computational complexity and to avoid the cumbersome task of simulating the quantum walk evolution explicitly. Next, we compare the mixed states represented by the density matrices using the quantum Jensen–Shannon divergence. With the quantum states for a pair of graphs described by their density matrices to hand, the quantum graph kernel between the pair of graphs is defined using the quantum Jensen–Shannon divergence between the graph density matrices. We evaluate the performance of our kernel on several standard graph datasets from both bioinformatics and computer vision. The experimental results demonstrate the effectiveness of the proposed quantum graph kernel.

EM-type method for measuring graph dissimilarity

Graph dissimilarity (or similarity) computation is one of the key intermediate operations for intelligent information processing. In this paper, a new dissimilarity measure is proposed to gauge the pairwise structural difference between graphs through constrained graph matching. The measure is obtained as a solution to an optimization problem that we formalize by a new objective function in matrix representation, based on the adjacency matrices of graphs and binary constraints. An EM-type algorithm based on the minimum-weight bipartite graph method is proposed to optimize the matrix in polynomial time. The suitability of the proposed measure is evaluated in nearest-neighbor classification on four real-world graph datasets and on synthetic datasets.

Exploitation of genetic interaction network topology for the prediction of epistatic behavior

Genetic interaction (GI) detection impacts the understanding of human disease and the ability to design personalized treatment. The mapping of every GI in most organisms is far from complete due to the combinatorial amount of gene deletions and knockdowns required. Computational techniques to predict new interactions based only on network topology have been developed in network science but never applied to GI networks.We show that topological prediction of GIs is possible with high precision and propose a graph dissimilarity index that is able to provide robust prediction in both dense and sparse networks.Computational prediction of GIs is a strong tool to aid high-throughput GI determination. The dissimilarity index we propose in this article is able to attain precise predictions that reduce the universe of candidate GIs to test in the lab.

Learning the shape manifold to improve object recognition

This paper presents an approach for object recognition and shape retrieval in binary images. Object description is accomplished by contour and region-based solutions. Fusion of two categories of shape descriptors causes a considerable performance improvement in retrieval. In order to improve the recognition rate, it is important to extract the intrinsic object feature vectors in such a way that the geometrical distance between two such vectors matches the semantic relation between the two objects from which they were extracted. The main contribution in this paper is learning how to map the samples in high-dimensional observation space into the new manifold space so that the geometrically closer vectors belong to near semantics. Feature extraction is done by Isomap that belongs to a non-linear feature extraction algorithm. In the proposed method, the shortest path in the dissimilarity graph leads to less value between the samples in the same category by denoising the edge values. Experiments show that the geometrical distance between the samples on the manifold space are more compatible to the semantic distance of them. The proposed method has been compared to some well-known approaches by a variety of shape databases which includes MPEG-7 Part B, Kimia silhouettes, and Fish. The results show the effectiveness and validity of the method.

Graph averaging as a means to compare multichannel EEG coherence networks and its application to the study of mental fatigue and neurodegenerative disease

A method is proposed for quantifying differences between multichannel EEG coherence networks represented by functional unit (FU) maps. The approach is based on inexact graph matching for attributed relational graphs and graph averaging, adapted to FU-maps. The mean of a set of input FU-maps is defined in such a way that it not only represents the mean group coherence during a certain task or condition but also to some extent displays individual variations in brain activity. The definition of a mean FU-map relies on a graph dissimilarity measure which takes into account both node positions and node or edge attributes. A visualization of the mean FU-map is used with a visual representation of the frequency of occurrence of nodes and edges in the input FUs. This makes it possible to investigate which brain regions are more commonly involved in a certain task, by analysing the occurrence of a FU of the mean graph in the input FUs. Furthermore, our method gives the possibility to quantitatively compare individual FU-maps by computing their distance to the mean FU-map. The method is applied to the analysis of EEG coherence networks in two case studies, one on mental fatigue and one on patients with corticobasal ganglionic degeneration (CBGD). The method is proposed as a preliminary step towards a complete quantitative comparison, and the real benefit of its application is still to be proven.

A novel swarm intelligence algorithm for finding DNA motifs

Discovering DNA motifs from co-expressed or co-regulated genes is an important step towards deciphering complex gene regulatory networks and understanding gene functions. Despite significant improvement in the last decade, it still remains one of the most challenging problems in computational molecular biology. In this work, we propose a novel motif finding algorithm that finds consensus patterns using a population-based stochastic optimisation technique called Particle Swarm Optimisation (PSO), which has been shown to be effective in optimising difficult multidimensional problems in continuous domains. We propose to use a word dissimilarity graph to remap the neighborhood structure of the solution space of DNA motifs, and propose a modification of the naive PSO algorithm to accommodate discrete variables. In order to improve efficiency, we also propose several strategies for escaping from local optima and for automatically determining the termination criteria. Experimental results on simulated challenge problems show that our method is both more efficient and more accurate than several existing algorithms. Applications to several sets of real promoter sequences also show that our approach is able to detect known transcription factor binding sites, and outperforms two of the most popular existing algorithms.

Some measures of dissimilarity in intuitionistic fuzzy structures

The concept of an intuitionistic fuzzy set introduced by K. Atanassov is a generalization of that of a fuzzy set. Many similarity/dissimilarity measures have been proposed for measuring the degree of similarity/dissimilarity between fuzzy sets. However, it seems that there is little investigation on dissimilarity measures for measuring the degree of dissimilarity between intuitionistic fuzzy sets. In this paper, two dissimilarity measures between intuitionistic fuzzy sets of a finite set are defined, and it is proved that both of the measures are metrical. The measures can also be generalized to intuitionistic fuzzy structures, such as intuitionistic fuzzy subgraphs, defined on a finite set. The generalization of intuitionistic fuzzy subgraph is of important to investigation on graph dissimilarity.

Dissimilarity measures between fuzzy sets or fuzzy structures

Two simple dissimilarity measures between fuzzy subsets of a finite set S are defined, and it is shown that both of the measures are metrics. The measures can also be generalized to fuzzy structures, such as fuzzy graphs, defined on S; the graph generalization is closely related to a recently defined measure of graph dissimilarity.