Sequence Of Graphs Research Articles

Weakly interacting oscillators on dense random graphs

AbstractWe consider a class of weakly interacting particle systems of mean-field type. The interactions between the particles are encoded in a graph sequence, i.e. two particles are interacting if and only if they are connected in the underlying graph. We establish a law of large numbers for the empirical measure of the system that holds whenever the graph sequence is convergent to a graphon. The limit is the solution of a non-linear Fokker–Planck equation weighted by the (possibly random) graphon limit. In contrast with the existing literature, our analysis focuses on both deterministic and random graphons: no regularity assumptions are made on the graph limit and we are able to include general graph sequences such as exchangeable random graphs. Finally, we identify the sequences of graphs, both random and deterministic, for which the associated empirical measure converges to the classical McKean–Vlasov mean-field limit.

A fourth‐moment phenomenon for asymptotic normality of monochromatic subgraphs

Given a graph sequence and a simple connected subgraph , we denote by the number of monochromatic copies of in a uniformly random vertex coloring of with colors. We prove a central limit theorem for (we denote the appropriately centered and rescaled statistic as ) with explicit error rates. The error rates arise from graph counts of collections formed by joining copies of which we callgood joins. Good joins are closely related to the fourth moment of , which allows us to show afourth moment phenomenonfor the central limit theorem. For , we show that converges in distribution to whenever its fourth moment converges to 3. We show the convergence of the fourth moment is necessary to obtain a normal limit when .

Self-Supervised Continual Graph Learning in Adaptive Riemannian Spaces

Continual graph learning routinely finds its role in a variety of real-world applications where the graph data with different tasks come sequentially. Despite the success of prior works, it still faces great challenges. On the one hand, existing methods work with the zero-curvature Euclidean space, and largely ignore the fact that curvature varies over the com- ing graph sequence. On the other hand, continual learners in the literature rely on abundant labels, but labeling graph in practice is particularly hard especially for the continuously emerging graphs on-the-fly. To address the aforementioned challenges, we propose to explore a challenging yet practical problem, the self-supervised continual graph learning in adaptive Riemannian spaces. In this paper, we propose a novel self-supervised Riemannian Graph Continual Learner (RieGrace). In RieGrace, we first design an Adaptive Riemannian GCN (AdaRGCN), a unified GCN coupled with a neural curvature adapter, so that Riemannian space is shaped by the learnt curvature adaptive to each graph. Then, we present a Label-free Lorentz Distillation approach, in which we create teacher-student AdaRGCN for the graph sequence. The student successively performs intra-distillation from itself and inter-distillation from the teacher so as to consolidate knowledge without catastrophic forgetting. In particular, we propose a theoretically grounded Generalized Lorentz Projection for the contrastive distillation in Riemannian space. Extensive experiments on the benchmark datasets show the superiority of RieGrace, and additionally, we investigate on how curvature changes over the graph sequence.

Privacy-Preserved Evolutionary Graph Modeling via Gromov-Wasserstein Autoregression

Real-world graphs like social networks are often evolutionary over time, whose observations at different timestamps lead to graph sequences. Modeling such evolutionary graphs is important for many applications, but solving this problem often requires the correspondence between the graphs at different timestamps, which may leak private node information, e.g., the temporal behavior patterns of the nodes. We proposed a Gromov-Wasserstein Autoregressive (GWAR) model to capture the generative mechanisms of evolutionary graphs, which does not require the correspondence information and thus preserves the privacy of the graphs' nodes. This model consists of two autoregressions, predicting the number of nodes and the probabilities of nodes and edges, respectively. The model takes observed graphs as its input and predicts future graphs via solving a joint graph alignment and merging task. This task leads to a fused Gromov-Wasserstein (FGW) barycenter problem, in which we approximate the alignment of the graphs based on a novel inductive fused Gromov-Wasserstein (IFGW) distance. The IFGW distance is parameterized by neural networks and can be learned under mild assumptions, thus, we can infer the FGW barycenters without iterative optimization and predict future graphs efficiently. Experiments show that our GWAR achieves encouraging performance in modeling evolutionary graphs in privacy-preserving scenarios.

Asymptotic enumeration of graphs by degree sequence, and the degree sequence of a random graph

In this paper we relate a fundamental parameter of a random graph, its degree sequence, to a simple model of nearly independent binomial random variables. As a result, many interesting functions of the joint distribution of graph degrees, such as the distribution of the median degree, become amenable to estimation. Our result is established by proving an asymptotic formula conjectured in 1990 for the number of graphs with given degree sequence. In particular, this gives an asymptotic formula for the number of d -regular graphs for all d , as n\to\infty . The key to our results is a new approach to estimating ratios between point probabilities in the space of degree sequences of the random graph, including analysis of fixed points of the associated operators.

Spectral dynamics of guided edge removals and identifying transient amplifiers for death–Birth updating

The paper deals with two interrelated topics: (1) identifying transient amplifiers in an iterative process, and (2) analyzing the process by its spectral dynamics, which is the change in the graph spectra by edge manipulation. Transient amplifiers are networks representing population structures which shift the balance between natural selection and random drift. Thus, amplifiers are highly relevant for understanding the relationships between spatial structures and evolutionary dynamics. We study an iterative procedure to identify transient amplifiers for death–Birth updating. The algorithm starts with a regular input graph and iteratively removes edges until desired structures are achieved. Thus, a sequence of candidate graphs is obtained. The edge removals are guided by quantities derived from the sequence of candidate graphs. Moreover, we are interested in the Laplacian spectra of the candidate graphs and analyze the iterative process by its spectral dynamics. The results show that although transient amplifiers for death–Birth updating are generally rare, a substantial number of them can be obtained by the proposed procedure. The graphs identified share structural properties and have some similarity to dumbbell and barbell graphs. We analyze amplification properties of these graphs and also two more families of bell-like graphs and show that further transient amplifiers for death–Birth updating can be found. Finally, it is demonstrated that the spectral dynamics possesses characteristic features useful for deducing links between structural and spectral properties. These feature can also be taken for distinguishing transient amplifiers among evolutionary graphs in general.

Detecting gene breakpoints in noisy genome sequences using position-annotated colored de-Bruijn graphs

BackgroundIdentifying the locations of gene breakpoints between species of different taxonomic groups can provide useful insights into the underlying evolutionary processes. Given the exact locations of their genes, the breakpoints can be computed without much effort. However, often, existing gene annotations are erroneous, or only nucleotide sequences are available. Especially in mitochondrial genomes, high variations in gene orders are usually accompanied by a high degree of sequence inconsistencies. This makes accurately locating breakpoints in mitogenomic nucleotide sequences a challenging task.ResultsThis contribution presents a novel method for detecting gene breakpoints in the nucleotide sequences of complete mitochondrial genomes, taking into account possible high substitution rates. The method is implemented in the software package DeBBI. DeBBI allows to analyze transposition- and inversion-based breakpoints independently and uses a parallel program design, allowing to make use of modern multi-processor systems. Extensive tests on synthetic data sets, covering a broad range of sequence dissimilarities and different numbers of introduced breakpoints, demonstrate DeBBI ’s ability to produce accurate results. Case studies using species of various taxonomic groups further show DeBBI ’s applicability to real-life data. While (some) multiple sequence alignment tools can also be used for the task at hand, we demonstrate that especially gene breaks between short, poorly conserved tRNA genes can be detected more frequently with the proposed approach.ConclusionThe proposed method constructs a position-annotated de-Bruijn graph of the input sequences. Using a heuristic algorithm, this graph is searched for particular structures, called bulges, which may be associated with the breakpoint locations. Despite the large size of these structures, the algorithm only requires a small number of graph traversal steps.

Efficient short read mapping to a pangenome that is represented by a graph of ED strings.

A pangenome represents many diverse genome sequences of the same species. In order to cope with small variations as well as structural variations, recent research focused on the development of graph-based models of pangenomes. Mapping is the process of finding the original location of a DNA read in a reference sequence, typically a genome. Using a pangenome instead of a (linear) reference genome can, e.g. reduce mapping bias, the tendency to incorrectly map sequences that differ from the reference genome. Mapping reads to a graph, however, is more complex and needs more resources than mapping to a reference genome. Reducing the complexity of the graph by encoding simple variations like SNPs in a simple way can accelerate read mapping and reduce the memory requirements at the same time. We introduce graphs based on elastic-degenerate strings (ED strings, EDS) and the linearized form of these EDS graphs as a new representation for pangenomes. In this representation, small variations are encoded directly in the sequence. Structural variations are encoded in a graph structure. This reduces the size of the representation in comparison to sequence graphs. In the linearized form, mapping techniques that are known from ordinary strings can be applied with appropriate adjustments. Since most variations are expressed directly in the sequence, the mapping process rarely has to take edges of the EDS graph into account. We developed a prototypical software tool GED-MAP that uses this representation together with a minimizer index to map short reads to the pangenome. Our experiments show that the new method works on a whole human genome scale, taking structural variants properly into account. The advantage of GED-MAP, compared with other pangenomic short read mappers, is that the new representation allows for a simple indexing method. This makes GED-MAP fast and memory efficient. Sources are available at: https://github.com/thomas-buechler-ulm/gedmap.

Continuum limit ofp-Laplacian evolution problems on graphs:Lqgraphons and sparse graphs

In this paper we study continuum limits of the discretizedp-Laplacian evolution problem on sparse graphs with homogeneous Neumann boundary conditions. This goes far beyond known results by handling much more general class of kernels, possibly singular, and graph sequences whose limit are the so-calledLq-graphons. More precisely, we derive a bound on the distance between two continuous-in-time trajectories defined by two different evolution systems (i.e., with different kernels, second member and initial data). Similarly, we provide a bound in the case that one of the trajectories is discrete-in-time and the other is continuous. In turn, these results lead us to establish error estimates of the full discretization of thep-Laplacian problem on sparse random graphs. In particular, we provide rate of convergence of solutions for the discrete models to the solution of the continuous problem as the number of vertices grows.

Connected Grundy Hop Dominating Sequences in Graphs

In this paper, we introduce and do an initial investigation of a variant of Grundy hop domination in a graph called the connected Grundy hop domination. We show that the connected Grundy hop domination number lies between the connected hop domination and Grundy hop domination number of a graph. In particular, we give realization results involving connected hop domination, connected Grundy hop domination, and Grundy hop domination numbers. Moreover, we determine the connected Grundy hop domination numbers of some graphs.

Grundy Dominating and Grundy Hop Dominating Sequences in Graphs: Relationships and Some Structural Properties

In this paper, we revisit the concepts of Grundy domination and Grundy hop domination in graphs and give some realization results involving these parameters. We show that the Grundy domination number and Grundy hop domination number of a graph G are generally incomparable (one is not always less than or equal the other). It is shown that their absolute difference can be made arbitrarily large. Moreover, the Grundy domination numbers of some graphs are determined.

Dynamic ranking and translation synchronization

Abstract In many applications, such as sport tournaments or recommendation systems, we have at our disposal data consisting of pairwise comparisons between a set of $n$ items (or players). The objective is to use these data to infer the latent strength of each item and/or their ranking. Existing results for this problem predominantly focus on the setting consisting of a single comparison graph $G$. However, there exist scenarios (e.g. sports tournaments) where the pairwise comparison data evolve with time. Theoretical results for this dynamic setting are relatively limited, and are the focus of this paper. We study an extension of the translation synchronization problem, to the dynamic setting. In this set-up, we are given a sequence of comparison graphs $(G_t)_{t\in{{\mathscr{T}}}}$, where $ {{\mathscr{T}}} \subset [0,1]$ is a grid representing the time domain, and for each item $i$ and time $t\in{{\mathscr{T}}}$ there is an associated unknown strength parameter $z^*_{t,i}\in{{\mathbb{R}}}$. We aim to recover, for $t\in{{\mathscr{T}}}$, the strength vector $z^*_t=(z^*_{t,1},\dots ,z^*_{t,n})$ from noisy measurements of $z^*_{t,i}-z^*_{t,j}$, where $\left \{{i,j}\right \}$ is an edge in $G_t$. Assuming that $z^*_t$ evolves smoothly in $t$, we propose two estimators—one based on a smoothness-penalized least squares approach and the other based on projection onto the low-frequency eigenspace of a suitable smoothness operator. For both estimators, we provide finite sample bounds for the $\ell _2$ estimation error under the assumption that $G_t$ is connected for all $t\in{{\mathscr{T}}}$, thus proving the consistency of the proposed methods in terms of the grid size $|\mathscr{T}|$. We complement our theoretical findings with experiments on synthetic and real data.

Synthesis of Pneumatic Systems in the Control of the Transport Line of Rolling Elements

Abstract This paper presents the synthesis of a pneumatic control system for a selected configuration of the transport path for the delivery of rolling elements to spiral storage in inter-operational transport. The sequential control system sets the state of the manifolds to ensure a flow of workpieces to serve the subsequent storage. The essential module of the control system is the memory block. It is developed based on a storage filling sequence graph. The filling level of the storages can be monitored in one or two points using sensors. The rolling element displacement control sensors work together with appropriately designed systems to execute the delay of the rising and falling edge input signal. By using a two-level control of the filling level of the storages, it is possible to control the emptying status of the storages as a function of the technological time of removal of the items from the storage between the two control points. Control systems were synthesised and verified using Festo’s FluidSim computer programme.

Forcibly bipartite and acyclic (uni-)graphic sequences

The paper concerns the question of which properties of a graph are already determined by its degree sequence. The classic degree realization problem asks to characterize graphic sequences, i.e., sequences of positive integers, which are the degree sequence of some simple graph. Erdős and Gallai [11] solved this problem. Havel and Hakimi [14,17] provide a different characterization implying an algorithm to generate a realizing graph (if one exists). It is known that a graphic sequence can have several non-isomorphic realizations.We characterize graphic sequences where every realization has some given graph property P. Such sequences are called forcibly P-graphic. In particular, we present complete results characterizing forcibly (connected) bipartite, forcibly acyclic, and forcibly tree-graphic sequences. In those four models, we also characterize the sequences with a unique realizing graph, called unigraphic sequences. Finally, we address the problem of counting the number of sequences in each model.

C-BEDIM and S-BEDIM: Lateral movement detection in enterprise network through behavior deviation measurement

Lateral movement plays a vital role in a network attack campaign. After breaking into the intranet, perpetrators penetrate to their final target through this procedure. In order to protect the crucial resources of an enterprise, it is of significant importance to identify lateral movement traces. Previous studies related to this area have proposed several methods. However, most of them failed to raise high-quality alerts, and security operators cannot identify the real threat from raised massive alerts and make a response in time. To fill this gap, in this paper, we propose a novel approach based on Behavior Deviation Measurement (BEDIM) to raise a few effective alerts for lateral movement detection. By modeling the behavior deviation level of each machine on the connection expanded graph sequences, BEDIM can locate unusual connections from massive logs as initial abnormal records. Apart from this, BEDIM also applies a strategy to filter benign records from initial abnormal connections to further reduce false alerts. Specifically, we propose two implementations of BEDIM, which are S-BEDIM (Simple) and C-BEDIM (Complex). Compared to related state-of-the-arts, both S-BEDIM and C-BEDIM are more precise, robust, and efficient. To demonstrate this, we conduct experiments on two datasets, which are collected from the intranet of two enterprises. The experiment results show that both S-BEDIM and C-BEDIM can raise a few effective alerts. Especially for C-BEDIM, with taking proper period, it can achieve 100% accuracy on both datasets.

The potential-Ramsey numbers [formula omitted] and [formula omitted

A nonincreasing sequence ρ=(ρ1,…,ρn) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. In this case, G is referred to as a realization of ρ. Given two graphs G1 and G2, Busch et al. introduced the potential-Ramsey number of G1 and G2, denoted rpot(G1,G2), is the smallest nonnegative integer m such that for every m-term graphic sequence ρ, there is a realization G of ρ with G1⊆G or with G2⊆G¯, where G¯ is the complement of G. For t≥2 and 0≤k≤⌊t2⌋, let Kt−k be the graph obtained from Kt by deleting k independent edges. Busch et al. determined rpot(Cn,Kt−k) and rpot(Pn,Kt−k) for k=0. Du and Yin determined rpot(Cn,Kt−k) and rpot(Pn,Kt−k) for k=1. In this paper, we further determine rpot(Cn,Kt−k) and rpot(Pn,Kt−k) for 2≤k≤⌊t2⌋.

Local limits of spatial inhomogeneous random graphs

AbstractConsider a set of n vertices, where each vertex has a location in $\mathbb{R}^d$ that is sampled uniformly from the unit cube in $\mathbb{R}^d$ , and a weight associated to it. Construct a random graph by placing edges independently for each vertex pair with a probability that is a function of the distance between the locations and the vertex weights.Under appropriate integrability assumptions on the edge probabilities that imply sparseness of the model, after appropriately blowing up the locations, we prove that the local limit of this random graph sequence is the (countably) infinite random graph on $\mathbb{R}^d$ with vertex locations given by a homogeneous Poisson point process, having weights which are independent and identically distributed copies of limiting vertex weights. Our set-up covers many sparse geometric random graph models from the literature, including geometric inhomogeneous random graphs (GIRGs), hyperbolic random graphs, continuum scale-free percolation, and weight-dependent random connection models.We prove that the limiting degree distribution is mixed Poisson and the typical degree sequence is uniformly integrable, and we obtain convergence results on various measures of clustering in our graphs as a consequence of local convergence. Finally, as a byproduct of our argument, we prove a doubly logarithmic lower bound on typical distances in this general setting.

Mixing of the averaging process and its discrete dual on finite-dimensional geometries

We analyze the $L^1$-mixing of a generalization of the Averaging process introduced by Aldous. The process takes place on a growing sequence of graphs which we assume to be finite-dimensional, in the sense that the random walk on those geometries satisfies a family of Nash inequalities. As a byproduct of our analysis, we provide a complete picture of the total variation mixing of a discrete dual of the Averaging process, which we call Binomial Splitting process. A single particle of this process is essentially the random walk on the underlying graph. When several particles evolve together, they interact by synchronizing their jumps when placed on neighboring sites. We show that, given $k$ the number of particles and $n$ the (growing) size of the underlying graph, the system exhibits cutoff in total variation if $k\to\infty$ and $k=O(n^2)$. Finally, we exploit the duality between the two processes to show that the Binomial Splitting satisfies a version of Aldous' spectral gap identity, namely, the relaxation time of the process is independent of the number of particles.

Local weak convergence for sparse networks of interacting processes

We study the limiting behavior of interacting particle systems indexed by large sparse graphs, which evolve either according to a discrete time Markov chain or a diffusion, in which particles interact directly only with their nearest neighbors in the graph. To encode sparsity we work in the framework of local weak convergence of marked (random) graphs. We show that the joint law of the particle system varies continuously with respect to local weak convergence of the underlying graph marked with the initial conditions. In addition, we show that the global empirical measure converges to a nonrandom limit for a large class of graph sequences including sparse Erdős–Rényi graphs and configuration models, whereas the empirical measure of the connected component of a uniformly random vertex converges to a random limit. Along the way, we develop some related results on the time-propagation of ergodicity and empirical field convergence, as well as some general results on local weak convergence of Gibbs measures in the uniqueness regime which appear to be new. The results obtained here are also useful for obtaining autonomous descriptions of marginal dynamics of interacting diffusions and Markov chains on sparse graphs. While limits of interacting particle systems on dense graphs have been extensively studied, there are relatively few works that have studied the sparse regime in generality.

Fusion of CCTV Video and Spatial Information for Automated Crowd Congestion Monitoring in Public Urban Spaces

Crowd congestion is one of the main causes of modern public safety issues such as stampedes. Conventional crowd congestion monitoring using closed-circuit television (CCTV) video surveillance relies on manual observation, which is tedious and often error-prone in public urban spaces where crowds are dense, and occlusions are prominent. With the aim of managing crowded spaces safely, this study proposes a framework that combines spatial and temporal information to automatically map the trajectories of individual occupants, as well as to assist in real-time congestion monitoring and prediction. Through exploiting both features from CCTV footage and spatial information of the public space, the framework fuses raw CCTV video and floor plan information to create visual aids for crowd monitoring, as well as a sequence of crowd mobility graphs (CMGraphs) to store spatiotemporal features. This framework uses deep learning-based computer vision models, geometric transformations, and Kalman filter-based tracking algorithms to automate the retrieval of crowd congestion data, specifically the spatiotemporal distribution of individuals and the overall crowd flow. The resulting collective crowd movement data is then stored in the CMGraphs, which are designed to facilitate congestion forecasting at key exit/entry regions. We demonstrate our framework on two video data, one public from a train station dataset and the other recorded at a stadium following a crowded football game. Using both qualitative and quantitative insights from the experiments, we demonstrate that the suggested framework can be useful to help assist urban planners and infrastructure operators with the management of congestion hazards.