Graphs Of Different Sizes Research Articles

Solving the non-submodular network collapse problems via Decision Transformer

Given a graph G, the network collapse problem (NCP) selects a vertex subset S of minimum cardinality from G such that the difference in the values of a given measure function f(G)−f(G∖S) is greater than a predefined collapse threshold. Many graph analytic applications can be formulated as NCPs with different measure functions, which often pose a significant challenge due to their NP-hard nature. As a result, traditional greedy algorithms, which select the vertex with the highest reward at each step, may not effectively find the optimal solution. In addition, existing learning-based algorithms do not have the ability to model the sequence of actions taken during the decision-making process, making it difficult to capture the combinatorial effect of selected vertices on the final solution. This limits the performance of learning-based approaches in non-submodular NCPs.To address these limitations, we propose a unified framework called DT-NC, which adapts the Decision Transformer to the Network Collapse problems. DT-NC takes into account the historical actions taken during the decision-making process and effectively captures the combinatorial effect of selected vertices. The ability of DT-NC to model the dependency among selected vertices allows it to address the difficulties caused by the non-submodular property of measure functions in some NCPs effectively. Through extensive experiments on various NCPs and graphs of different sizes, we demonstrate that DT-NC outperforms the state-of-the-art methods and exhibits excellent transferability and generalizability.

Learn to solve dominating set problem with GNN and reinforcement learning

The Dominating Set Problem has a wide range of applications in many industrial areas and the problem has been proven to be NP-hard. The idea using neural networks to solve combinatorial optimization problems has been shown to be effective and time-saving in recent years. Inspired by these studies, to solve the Dominating Set Problem, we train a neural network by Double Deep Q-Networks (DDQN). To better capture the features and structures of the graph, we use a message passing network for the graph representation. We validate our model on random graphs of different sizes, and even on several different lattice graphs, which show our model is effective.

Graph GOSPA metric: a metric to measure the discrepancy between graphs of different sizes

Graph GOSPA metric: a metric to measure the discrepancy between graphs of different sizes

Ramsey numbers for a large tree versus multiple copies of complete graphs of different sizes

Ramsey numbers for a large tree versus multiple copies of complete graphs of different sizes

Ship route simulation based on the cluster analysis

The problem of safe ship’s route planning at the variable hydrometeorological situation along the route is considered in the paper. The problem solved in the paper is based on the division of the water area into separate clusters, depending on its characteristics. The route from Busan Port to Kushiro Port is used as an example. The specifics of this route is that it runs through the open sea, and through the straits and archipelagos. The aim of the work is to automate the process of planning the route and adjusting it during the trip, depending on changes in external conditions. The graph theory for modeling a route is proposed in this paper. The construction of graphs is implemented using the cluster analysis method. In the analysis, the water area of the route is divided into separate subareas, depending on the distance to the coast and the depths difference. Open water areas are separated into larger clusters. Near the coast and at shallow depths, clustering is shorter. The cluster centroids are the vertices of the graphs of the future route. As result it is a simulation of graphs of different sizes. The union of graphs forms the hypergraph. The distance is used as the weight of the graph edges. To determine the most preferable route in criteria of speed, cost or safety, weights are added depending on weather and other factors. As a result of this approach, several routes are planning on the hypergraph. Depending on the weight of the priority criteria, a route is selected automatically. The proposed method can be used to create systems for automated planning of optimal routes, taking into account the selected criteria under changing environmental conditions in the process of voyage and replanning it if necessary.

Strong SDP based bounds on the cutwidth of a graph

Given a linear ordering of the vertices of a graph, the cutwidth of a vertex v with respect to this ordering is the number of edges from any vertex before v (including v) to any vertex after v in this ordering. The cutwidth of an ordering is the maximum cutwidth of any vertex with respect to this ordering. We are interested in finding the cutwidth of a graph, that is, the minimum cutwidth over all orderings, which is an NP-hard problem. In order to approximate the cutwidth of a given graph, we present a semidefinite relaxation. We identify several classes of valid inequalities and equalities that we use to strengthen the semidefinite relaxation. These classes are on the one hand the well-known 3-dicycle equations and the triangle inequalities and on the other hand we obtain inequalities from the squared linear ordering polytope and via lifting the linear ordering polytope. The solution of the semidefinite program serves to obtain a lower bound and also to construct a feasible solution and thereby having an upper bound on the cutwidth.In order to evaluate the quality of our bounds, we perform numerical experiments on graphs of different sizes and densities. It turns out that we produce high quality bounds for graphs of medium size independent of their density in reasonable time. Compared to that, obtaining bounds for dense instances of the same quality is out of reach for solvers using integer linear programming techniques.

A centrality based genetic algorithm for the graph burning problem

Information spread is an intriguing topic to study in network science, which investigates how information, influence, or contagion propagate through networks. Graph burning is a simplified deterministic model for how information spreads within networks. The complicated NP-complete nature of the problem makes it computationally difficult to solve using exact algorithms. Accordingly, a number of heuristics and approximation algorithms have been proposed in the literature for the graph burning problem. In this paper, we propose an efficient genetic algorithm called Centrality BAsed Genetic-algorithm (CBAG) for solving the graph burning problem. Considering the unique characteristics of the graph burning problem, we introduce novel genetic operators, chromosome representation, and evaluation method. In the proposed algorithm, the well-known betweenness centrality is used as the backbone of our chromosome initialization procedure. The proposed algorithm is implemented and compared with previous heuristics and approximation algorithms on 15 benchmark graphs of different sizes. Based on the results, it can be seen that the proposed algorithm achieves better performance in comparison to the previous state-of-the-art heuristics. The complete source code is available online and can be used to find optimal or near-optimal solutions for the graph burning problem.

Spectral density-based clustering algorithms for complex networks.

Clustering is usually the first exploratory analysis step in empirical data. When the data set comprises graphs, the most common approaches focus on clustering its vertices. In this work, we are interested in grouping networks with similar connectivity structures together instead of grouping vertices of the graph. We could apply this approach to functional brain networks (FBNs) for identifying subgroups of people presenting similar functional connectivity, such as studying a mental disorder. The main problem is that real-world networks present natural fluctuations, which we should consider. In this context, spectral density is an exciting feature because graphs generated by different models present distinct spectral densities, thus presenting different connectivity structures. We introduce two clustering methods: k-means for graphs of the same size and gCEM, a model-based approach for graphs of different sizes. We evaluated their performance in toy models. Finally, we applied them to FBNs of monkeys under anesthesia and a dataset of chemical compounds. We show that our methods work well in both toy models and real-world data. They present good results for clustering graphs presenting different connectivity structures even when they present the same number of edges, vertices, and degree of centrality. We recommend using k-means-based clustering for graphs when graphs present the same number of vertices and the gCEM method when graphs present a different number of vertices.

Deep reinforcement learning meets graph neural networks: Exploring a routing optimization use case

Deep Reinforcement Learning (DRL) has shown a dramatic improvement in decision-making and automated control problems. Consequently, DRL represents a promising technique to efficiently solve many relevant optimization problems (e.g., routing) in self-driving networks. However, existing DRL-based solutions applied to networking fail to generalize, which means that they are not able to operate properly when applied to network topologies not observed during training. This lack of generalization capability significantly hinders the deployment of DRL technologies in production networks. This is because state-of-the-art DRL-based networking solutions use standard neural networks (e.g., fully connected, convolutional), which are not suited to learn from information structured as graphs.In this paper, we integrate Graph Neural Networks (GNN) into DRL agents and we design a problem specific action space to enable generalization. GNNs are Deep Learning models inherently designed to generalize over graphs of different sizes and structures. This allows the proposed GNN-based DRL agent to learn and generalize over arbitrary network topologies. We test our DRL+GNN agent in a routing optimization use case in optical networks and evaluate it on 180 and 232 unseen synthetic and real-world network topologies respectively. The results show that the DRL+GNN agent is able to outperform state-of-the-art solutions in topologies never seen during training.

Solving maximum weighted matching on large graphs with deep reinforcement learning

Maximum weighted matching (MWM), which finds a subset of vertex-disjoint edges with maximum weight, is a fundamental topic with a wide spectrum of applications in various fields, such as biotechnology, social analysis, web services, etc. However, traditional methods cannot achieve a good balance between cost and quality. Inspired by the reinforcement learning techniques, we propose an MWM framework, L2M, with a Deep Reinforcement Learning (DRL) model, which can efficiently and effectively solve the MWM problem on large-scale general graphs. First, in contrast to traditional DRL methods that define the Markov Decision Process (MDP) without considering the efficiency issue, we represent MWM as an MDP that is carefully designed to accelerate computation. In particular, this MDP supports selecting multiple edges for each action and uses a pruning method to reduce search space efficiently. Second, since none of the existing DRL methods can support edge selection on large-scale graphs efficiently, we propose an edge-message-passing network EEN to generate edge embedding. To the best of our knowledge, this is the first attempt that uses DRL model for solving the MWM problem. Experimental results show that L2M outperforms state-of-the-art algorithms and generalizes well on graphs of different sizes and distributions.

Partial Lasserre relaxation for sparse Max-Cut

A common approach to solve or find bounds of polynomial optimization problems like Max-Cut is to use the first level of the Lasserre hierarchy. Higher levels of the Lasserre hierarchy provide tighter bounds, but solving these relaxations is usually computationally intractable. We propose to strengthen the first level relaxation for sparse Max-Cut problems using constraints from the second order Lasserre hierarchy. We explore a variety of approaches for adding a subset of the positive semidefinite constraints of the second order sparse relaxation obtained by using the maximum cliques of the graph’s chordal extension. We apply this idea to sparse graphs of different sizes and densities, and provide evidence of its strengths and limitations when compared to the state-of-the-art Max-Cut solver BiqCrunch and the alternative sparse relaxation CS-TSSOS.

Pool Compression for Undirected Graphs

We present a new graph compression scheme that intrinsically exploits the similarity and locality of references in a graph by first ordering the nodes and then merging the contiguous adjacency lists of the graph into blocks to create a pool of nodes. The nodes in the adjacency lists of the graph are encoded by their position in the pool. This simple yet powerful scheme achieves compression ratios better than the previous methods for many datasets tested in this paper and, on average, surpasses all the previous methods. The scheme also provides an easy and efficient access to neighbor queries, e.g., finding the neighbors of a node, and reachability queries, e.g., finding if node u is reachable from node v. We test our scheme on publicly available graphs of different sizes and show a significant improvement in the compression ratio and query access time compared to the previous approaches.

On Local Distributions in Graph Signal Processing

Graph filtering is the cornerstone operation in graph signal processing (GSP). Thus, understanding it is key in developing potent GSP methods. Graph filters are local and distributed linear operations, whose output depends only on the local neighborhood of each node. Moreover, a graph filter's output can be computed separately at each node by carrying out repeated exchanges with immediate neighbors. Graph filters can be compactly written as polynomials of a graph shift operator (typically, a sparse matrix description of the graph). This has led to relating the properties of the filters with the spectral properties of the corresponding matrix – which encodes global structure of the graph. In this work, we propose a framework that relies solely on the local distribution of the neighborhoods of a graph. The crux of this approach is to describe graphs and graph signals in terms of a measurable space of rooted balls. Leveraging this, we are able to seamlessly compare graphs of different sizes and coming from different models, yielding results on the convergence of spectral densities, transferability of filters across arbitrary graphs, and continuity of graph signal properties with respect to the distribution of local substructures.

A diagnostic unified classification model for classifying multi-sized and multi-modal brain graphs using graph alignment

BackgroundPresence of multimodal brain graphs derived from different neuroimaging modalities is inarguably one of the most critical challenges in building unified classification models that can be trained and tested on any brain graph regardless of its size and the modality it was derived from. Existing methodsOne solution is to learn a model for each modality independently, which is cumbersome and becomes more time-consuming as the number of modalities increases. Another traditional solution is to build a model inputting multimodal brain graphs for the target prediction task; however, this is only applicable to datasets where all samples have joint neuro-modalities. New methodIn this paper, we propose to build a unified brain graph classification model trained on unpaired multimodal brain graphs, which can classify any brain graph of any size. This is enabled by incorporating a graph alignment step where all multi-modal graphs of different sizes and heterogeneous distributions are mapped to a common template graph. Next, we design a graph alignment strategy to the target fixed-size template and further apply linear discriminant analysis (LDA) to the aligned graphs as a supervised dimensionality reduction technique for the target classification task. ResultsWe tested our method on unpaired autistic and healthy brain connectomes derived from functional and morphological MRI datasets (two modalities). ConclusionOur results showed that our unified model method not only has great promise in solving such a challenging problem but achieves comparable performance to models trained on each modality independently.

Controllability and Stabilization for Herding a Robotic Swarm Using a Leader: A Mean-Field Approach

In this article, we introduce a model and a control approach for herding a swarm of “follower” agents to a target distribution among a set of states using a single “leader” agent. The follower agents evolve on a finite state space that is represented by a graph and transition between states according to a continuous-time Markov chain (CTMC), whose transition rates are determined by the location of the leader agent. The control problem is to define a sequence of states for the leader agent that steers the probability density of the forward equation of the Markov chain. For the case, when the followers are possibly interacting, we prove local approximate controllability of the system about equilibrium probability distributions. For the case, when the followers are noninteracting, we design two switching control laws for the leader that drive the swarm of follower agents asymptotically to a target probability distribution that is positive for all states. The first strategy is open-loop in nature, and the switching times of the leader are independent of the follower distribution. The second strategy is of feedback type, and the switching times of the leader are functions of the follower density in the leader's current state. We validate our control approach through numerical simulations with varied numbers of follower agents that evolve on graphs of different sizes, through a 3-D multirobot simulation in which a quadrotor is used to control the spatial distribution of eight ground robots over four regions, and through a physical experiment in which a swarm of ten robots is herded by a virtual leader over four regions.

Efficient manipulation and generation of Kirchhoff polynomials for the analysis of non-equilibrium biochemical reaction networks.

Kirchhoff polynomials are central for deriving symbolic steady-state expressions of models whose dynamics are governed by linear diffusion on graphs. In biology, such models have been unified under a common linear framework subsuming studies across areas such as enzyme kinetics, G-protein coupled receptors, ion channels and gene regulation. Due to 'history dependence' away from thermodynamic equilibrium, these models suffer from a (super) exponential growth in the size of their symbolic steady-state expressions and, respectively, Kirchhoff polynomials. This algebraic explosion has limited applications of the linear framework. However, recent results on the graph-based prime factorization of Kirchhoff polynomials may help subdue the combinatorial complexity. By prime decomposing the graphs contained in an expression of Kirchhoff polynomials and identifying the graphs giving rise to equal polynomials, we formulate a coarse-grained variant of the expression suitable for symbolic simplification. We devise criteria to efficiently test the equality of Kirchhoff polynomials and propose two heuristic algorithms to explicitly generate individual Kirchhoff polynomials in a compressed form; they are inspired by algebraic simplifications but operate on the corresponding graphs. We illustrate the practicality of the developed theory and algorithms for a diverse set of graphs of different sizes and for non-equilibrium gene regulation analyses.

Shortest Path Computing in Directed Graphs with Weighted Edges Mapped on Random Networks of Memristors

To accelerate the execution of advanced computing tasks, in-memory computing with resistive memory provides a promising solution. In this context, networks of memristors could be used as parallel computing medium for the solution of complex optimization problems. Lately, the solution of the shortest-path problem (SPP) in a two-dimensional memristive grid has been given wide consideration. Some still open problems in such computing approach concern the time required for the grid to reach to a steady state, and the time required to read the result, stored in the state of a subset of memristors that represent the solution. This paper presents a circuit simulation-based performance assessment of memristor networks as SPP solvers. A previous methodology was extended to support weighted directed graphs. We tried memristor device models with fundamentally different switching behavior to check their suitability for such applications and the impact on the timely detection of the solution. Furthermore, the requirement of binary vs. analog operation of memristors was evaluated. Finally, the memristor network-based computing approach was compared to known algorithmic solutions to the SPP over a large set of random graphs of different sizes and topologies. Our results contribute to the proper development of bio-inspired memristor network-based SPP solvers.

Machine Learning Transfer Efficiencies for Noisy Quantum Walks

AbstractQuantum effects are known to provide an advantage in particle transfer across networks. In order to achieve this advantage, requirements on both a graph type and a quantum system coherence must be found. Here, it is shown that the process of finding these requirements can be automated by learning from simulated examples. The automation is done by using a convolutional neural network of a particular type that learns to understand with which network and under which coherence requirements quantum advantage is possible. The machine learning approach is applied to study noisy quantum walks on cycle graphs of different sizes. It is found that it is possible to predict the existence of quantum advantage for the entire decoherence parameter range, even for graphs outside of the training set. The results are of importance for demonstration of advantage in quantum experiments and pave the way toward automating scientific research and discoveries.

A Reinforcement Learning Framework for Efficient Informative Sensing

Large-scale spatial data can be collected using mobile robots with sensing and navigation capabilities. Due to limited battery lifetime and scarcity of charging stations, it is important to plan informative paths so as to maximize the utility of data given a limited travel budget, which is known as the informative path planning (IPP) problem. IPP is NP-hard, and existing solutions suffer from high complexity or low optimality. In this paper, we present a novel IPP solution based on reinforcement learning (RL). The basic idea is to learn the structural characteristics of informative paths, so informative paths can be predicted. As such, when budgets change, we avoid solving the problem from scratch and thus path planning efficiency can be improved dramatically. Among the 20 path planning experiments in two areas, the proposed RL based solution achieves the best path utility in 15 experiments, compared with state-of-the-art algorithms. More importantly, the inference complexity is linear with respect to the budget (equivalently, the maximum number of steps in RL), which is lower than other solutions. Despite the NP-hardness, the path planning process can be finished within a few seconds in our experiments on two graphs of different sizes.

Scalable Method to Find the Shortest Path in a Graph with Circuits of Memristors.

Finding the shortest path in a graph has applications in a wide range of optimization problems. However, algorithmic methods scale with the size of the graph in terms of time and energy. We propose a method to solve the shortest-path problem using circuits of nanodevices called memristors and validate it on graphs of different sizes and topologies. It is both valid for an experimentally derived memristor model and robust to device variability. The time and energy of the computation scale with the length of the shortest path rather than with the size of the graph, making this method particularly attractive for solving large graphs with small path lengths.