Independent Graph Research Articles

Transfer graph feature alignment guided multi-source domain adaptation network for machinery fault diagnosis

In recent years, the application of unsupervised multi-source domain adaptation (MSDA) techniques for fault diagnosis has gained significant traction. Current research typically overlooks or fails to effectively capture critical data structure information during feature extraction. Another challenge is optimising the integration of information from multiple source domains to diagnose the target domain while avoiding negative transfer. To address these challenges, this study proposes a transfer graph feature alignment guided multi-source domain adaptation network (MDTGAL). In the proposed method, a transfer graph sample generator module (GSG) is constructed to model the data structure between source and target domains, and multiple graph feature extractors are employed to learn the data structure information from different domain combinations. A regularisation technique is introduced to extract the domain-invariant features by aligning the parameters across multiple independent graph feature extractors. In addition, a weighted soft-voting mechanism based on the polynomial kernel-induced maximum mean difference metric (PK-MMD) is designed to fuse the outputs from multiple classifiers, to comprehensively account for the influence of each source domain. The proposed method was tested on multi-source domain transfer tasks involving various operating conditions of rotating machinery. The experimental results demonstrate that the MDTGAL exhibits superior cross-domain diagnostic performance, outperforming existing mainstream methods. In addition, this study explored the impact of varying numbers of source domains on the diagnostic accuracy of the target domain, providing insights into selecting the correct number of source domains for specific MSDA tasks.

Gating-Enhanced Hierarchical Structure Learning in Hyperbolic Space and Multi-scale Neighbor Topology Learning in Euclidean Space for Prediction of Microbe-Drug Associations.

Identifying drug-related microbes may help us explore how the microbes affect the functions of drugs by promoting or inhibiting their effects. Most previous methods for the prediction of microbe-drug associations focused on integrating the attributes and topologies of microbe and drug nodes in Euclidean space. The heterogeneous network composed of microbes and drugs has a hierarchical structure, and the hyperbolic space is helpful for reflecting the structure. However, the previous methods did not fully exploit the structure. We propose a multi-space feature learning enhanced microbe-drug association prediction method, MFLP, to fuse the hierarchical structure of microbe and drug nodes in hyperbolic space and the multiscale neighbor topologies in Euclidean space. First, we project the nodes of the microbe-drug heterogeneous network on the sphere in hyperbolic space and then construct a topology which implies hierarchical structure and forms a hierarchical attribute embedding. The node information from multiple types of neighbor nodes with the new topological structure in the tangent plane space of a sphere is aggregated by the designed gating-enhanced hyperbolic graph neural network. Second, the gate at the node feature level is constructed to adaptively fuse the hierarchical features of microbe and drug nodes from two adjacent graph neural encoding layers. Third, multiple neighbor topological embeddings for each microbe and drug node are formed by neighborhood random walks on the microbe-drug heterogeneous network, and they cover neighborhood topologies with multiple scales, respectively. Finally, as each scale of topological embedding contains its specific neighborhood topology, we establish an independent graph convolutional neural network for the topology and form the topological representations of microbe and drug nodes in Euclidean space. The comparison experiments based on cross validation showed that MFLP outperformed several advanced prediction methods, and the ablation experiments verified the effectiveness of MFLP's major innovations. The case studies on three drugs further demonstrated MFLP's ability in being applied to discover potential candidate microbes for the given drugs.

A binary-domain recurrent-like architecture-based dynamic graph neural network

The integration of Dynamic Graph Neural Networks (DGNNs) with Smart Manufacturing is crucial as it enables real-time, adaptive analysis of complex data, leading to enhanced predictive accuracy and operational efficiency in industrial environments. To address the problem of poor combination effect and low prediction accuracy of current dynamic graph neural networks in spatial and temporal domains, and over-smoothing caused by traditional graph neural networks, a dynamic graph prediction method based on spatiotemporal binary-domain recurrent-like architecture is proposed: Binary Domain Graph Neural Network (BDGNN). The proposed model begins by utilizing a modified Graph Convolutional Network (GCN) without an activation function to extract meaningful graph topology information, ensuring non-redundant embeddings. In the temporal domain, Recurrent Neural Network (RNN) and residual systems are employed to facilitate the transfer of dynamic graph node information between learner weights, aiming to mitigate the impact of noise within the graph sequence. In the spatial domain, the AdaBoost (Adaptive Boosting) algorithm is applied to replace the traditional approach of stacking layers in a graph neural network. This allows for the utilization of multiple independent graph learners, enabling the extraction of higher-order neighborhood information and alleviating the issue of over-smoothing. The efficacy of BDGNN is evaluated through a series of experiments, with performance metrics including Mean Average Precision (MAP) and Mean Reciprocal Rank (MRR) for link prediction tasks, as well as metrics for traffic speed regression tasks across diverse test sets. Compared with other models, the better experiments results demonstrate that BDGNN model can not only better integrate the connection between time and space information, but also extract higher-order neighbor information to alleviate the over-smoothing phenomenon of the original GCN.

Stage-dependent differential impact of network communication on cognitive function across the continuum of cognitive decline in Parkinson's disease

ObjectiveOur objective was to explore the patterns of resting-state network (RSN) connectivity alterations and investigate how the influences of individual-level network connections on cognition varied across clinical stages without assuming a constant relationship. Methods108 PD patients with continuum of cognitive decline (PD-NC = 46, PD-MCI = 43, PDD = 19) and 34 healthy controls (HCs) underwent resting-state functional MRI and neuropsychological tests. Independent component analysis (ICA) and graph theory analyses (GTA) were employed to explore RSN connection changes. Additionally, stage-dependent differential impact of network communication on cognitive performance were examined using sparse varying coefficient modeling. ResultsCompared to HCs, the dorsal attention network (DAN) and dorsal sensorimotor network (dSMN) were central networks with decreased connections in PD-NC and PD-MCI stage, while the lateral visual network (LVN) emerged as a central network in patients with dementia. Additionally, connectivity of the cerebellum network (CBN) increased in the PD-NC and PD-MCI stages. GTA demonstrated decreased nodal metrics for DAN and dSMN, coupled with an increase for CBN. Moreover, the degree centrality (DC) values of DAN and dSMN exhibited a stage-dependent differential impact on cognitive performance across the continuum of cognitive decline. ConclusionOur findings suggest that across the progression of cognitive impairment, the LVN gradually transitions into a core node with reduced connectivity, while the enhancement of connections in CBN diminishes. Furthermore, the non-linear relationship between the DC values of RSNs and cognitive decline indicates the potential for tailored interventions targeting specific stages.

Heterogeneous graph convolutional network for multi-view semi-supervised classification

This paper proposes a novel approach to semantic representation learning from multi-view datasets, distinct from most existing methodologies which typically handle single-view data individually, maintaining a shared semantic link across the multi-view data via a unified optimization process. Notably, even recent advancements, such as Co-GCN, continue to treat each view as an independent graph, subsequently aggregating the respective GCN representations to form output representations, which ignores the complex semantic interactions among heterogeneous data. To address the issue, we design a unified framework to connect multi-view data with heterogeneous graphs. Specifically, our study envisions multi-view data as a heterogeneous graph composed of shared isomorphic nodes and multi-type edges, wherein the same nodes are shared across different views, but each specific view possesses its own unique edge type. This perspective motivates us to utilize the heterogeneous graph convolutional network (HGCN) to extract semantic representations from multi-view data for semi-supervised classification tasks. To the best of our knowledge, this is an early attempt to transfigure multi-view data into a heterogeneous graph within the realm of multi-view semi-supervised learning. In our approach, the original input of the HGCN is composed of concatenated multi-view matrices, and its convolutional operator (the graph Laplacian matrix) is adaptively learned from multi-type edges in a data-driven fashion. After rigorous experimentation on eight public datasets, our proposed method, hereafter referred to as HGCN-MVSC, demonstrated encouraging superiority over several state-of-the-art competitors for semi-supervised classification tasks.

Independent Dual Graph Attention Convolutional Network for Skeleton-Based Action Recognition

Graph convolutional networks (GCNs) have been widely adopted in skeleton-based action recognition, achieving impressive outcomes. However, the convolution operations in GCNs fail to make full use of the original input data, which restricts its ability to accurately capture the correlation within the skeleton. To solve this issue, this study introduces an independent dual graph attention convolutional network (IDGAN). Specifically, IDGAN additionally incorporates an instinctive attention module that leverages self-attention to capture the correlation among the joints in the original input skeleton. In addition, two independent convolutional operations are used to process two self-attention modules, respectively, to further refine the relationship between skeleton joints. Extensive experiments on several publicly available datasets show that IDGAN outperforms most state-of-the-art algorithms.

Graph theoretical analysis and independent component analysis of diabetic optic neuropathy: A resting-state functional magnetic resonance imaging study.

This study aimed to investigate the resting-state functional connectivity and topologic characteristics of brain networks in patients with diabetic optic neuropathy (DON). Resting-state functional magnetic resonance imaging scans were performed on 23 patients and 41 healthy control (HC) subjects. We used independent component analysis and graph theoretical analysis to determine the topologic characteristics of the brain and as well as functional network connectivity (FNC) and topologic properties of brain networks. Compared with HCs, patients with DON showed altered global characteristics. At the nodal level, the DON group had fewer nodal degrees in the thalamus and insula, and a greater number in the right rolandic operculum, right postcentral gyrus, and right superior temporal gyrus. In the internetwork comparison, DON patients showed significantly increased FNC between the left frontoparietal network (FPN-L) and ventral attention network (VAN). Additionally, in the intranetwork comparison, connectivity between the left medial superior frontal gyrus (MSFG) of the default network (DMN) and left putamen of auditory network was decreased in the DON group. DON patients altered node properties and connectivity in the DMN, auditory network, FPN-L, and VAN. These results provide evidence of the involvement of specific brain networks in the pathophysiology of DON.

Temporal and topological properties of dynamic networks reflect disability in patients with neuromyelitis optica spectrum disorders

Approximately 36% of patients with neuromyelitis optica spectrum disorders (NMOSD) suffer from severe visual and motor disability (blindness or light perception or unable to walk) with abnormalities of whole-brain functional networks. However, it remains unclear how whole-brain functional networks and their dynamic properties are related to clinical disability in patients with NMOSD. Our study recruited 30 NMOSD patients (37.70 ± 11.99 years) and 45 healthy controls (HC, 41.84 ± 11.23 years). The independent component analysis, sliding-window approach and graph theory analysis were used to explore the static strength, time-varying and topological properties of large-scale functional networks and their associations with disability in NMOSD. Compared to HC, NMOSD patients showed significant alterations in dynamic networks rather than static networks. Specifically, NMOSD patients showed increased occurrence (fractional occupancy; P < 0.001) and more dwell times of the low-connectivity state (P < 0.001) with fewer transitions (P = 0.028) between states than HC, and higher fractional occupancy, increased dwell times of the low-connectivity state and lower transitions were related to more severe disability. Moreover, NMOSD patients exhibited altered small-worldness, decreased degree centrality and reduced clustering coefficients of hub nodes in dynamic networks, related to clinical disability. NMOSD patients exhibited higher occurrence and more dwell time in low-connectivity states, along with fewer transitions between states and decreased topological organizations, revealing the disrupted communication and coordination among brain networks over time. Our findings could provide new perspective to help us better understand the neuropathological mechanism of the clinical disability in NMOSD.

Changes in Intrinsic Connectivity Networks Topology Across Levels of Dexmedetomidine-Induced Alteration of Consciousness.

Human consciousness is generally thought to emerge from the activity of intrinsic connectivity networks (resting-state networks [RSNs]) of the brain, which have topological characteristics including, among others, graph strength and efficiency. So far, most functional brain imaging studies in anesthetized subjects have compared wakefulness and unresponsiveness, a state considered as corresponding to unconsciousness. Sedation and general anesthesia not only produce unconsciousness but also phenomenological states of preserved mental content and perception of the environment (connected consciousness), and preserved mental content but no perception of the environment (disconnected consciousness). Unresponsiveness may be seen during unconsciousness, but also during disconnectedness. Deep dexmedetomidine sedation is frequently a state of disconnected consciousness. In this study, we were interested in characterizing the RSN topology changes across 4 different and steady-state levels of dexmedetomidine-induced alteration of consciousness, namely baseline (Awake, drug-free state), Mild sedation (drowsy, still responding), Deep sedation (unresponsive), and Recovery, with a focus on changes occurring between a connected consciousness state and an unresponsiveness state. A functional magnetic resonance imaging database acquired in 14 healthy volunteers receiving dexmedetomidine sedation was analyzed using a method combining independent component analysis and graph theory, specifically looking at changes in connectivity strength and efficiency occurring during the 4 above-mentioned dexmedetomidine-induced altered consciousness states. Dexmedetomidine sedation preserves RSN architecture. Unresponsiveness during dexmedetomidine sedation is mainly characterized by a between-networks graph strength alteration and within-network efficiency alteration of lower-order sensory RSNs, while graph strength and efficiency in higher-order RSNs are relatively preserved. The differential dexmedetomidine-induced RSN topological changes evidenced in this study may be the signature of inadequate processing of sensory information by lower-order RSNs, and of altered communication between lower-order and higher-order networks, while the latter remain functional. If replicated in an experimental paradigm distinguishing, in unresponsive subjects, disconnected consciousness from unconsciousness, such changes would sustain the hypothesis that disconnected consciousness arises from altered information handling by lower-order sensory networks and altered communication between lower-order and higher-order networks, while the preservation of higher-order networks functioning allows for an internally generated mental content (or dream).

IndRT-GCNets: Knowledge Reasoning with Independent Recurrent Temporal Graph Convolutional Representations

IndRT-GCNets: Knowledge Reasoning with Independent Recurrent Temporal Graph Convolutional Representations

Underlying mechanisms, clinical manifestation, and methodological considerations of functional connectivity in Dementia with Lewy Bodies

AbstractBackgroundFunctional connectivity disturbances are potential disease mechanisms related to observable symptoms in Dementia with Lewy Bodies (DLB). Despite several existing theories, the exact relationship between functional connectivity, underlying mechanisms of the DLB, and symptomatology is yet not clear. The present review addresses this gap by summarising research findings of functional magnetic resonance imaging (fMRI) and electroencephalography (EEG) research while considering the impact of different methodological choices causing mixed findings.MethodWe performed a systematic literature review of resting‐state functional connectivity in DLB following PRISMA. We searched multiple databases from January 2000 to November 2022. Keywords included “DLB”, “connectivity”, “EEG”, and “fMRI”, their synonyms and acronyms, and related terms of network analysis. The final sample contained 6 EEG and 14 fMRI studies.ResultThe main methodological choices of fMRI articles included a data‐driven approach (i.e., Independent Component Analysis or Graph Theory; 43%), seed‐based or regions of interest (50%), or their combination (7%) with most studies focusing on large‐scale resting‐state networks (71%). The overall trend of connectivity findings showed a decrease, yet more subtle changes in the directionality of connectivity were observed when using a hypothesis‐driven approach (Fig 1A). The Default Mode Network (DMN) was the most studied network with 75% of relevant articles reporting decreased connectivity. Other networks with a consistent decrease in connectivity included Executive and Sensorimotor Networks (80% and 100% of relevant studies respectively). The inter‐network connectivity revealed the mixed relationship between the DMN, executive, attentional and visual networks (VN; Fig 1B). EEG studies showed a consistent decrease in connectivity on alpha and beta bands (100% and 66% of relevant studies respectively) and disorganisation in graph properties in regions overlapping with findings from large‐scale networks.ConclusionDysfunctions in large‐scale networks likely contribute to progressive degenerative mechanisms and are consistent with visual attention and perception processing theories of psychotic symptoms in the DLB. Hence, connectivity measures can capture brain disturbances in the DLB and remain crucial in uncovering underlying relationships between the networks’ disorganisation and disease mechanisms resulting in psychotic, motor, and cognitive symptoms in DLB. The choice of analytical approach might contribute to the directionality and mixed findings in the literature.

Eventually lattice-linear algorithms

Lattice-linear systems allow nodes to execute asynchronously. We introduce eventually lattice-linear algorithms, where lattices are induced only among the states in a subset of the state space. The algorithm guarantees that the system transitions to a state in one of the lattices. Then, the algorithm behaves lattice linearly while traversing to an optimal state through that lattice.We present a lattice-linear self-stabilizing algorithm for service demand based minimal dominating set (SDMDS) problem. Using this as an example, we elaborate the working of, and define, eventually lattice-linear algorithms. Then, we present eventually lattice-linear self-stabilizing algorithms for minimal vertex cover (MVC), maximal independent set (MIS), graph colouring (GC) and 2-dominating set problems (2DS).Algorithms for SDMDS, MVC and MIS converge in 1 round plus n moves (within 2n moves), GC in n+4m moves, and 2DS in 1 round plus 2n moves (within 3n moves). These results are an improvement over the existing literature. We also present experimental results to show performance gain demonstrating the benefit of lattice-linearity.

Altered functional brain networks in coronary heart disease: independent component analysis and graph theoretical analysis.

Coronary heart disease (CHD) confers a high risk of cognitive and mental impairments in patients. This study aimed to explore the association of CHD with functional connectivity and topological properties of brain networks. A total of 27 patients with CHD and 44 healthy controls (HCs) participated in this study and underwent a resting-state functional magnetic resonance imaging (rs-fMRI) scan. Intra- and internetwork functional connectivity alterations were explored using independent component analysis in CHD patients. Furthermore, graph theoretical analysis was adopted to assess abnormalities in small-world properties and network efficiency metrics of brain networks. Compared to HCs, CHD patients exhibited increased functional connectivity between the posterior default mode network and posterior visual network, as well as decreased functional connectivity between the left frontoparietal network and auditory network. In terms of graph theoretical analysis, small-world network topology was identified in both CHD patients and HCs. Furthermore, the nodal local efficiency of the left putamen was significantly decreased in CHD patients compared to HCs. This study revealed alterations in brain functional connectivity and topological properties in CHD patients, shedding light on the potential neurological mechanism underlying cognitive and mental impairments in these patients and suggesting unexplored connections between CHD and higher order cognitive processing.

Anchor Link Prediction for Privacy Leakage via De-Anonymization in Multiple Social Networks

Anchor link prediction exacerbates the risk of privacy leakage via the de-anonymization of social network data. Embedding-based methods for anchor link prediction are limited by the excessive similarity of the associated nodes in a latent feature space and the variation between latent feature spaces caused by the semantics of different networks. In this paper, we propose a novel method which reduces the impact of semantic discrepancies between different networks in the latent feature space. The proposed method consists of two phases. First, graph embedding focuses on the network structural roles of nodes and increases the distinction between the associated nodes in the embedding space. Second, a federated adversarial learning framework which performs graph embedding on each social network and an adversarial learning model on the server according to the observable anchor links is used to associate independent graph embedding approaches on different social networks. The combination of distinction enhancement and the association of graph embedding approaches alleviates variance between the latent feature spaces caused by the semantics of different social networks. Extensive experiments on real social networks demonstrate that the proposed method significantly outperforms the state-of-the-art methods in terms of both precision and robustness.

Intellectual Capital Valuation of DSE-Listed Non-financial Companies in Bangladesh

Intellectual capital (IC) is an organization's vital resource that helps the firm survive by confirming competitive edges. However, the valuation of IC resources is a challenging task. IC valuation is essential to employ the knowledge resource efficiently. Hence, the study aims to evaluate the IC resources of non-financial companies in Bangladesh and compare the IC performance between manufacturing and service firms. The study also explores the year-wise trend study and industry-based IC performance study. Data have been collected from secondary sources from 69 DSE-listed non-financial companies in Bangladesh from 2017 to 2021, comprising 345 company-year observations. The Modified Value Added Intellectual Coefficient (MVAIC) methodology has been adopted to reflect IC value. Independent Sample t-tests and graph and chart analyses have been performed to fulfill the study objectives. Results have shown that human capital is the primary driver of IC resources, and it dominates Intellectual Capital Efficiency (ICE) and then the overall IC efficiency (MVAIC) score. Human and structural resources utilized by Bangladeshi companies are better than those of relational and tangible capital. Service companies have better handled human and structural resources, but manufacturing companies have better used relational and employed capital. Research has also observed a declining trend of IC resources of the sampled companies. Telecommunication, Fuel and power, and Textile industries have better used IC resources. However, the tannery industry still needs to do so. The research contributes to resources-based theory by confirming the value of invisible resources, which supports business executives, policy-makers, managers, and potential and existing investors to make the right decisions. Further research may be conducted using an extended sample size, comparing financial and non-financial sectors or comparing among industries or valuation techniques like EVA, KCE, and CIV.

Intellectual Capital Valuation of DSE-Listed Non-financial Companies in Bangladesh

Intellectual capital (IC) is an organization's vital resource that helps the firm survive by confirming competitive edges. However, the valuation of IC resources is a challenging task. IC valuation is essential to employ the knowledge resource efficiently. Hence, the study aims to evaluate the IC resources of non-financial companies in Bangladesh and compare the IC performance between manufacturing and service firms. The study also explores the year-wise trend study and industry-based IC performance study. Data have been collected from secondary sources from 69 DSE-listed non-financial companies in Bangladesh from 2017 to 2021, comprising 345 company-year observations. The Modified Value Added Intellectual Coefficient (MVAIC) methodology has been adopted to reflect IC value. Independent Sample t-tests and graph and chart analyses have been performed to fulfill the study objectives. Results have shown that human capital is the primary driver of IC resources, and it dominates Intellectual Capital Efficiency (ICE) and then the overall IC efficiency (MVAIC) score. Human and structural resources utilized by Bangladeshi companies are better than those of relational and tangible capital. Service companies have better handled human and structural resources, but manufacturing companies have better used relational and employed capital. Research has also observed a declining trend of IC resources of the sampled companies. Telecommunication, Fuel and power, and Textile industries have better used IC resources. However, the tannery industry still needs to do so. The research contributes to resources-based theory by confirming the value of invisible resources, which supports business executives, policy-makers, managers, and potential and existing investors to make the right decisions. Further research may be conducted using an extended sample size, comparing financial and non-financial sectors or comparing among industries or valuation techniques like EVA, KCE, and CIV.

Deep graph reconstruction for multi-view clustering

Graph-based multi-view clustering methods have achieved impressive success by exploring a complemental or independent graph embedding with low-dimension among multiple views. The majority of them, however, are shallow models with limited ability to learn the nonlinear information in multi-view data. To this end, we propose a novel deep graph reconstruction (DGR) framework for multi-view clustering, which contains three modules. Specifically, a Multi-graph Fusion Module (MFM) is employed to obtain the consensus graph. Then node representation is learned by the Graph Embedding Network (GEN). To assign clusters directly, the Clustering Assignment Module (CAM) is devised to obtain the final low-dimensional graph embedding, which can serve as the indicator matrix. In addition, a simple and powerful loss function is designed in the proposed DGR. Extensive experiments on seven real-world datasets have been conducted to verify the superior clustering performance and efficiency of DGR compared with the state-of-the-art methods.

Interaction-Aware Graph Neural Networks for Fault Diagnosis of Complex Industrial Processes.

Fault diagnosis of complex industrial processes becomes a challenging task due to various fault patterns in sensor signals and complex interactions between different units. However, how to explore the interactions and integrate with sensor signals remains an open question. Considering that the sensor signals and their interactions in an industrial process with the form of nodes and edges can be represented as a graph, this article proposes a novel interaction-aware and data fusion method for fault diagnosis of complex industrial processes, named interaction-aware graph neural networks (IAGNNs). First, to describe the complex interactions in an industrial process, the sensor signals are transformed into a heterogeneous graph with multiple edge types, and the edge weights are learned by the attention mechanism, adaptively. Then, multiple independent graph neural network (GNN) blocks are employed to extract the fault feature for each subgraph with one edge type. Finally, each subgraph feature is concatenated or fused by a weighted summation function to generate the final graph embedding. Therefore, the proposed method can learn multiple interactions between sensor signals and extract the fault feature from each subgraph by message passing operation of GNNs. The final fault feature contains the information from raw data and implicit interactions between sensor signals. The experimental results on the three-phase flow facility and power system (PS) demonstrate the reliable and superior performance of the proposed method for fault diagnosis of complex industrial processes.

An Independent Cascade Model of Graph Burning

Graph burning was introduced to simulate the spreading of news/information/rumors in social networks. The symmetric undirected graph is considered here. That is, vertex u can spread the information to vertex v, and symmetrically vertex v can also spread information to vertex u. When it is modeled as a graph burning process, a vertex can be set on fire directly or burned by its neighbor. Thus, the task is to find the minimum sequence of vertices chosen as sources of fire to burn the entire graph. This problem has been proved to be NP-hard. In this paper, from a new perspective, we introduce a generalized model called the Independent Cascade Graph Burning model, where a vertex v can be burned by one of its burning neighbors u only if the influence that u gives to v is larger than a given threshold β≥0. We determine the graph burning number with this new Independent Cascade Graph Burning model for several graphs and operation graphs and also discuss its upper and lower bounds.

Analytical framework for quantum alternating operator ansätze

We develop a framework for analyzing layered quantum algorithms such as quantum alternating operator ansätze (QAOA). In the context of combinatorial optimization, our framework relates quantum cost gradient operators, derived from the cost and mixing Hamiltonians, to classical cost difference functions that reflect cost function neighborhood structure. By considering QAOA circuits from the Heisenberg picture, we derive exact general expressions for expectation values as series expansions in the algorithm parameters, cost gradient operators, and cost difference functions. This enables novel interpretability and insight into QAOA behavior in various parameter regimes. For single-level QAOA1 we show the leading-order changes in the output probabilities and cost expectation value explicitly in terms of classical cost differences, for arbitrary cost functions. This demonstrates that, for sufficiently small positive parameters, probability flows from lower to higher cost states on average. By selecting signs of the parameters, we can control the direction of flow. We use these results to derive a classical random algorithm emulating QAOA1 in the small-parameter regime, i.e. that produces bitstring samples with the same probabilities as QAOA1 up to small error. For deeper QAOA p circuits we apply our framework to derive analogous and additional results in several settings. In particular we show QAOA always beats random guessing. We describe how our framework incorporates cost Hamiltonian locality for specific problem classes, including causal cone approaches, and applies to QAOA performance analysis with arbitrary parameters. We illuminate our results with a number of examples including applications to QUBO problems, MaxCut, and variants of MaxSAT. We illustrate the generalization of our framework to QAOA circuits using mixing unitaries beyond the transverse-field mixer through two examples of constrained optimization problems, Max Independent Set and Graph Coloring. We conclude by outlining some of the further applications we envision for the framework.