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Control using higher order Laplacians in network topologies
Abstract
This paper establishes the proper notation and precise interpretation for Laplacian flows on simplicial complexes. In particular, we have shown how to interpret these flows as time-varying discrete differential forms that converge to harmonic forms. The stability properties of the corresponding dynamical system are shown to be related to the topological structure of the underlying simplicial complex. Finally, we discuss the relevance of these results in the context of networked control and sensing.
... We first define higher-order adjacency for a simplicial complex, as previously studied [27][28][29]. Two n-simplices σ and σ (n ≥ 0) are upper adjacent if there exists an (n + 1)-simplex γ such that σ ⊂ γ and σ ⊂ γ , that is, they are both faces of a E. Song common (n + 1)-simplex. ...
... where L up n = B n+1 B T n+1 and L down n = B T n B n represent diffusion through upper adjacent simplices and lower adjacent simplices, respectively. For n > 0 and n-simplices σ, σ ∈ K, the matrix elements of L up n and L down n can be written as follows [29]: ...
... Most elements of the first 1-down-community (red edges in Fig. 1) consist of the edges connected within the Mr. Hi group, whereas most elements of the second 1-down-community (blue edges in Fig. 1) consist of the edges connected within the Officer group. There are several edges (1-simplices) connecting the Officer group and the Mr. Hi group: (1, 32), (2,31), (3,10), (3,28), and (3,29) in the first 1-down-community, and (3, 33), (9, 31), (9, 33), (9, 34), (14,34), and (20, 34) in the second 1-down-community. The simplicial modularity (Eq. ...
Quantum walks have emerged as a transformative paradigm in quantum information processing and can be applied to various graph problems. This study explores discrete-time quantum walks on simplicial complexes, a higher-order generalization of graph structures. Simplicial complexes, encoding higher-order interactions through simplices, offer a richer topological representation of complex systems. Since the conventional classical random walk cannot directly detect community structures, we present a quantum walk algorithm to detect higher-order community structures called simplicial communities. We utilize the Fourier coin to produce entangled translation states among adjacent simplices in a simplicial complex. The potential of our quantum algorithm is tested on Zachary’s karate club network. This study may contribute to understanding complex systems at the intersection of algebraic topology and quantum walk algorithms.
... where B k and B k+1 are the boundary matrices of k-simplices and (k + 1)-simplices, respectively. The combinatorial Laplacian can be calculated as follows [23]: ...
... So it can be computed in a distributed fashion. The nonzero elements in ker L k are representatives of k-dimensional holes in the simplicial complexes [23]. Let X be a simplicial complex, and v be the first vector in the null space of L k . ...
... Take any cyclic path on X and accumulate the corresponding values of v along this path. If the accumulation sums up to zero, the path does not bound a hole in X [23]. As an example, consider the simplicial complex depicted in Fig. 2. Its first combinatorial Laplacian matrix is given by matrix L 1 . ...
Coverage is one of the fundamental problems in directional sensor networks (DSNs). This problem is more complicated when we deal with heterogeneous DSNs (HDSNs). Prolonging the network lifetime is another important problem in this area. The problem of finding k disjoint cover sets known as the Set k-Cover problem can solve both the coverage and lifetime issues. In this paper a distributed algorithm is proposed for the Set k-Cover problem in HDSNs and then the method is applied for target k-tracking problem. In the Set k-Cover problem, directional sensors are partitioned into k disjoint sets where each set covers the entire area, and in object k-tracking problem, the object must be tracked by at least k sensors. The proposed algorithms are based on the notion of homology in algebraic topology. We consider the nerve complex corresponding to the HDSN and demonstrate how topological properties of the nerve complex of the network can be used to formulate the Set k-Cover problem as an integer linear programming problem. Then, we propose a distributed algorithm based on the subgradient method for this problem. After that, we propose a distributed algorithm for object k-tracking based on the solution of the Set k-Cover problem. Finally, we evaluate the performance of the proposed algorithms by conducting simulation experiments.
... Simplicial descriptions are very powerful because they come equipped with many nice mathematical gadgets. It is in fact straight-forward to define Laplacian operators for any dimension on simplicial complexes [85,86], they can approximate both regular manifolds and highly irregular structures [87,88], and they come naturally equipped with boundary operators stringing together simplices with different dimensions. Crucially, these operators describe the topology and shape of simplicial complexes in terms of their cycles, cavities and higher-order topological holes [89] and are naturally related to the combinatorial Laplacians [86]. ...
... It is in fact straight-forward to define Laplacian operators for any dimension on simplicial complexes [85,86], they can approximate both regular manifolds and highly irregular structures [87,88], and they come naturally equipped with boundary operators stringing together simplices with different dimensions. Crucially, these operators describe the topology and shape of simplicial complexes in terms of their cycles, cavities and higher-order topological holes [89] and are naturally related to the combinatorial Laplacians [86]. In the following sections, we will describe many of these properties in greater detail, because they represent some of the most powerful tools currently available and are the foundation of recent advances in topological data analysis [90][91][92]. ...
... Then (A k L ) αβ = 1 only if the k-simplices α and β are lower adjacent, while (A k U ) αβ = 1 only if the k-simplices α and β are upper adjacent [113]. Another way of defining adjacency is to construct a single adjacency matrix A k that isolates lower adjacent interactions that are not involved in upper ones, that is, A k αβ = 1 only if the k-simplices α and β are lower adjacent but not upper adjacent [86,113]. Both these definitions of adjacency will be instrumental in the study of node shortest path centrality defined on paths on k-dimensional simplices (Section 3.2). ...
The complexity of many biological, social and technological systems stems from the richness of the interactions among their units. Over the past decades, a great variety of complex systems has been successfully described as networks whose interacting pairs of nodes are connected by links. Yet, in face-to-face human communication, chemical reactions and ecological systems, interactions can occur in groups of three or more nodes and cannot be simply described just in terms of simple dyads. Until recently, little attention has been devoted to the higher-order architecture of real complex systems. However, a mounting body of evidence is showing that taking the higher-order structure of these systems into account can greatly enhance our modeling capacities and help us to understand and predict their emerging dynamical behaviors. Here, we present a complete overview of the emerging field of networks beyond pairwise interactions. We first discuss the methods to represent higher-order interactions and give a unified presentation of the different frameworks used to describe higher-order systems, highlighting the links between the existing concepts and representations. We review both the measures designed to characterize the structure of these systems, and the models proposed in the literature to generate synthetic structures, such as random and growing simplicial complexes, bipartite graphs and hypergraphs. We then introduce and discuss the rapidly growing research on higher-order dynamical systems and on dynamical topology. We focus on novel emergent phenomena characterizing landmark dynamical processes, such as diffusion, spreading, synchronization and games, when extended beyond pairwise interactions. We elucidate the relations between higher-order topology and dynamical properties, and conclude with a summary of empirical applications, providing an outlook on current modeling and conceptual frontiers.
... There is a recent interest in the study of simplicial complexes for representing complex systems and we should mention here their applications to study brain networks [3,15,21,30,31], social systems [19,23,25], biological networks [2,36,37], and infrastructural systems [4,5,14,26,35]. ...
... Here we introduce the main concepts and terminology used in this paper (see also [16,17,24,26,27,34,35]). Definition 1. ...
... A consequence of this is that it allows us to analyze the relationships between the centralities of simplices and their faces which we are particularly interested in at the node level. Secondly, this notion of adjacency lines up nicely with the extensively studied higher order Laplacians of simplicial complexes [26]. An off-diagonal entry of the higher order Laplacian matrix is non zero if and only if the corresponding off-diagonal entry of A k l − A k u is non-zero. ...
Complex networks are graph representation of complex systems from the real-world. They are ubiquitous in biological, ecological, social and infrastructural systems. Here we study a transformation of these complex networks into simplicial complexes, where cliques represent the simplicies of the complex. We extend the concept of node centrality to that of simplicial centrality and study several mathematical properties of degree, clossenness, betweenness, eigenvector, Katz, and subgraph centrality for simplicial complexes. We also define the communicability function between pairs of simplices in a simplicial complex. Using a scaling between the simplicial communicability and the simplicial eigenvector centrality we develop a topological classification of simplicial complexes into four universal classes. We then study a series of simplicial complexes representing real-world complex systems. We show fundamental differences between the centralities at the different levels of the complexes---nodes, edges and triangles, which may have important implications for the dynamical processes taking places on these systems. Finally, we study the invasion rate of invasive species into a series of 14 ecological systems. Using the simplicial spectral scaling previously defined we describe quantitatively the factors determining the invasion rate of species to these ecosystems. Mainly, the existence of topological holes at the levels of nodes and triangles explain 95% of the variance in the invasion rate of these ecosystems. The paper is written in a self-contained way, such that it can be used by practitioners of network theory as a basis for further developments.
... Recently, the notion of homology in algebraic topology has been used to model the coverage problem of sensor networks in the absence of location information [6][7][8][9]. In this direction, the proposed algorithms can be categorized in algorithms for building the appropriate simplicial complex of the network [10,11], coverage verification [12,13], and sensor selection [13,14]. In this paper, we present a novel algorithm named DHSS (distributed homological sensor selection), for selecting the minimum number of sensors which are sufficient to cover the entire area. ...
... where n 2 is the number of 2-simplices. (11) and (12) show that for converging in the kth , where e 1 is Y ðkþ1Þ À Y ðkÞ 1 and e 2 is f ðY kþ1 Þ À f ðY k Þ . ...
... For a given area, T 1 is less than the minimum number of required 2-simplices and so it is constant and also is independent of the number of deployed sensors. Let a t = 1/t, according to (12), T 2 ðThe number of 2 -simplicesÞ=e, therefore T 2 = O(n 2 ), where n is the number of sensors. Consequently, the time complexity of DHSS is O(n 2 D 2 ). ...
Area coverage is one of the fundamental problems in wireless sensor networks. The main objective is to minimize the number of active sensors to conserve energy consumption, while complete area coverage is achieved. Some of the existing solutions are based on the location information of sensors which are not applicable to scenarios where GPS modules are not available and there is no reliable location information of sensors. In this paper, we present a distributed homological sensor selection algorithm, namely DHSS, to select the least number of sensors to cover the entire area in the case that no location information is available. We consider the Rips complex of the network and formulate the selection problem as an optimization problem. DHSS tries to find the least number of 2-simplices of the Rips complex of the network to cover the entire area in a distributed manner. Finally, we evaluate the performance of the proposed algorithm by conducting simulation experiments.
... The investigation of the dynamical state of simplicial complexes has instead revealed that this is only a special case and that in general each simplex (higherorder interaction) can be associated with a dynamical variable leading to the notion of topological signals. This change of paradigm has lead to novel understanding of topological synchronization [30][31][32][33][34] and higher-order diffusion dynamics [35][36][37][38] and to novel signal processing [39][40][41] and topological neural network algorithms [42,43]. In particular, higher-order diffusion dynamics is among the most basic topological dynamical processes, describing diffusion from n-dimensional simplices to n-dimensional simplices going either one dimension up or one dimension down. ...
... The Connection Laplacians defined in the previous section can be used to define higher-order diffusion processes that will reflect the directionality of the simplicial complex extending previous work on higher-order diffusion over undirected simplicial complexes [35][36][37][38]. If we focus on the diffusion induced by the 1-Connection Laplacian we can define three types of dynamical processes describing diffusion from edge to edge going exclusively through triangles (upper diffusion), going exclusively through nodes (lower diffusion), or going either through triangles or nodes (combined diffusion). ...
Higher-order networks encode the many-body interactions existing in complex systems, such as the brain, protein complexes, and social interactions. Simplicial complexes are higher-order networks that allow a comprehensive investigation of the interplay between topology and dynamics. However, simplicial complexes have the limitation that they only capture undirected higher-order interactions while in real-world scenarios, often there is a need to introduce the direction of simplices, extending the popular notion of direction of edges. On graphs and networks the Magnetic Laplacian, a special case of Connection Laplacian, is becoming a popular operator to treat edge directionality. Here we tackle the challenge of treating directional simplicial complexes by formulating Higher-order Connection Laplacians taking into account the configurations induced by the simplices' directions. Specifically, we define all the Connection Laplacians of directed simplicial complexes of dimension two and we discuss the induced higher-order diffusion dynamics by considering instructive synthetic examples of simplicial complexes. The proposed higher-order diffusion processes can be adopted in real scenarios when we want to consider higher-order diffusion displaying non-trivial frustration effects due to conflicting directionalities of the incident simplices.
... The algebraic sum of the flows over any filled-in region is equal to zero. However, for any holes, this algebraic sum is greater than zero [27]. Consquently, a directional cycle represents a hole in the network. ...
... Let Y be the vector representation of edges. The magnitudes of faces in W increases significantly near the hole [27]. Therefore, the edge with the largest magnitude is on the minimum-length cycle, so its corresponding element in Y is 1. ...
... The spectral simplicial complex model is derived from combinatorial Laplacian (or Hodge-Laplacian) matrixes, constructed based on a simplicial complex [59][60][61][62][63][64][65][66]. ...
... It can be seen that this expression is exactly the graph Laplacian as in Eq. 3. Further, when k > 0, L k can be expressed as [60], ...
Molecular representations are of great importance for machine learning models in RNA data analysis. Essentially, efficient molecular descriptors or fingerprints that characterize the intrinsic structural and interactional information of RNAs can significantly boost the performance of all learning modeling. In this paper, we introduce two persistent models, including persistent homology and persistent spectral, for RNA structure and interaction representations and their applications in RNA data analysis. Different from traditional geometric and graph representations, persistent homology is built on simplicial complex, which is a generalization of graph models to higher-dimensional situations. Hypergraph is a further generalization of simplicial complexes and hypergraph-based embedded persistent homology has been proposed recently. Moreover, persistent spectral models, which combine filtration process with spectral models, including spectral graph, spectral simplicial complex, and spectral hypergraph, are proposed for molecular representation. The persistent attributes for RNAs can be obtained from these two persistent models and further combined with machine learning models for RNA structure, flexibility, dynamics, and function analysis.
... Similarly, in a simplicial complex the higher-order Laplacian L [ ] (with > 0) [34,35,14] is the [ ] × [ ] matrix defined as ...
... withˆ [ 3] indicating the three-body interaction [3] = [13] + [23] − [34] . ...
Simplicial synchronization reveals the role that topology and geometry have in determining the dynamical properties of simplicial complexes. Simplicial network geometry and topology are naturally encoded in the spectral properties of the graph Laplacian and of the higher-order Laplacians of simplicial complexes. Here we show how the geometry of simplicial complexes induces spectral dimensions of the simplicial complex Laplacians that are responsible for changing the phase diagram of the Kuramoto model. In particular, simplicial complexes displaying a non-trivial simplicial network geometry cannot sustain a synchronized state in the infinite network limit if their spectral dimension is smaller or equal to four. This theoretical result is here verified on the Network Geometry with Flavor simplicial complex generative model displaying emergent hyperbolic geometry. On its turn simplicial topology is shown to determine the dynamical properties of the higher-order Kuramoto model. The higher-order Kuramoto model describes synchronization of topological signals, i.e., phases not only associated to the nodes of a simplicial complexes but associated also to higher-order simplices, including links, triangles and so on. This model displays discontinuous synchronization transitions when topological signals of different dimension and/or their solenoidal and irrotational projections are coupled in an adaptive way.
... A topological space, i.e. a set of elements V along with a set of multiway relations among them, is an abstract simplicial complex if it can be represented by a set S containing subsets of various cardinality of the elements of V satisfying the inclusion property, meaning that if a set A belongs to S, then any subset of A also belongs to S. The framework of Topological Data Analysis (TDA) [2] has been proposed to extract information from data by using algebraic topological tools applied to the domain representing relations among the data [14], [15]. Methods based on simplicial complexes have been already applied in many fields, such as statistical ranking [16], control systems [17], tumor progression analysis [18], biological [19] and brain networks [20]. The two books [21], [22] focus on topological and geometric methods to analyze signals and images. ...
... The solenoidal and irrotational components refer, by construction, to eigenvalues different from zero. Hence, it makes sense to define the corresponding filters by setting α 0 = 0 in (16) and (17). In this section, we will provide theoretical conditions under which the design of two separate FIR filters based on the lower and upper Laplacians may offer performance gains with respect to a single FIR filter handling jointly the solenoidal and irrotational signals. ...
Topological Signal Processing (TSP) over simplicial complexes is a framework that has been recently proposed, as a generalization of graph signal processing (GSP), aimed to analyze signals defined over sets of any order (i.e. not only vertices of a graph) and to capture relations of any order present in the observed data. Our goal in this paper is to extend the TSP framework to deal with signals defined over cell complexes, i.e. topological spaces that are not constrained to satisfy the inclusion property of simplicial complexes, namely the condition that, if a set belongs to the complex, then all its subsets belong to the complex as well. We start showing how to translate the algebraic topological tools to deal with signals defined over cell complexes and then we propose a method to infer the structure of the cell complex from data. Then, we address the filtering problem over cell complexes and we provide the theoretical conditions under which the independent filtering of the solenoidal and irrotational components of edge signals brings a performance improvement with respect to a common filtering strategy. Furthermore, we propose a distributed strategy to filter the harmonic signals with the aim of retrieving the sparsest representation of the harmonic components. Finally, we quantify the advantages of using cell complexes instead of simplicial complexes, in terms of the sparsity/accuracy trade-off and of the signal recovery accuracy from sparse samples, using both simulated and real flows measured on data traffic and transportation networks.
... Persistent homologies represent one of the fundamental tools in this framework [10]. Methods based on simplicial complexes have been already applied in many fields, such as statistical ranking [11], control systems [12], tumor progression analysis [13], and brain networks [14]. The two books [15], [16] focus on topological and geometric methods to analyze signals and images. ...
Topological Signal Processing (TSP) over simplicial complexes is a framework that has been recently proposed, as a generalization of graph signal processing (GSP), to extend GSP to analyzing signals defined over sets of any order (i.e., not only vertices of a graph) and to capture multiway relations of any order among the data. However, simplicial complexes are required to satisfy the so-called inclusion property, according to which, if a set belongs to the complex, then all its subsets must also belong to the complex. In some applications, this is a severe limitation. To overcome this limit, in this paper we extend TSP to deal with signals defined over cell complexes and we also generalize the concept of cell complexes to include hollow cells. We show that, even if the algebraic formulation does not change significantly, the extension to the generalized cell complexes considerably broadens the number of applications. Most important, the new representation provides a much better trade-off between the complexity of the representation and its accuracy. In addition, we propose a method to infer the structure of the cell complex from data and we propose distributed filtering strategies, including a method to retrieve the sparsest representation of the harmonic component. We quantify the advantages of using cell complexes instead of simplicial complexes, in terms of the complexity/accuracy trade-off, for different applications such image segmentation and recovering of real flows measured on data traffic and transportation networks.
... The operator L k : C k (X) → C k (X) is called the k th combinatorial Laplacian of the simplicial complex X that is defined as where B k and B k+1 are the boundary matrices of k-simplices and (k + 1)-simplices, respectively. The nonzero elements in ker L k are representatives of k-dimensional holes in the simplicial complexes [29]. [30]: if a collection of sets and all their nonempty finite intersections are contractible, then the union of those sets has the homotopy type as the nerve complex. ...
One of the fundamental problems in randomly deployed sensor networks is enhancing the network lifetime while providing full area coverage. The problem is more challenging when location information is not available. Scheduling the activities of sensor nodes in a way that each point of the area of interest is covered by at least one sensor node is a promising way when a smaller set of sensor nodes is scheduled autonomously. The autonomous sleep scheduling of sensor nodes can be efficiently achieved based on the topological properties of the sensor network in a distributed fashion. In this paper, we address the problem of autonomously scheduling of sensor nodes to provide full area coverage in wireless sensor networks, even when location information is unavailable. The goal is to prolong the network lifetime. The proposed method is based on homology. The idea is autonomous selection of the minimum number of active sensors with the highest level of energy based on the properties of the simplicial complex of the network. We formulate this problem as an integer programming problem. Then, we propose a distributed algorithm, which does not require the knowledge of the location of nodes or distance between them. Finally, we provide simulation results demonstrating the performance of the proposed algorithm.
... Similarly to the 0-order case L 0 , one can describe the entries of L 1 in terms of the structure of the simplicial complex, see e.g. [23]. ...
Simplicial complexes are generalizations of classical graphs. Their homology groups are widely used to characterize the structure and the topology of data in e.g. chemistry, neuroscience, and transportation networks. In this work we assume we are given a simplicial complex and that we can act on its underlying graph, formed by the set of 1-simplices, and we investigate the stability of its homology with respect to perturbations of the edges in such graph. Precisely, exploiting the isomorphism between the homology groups and the higher-order Laplacian operators, we propose a numerical method to compute the smallest graph perturbation sufficient to change the dimension of the simplex’s Hodge homology. Our approach is based on a matrix nearness problem formulated as a matrix differential equation, which requires an appropriate weighting and normalizing procedure for the boundary operators acting on the Hodge algebra’s homology groups. We develop a bilevel optimization procedure suitable for the formulated matrix nearness problem and illustrate the method’s performance on a variety of synthetic quasi-triangulation datasets and real-world transportation networks.
... From Eq. (29), Ω, h and ϵ are defined as [Eqs. 30,31] Ωh ...
Phase synchronizations in models of coupled oscillators such as the Kuramoto model have been widely studied with pairwise couplings on arbitrary topologies, showing many unexpected dynamical behaviors. Here, based on a recent formulation the Kuramoto model on weighted simplicial complexes with phases supported on simplices of any order k, we introduce linear and non-linear frustration terms independent of the orientation of the k + 1 simplices, as a natural generalization of the Sakaguchi-Kuramoto model to simplicial complexes. With increasingly complex simplicial complexes, we study the the dynamics of the edge simplicial Sakaguchi-Kuramoto model with nonlinear frustration to highlight the complexity of emerging dynamical behaviors. We discover various dynamical phenomena, such as the partial loss of synchronization in subspaces aligned with the Hodge subspaces and the emergence of simplicial phase re-locking in regimes of high frustration.
... Geometrically, homology generators (eigenvectors from zero eigenvalues) correspond to the cycle structures within the data. Non-homology generators can be used in clustering (spectral clustering) and community detection 17,21 . Furthermore, eigendecomposition-based HodgeRank model can be used in biomolecular structure folding analysis. ...
Hodge theory reveals the deep intrinsic relations of differential forms and provides a bridge between differential geometry, algebraic topology, and functional analysis. Here we use Hodge Laplacian and Hodge decomposition models to analyze biomolecular structures. Different from traditional graph-based methods, biomolecular structures are represented as simplicial complexes, which can be viewed as a generalization of graph models to their higher-dimensional counterparts. Hodge Laplacian matrices at different dimensions can be generated from the simplicial complex. The spectral information of these matrices can be used to study intrinsic topological information of biomolecular structures. Essentially, the number (or multiplicity) of k-th dimensional zero eigenvalues is equivalent to the k-th Betti number, i.e., the number of k-th dimensional homology groups. The associated eigenvectors indicate the homological generators, i.e., circles or holes within the molecular-based simplicial complex. Furthermore, Hodge decomposition-based HodgeRank model is used to characterize the folding or compactness of the molecular structures, in particular, the topological associated domain (TAD) in high-throughput chromosome conformation capture (Hi-C) data. Mathematically, molecular structures are represented in simplicial complexes with certain edge flows. The HodgeRank-based average/total inconsistency (AI/TI) is used for the quantitative measurements of the folding or compactness of TADs. This is the first quantitative measurement for TAD regions, as far as we know.
... 21,22 In this pursuit, a variety of mathematical tools have been developed ranging from mean field theory 23 and probability theory 24 to persistent homology, [25][26][27] algebraic topology, 28,29 and Hodge theory. 30 Thus, new problems in modeling and understanding the dynamical systems on simplicial complexes are revitalizing interest in analytical methods from algebraic topology and Hodge theory. ...
Despite the vast literature on network dynamics, we still lack basic insights into dynamics on higher-order structures (e.g., edges, triangles, and more generally, [Formula: see text]-dimensional “simplices”) and how they are influenced through higher-order interactions. A prime example lies in neuroscience where groups of neurons (not individual ones) may provide building blocks for neurocomputation. Here, we study consensus dynamics on edges in simplicial complexes using a type of Laplacian matrix called a Hodge Laplacian, which we generalize to allow higher- and lower-order interactions to have different strengths. Using techniques from algebraic topology, we study how collective dynamics converge to a low-dimensional subspace that corresponds to the homology space of the simplicial complex. We use the Hodge decomposition to show that higher- and lower-order interactions can be optimally balanced to maximally accelerate convergence and that this optimum coincides with a balancing of dynamics on the curl and gradient subspaces. We additionally explore the effects of network topology, finding that consensus over edges is accelerated when two-simplices are well dispersed, as opposed to clustered together.
... Motivated by the great success, we have proposed persistent spectral based machine learning models (PerSpect-ML) and use them in drug design [33,36]. Mathematically, spectral models, including spectral graph theory [11,56], spectral simplicial complex [1,14,23,41] and spectral hypergraph [33], study the topological properties with algebraic tools, including characteristic polynomial, eigenvalues, eigenvectors and other eigenspectrum properties. The spectral information is used for the characterization of biomolecular structures and interactions [33,36]. ...
Protein–protein interactions (PPIs) play a significant role in nearly all cellular and biological activities. Data-driven machine learning models have demonstrated great power in PPIs. However, the design of efficient molecular featurization poses a great challenge for all learning models for PPIs. Here, we propose persistent spectral (PerSpect) based PPI representation and featurization, and PerSpect-based ensemble learning (PerSpect-EL) models for PPI binding affinity prediction, for the first time. In our model, a sequence of Hodge (or combinatorial) Laplacian (HL) matrices at various different scales are generated from a specially designed filtration process. PerSpect attributes, which are statistical and combinatorial properties of spectrum information from these HL matrices, are used as features for PPI characterization. Each PerSpect attribute is input into a 1D convolutional neural network (CNN), and these CNN networks are stacked together in our PerSpect-based ensemble learning models. We systematically test our model on the two most commonly used datasets, i.e. SKEMPI and AB-Bind. It has been found that our model can achieve state-of-the-art results and outperform all existing models to the best of our knowledge.
... A simplicial complex is a generalization of graphs into their higher-dimensional counterparts. Moreover, spectral graph can be generalized into spectral simplicial complex, which studies the spectral properties of combinatorial Laplacian (or Hodge Laplacian) matrixes [65][66][67][68][69][70][71][72]. ...
Artificial intelligence (AI)-based drug design has great promise to fundamentally change the landscape of the pharmaceutical industry. Even though there are great progress from handcrafted feature-based machine learning models, 3D convolutional neural networks (CNNs) and graph neural networks, effective and efficient representations that characterize the structural, physical, chemical and biological properties of molecular structures and interactions remain to be a great challenge. Here, we propose an equal-sized molecular 2D image representation, known as the molecular persistent spectral image (Mol-PSI), and combine it with CNN model for AI-based drug design. Mol-PSI provides a unique one-to-one image representation for molecular structures and interactions. In general, deep models are empowered to achieve better performance with systematically organized representations in image format. A well-designed parallel CNN architecture for adapting Mol-PSIs is developed for protein-ligand binding affinity prediction. Our results, for the three most commonly used databases, including PDBbind-v2007, PDBbind-v2013 and PDBbind-v2016, are better than all traditional machine learning models, as far as we know. Our Mol-PSI model provides a powerful molecular representation that can be widely used in AI-based drug design and molecular data analysis.
... Cliques in networks are the basis for analysis in terms of simplicial complexes as used in algebraic topology. In network analysis, simplicial complexes [25][26][27] have been used to analyse network geometry 28 , to model structure in temporal networks 29 , investigate synchronization phenomenon 30-33 , social contagion 34 , epidemic spreading 35 , and neuroscience 10,11 . One area where network analysis is less well developed is the temporal evolution of networks. ...
We study the evolution of networks through ‘triplets’—three-node graphlets. We develop a method to compute a transition matrix to describe the evolution of triplets in temporal networks. To identify the importance of higher-order interactions in the evolution of networks, we compare both artificial and real-world data to a model based on pairwise interactions only. The significant differences between the computed matrix and the calculated matrix from the fitted parameters demonstrate that non-pairwise interactions exist for various real-world systems in space and time, such as our data sets. Furthermore, this also reveals that different patterns of higher-order interaction are involved in different real-world situations. To test our approach, we then use these transition matrices as the basis of a link prediction algorithm. We investigate our algorithm’s performance on four temporal networks, comparing our approach against ten other link prediction methods. Our results show that higher-order interactions in both space and time play a crucial role in the evolution of networks as we find our method, along with two other methods based on non-local interactions, give the best overall performance. The results also confirm the concept that the higher-order interaction patterns, i.e., triplet dynamics, can help us understand and predict the evolution of different real-world systems.
... These methods generally preserve conservation properties and spectral representations of operators, provide a coordinate-free means of prescribing physics on manifolds, and allow handling of the non-trivial null-spaces required in electromagnetics. In topological data analysis, combinatorial Hodge theory has emerged as a tool for analyzing flows on graphs along with their spectral and homological properties, e.g., [46,37,45,8,11,44,58,62]. These techniques are supported by a graph calculus providing generalizations of gradient, curl, and divergence operators admitting interpretation as discrete exterior derivatives. ...
As traditional machine learning tools are increasingly applied to science and engineering applications, physics-informed methods have emerged as effective tools for endowing inferences with properties essential for physical realizability. While promising, these methods generally enforce physics weakly via penalization. To enforce physics strongly, we turn to the exterior calculus framework underpinning combinatorial Hodge theory and physics-compatible discretization of partial differential equations (PDEs). Historically, these two fields have remained largely distinct, as graphs are strictly topological objects lacking the metric information fundamental to PDE discretization. We present an approach where this missing metric information may be learned from data, using graphs as coarse-grained mesh surrogates that inherit desirable conservation and exact sequence structure from the combinatorial Hodge theory. The resulting data-driven exterior calculus (DDEC) may be used to extract structure-preserving surrogate models with mathematical guarantees of well-posedness. The approach admits a PDE-constrained optimization training strategy which guarantees machine-learned models enforce physics to machine precision, even for poorly trained models or small data regimes. We provide analysis of the method for a class of models designed to reproduce nonlinear perturbations of elliptic problems and provide examples of learning H(div)/H(curl) systems representative of subsurface flows and electromagnetics.
... Popular choices for modelling genuine group interactions include hypergraphs [14][15][16] and simplicial complexes. The latter has opened the doors to the use of algebraic topology in the analysis and study of complex systems [17][18][19][20][21][22][23]. With regard to dynamical processes on higher-order models, several works have attempted to apply social dynamics to either of these structures [22,24,25]. ...
The basic interaction unit of many dynamical systems involves more than two nodes. In such situations where networks are not an appropriate modelling framework, it has recently become increasingly popular to turn to higher-order models, including hypergraphs. In this paper, we explore the non-linear dynamics of consensus on hypergraphs, allowing for interactions within hyperedges of any cardinality. After discussing the different ways in which nonlinearities can be incorporated in the dynamical model, building on different sociological theories, we explore its mathematical properties and perform simulations to investigate them numerically. After focussing on synthetic hypergraphs, namely on block hypergraphs, we investigate the dynamics on real-world structures, and explore in detail the role of involvement and stubbornness on polarisation.
... [13], has exactly this goal. Interesting applications of algebraic topology tools have been proposed to control systems [14], statistical ranking from incomplete data [15], [16], distributed coverage control of sensor networks [17]- [19], wheeze detection [20]. One of the fundamental tools of TDA is the analysis of persistent homologies extracted from data [21], [22]. ...
The goal of this paper is to establish the fundamental tools to analyze signals defined over a topological space, i.e. a set of points along with a set of neighborhood relations. This setup does not require the definition of a metric and then it is especially useful to deal with signals defined over non-metric spaces. We focus on signals defined over simplicial complexes. Graph Signal Processing (GSP) represents a very simple case of Topological Signal Processing (TSP), referring to the situation where the signals are associated only with the vertices of a graph. We are interested in the most general case, where the signals are associated with vertices, edges and higher order complexes. After reviewing the basic principles of algebraic topology, we show how to build unitary bases to represent signals defined over sets of increasing order, giving rise to a spectral simplicial complex theory. Then we derive a sampling theory for signals of any order and emphasize the interplay between signals of different order. After having established the analysis tools, we propose a method to infer the topology of a simplicial complex from data. We conclude with applications to real edge signals to illustrate the benefits of the proposed methodologies.
... Then, concepts from algebraic topology, such as homology, have been used to detect holes in sensor networks [9,10]. Distributed algorithms to localize holes in sensor networks using related concepts have been addressed in [11] and in [12]. Recently such methods have been also used for filtering and position estimation in [13,14]. ...
In this paper, we cast the classic problem of achieving k-anonymity for a given database as a problem in algebraic topology. Using techniques from this field of mathematics, we propose a framework for k-anonymity that brings new insights and algorithms to anonymize a database. We begin by addressing the simpler case when the data lies in a metric space. This case is instrumental to introduce the main ideas and notation. Specifically, by mapping a database to the Euclidean space and by considering the distance between datapoints, we introduce a simplicial representation of the data and show how concepts from algebraic topology, such as the nerve complex and persistent homology, can be applied to efficiently obtain the entire spectrum of k-anonymity of the database for various values of k and levels of generalization. For this representation, we provide an analytic characterization of conditions under which a given representation of the dataset is k-anonymous. We introduce a weighted barcode diagram which, in this context, becomes a computational tool to tradeoff data anonymity with data loss expressed as level of generalization. Some simulations results are used to illustrate the main idea of the paper. We conclude the paper with a discussion on how to extend this method to address the general case of a mix of categorical and metric data.
... Distributed algorithms for the coverage of sensor networks have been developed using homology methods (Tahbaz-Salehi and Jadbabaie, 2010;Yan et al., 2011;Dłotko et al., 2012). A network hole detection, or rather 'fragility of simplicial complexes' based on eigenvalue/eigenvector of combinatorial Laplacian of simplicial complexes is presented in Muhammad and Egerstedt (2006). A distributed randomised scheduling algorithm which does not require accurate location information of the sensors is designed in Liu et al. (2006) to address sensing coverage. ...
Wireless sensor network (WSN) coverage completeness is an important quality of service (QoS). It is frequently assumed that events occurring in the sensor field can always be detected. However, this is not necessarily the case, particularly if there are holes in the sensor network coverage. This paper introduces a novel centralised method for detection and relative localisation of WSN coverage holes in coordinate-free networks. We identify sensor nodes that surround or bound coverage holes, called 'hole boundary-nodes', by processing information embedded in a communication graph, which is non-planar in general. We create a hole-equivalent planar graph that has the same number and position of holes as the original sensor network graph while preserving the embedding. Finally, we build a plane simplicial complex, called maximal simplicial complex, which contains the information regarding coverage holes. Simulation results are presented to illustrate the proposed method and evaluate its accuracy and run time.
... Several familiar concepts from standard graph theory, and of frequent use in application to social network theory, sociology, and biology have been generalized to simplicial sets. We give the following generalizations of the familiar concepts of degree and adjacency, as detailed in (Muhammad and Egerstedt, 2006). The upper degree of a k-simplex k i X X is the number of (k+1)-simplices in X of which k i X is a face. ...
A predominant benefit of social living is the ability to share knowledge that cannot be gained through the information an individual accumulates based on its personal experience alone. Traditional computational models have portrayed sharing knowledge through interactions among members of social groups via dyadic networks. Such models aim at understanding the percolation of information among individuals and groups to identify potential limitations to successful knowledge transfer. However, because many real-world interactions are not solely pairwise, i.e., several group members may obtain information from one another simultaneously, it is necessary to understand more than dyadic communication and learning processes to capture their full complexity. We detail a modeling framework based on the simplicial set, a concept from algebraic topology, which allows elegant encapsulation of multi-agent interactions. Such a model system allows us to analyze how individual information within groups accumulates as the group's collective set of knowledge, which may be different than the simple union of individually contained information. Furthermore, the simplicial modeling approach we propose allows us to investigate how information accumulates via sub-group interactions, offering insight into complex aspects of multi-way communication systems. The fundamental change in modeling strategy we offer here allows us to move from portraying knowledge as a "token", passed from signaler to receiver, to portraying knowledge as a set of accumulating building blocks from which novel ideas can emerge. We provide an explanation of relevant mathematical concepts in a way that promotes accessibility to a general audience
This paper investigates a higher-order dynamical network on simplicial complexes, and the evolution of topological signals residing on simplices (e.g. nodes, links, triangles, etc) is governed by coupled higher-dimensional linear oscillators. We uncover the stability conditions of cluster synchronization of both the higher-order network and its projected systems corresponding to solenoidal and irrotational component of the dynamics, and the synchronization phenomena of the higher-order network and its projected systems can co-exist (i.e., symbiotic synchronization). In the framework of simplicial complexes, this study indicates that there are essential differences between the collective behaviours of phase oscillators and higher-dimensional oscillators.
We consider the general model for dynamical systems defined on a simplicial complex. We describe the conjugacy classes of these systems and show how symmetries in a given simplicial complex manifest in the dynamics defined thereon, especially with regard to invariant subspaces in the dynamics.
The analysis of networks, aimed at suitably defined functionality, often focuses on partitions into subnetworks that capture desired features. Chief among the relevant concepts is a 2-partition; this underlies the classical Cheeger inequality and highlights a “constriction” (bottleneck) that limits accessibility between the respective parts of the network. In a similar spirit, we explore a notion of
global circulation
which necessitates a concept of a 3-partition that exposes this macroscopic feature of network flows. Graph circulation is often present in transportation networks as well as in certain biological networks. We introduce a notion of circulation for general graphs and then focus on planar graphs. For the case of the latter, we explain that a scalar potential characterizes circulation in complete analogy with the curl of planar vector fields and we present an algorithm for determining values of the potential and, hence, quantify circulation. We then discuss alternative notions of circulation, explain how these may depend on graph embedding, draw parallels between networks and Helmholtz–Hodge decomposition of vector fields, and conclude with a suggestive application of the framework in detecting abnormalities in cardiac circulatory physiology.
Higher-order networks have so far been considered primarily in the context of studying the structure of complex systems, i.e., the higher-order or multi-way relations connecting the constituent entities. More recently, a number of studies have considered dynamical processes that explicitly account for such higher-order dependencies, e.g., in the context of epidemic spreading processes or opinion formation. In this chapter, we focus on a closely related, but distinct third perspective: how can we use higher-order relationships to process signals and data supported on higher-order network structures. In particular, we survey how ideas from signal processing of data supported on regular domains, such as time series or images, can be extended to graphs and simplicial complexes. We discuss Fourier analysis, signal denoising, signal interpolation, and nonlinear processing through neural networks based on simplicial complexes. Key to our developments is the Hodge Laplacian matrix, a multi-relational operator that leverages the special structure of simplicial complexes and generalizes desirable properties of the Laplacian matrix in graph signal processing.
In this chapter, we derive and analyse models for consensus dynamics on hypergraphs. As we discuss, unless there are nonlinear node interaction functions, it is always possible to rewrite the system in terms of a new network of effective pairwise node interactions, regardless of the initially underlying multi-way interaction structure. We thus focus on dynamics based on a certain class of non-linear interaction functions, which can model different sociological phenomena such as peer pressure and stubbornness. Unlike for linear consensus dynamics on networks, we show how our nonlinear model dynamics can cause shifts away from the average system state. We examine how these shifts are influenced by the distribution of the initial states, the underlying hypergraph structure and different forms of non-linear scaling of the node interaction function.
As traditional machine learning tools are increasingly applied to science and engineering applications, physics-informed methods have emerged as effective tools for endowing inferences with properties essential for physical realizability. While promising, these methods generally enforce physics weakly via penalization. To enforce physics strongly, we turn to the exterior calculus framework underpinning combinatorial Hodge theory and physics-compatible discretization of partial differential equations (PDEs). Historically, these two fields have remained largely distinct, as graphs are strictly topological objects lacking the metric information fundamental to PDE discretization. We present an approach where this missing metric information may be learned from data, using graphs as coarse-grained mesh surrogates that inherit desirable conservation and exact sequence structure from the combinatorial Hodge theory. The resulting data-driven exterior calculus (DDEC) may be used to extract structure-preserving surrogate models with mathematical guarantees of well-posedness. The approach admits a PDE-constrained optimization training strategy which guarantees machine-learned models enforce physics to machine precision, even for poorly trained models or small data regimes. We provide analysis of the method for a class of models designed to reproduce nonlinear perturbations of elliptic problems and provide examples of learning H(div)/H(curl) systems representative of subsurface flows and electromagnetics.
We introduce a high-dimensional analogue of the Kirchhoff index which is also known as total effective resistance. This analogue, which we call the simplicial Kirchhoff index Kf(X), is defined to be the sum of the simplicial effective resistances of all (d+1)-subsets of the vertex set of a simplicial complex X of dimension d. This definition generalizes the Kirchoff index of a graph G defined as the sum of the effective resistance between all pairs of vertices of G. For a d-dimensional simplicial complex X with n vertices, we give formulas for the simplicial Kirchhoff index in terms of the pseudoinverse of the Laplacian LX in dimension d−1 and its eigenvalues:Kf(X)=n⋅trLX+=n⋅∑λ∈Λ+1λ, where LX+ is the pseudoinverse of LX, and Λ+ is the multi-set of non-zero eigenvalues of LX. Using this formula, we obtain an inequality for a high-dimensional analogue of algebraic connectivity and Kirchhoff index, and propose these quantities as measures of robustness of simplicial complexes. In addition, we derive its integral formula and relate this index to a simplicial dynamical system. We present an open problem for a combinatorial proof of our formula by relating the combinatorial interpretation of Rσ to rooted forests in higher dimensions.
We consider a nonlinear flow on simplicial complexes related to the simplicial Laplacian and show that it is a generalization of various consensus and synchronization models commonly studied on networks. In particular, our model allows us to formulate flows on simplices of any dimension so that it includes edge flows, triangle flows, etc. We show that the system can be represented as the gradient flow of an energy functional and use this to deduce the stability of various steady states of the model. Finally, we demonstrate that our model contains higher-dimensional analogs of structures seen in related network models.
Uncovering hidden relations in complex data sets is a key step to making sense of the data, which is a hot topic in our era of data deluge. Graph-based representations are examples of tools able to encode relations in a mathematical structure enabling the uncovering of patterns like clusters and paths. However, graphs only capture pairwise relations encoded in the presence of edges, but there are many forms of interaction that cannot be reduced to pairwise relations. To overcome the limitations of graph-based representations, it is necessary to incorporate multiway relations. In this article, we exploit tools from algebraic topology to handle multiway relations. Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study a topological space, that is, a set of points, along with a set of neighborhoods. More specifically, we illustrate topological signal processing (TSP), a framework encompassing a class of methods for analyzing signals defined over a topological space. Given its generality, TSP incorporates graph signal processing (GSP) as a particular case. After motivating the use of topological and geometrical methods for detecting patterns in the data, we present the signal processing tools based on algebraic topology and then illustrate their advantages with respect to graph-based methodology.
Network Science provides a universal formalism for modelling and studying complex systems based on pairwise interactions between agents. However, many real networks in the social, biological or computer sciences involve interactions among more than two agents, having thus an inherent structure of a simplicial complex. The relevance of an agent in a graph network is given in terms of its degree, and in a simplicial network there are already notions of adjacency and degree for simplices that, as far as we know, are not valid for comparing simplices in different dimensions. We propose new notions of higher-order degrees of adjacency for simplices in a simplicial complex, allowing any dimensional comparison among them and their faces. We introduce multi-parameter boundary and coboundary operators in an oriented simplicial complex and also a novel multi-combinatorial Laplacian is defined. As for the graph or combinatorial Laplacian, the multi-combinatorial Laplacian is shown to be an effective tool for calculating the higher-order degrees presented here. To illustrate the potential applications of these theoretical results, we perform a structural analysis of higher-order connectivity in simplicial-complex networks by studying the associated distributions with these simplicial degrees in 17 real-world datasets coming from different domains such as coauthor networks, cosponsoring Congress bills, contacts in schools, drug abuse warning networks, e-mail networks or publications and users in online forums. We find rich and diverse higher-order connectivity structures and observe that datasets of the same type reflect similar higher-order collaboration patterns. Furthermore, we show that if we use what we have called the maximal simplicial degree (which counts the distinct maximal communities in which our simplex and all its strict sub-communities are contained), then its degree distribution is, in general, surprisingly different from the classical node degree distribution.
Many real networks in social sciences, biological and biomedical sciences or computer science have an inherent structure of simplicial complexes reflecting many-body interactions. Therefore, to analyse topological and dynamical properties of simplicial complex networks centrality measures for simplices need to be proposed. Many of the classical complex networks centralities are based on the degree of a node, so in order to define degree centrality measures for simplices (which would characterise the relevance of a simplicial community in a simplicial network), a different definition of adjacency between simplices is required, since, contrarily to what happens in the vertex case (where there is only upper adjacency), simplices might also have other types of adjacency. The aim of these notes is threefold: first we will use the recently introduced notions of higher order simplicial degrees to propose new degree based centrality measures in simplicial complexes. These theoretical centrality measures, such as the simplicial degree centrality or the eigenvector centrality would allow not only to study the relevance of a simplicial community and the quality of its higher-order connections in a simplicial network, but also they might help to elucidate topological and dynamical properties of simplicial networks; sencond, we define notions of walks and distances in simplicial complexes in order to study connectivity of simplicial networks and to generalise, to the simplicial case, the well known closeness and betweenness centralities (needed for instance to study the relevance of a simplicial community in terms of its ability of transmitting information); third, we propose a new clustering coefficient for simplices in a simplicial network, different from the one knows so far and which generalises the standard graph clustering of a vertex. This measure should be essential to know the density of a simplicial network in terms of its simplicial communities.
The goal of this paper is to establish the fundamental tools to analyze signals defined over a topological space, i.e. a set of points along with a set of neighborhood relations. This setup does not require the definition of a metric and then it is especially useful to deal with signals defined over non-metric spaces. We focus on signals defined over simplicial complexes. Graph Signal Processing (GSP) represents a special case of Topological Signal Processing (TSP), referring to the situation where the signals are associated only with the vertices of a graph. Even though the theory can be applied to signals of any order, we focus on signals defined over the edges of a graph and show how building a simplicial complex of order two, i.e. including triangles, yields benefits in the analysis of edge signals. After reviewing the basic principles of algebraic topology, we derive a sampling theory for signals of any order and emphasize the interplay between signals of different order. Then we propose a method to infer the topology of a simplicial complex from data. We conclude with applications to real edge signals and to the analysis of discrete vector fields to illustrate the benefits of the proposed methodologies.
Spectral graph analysis based on Laplace operators has been ubiquitously applied in graph signal processing. Extending these tools to a generalization of graphs is helpful for several applications, such as network analysis, sensor network coverage problem and other fields where the relationships between two or more vertices should be modeled. Such mathematical theories already exist, but the associated algorithms are still in their infancy. In this article, we propose a new algorithm to compute harmonic forms, i.e. solutions of the Laplace equation, whose implementation is simple enough to be distributed among a network without central control center.
Present approaches to achieve k-coverage for Wireless Sensor Networks still rely on centralized techniques. In this paper, we devise a distributed method for this problem, namely Distributed VOronoi based Cooperation scheme (DVOC), where nodes cooperate in hole detection and recovery. In previous Voronoi based schemes, each node only monitors its own critical points. Such methods are inefficient for k-coverage because the critical points are far away from their generating nodes in k-order Voronoi diagram, causing high cost for transmission and computing. As a solution, DVOC enables nodes to monitor others' critical points around themselves by building local Voronoi diagrams (LVDs). Further, DVOC constrains the movement of every node to avoid generating new holes. If a node cannot reach its destination due to the constraint, its hole healing responsibility will fall to other cooperating nodes. The experimental results from the real world testbed demonstrate that DVOC outperforms the previous schemes.
In this chapter we review conventional vector calculus from the standpoint of a generalized exposition in terms of exterior calculus and the theory of forms. This generalization allows us to distill the important elements necessary to operate the basic machinery of conventional vector calculus. This basic machinery is then redefined in a discrete setting to produce appropriate definitions of the domain, boundary, functions, integrals, metric and derivative. These definitions are then employed to demonstrate how the structure of the discrete calculus behaves analogously to the conventional vector calculus in many different ways.
Clustering algorithms are used to find communities of nodes that all belong to the same group. This grouping process is also known as image segmentation in image processing. The clustering problem is also deeply connected to machine learning because a solution to the clustering problem may be used to propagate labels from observed data to unobserved data. In general network analysis, the identification of a grouping allows for the analysis of the nodes within each group as separate entities. In this chapter, we use the tools of discrete calculus to examine both the targeted clustering problem (i.e., finding a specific group) and the untargeted clustering problem (i.e., discovering all groups). We additionally show how to apply these clustering models to the clustering of higher-order cells, e.g., to cluster edges.
We consider the problem of deploying a swarm of mobile robots into an unknown environment for attaining complete sensor coverage of the environment. The robots have limited and noisy sensing capabilities and no metric or global information available to them. Using tools from algebraic topology, we formally describe the sensor coverage as a simplicial complex, deploy robots through the complex using bearing-based local controllers, and attain coverage while identifying and removing sensor redundancies. Despite the highly limited sensing capabilities and complete lack of global localization and metric information, we demonstrate that the proposed algorithm is complete, always terminates in a finite-sized environment, is guaranteed to attain complete coverage and is robust to sensor failures. The algorithm presented in this paper was demonstrated through simulation and proves to effectively cover and explore unknown indoor environments.
Recent studies have found that the modular structure of functional brain network is disrupted during the progress of Alzheimer's is the most basic topological disease. The modular structure of network invariant in determining the shape of network in the view of algebraic topology. In this study, we propose a new method to find another higher order topological invariant, hole, based on persistent homology. If a hole exists in the network, the information can be inefficiently delivered between regions. If we can localize the hole in the network, we can infer the reason of network inefficiency. We propose to detect the persistent hole using the spectrum of kappa-Laplacian, which is the generalized version of graph Laplacian. The method is applied to the metabolic network based on FDG-PET data of Alzheimer disease (AD), mild cognitive impairment (MCI) and normal control (NC) groups. The experiments show that the persistence of hole can be used as a biological marker of disease progression to AD. The localized hole may help understand the brain network abnormality in AD, revealing that the limbic-temporo-parietal association regions disturb direct connections between other regions.
This note analyzes the stability properties of a group of mobile agents that align their velocity vectors, and stabilize their inter-agent distances, using decentralized, nearest-neighbor interaction rules, exchanging information over networks that change arbitrarily (no dwell time between consecutive switches). These changes introduce discontinuities in the agent control laws. To accommodate for arbitrary switching in the topology of the network of agent interactions we employ nonsmooth analysis. The main result is that regardless of switching, convergence to a common velocity vector and stabilization of inter-agent distances is still guaranteed as long as the network remains connected at all times.
We show that the persistent homology of a filtered d-dimensional
simplicial complex is simply the standard homology of a
particular graded module over a polynomial ring.
Our analysis establishes the existence of a simple description of
persistent homology groups over arbitrary fields.
It also enables us to derive a natural
algorithm for computing persistent homology of spaces in
arbitrary dimension over any field.
This result generalizes and extends the previously known
algorithm that was restricted to subcomplexes of S3 and
Z2 coefficients.
Finally, our study implies the lack of a simple
classification over non-fields.
Instead, we give an algorithm for computing individual
persistent homology groups over an arbitrary principal ideal domain
in any dimension.
We consider distributed dynamic systems operating over state-dependent dynamic graphs or networks. Various aspects of the resulting abstract structure, having a mixture of combinatorial and control theoretic features, are then studied. Along the way, we explore an interplay between notions from the theory of graphs (elementary, extremal, and random), and system theory (controllability, invariance, etc.).
We present gradient landmark-based distributed routing (GLIDER), a novel naming/addressing scheme and associated routing algorithm, for a network of wireless communicating nodes. We assume that the nodes are fixed (though their geographic locations are not necessarily known), and that each node can communicate wirelessly with some of its geographic neighbors - a common scenario in sensor networks. We develop a protocol which in a preprocessing phase discovers the global topology of the sensor field and, as a byproduct, partitions the nodes into routable tiles - regions where the node placement is sufficiently dense and regular that local greedy methods can work well. Such global topology includes not just connectivity but also higher order topological features, such as the presence of holes. We address each node by the name of the tile containing it and a set of local coordinates derived from connectivity graph distances between the node and certain landmark nodes associated with its own and neighboring tiles. We use the tile adjacency graph for global route planning and the local coordinates for realizing actual inter- and intra-tile routes. We show that efficient load-balanced global routing can be implemented quite simply using such a scheme.
This note analyzes the stability properties of a group of mobile agents that align their velocity vectors, and stabilize their inter-agent distances, using decentralized, nearest-neighbor interaction rules, exchanging information over networks that change arbitrarily (no dwell time between consecutive switches). These changes introduce discontinuities in the agent control laws. To accommodate for arbitrary switching in the topology of the network of agent interactions we employ nonsmooth analysis. The main result is that regardless of switching, convergence to a common velocity vector and stabilization of inter-agent distances is still guaranteed as long as the network remains connected at all times
The feasibility problem is studied of achieving a specified formation among a group of autonomous unicycles by local distributed control. The directed graph defined by the information flow plays a key role. It is proved that formation stabilization to a point is feasible if and only if the sensor digraph has a globally reachable node. A similar result is given for formation stabilization to a line and to more general geometric arrangements.
Within a framework of noncommutative geometry, we develop an analogue of (pseudo) Riemannian geometry on finite and discrete sets. On a finite set, there is a counterpart of the continuum metric tensor with a simple geometric interpretation. The latter is based on a correspondence between first order differential calculi and digraphs. Arrows originating from a vertex span its (co)tangent space. If the metric is to measure length and angles at some point, it has to be taken as an element of the left-linear tensor product of the space of 1-forms with itself, and not as an element of the (non-local) tensor product over the algebra of functions. It turns out that linear connections can always be extended to this left tensor product, so that metric compatibility can be defined in the same way as in continuum Riemannian geometry. In particular, in the case of the universal differential calculus on a finite set, the Euclidean geometry of polyhedra is recovered from conditions of metric compatibility and vanishing torsion. In our rather general framework (which also comprises structures which are far away from continuum differential geometry), there is in general nothing like a Ricci tensor or a curvature scalar. Because of the non-locality of tensor products (over the algebra of functions) of forms, corresponding components (with respect to some module basis) turn out to be rather non-local objects. But one can make use of the parallel transport associated with a connection to `localize' such objects and in certain cases there is a distinguished way to achieve this. This leads to covariant components of the curvature tensor which then allow a contraction to a Ricci tensor. In the case of a differential calculus associated with a hypercubic lattice we propose a new discrete analogue of the (vacuum) Einstein equations. Comment: 34 pages, 1 figure (eps), LaTeX, amssymb, epsfig
In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disagreement function that is a common Lyapunov function for the disagreement dynamics of a directed network with switching topology. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results.
We consider distributed dynamic systems operating over a graph or a network. The geometry of the network is assumed to be a function of the underling system's states-giving it a unique dynamic character. Certain aspects of the resulting abstract structure, having a mixture of combinatorial and system theoretic features, are then studied. In this venue, we will explore an interplay between notions from extremal graph theory and system theory by considering a controllability framework for such state-dependent dynamic graphs.
This paper proves a discrete analogue of the Poincar´e lemma in the context of a discrete exterior calculus based on simplicial cochains. The proof requires the construction of a generalized cone operator, p : Ck(K) -> Ck+1(K), as the geometric cone of a simplex cannot, in general, be interpreted as a chain in the simplicial complex. The corresponding cocone operator H : Ck(K) -> Ck−1(K) can be shown to be a homotopy operator, and this yields the discrete Poincar´e lemma.
The generalized cone operator is a combinatorial operator that can be constructed for any simplicial complex that can be grown by a process of local augmentation. In particular, regular triangulations and tetrahedralizations of R2 and R3 are presented, for which the discrete Poincar´e lemma is globally valid.
In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disa
We consider coverage problems in robot sensor networks with minimal sensing capabilities. In particular, we demonstrate that a "blind" swarm of robots with no localization and only a weak form of distance estimation can rigorously determine coverage in a bounded planar domain of unknown size and shape. The methods we introduce come from algebraic topology.
Elements of Algebraic Topology provides the most concrete approach to the subject. With coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, and applications to classical theorems of point-set topology, this book is perfect for comunicating complex topics and the fun nature of algebraic topology for beginners.
* Established textbook
* Continues to lead its readers to some of the hottest topics of contemporary mathematical research
This established reference work continues to lead its readers to some of the hottest topics of contemporary mathematical research.This new edition introduces and explains the ideas of the parabolic methods that have recently found such a spectacular success in the work of Perelman at the examples of closed geodesics and harmonic forms. It also discusses further examples of geometric variational problems from quantum field theory, another source of profound new ideas and methods in geometry.
From the reviews:
"This book provides a very readable introduction to Riemannian geometry and geometric analysis. The author focuses on using analytic methods in the study of some fundamental theorems in Riemannian geometry, e.g., the Hodge theorem, the Rauch comparison theorem, the Lyusternik and Fet theorem and the existence of harmonic mappings. With the vast development of the mathematical subject of geometric analysis, the present textbook is most welcome. [..] The book is made more interesting by the perspectives in various sections." Mathematical Reviews
We consider coverage problems in sensor networks of stationary nodes with minimal geometric data. In particular, there are no coordinates and no localization of nodes. We introduce a new technique for detecting holes in coverage by means of homology, an algebraic topological invariant. The impetus for these techniques is a completion of network communication graphs to two types of simplicial complexes: the nerve complex and the Rips complex. The former gives information about coverage intersection of individual sensor nodes, and is very difficult to compute. The latter captures connectivity in terms of inter-node communication: it is easy to compute but does not in itself yield coverage data. We obtain coverage data by using persistence of homology classes for Rips complexes. These homological invariants are computable: we provide simulation results.
This paper demonstrates how discrete exterior
calculus tools may be useful in computer vision and graphics.
A variational approach provides a link with mechanics.
Vicsek et al. proposed (1995) a simple but compelling discrete-time model of n autonomous agents {i.e., points or particles} all moving in the plane with the same speed but with different headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its "neighbors". In their paper, Vicsek et al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models. The Vicsek model proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function.
In a recent Physical Review Letters article, Vicsek et al. propose a simple but compelling discrete-time model of n autonomous agents (i.e., points or particles) all moving in the plane with the same speed but with different headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its "neighbors." In their paper, Vicsek et al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models. The Vicsek model proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function.
A striking result of Bouc gives the decomposition of the representation of the symmetric group on the homology of the matching complex into irreducibles that are self-conjugate. We show how the combinatorial Laplacian can be used to give an elegant proof of this result. We also show that the spectrum of the Laplacian is integral. 1
. We show that the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra. We give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences, generalizing a theorem of Merris for graphs. We also conjecture a majorization inequality for the spectra of these Laplace operators in an arbitrary simplicial complex, with equality achieved if and only if the complex is shifted. This generalizes a conjecture of Grone and Merris for graphs. 1. Introduction This paper is about spectra of combinatorial Laplace operators associated to simplicial complexes. We begin with some history. The theory of graph Laplacians goes back to Kirchhoff [19] in his study of electrical networks, and his celebrated matrix-tree theorem (see e.g. [17]). The spectra of graph Laplacians gained attention in the early 1970's after work of Fiedler [10] (and also independent work of Anderson and Morley, and of Kelmans -- see [2...
A variety of questions in combinatorics lead one to the task of analyzing the topology of a simplicial complex, or a more general cell complex. However, there are few general techniques to aid in this investigation. On the other hand, the subjects of differential topology and geometry are devoted to precisely this sort of problem, except that the topological spaces in question are smooth manifolds. In this paper we show how two standard techniques from the study of smooth manifolds, Morse theory and Bochner’s method, can be adapted to aid in the investigation of combinatorial spaces.
- M Wachs
- X Dong
M.Wachs and X. Dong, " Combinatorial Laplacian of the Matching Complex, " Electronic Journal of Combinatorics, Vol. 9, 2002.
- P Antsaklis
- A Michel
P. Antsaklis and A. Michel, Linear Systems, Birkhuser, 1997.