An algebra for patterns on a complex, II

Atkin, R.H.

doi:10.1016/s0020-7373(76)80015-6

Cited by 45 publications

(39 citation statements)

References 4 publications

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“…This theory is based on ideas that go back to the work of Atkin [2], [3] indeed, the letter A is in honour of Atkin -and that were already re-explored in [12]. This theory is based on a notion of continuous path that makes sense for all locally finite metric spaces 1 .…”

Section: Introduction Main Results and Motivationsmentioning

confidence: 99%

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An Algebra for Patterns on a Complex, I

Atkin

1974

International Journal of Man-Machine Studies

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We propose a notion of continuous path for locally finite metric spaces, taking inspiration from the recent development of A-theory for locally finite connected graphs. We use this notion of continuity to derive an analogue in Z 2 of the Jordan curve theorem and to extend to a quite large class of locally finite metric spaces (containing all finite metric spaces) an inequality for the ℓ p -distortion of a metric space that has been recently proved by Pierre-Nicolas Jolissaint and Alain Valette for finite connected graphs.2010 MSC: Primary 52A01; Secondary 46L36.

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Section: Introduction Main Results and Motivationsmentioning

confidence: 99%

“…Our definition of simplicity is exactly the one that makes this procedure working in our discrete world of Z 2 . Indeed, if now 3 Recall that we are working inside a square and therefore adjacent vertices are exactly the extremal point of a side of the square and opposite vertices are the extremal points of a diagonal of the square.…”

Section: Now Suppose Thatδ(mentioning

confidence: 99%

An Algebra for Patterns on a Complex, I

Atkin

1974

International Journal of Man-Machine Studies

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“…The appearance of higher-order structures, i.e., simplicial complexes, in networks is suitably studied by Q-analysis [41][42][43] based on the algebraic topology of graphs [44][45][46]. In the clique complex method [47,48] that we use here, the elementary geometrical descriptors of the network structure are identified as cliques of different orders q = 0, 1, 2 · · · q max , i.e., nodes, links, triangles, tetrahedra and higher-order cliques up to the largest size q max + 1 that occurs in the network.…”

Section: Structure Of Simplicial Complexes In Bbrain-to-brain Coordinmentioning

confidence: 99%

“…Moreover, the method identifies the nodes that make a particular clique, which allows determining the ways that different cliques interconnect with each other via shared faces-cliques of the lower order, to form a simplicial complex. Consequently, based on Q-analysis [41][42][43], these combinatorial topologies are adequately quantified by the structure vectors of the graph, specifically:…”

Section: Structure Of Simplicial Complexes In Bbrain-to-brain Coordinmentioning

confidence: 99%

Origin of Hyperbolicity in Brain-to-Brain Coordination Networks

Tadić¹,

Andjelković²,

Šuvakov³

2018

Front. Phys.

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Hyperbolicity or negative curvature of complex networks is the intrinsic geometric proximity of nodes in the graph metric space, which implies an improved network function. Here, we investigate hidden combinatorial geometries in brain-to-brain coordination networks arising through social communications. The networks originate from correlations among EEG signals previously recorded during spoken communications comprising of 14 individuals with 24 speaker-listener pairs. We find that the corresponding networks are δ-hyperbolic with δ max = 1 and the graph diameter D = 3 in each brain. While the emergent hyperbolicity in the two-brain networks varies satisfying δ max /D/2 ≤ 1 and can be attributed to the topology of the subgraph formed around the cross-brains linking channels. We identify these subgraphs in each studied two-brain network and decompose their structure into simple geometric descriptors (triangles, tetrahedra and cliques of higher orders) that contribute to hyperbolicity. Considering topologies that exceed two separate brain networks as a measure of coordination synergy between the brains, we identify different neural correlation patterns ranging from weak coordination to super-brain structure. These topology features are in qualitative agreement with the listener's self-reported ratings of own experience and quality of the speaker, suggesting that studies of the cross-brain connector networks can reveal new insight into the neural mechanisms underlying human social behavior.

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“…The first is the dimension of the complex. The next one is the so called an f -vector (also known as the second structure vector ) [2], [3], [4], [5], which is an integer vector with dim(K) + 1 components, the i-th one being equal to the number of i-dimensional simplices in K. An invariant is also a Q-vector (f irst structure vector), an integer vector of the same length as the f -vector, whose i-th component is equal to the number of i-connectivity classes. The structure vector, illustrated in Fig.…”

Section: Invariants Of Simplicial Complexesmentioning

confidence: 99%

Simplicial Complexes of Networks and Their Statistical Properties

Maletić

Rajković

Vasiljević

2008

Computational Science – ICCS 2008

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Abstract. Topological, algebraic and combinatorial properties of simplicial complexes which are constructed from networks (graphs) are examined from the statistical point of view. We show that basic statistical features of scale free networks are preserved by topological invariants of simplicial complexes and similarly statistical properties pertaining to topological invariants of other types of networks are preserved as well. Implications and advantages of such an approach to various research areas involving network concepts are discussed.

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An algebra for patterns on a complex, II

Cited by 45 publications

References 4 publications

An Algebra for Patterns on a Complex, I

An Algebra for Patterns on a Complex, I

Origin of Hyperbolicity in Brain-to-Brain Coordination Networks

Simplicial Complexes of Networks and Their Statistical Properties

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