On the phase transition in random simplicial complexes

Linial, Nathan; Peled, Yuval

doi:10.4007/annals.2016.184.3.3

Cited by 81 publications

(170 citation statements)

References 29 publications

(69 reference statements)

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“…A detailed analysis of the Linial–Meshulam complex

{scriptK}^{(d)} (t)

at time

t = c / n (c \geq 0)

has recently been reported in . By applying their results, we formally show that the limit

I_{d - 1} : = \lim_{n \to \infty} \frac{1}{n^{d - 1}} E [L_{d - 1}]

can be expressed by an integral form which recovers

I_{0} = ζ (3) f o r d = 1

.…”

Section: Discussionmentioning

confidence: 72%

“…A detailed analysis of the Linial-Meshulam complex K (d) (t) at time t = c/n (c ≥ 0) has recently been reported in [21]. By applying their results, we formally show that the limit…”

Section: Discussionmentioning

confidence: 75%

“…As was studied in the Erdős–Rényi graph, Poisson trees play an important role to characterize the Linial–Meshulam complex at

t = c / n

. By using the spectral measure of the upper

(d - 1)

‐dimensional Laplacian

\partial_{d} \partial_{d}^{T}

obtained from the boundary operator, Linial and Peled basically show the following: let

t_{d}^{*}

be the unique root in

(0, 1)

of the following equation

(d + 1) (1 - t) + (1 + d t) \log t = 0

and set

c_{d}^{*} = ψ_{d} (t_{d}^{*})

ψ_{d} (t) = \frac{- \log t}{{(1 - t)}^{d}}, t \in (0, 1) .

…”

Section: Discussionmentioning

confidence: 99%

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Minimum spanning acycle and lifetime of persistent homology in the Linial-Meshulam process

Hiraoka

Shirai

2017

Random Struct. Alg.

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Abstract. This paper studies a higher dimensional generalization of Frieze's ζ(3)-limit theorem in the Erdös-Rényi graph process. Frieze's theorem states that the expected weight of the minimum spanning tree converges to ζ(3) as the number of vertices goes to infinity. In this paper, we study the dLinial-Meshulam process as a model for random simplicial complexes, where d = 1 corresponds to the Erdös-Rényi graph process. First, we define spanning acycles as a higher dimensional analogue of spanning trees, and connect its minimum weight to persistent homology. Then, our main result shows that the expected weight of the minimum spanning acycle behaves in O(n d−1 ).

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“…A detailed analysis of the Linial–Meshulam complex

{scriptK}^{(d)} (t)

at time

t = c / n (c \geq 0)

has recently been reported in . By applying their results, we formally show that the limit

I_{d - 1} : = \lim_{n \to \infty} \frac{1}{n^{d - 1}} E [L_{d - 1}]

can be expressed by an integral form which recovers

I_{0} = ζ (3) f o r d = 1

.…”

Section: Discussionmentioning

confidence: 72%

“…A detailed analysis of the Linial-Meshulam complex K (d) (t) at time t = c/n (c ≥ 0) has recently been reported in [21]. By applying their results, we formally show that the limit…”

Section: Discussionmentioning

confidence: 75%

“…As was studied in the Erdős–Rényi graph, Poisson trees play an important role to characterize the Linial–Meshulam complex at

t = c / n

. By using the spectral measure of the upper

(d - 1)

‐dimensional Laplacian

\partial_{d} \partial_{d}^{T}

obtained from the boundary operator, Linial and Peled basically show the following: let

t_{d}^{*}

be the unique root in

(0, 1)

of the following equation

(d + 1) (1 - t) + (1 + d t) \log t = 0

and set

c_{d}^{*} = ψ_{d} (t_{d}^{*})

ψ_{d} (t) = \frac{- \log t}{{(1 - t)}^{d}}, t \in (0, 1) .

…”

Section: Discussionmentioning

confidence: 99%

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Minimum spanning acycle and lifetime of persistent homology in the Linial-Meshulam process

Hiraoka

Shirai

2017

Random Struct. Alg.

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“…* f.nobregasantos@amsterdamumc.nl On the one hand, research in stochastic topology, which started with Erdos and Reyni in [2], investigated the problem of tracking the emergence of a giant component in a random graph for a critical probability threshold. The above idea was rigorously extended by Kahle [3] and Linial [4] to simplicial complexes using methods of algebraic topology. In the language of algebraic topology, the generalization of the giant component transition constitutes a major change in the distribution of ndimensional holes, the so-called Betti numbers of a simplicial complex [3,4], which is a generalization of a graph.…”

Section: Introductionmentioning

confidence: 99%

The Euler Characteristic and Topological Phase Transitions in Complex Systems

Amorim

Moreira

Santos

2019

Preprint

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In this work, we use methods and concepts of applied algebraic topology to comprehensively explore topological phase transitions in complex systems. Topological phase transitions are characterized by the zeros of the Euler characteristic (EC) or by singularities of the Euler entropy and also indicate signal changes in the mean node curvature of networks. Here, we provide strong evidence that the zeros of the Euler characteristic can be interpreted as a complex network's intrinsic fingerprint. We theoretically and empirically illustrate this across different biological networks: We first target our investigation to protein-protein interaction networks (PPIN). To do so, we used methods of topological data analysis to compute the Euler characteristic analytically, and the Betti numbers numerically as a function of the attachment probability for two variants of the Duplication Divergence model, namely the totally asymmetric model and the heterodimerization model. We contrast our theoretical results with experimental data freely available for gene co-expression networks (GCN) of Saccharomyces cerevisiae, also known as baker's yeast, as well as of the nematode Caenorhabditis elegans. Supporting our theoretical expectations, we are able to detect topological phase transitions in both networks obtained according to different similarity measures. Later, we theoretically illustrate the emergence of topological phase transitions in three classical network models, namely the Watts-Strogratz model, the Random Geometric Graph, and the Barabasi-Albert model. Given the universality and wide use of those models across disciplines, our results indicate that topological phase transitions may permeate across a wide range of theoretical and empirical networks. Hereby, our paper reinforces the idea of using topological phase transitions to advance the understanding of complex systems more generally.

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“…For d = 1, the Linial‐Meshulam model reduces to the Erdős‐Rényi model G ( n, p ). Following its introduction in , the Linial‐Meshulam has been extensively studied in . The notion of adjacency matrix has a natural extension to simplicial complexes, whereby the adjacency operator of a complex X is a self‐adjoint operator that encodes the information whether two

(d - 1)

‐cells belong to a common d ‐cell or not.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue confinement and spectral gap for random simplicial complexes

Knowles

Rosenthal

2017

Random Struct Algorithms

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We consider the adjacency operator of the Linial-Meshulam model for random simplicial complexes on n vertices, where each d-cell is added independently with probability p to the complete (d − 1)-skeleton. Under the assumption np(1 − p) log 4 n, we prove that the spectral gap between the n−1 d smallest eigenvalues and the remaining (1)) with high probability. This estimate follows from a more general result on eigenvalue confinement. In addition, we prove that the global distribution of the eigenvalues is asymptotically given by the semicircle law. The main ingredient of the proof is a Füredi-Komlós-type argument for random simplicial complexes, which may be regarded as sparse random matrix models with dependent entries.

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On the phase transition in random simplicial complexes

Cited by 81 publications

References 29 publications

Minimum spanning acycle and lifetime of persistent homology in the Linial-Meshulam process

Minimum spanning acycle and lifetime of persistent homology in the Linial-Meshulam process

The Euler Characteristic and Topological Phase Transitions in Complex Systems

Eigenvalue confinement and spectral gap for random simplicial complexes

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