Random Simplicial Complexes: Models and Phenomena

Bobrowski, Omer; Krioukov, Dmitri

doi:10.48550/arxiv.2105.12914

Cited by 3 publications

(8 citation statements)

References 105 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For instance, one can find multiple applications of algebraic topology in neuroscience, e.g., in psychedelics [53], neuronal dynamics [54], brain artery trees [55], epilepsy [56], and schizophrenia [57], to name a few. Yet, numerous significant concepts from algebraic topology and discrete geometry, especially stochastic topology [18], remain to be explored in neuroscience.…”

Section: Tpts In Functional Brain Networkmentioning

confidence: 99%

“…Research on phase transitions in the context of stochastic topology started with the work of Erdős and Rényi [15], which investigates the problem of tracking the emergence of a giant component in a random graph for a critical probability threshold. Using methods from algebraic topology, one can generalize the basic concept of graphs to simplicial complexes, i.e., a set of points, edges, triangles, tetrahedrons and their n-dimensional analogues, which was later used by Kahle [16] and Linial and Peled [17], among others [18], to rigorously extend the giant component transition to a simplicial complex. In this new language, the generalization of the giant component transition constitutes a major change in the distribution of k-dimensional holes, i.e., the emergence of non-trivial homology groups in a simplicial complex, which are characterized by their k-Betti numbers, β k .…”

Section: Introductionmentioning

confidence: 99%

“…This intuitively led to the introduction of TPTs in simplicial complexes as the loci of the zeros of the EC [27], where it was observed that the zeros of the EC also emerge in the vicinity of the generalization of the giant component transition [17]. In particular, it was observed that the zeros of the EC are also associated with the emergence of giant cycles in a simplicial complex [18,28,29]. Those transitions can be seen as one possible generalization of percolation transitions to higher-dimensional objects and were observed for the random clique complex of the Erdős-Rényi graph, as well as in functional brain networks [27].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Euler characteristic and topological phase transitions in complex systems

Amorim¹,

Moreira²,

Santos³

2022

J. Phys. Complex.

View full text Add to dashboard Cite

In this work, we use methods and concepts of applied algebraic topology to comprehensively explore the recent idea of topological phase transitions (TPT) in complex systems. TPTs are characterized by the emergence of nontrivial homology groups as a function of a threshold parameter. Under certain conditions, one can identify TPT's via the zeros of the Euler characteristic or by singularities of the Euler entropy. Recent works provide strong evidence that TPTs can be interpreted as a complex network's intrinsic fingerprint. This work illustrates this possibility by investigating some classic network and empirical protein interaction networks under a topological perspective. We first investigate TPT in protein-protein interaction networks (PPIN) using methods of topological data analysis for two variants of the Duplication-Divergence model, namely, the totally asymmetric model and the heterodimerization model. We compare our theoretical and computational results to 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 can detect topological phase transitions in both networks obtained according to different similarity measures. Later, we perform numerical simulations of TPTs in four classical network models: the Erdos-Renyi, the Watts-Strogatz model, the Random Geometric Graph, and the Barabasi-Albert. Finally, we discuss some perspectives and insights on the topic. Given the universality and wide use of those models across disciplines, our work indicates that TPT permeates a wide range of theoretical and empirical networks.

show abstract

Section: Tpts In Functional Brain Networkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Euler characteristic and topological phase transitions in complex systems

Amorim¹,

Moreira²,

Santos³

2022

J. Phys. Complex.

View full text Add to dashboard Cite

show abstract

“…This random variable, which arises very naturally in stochastic topology [7,6], has been poorly studied from a distributional approximation perspective. To the best of our knowledge, only the expected value has been calculated [4, Section 8].…”

Section: This Papermentioning

confidence: 99%

Multivariate Central Limit Theorems for Random Clique Complexes

Temčinas¹,

Nanda²,

Reinert³

2021

Preprint

View full text Add to dashboard Cite

Acyclic partial matchings on simplicial complexes play an important role in topological data analysis by facilitating efficient computation of (persistent) homology groups. Here we describe probabilistic properties of critical simplex counts for such matchings on clique complexes of Bernoulli random graphs. In order to accomplish this goal, we generalise the notion of a dissociated sum to a multivariate setting and prove an abstract multivariate central limit theorem using Stein's method. As a consequence of this general result, we are able to extract central limit theorems not only for critical simplex counts, but also for generalised U -statistics (and hence for clique counts in Bernoulli random graphs) as well as simplex counts in the link of a fixed simplex in a random clique complex.

show abstract

“…Different models of higher-order networks [27] have been developed so far. Detailed analysis of models of growing simplicial complexes [28][29][30] is presented, built upon the concept of 'network geometry with flavor (NGF)' [31,32].…”

Section: Introductionmentioning

confidence: 99%

Dynamics on higher-order networks: A review

Majhi,

Perc,

Ghosh

2022

Preprint

View full text Add to dashboard Cite

Network science has evolved into an indispensable platform for studying complex systems. But recent research has identified limits of classical networks, where links connect pairs of nodes, to comprehensively describe group interactions. Higher-order networks, where a link can connect more than two nodes, have therefore emerged as a new frontier in network science. Since group interactions are common in social, biological, and technological systems, higher-order networks have recently led to important new discoveries across many fields of research. We here review these works, focusing in particular on the novel aspects of the dynamics that emerges on higher-order networks. We cover a variety of dynamical processes that have thus far been studied, including different synchronization phenomena, contagion processes, the evolution of cooperation, and consensus formation. We also outline open challenges and promising directions for future research.

show abstract

Random Simplicial Complexes: Models and Phenomena

Cited by 3 publications

References 105 publications

The Euler characteristic and topological phase transitions in complex systems

The Euler characteristic and topological phase transitions in complex systems

Multivariate Central Limit Theorems for Random Clique Complexes

Dynamics on higher-order networks: A review

Contact Info

Product

Resources

About