Exact and asymptotic results on coarse Ricci curvature of graphs

Bhattacharya, Bhaswar B.; Mukherjee, Sumit

doi:10.1016/j.disc.2014.08.012

Cited by 43 publications

(68 citation statements)

References 30 publications

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“…Cho and Paeng [41] slightly generalise the result on trees to graphs of girth at least 6, where the two expressions are in fact equivalent due to the locality property discussed above (graphs of girth at least 6 are locally tree-like). These results were significantly extended by Bhattacharya and Mukherjee in [42] to graphs of girth at least 5 and to bipartite graphs. Loisel and Romon [43] demonstrate that the Ollivier curvature depends only on the degree of the vertices for edges in a surface triangulation satisfying certain genericity assumptions and consequently calculate the Ollivier curvature for many polyhedral surfaces.…”

Section: The Ollivier Curvature In Graphsmentioning

confidence: 64%

Self-assembly of geometric space from random graphs

Kelly

Trugenberger

Biancalana

2019

Class. Quantum Grav.

117

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We present a Euclidean quantum gravity model in which random graphs dynamically self-assemble into discrete manifold structures. Concretely, we consider a statistical model driven by a discretisation of the Euclidean Einstein-Hilbert action; contrary to previous approaches based on simplicial complexes and Regge calculus our discretisation is based on the Ollivier curvature, a coarse analogue of the manifold Ricci curvature defined for generic graphs. The Ollivier curvature is generally difficult to evaluate due to its definition in terms of optimal transport theory, but we present a new exact expression for the Ollivier curvature in a wide class of relevant graphs purely in terms of the numbers of short cycles at an edge. This result should be of independent intrinsic interest to network theorists. Action minimising configurations prove to be cubic complexes up to defects; there are indications that such defects are dynamically suppressed in the macroscopic limit. Closer examination of a defect free model shows that certain classical configurations have a geometric interpretation and discretely approximate vacuum solutions to the Euclidean Einstein-Hilbert action. Working in a configuration space where the geometric configurations are stable vacua of the theory, we obtain direct numerical evidence for the existence of a continuous phase transition; this makes the model a UV completion of Euclidean Einstein gravity. Notably, this phase transition implies an area-law for the entropy of emerging geometric space. Certain vacua of the theory can be interpreted as baby universes; we find that these configurations appear as stable vacua in a mean field approximation of our model, but are excluded dynamically whenever the action is exact indicating the dynamical stability of geometric space. The model is intended as a setting for subsequent studies of emergent time mechanisms.

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Section: The Ollivier Curvature In Graphsmentioning

confidence: 64%

Self-assembly of geometric space from random graphs

Kelly

Trugenberger

Biancalana

2019

Class. Quantum Grav.

117

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“…Because of their random character, the Regge formulation of curvature is no more applicable, a purely combinatorial version of Ricci curvature is needed. Recently, exactly such a combinatorial Ricci curvature has been proposed by Ollivier [10,11] and further elaborated on in [12][13][14][15].…”

Section: Jhep09(2017)045mentioning

confidence: 99%

“…Fortunately, it becomes much simpler for bipartite graphs [15], which have no odd cycles. Since the Ollivier curvature of an edge depends only on the triangles, squares and pentagrams supported on that edge (a discrete form of locality) [12][13][14] and there are no triangles and pentagrams on graphs in the configuration space, one can use for all practical…”

Section: Jhep09(2017)045mentioning

confidence: 99%

“…one of the distinguishing features of random graphs vs. random geometric graphs [8,9]. As a driver of the quantum phase transition I will consider the combinatorial Ollivier-Ricci curvature [10][11][12][13][14][15], which becomes the standard Ricci curvature scalar in the ordered phase. Note that this program is totally different from previous approaches to quantum gravity based on graph structures [16,17]: there is no need of auxiliary group variables and the action is a purely combinatorial version of the Einstein-Hilbert action.…”

Section: Jhep09(2017)045mentioning

confidence: 99%

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Combinatorial quantum gravity: geometry from random bits

Trugenberger

2017

J. High Energ. Phys.

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I propose a quantum gravity model in which geometric space emerges from random bits in a quantum phase transition driven by the combinatorial Ollivier-Ricci curvature and corresponding to the condensation of short cycles in random graphs. This quantum critical point defines quantum gravity non-perturbatively. In the ordered geometric phase at large distances the action reduces to the standard Einstein-Hilbert term.

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“…This has two immediate consequences, there are no triangles and there is a perfect matching between the unit spheres of neighboring vertices. These properties imply that both recently developed concepts of combinatorial Ricci curvature for graphs, the Ollivier curvature [15][16][17] and the Knill curvature [18] vanish and so does then also the Ricci curvature obtained in the continuum limit. These "large world" ground-state graphs are thus Ricci flat.…”

Section: Introductionmentioning

confidence: 95%

Random holographic “large worlds” with emergent dimensions

Trugenberger¹

2016

Phys. Rev. E

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I propose a random network model governed by a Gaussian weight corresponding to Ising link antiferromagnetism as a model for emergent quantum space-time. In this model, discrete space is fundamental, not a regularization; its spectral dimension d_{s} is not a model input but is, rather, completely determined by the antiferromagnetic coupling constant. Perturbative terms suppressing triangles and favoring squares lead to locally Euclidean ground states that are Ricci flat "large worlds" with power-law extension. I then consider the quenched graphs of lowest energy for d_{s}=2 and d_{s}=3, and I show how quenching leads to the spontaneous emergence of embedding spaces of Hausdorff dimension d_{H}=4 and d_{H}=5, respectively. One of the additional, spontaneous dimensions can be interpreted as time, causality being an emergent property that arises in the large N limit (with N the number of vertices). For d_{s}=2, the quenched graphs constitute a discrete version of a 5D-space-filling surface with a number of fundamental degrees of freedom scaling like N^{2/5}, a graph version of the holographic principle. These holographic degrees of freedom can be identified with the squares of the quenched graphs, which, being triangle-free, are the fundamental area (or loop) quanta.

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Exact and asymptotic results on coarse Ricci curvature of graphs

Cited by 43 publications

References 30 publications

Self-assembly of geometric space from random graphs

Self-assembly of geometric space from random graphs

Combinatorial quantum gravity: geometry from random bits

Random holographic “large worlds” with emergent dimensions

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