On small world non-Sunada twins and cellular Voronoi diagrams

Special infinite families of regular graphs of unbounded degree and of bounded diameter (small world graphs) are considered. Two families of small world graphs Gi and Hi form a family of non-Sunada twins if Gi and Hi are isospectral of bounded diameter but groups Aut(Gi) and Aut(Hi) are nonisomorphic. We say that a family of non-Sunada twins is unbalanced if each Gi is edge-transitive but each Hi is edge-intransitive. If all Gi and Hi are edge-transitive we have a balanced family of small world non-Sunada twins. We say that a family of non-Sunada twins is strongly unbalanced if each Gi is edge-transitive but each Hi is edge-intransitive. We use term edge disbalanced for the family of non-Sunada twins such that all graphs Gi and Hi are edge-intransitive. We present explicit constructions of the above defined families. Two new families of distance-regular—but not distance-transitive—graphs will be introduced.

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Efficient and Robust WiFi Indoor Positioning Using Hierarchical Navigable Small World Graphs

2018 IEEE 17th International Symposium on Network Computing and Applications (NCA) ◽

10.1109/nca.2018.8548076 ◽

2018 ◽

Cited By ~ 1

Author(s):

Max Willian Soares Lima ◽

Horacio A. B. Fernandes de Oliveira ◽

Eulanda Miranda dos Santos ◽

Edleno Silva de Moura ◽

Rafael Kohler Costa ◽

...

Keyword(s):

Small World ◽

Indoor Positioning ◽

Small World Graphs

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On Edge-Transitive Graphs of Square-Free Order

The Electronic Journal of Combinatorics ◽

10.37236/4573 ◽

2015 ◽

Vol 22 (3) ◽

Cited By ~ 3

Author(s):

Cai Heng Li ◽

Zai Ping Lu ◽

Gai Xia Wang

Keyword(s):

Automorphism Groups ◽

Edge Transitive ◽

Transitive Graphs ◽

Free Order

We study the class of edge-transitive graphs of square-free order and valency at most $k$. It is shown that, except for a few special families of graphs, only finitely many members in this class are basic (namely, not a normal multicover of another member). Using this result, we determine the automorphism groups of locally primitive arc-transitive graphs with square-free order.

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Dynamical properties of profinite actions

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385711000654 ◽

2011 ◽

Vol 32 (6) ◽

pp. 1805-1835 ◽

Cited By ~ 13

Author(s):

MIKLÓS ABÉRT ◽

GÁBOR ELEK

Keyword(s):

Finite Index ◽

Linear Groups ◽

Regular Graphs ◽

Weak Containment ◽

Residually Finite Groups ◽

The Family ◽

Property T ◽

Free Actions ◽

Theoretical Consequence ◽

Lattice Of Subgroups

AbstractWe study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky’s property (τ) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to the question of Lubotzky and Zuk: for families of subgroups, is property (τ) inherited by the lattice of subgroups generated by the family? On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicit estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expanding covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance.

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It’s a Small World: Disneyland, the Family and the Multiple Re-representations of American Childhood

Constructing and Reconstructing Childhood ◽

10.4324/9780203362600-12 ◽

2003 ◽

pp. 121-138

Keyword(s):

Small World ◽

The Family

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Critical synchronization dynamics of the Kuramoto model on connectome and small world graphs

Scientific Reports ◽

10.1038/s41598-019-54769-9 ◽

2019 ◽

Vol 9 (1) ◽

Cited By ~ 3

Author(s):

Géza Ódor ◽

Jeffrey Kelling

Keyword(s):

Control Parameter ◽

Mean Field ◽

Small World ◽

Kuramoto Model ◽

Crossover Point ◽

Small World Graphs ◽

Synchronization Phase ◽

Experimental Values ◽

The Kuramoto Model ◽

Parameter Dependent

AbstractThe hypothesis, that cortical dynamics operates near criticality also suggests, that it exhibits universal critical exponents which marks the Kuramoto equation, a fundamental model for synchronization, as a prime candidate for an underlying universal model. Here, we determined the synchronization behavior of this model by solving it numerically on a large, weighted human connectome network, containing 836733 nodes, in an assumed homeostatic state. Since this graph has a topological dimension d < 4, a real synchronization phase transition is not possible in the thermodynamic limit, still we could locate a transition between partially synchronized and desynchronized states. At this crossover point we observe power-law–tailed synchronization durations, with τt ≃ 1.2(1), away from experimental values for the brain. For comparison, on a large two-dimensional lattice, having additional random, long-range links, we obtain a mean-field value: τt ≃ 1.6(1). However, below the transition of the connectome we found global coupling control-parameter dependent exponents 1 < τt ≤ 2, overlapping with the range of human brain experiments. We also studied the effects of random flipping of a small portion of link weights, mimicking a network with inhibitory interactions, and found similar results. The control-parameter dependent exponent suggests extended dynamical criticality below the transition point.

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