Topological Analysis of Numerosity-Constrained Social Networks

2010 ◽  
Author(s):  
Nicole Abaid ◽  
Maurizio Porfiri
Keyword(s):  
Directed Graphs ◽  
Physical Phenomenon ◽  
Degree Sequence ◽  
Topological Analysis ◽  
Random Variable ◽  
Large Animal ◽  
Behavioral Experiments ◽  
Critical Limit ◽  
Bounded Degree ◽  
Fish Schools

In this study, we present a class of directed graphs with bounded degree sequences, which embodies the physical phenomenon of numerosity found in the collective behavior of large animal groups. Behavioral experiments show that an animal’s perception of number is capped by a critical limit, above which an individual perceives a nonspecific “many”. This species-dependent limit plays a pivotal role in the decision making process of large groups, such as fish schools and bird flocks. Here, we consider directed graphs whose edges model information-sharing between individual vertices. We incorporate the numerosity phenomenon as a critical limit on the intake of information by bounding the degree sequence and include the variability of cognitive processes by using a random variable in the network construction. We analytically compute measures of the expected structure of this class of graphs based on cycles, clustering, and sorting among vertices. Theoretical results are verified with numerical simulation.

scholarly journals Universality for Random Surfaces in Unconstrained Genus

10.37236/8623 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Thomas Budzinski ◽  
Nicolas Curien ◽  
Bram Petri
Keyword(s):  
Total Variation ◽  
Degree Sequence ◽  
Random Variable ◽  
Total Variation Distance ◽  
Alternative Proof ◽  
Arbitrary Sequence ◽  
Weak Version ◽  
Variation Distance ◽  
Planar Maps ◽  
Dirichlet Partition

Starting from an arbitrary sequence of polygons whose total perimeter is $2n$, we can build an (oriented) surface by pairing their sides in a uniform fashion. Chmutov & Pittel have shown that, regardless of the configuration of polygons we started with, the degree sequence of the graph obtained this way is remarkably constant in total variation distance and converges towards a Poisson–Dirichlet partition as $n \to \infty$. We actually show that several other geometric properties of the graph are universal. En route we provide an alternative proof of a weak version of the result of Chmutov & Pittel using probabilistic techniques and related to the circle of ideas around the peeling process of random planar maps. At this occasion we also fill a gap in the existing literature by surveying the properties of a uniform random map with $n$ edges. In particular we show that the diameter of a random map with $n$ edges converges in law towards a random variable taking only values in $\{2,3\}$.

Design and Topological Analysis of Complex Networks with Optimal Controllability

2014 ◽  
Vol 24 (07) ◽  
pp. 1450103 ◽  
Author(s):  
Cuili Yang ◽  
Wallace K. S. Tang
Keyword(s):  
Complexity Analysis ◽  
Degree Sequence ◽  
Topological Analysis ◽  
Pinning Control ◽  
Network Design Problem ◽  
Node Degree ◽  
Network Characteristics ◽  
Optimal Network Design ◽  
Optimal Network ◽  
Practical Reality

In this paper, we aim to identify a directed complex network with optimal controllability, for which pinning control is to be applied. Since the controllability of a network can be reflected by the smallest nonzero eigenvalue of a matrix related to its topology, an optimal network design problem is formulated based on the maximization of this eigenvalue. To better consider the practical reality, constraints on node degree sequence are specified. Based on the derived bounds of the eigenvalue, an effective rewiring scheme is designed and solutions close to or equal to the upper bound are obtained. Finally, the relationship between network characteristics and controllability is studied. Through complexity analysis, it is concluded that the network with high controllability should possess two properties, i.e. nodes with high out-degree for pinning and other nodes with uniform degree distribution.

scholarly journals A Parameterized Algorithmics Framework for Degree Sequence Completion Problems in Directed Graphs

Algorithmica ◽  
2018 ◽  
Vol 81 (4) ◽  
pp. 1584-1614 ◽  
Author(s):  
Robert Bredereck ◽  
Vincent Froese ◽  
Marcel Koseler ◽  
Marcelo Garlet Millani ◽  
André Nichterlein ◽  
...  
Keyword(s):  
Directed Graphs ◽  
Degree Sequence ◽  
Parameterized Algorithmics ◽  
Completion Problems

scholarly journals Fourier inversion formula for discrete nilpotent groups

1989 ◽  
Vol 46 (3) ◽  
pp. 415-422
Author(s):  
Tsuyoshi Kajiwara
Keyword(s):  
Random Variable ◽  
Nilpotent Groups ◽  
Positive Definite Function ◽  
Definite Function ◽  
Matrix Elements ◽  
Fourier Inversion Formula ◽  
Bounded Degree ◽  
The Matrix ◽  
The Fourier Transform ◽  
Torsion Free

AbstractLet G be a countable torsion free finitely generated nilpotent group. Then the Fourier transform can be considered as a map from the space of bounded degree 1 random operators to the Fourier algebra A(G). In this paper, we recover the matrix elements of a positive random variable from the corresponding positive definite function in A(G) for such a group.

scholarly journals Embedding Spanning Bipartite Graphs of Small Bandwidth

2012 ◽  
Vol 22 (1) ◽  
pp. 71-96 ◽  
Author(s):  
FIACHRA KNOX ◽  
ANDREW TREGLOWN
Keyword(s):  
Bipartite Graph ◽  
Minimum Degree ◽  
Degree Sequence ◽  
Bipartite Graphs ◽  
Type Result ◽  
Bounded Degree ◽  
Expansion Property ◽  
Small Bandwidth ◽  
Bipartite Case

Böttcher, Schacht and Taraz (Math. Ann., 2009) gave a condition on the minimum degree of a graph G on n vertices that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth o(n), thereby proving a conjecture of Bollobás and Komlós (Combin. Probab. Comput., 1999). We strengthen this result in the case when H is bipartite. Indeed, we give an essentially best-possible condition on the degree sequence of a graph G on n vertices that forces G to contain every bipartite graph H on n vertices of bounded degree and of bandwidth o(n). This also implies an Ore-type result. In fact, we prove a much stronger result where the condition on G is relaxed to a certain robust expansion property. Our result also confirms the bipartite case of a conjecture of Balogh, Kostochka and Treglown concerning the degree sequence of a graph which forces a perfect H-packing.

scholarly journals New Classes of Degree Sequences with Fast Mixing Swap Markov Chain Sampling

2017 ◽  
Vol 27 (2) ◽  
pp. 186-207
Author(s):  
PÉTER L. ERDŐS ◽  
ISTVÁN MIKLÓS ◽  
ZOLTÁN TOROCZKAI
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Directed Graphs ◽  
Degree Sequence ◽  
Degree Sequences ◽  
Network Modelling ◽  
Split Graph ◽  
Simple Graphs ◽  
Large Classes ◽  
Special Degree

In network modelling of complex systems one is often required to sample random realizations of networks that obey a given set of constraints, usually in the form of graph measures. A much studied class of problems targets uniform sampling of simple graphs with given degree sequence or also with given degree correlations expressed in the form of a Joint Degree Matrix. One approach is to use Markov chains based on edge switches (swaps) that preserve the constraints, are irreducible (ergodic) and fast mixing. In 1999, Kannan, Tetali and Vempala (KTV) proposed a simple swap Markov chain for sampling graphs with given degree sequence, and conjectured that it mixes rapidly (in polynomial time) for arbitrary degree sequences. Although the conjecture is still open, it has been proved for special degree sequences, in particular for those of undirected and directed regular simple graphs, half-regular bipartite graphs, and graphs with certain bounded maximum degrees. Here we prove the fast mixing KTV conjecture for novel, exponentially large classes of irregular degree sequences. Our method is based on a canonical decomposition of degree sequences into split graph degree sequences, a structural theorem for the space of graph realizations and on a factorization theorem for Markov chains. After introducing bipartite ‘splitted’ degree sequences, we also generalize the canonical split graph decomposition for bipartite and directed graphs.

Real-Time Feedback-Controlled Robotic Fish for Behavioral Experiments With Fish Schools

2012 ◽  
Vol 100 (1) ◽  
pp. 150-163 ◽  
Author(s):  
Daniel T. Swain ◽  
Iain D. Couzin ◽  
Naomi Ehrich Leonard
Keyword(s):  
Real Time ◽  
Robotic Fish ◽  
Behavioral Experiments ◽  
Fish Schools
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