Multistage Games with Pairwise Interactions on Complete Graph

AbstractWe provide a new proof of the existence of Gibbs point processes with infinite range interactions, based on the compactness of entropy levels. Our main existence theorem holds under two assumptions. The first one is the standard stability assumption, which means that the energy of any finite configuration is superlinear with respect to the number of points. The second assumption is the so-called intensity regularity, which controls the long range of the interaction via the intensity of the process. This assumption is new and introduced here since it is well adapted to the entropy approach. As a corollary of our main result we improve the existence results by Ruelle (1970) for pairwise interactions by relaxing the superstabilty assumption. Note that our setting is not reduced to pairwise interaction and can contain infinite-range multi-body counterparts.

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On the Generalized Distance Energy of Graphs

Mathematics ◽

10.3390/math8010017 ◽

2019 ◽

Vol 8 (1) ◽

pp. 17 ◽

Cited By ~ 4

Author(s):

Abdollah Alhevaz ◽

Maryam Baghipur ◽

Hilal A. Ganie ◽

Yilun Shang

Keyword(s):

Lower Bounds ◽

Complete Graph ◽

Connected Graph ◽

Distance Matrix ◽

Wiener Index ◽

Diagonal Matrix ◽

Upper And Lower Bounds ◽

Star Graph ◽

Extremal Graphs ◽

Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

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Higher-order temporal network effects through triplet evolution

Scientific Reports ◽

10.1038/s41598-021-94389-w ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Qing Yao ◽

Bingsheng Chen ◽

Tim S. Evans ◽

Kim Christensen

Keyword(s):

Real World ◽

Link Prediction ◽

Higher Order ◽

Prediction Algorithm ◽

Interaction Patterns ◽

Temporal Networks ◽

World Systems ◽

Order Interaction ◽

Space And Time ◽

Pairwise Interactions

AbstractWe study the evolution of networks through ‘triplets’—three-node graphlets. We develop a method to compute a transition matrix to describe the evolution of triplets in temporal networks. To identify the importance of higher-order interactions in the evolution of networks, we compare both artificial and real-world data to a model based on pairwise interactions only. The significant differences between the computed matrix and the calculated matrix from the fitted parameters demonstrate that non-pairwise interactions exist for various real-world systems in space and time, such as our data sets. Furthermore, this also reveals that different patterns of higher-order interaction are involved in different real-world situations. To test our approach, we then use these transition matrices as the basis of a link prediction algorithm. We investigate our algorithm’s performance on four temporal networks, comparing our approach against ten other link prediction methods. Our results show that higher-order interactions in both space and time play a crucial role in the evolution of networks as we find our method, along with two other methods based on non-local interactions, give the best overall performance. The results also confirm the concept that the higher-order interaction patterns, i.e., triplet dynamics, can help us understand and predict the evolution of different real-world systems.

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On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽

10.3390/math9050512 ◽

2021 ◽

Vol 9 (5) ◽

pp. 512

Author(s):

Maryam Baghipur ◽

Modjtaba Ghorbani ◽

Hilal A. Ganie ◽

Yilun Shang

Keyword(s):

Lower Bounds ◽

Complete Graph ◽

Distance Matrix ◽

Upper And Lower Bounds ◽

Laplacian Eigenvalue ◽

Signless Laplacian ◽

Graph Parameters ◽

Simple Connected Graph ◽

Second Largest Eigenvalue ◽

Reciprocal Distance Matrix

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

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Evolutionary Stability for Interactions among Kin under Quantitative Inheritance.

Genetics ◽

10.1093/genetics/121.4.877 ◽

1989 ◽

Vol 121 (4) ◽

pp. 877-889

Author(s):

A B Harper

Keyword(s):

Inclusive Fitness ◽

Fitness Function ◽

Quantitative Inheritance ◽

Evolutionarily Stable Strategies ◽

Environmental Variance ◽

Pairwise Interactions ◽

Simplifying Assumptions ◽

Multilocus Model ◽

Evolutionarily Stable

Abstract The theory of evolutionarily stable strategies (ESS) predicts the long-term evolutionary outcome of frequency-dependent selection by making a number of simplifying assumptions about the genetic basis of inheritance. I use a symmetrized multilocus model of quantitative inheritance without mutation to analyze the results of interactions between pairs of related individuals and compare the equilibria to those found by ESS analysis. It is assumed that the fitness changes due to interactions can be approximated by the exponential of a quadratic surface. The major results are the following. (1) The evolutionarily stable phenotypes found by ESS analysis are always equilibria of the model studied here. (2) When relatives interact, one of the two conditions for stability of equilibria differs between the two models; this can be accounted for by positing that the inclusive fitness function for quantitative characters is slightly different from the inclusive fitness function for characters determined by a single locus. (3) The inclusion of environmental variance will in general change the equilibrium phenotype, but the equilibria of ESS analysis are changed to the same extent by environmental variance. (4) A class of genetically polymorphic equilibria occur, which in the present model are always unstable. These results expand the range of conditions under which one can validly predict the evolution of pairwise interactions using ESS analysis.

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Undirected Complete Graph to Design New Public Key Cryptosystem

Journal of Physics Conference Series ◽

10.1088/1742-6596/1897/1/012045 ◽

2021 ◽

Vol 1897 (1) ◽

pp. 012045

Author(s):

Karrar Taher R. Aljamaly ◽

Ruma Kareem K. Ajeena

Keyword(s):

Complete Graph ◽

Public Key ◽

Public Key Cryptosystem ◽

New Public

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Magnetic Janssen effect

Nature Communications ◽

10.1038/s41467-021-22722-y ◽

2021 ◽

Vol 12 (1) ◽

Author(s):

L. Thorens ◽

K. J. Måløy ◽

M. Bourgoin ◽

S. Santucci

Keyword(s):

Magnetic Field ◽

Dynamic Properties ◽

Radial Force ◽

Fine Tuning ◽

Apparent Mass ◽

Confined Geometries ◽

Pairwise Interactions ◽

Janssen Effect ◽

Lateral Walls ◽

Granular Column

AbstractA pile of grains, even when at rest in a silo, can display fascinating properties. One of the most celebrated is the Janssen effect, named after the pioneering engineer who explained the pressure saturation at the bottom of a container filled with corn. This surprising behavior arises because of frictional interactions between the grains through a disordered network of contacts, and the vessel lateral walls, which partially support the weight of the column, decreasing its apparent mass. Here, we demonstrate control over frictional interactions using ferromagnetic grains and an external magnetic field. We show that the anisotropic pairwise interactions between magnetized grains result in a radial force along the walls, whose amplitude and direction is fully determined by the applied magnetic field. Such magnetic Janssen effect allows for the fine tuning of the granular column apparent mass. Our findings pave the way towards the design of functional jammed materials in confined geometries, via a further control of both their static and dynamic properties.

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Turán theorems for unavoidable patterns

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500412100027x ◽

2021 ◽

pp. 1-20

Author(s):

ANTÓNIO GIRÃO ◽

BHARGAV NARAYANAN

Keyword(s):

Complete Graph ◽

Blow Up ◽

Ramsey Numbers ◽

Order Of Magnitude ◽

Transitive Tournament ◽

Unavoidable Patterns

Abstract We prove Turán-type theorems for two related Ramsey problems raised by Bollobás and by Fox and Sudakov. First, for t ≥ 3, we show that any two-colouring of the complete graph on n vertices that is δ-far from being monochromatic contains an unavoidable t-colouring when δ ≫ n−1/t, where an unavoidable t-colouring is any two-colouring of a clique of order 2t in which one colour forms either a clique of order t or two disjoint cliques of order t. Next, for t ≥ 3, we show that any tournament on n vertices that is δ-far from being transitive contains an unavoidable t-tournament when δ ≫ n−1/[t/2], where an unavoidable t-tournament is the blow-up of a cyclic triangle obtained by replacing each vertex of the triangle by a transitive tournament of order t. Conditional on a well-known conjecture about bipartite Turán numbers, both our results are sharp up to implied constants and hence determine the order of magnitude of the corresponding off-diagonal Ramsey numbers.

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Branching processes with interactions: subcritical cooperative regime

Advances in Applied Probability ◽

10.1017/apr.2020.59 ◽

2021 ◽

Vol 53 (1) ◽

pp. 251-278

Author(s):

Adrián González Casanova ◽

Juan Carlos Pardo ◽

José Luis Pérez

Keyword(s):

Population Genetics ◽

Branching Processes ◽

Jump Diffusion ◽

Pairwise Interaction ◽

Pairwise Interactions ◽

Different Types ◽

Interesting Object ◽

The Moment ◽

Dynamics Of Populations

AbstractIn this paper, we introduce a family of processes with values on the nonnegative integers that describes the dynamics of populations where individuals are allowed to have different types of interactions. The types of interactions that we consider include pairwise interactions, such as competition, annihilation, and cooperation; and interactions among several individuals that can be viewed as catastrophes. We call such families of processes branching processes with interactions. Our aim is to study their long-term behaviour under a specific regime of the pairwise interaction parameters that we introduce as the subcritical cooperative regime. Under such a regime, we prove that a process in this class comes down from infinity and has a moment dual which turns out to be a jump-diffusion that can be thought as the evolution of the frequency of a trait or phenotype, and whose parameters have a classical interpretation in terms of population genetics. The moment dual is an important tool for characterizing the stationary distribution of branching processes with interactions whenever such a distribution exists; it is also an interesting object in its own right.

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A property of the colored complete graph

Discrete Mathematics ◽

10.1016/0012-365x(79)90020-7 ◽

1979 ◽

Vol 25 (2) ◽

pp. 175-178 ◽

Cited By ~ 25

Author(s):

J. Shearer

Keyword(s):

Complete Graph

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