This document discusses the evolution of random graphs as the number of edges increases. It identifies 5 phases of evolution:
1. For very few edges, the graph consists only of tree components.
2. As more edges are added, cycles begin to form but most components are still trees or contain a single cycle.
3. At about half the maximum number of edges, a "giant" component emerges that contains a large portion of vertices.
4. Shortly after, the graph becomes fully connected as the remaining small components are absorbed into the giant one.
5. With further edges, the graph converges to a random graph where any edge is possible.
This document summarizes the method of variational formulation for linear and nonlinear problems. It introduces Gateaux derivatives and symmetry conditions, and defines variational formulations in both the restricted and extended senses. It provides an example applying these concepts to a first-order nonlinear differential equation. The key points are:
1) Gateaux derivatives generalize the concept of derivatives to nonlinear operators.
2) A variational principle exists if the Gateaux differential is symmetric.
3) Variational problems can be formulated in both a restricted sense, where the solutions are critical points of a functional, and an extended sense, where an equivalent functional exists.
4) An example applies these concepts to derive a variational formulation for
Reading the Lindley-Smith 1973 paper on linear Bayes estimatorsChristian Robert
The document outlines a seminar on Bayes estimates for the linear model. It introduces the linear model and Bayesian methods. It then discusses exchangeability, providing an example of an exchangeable distribution. It also discusses the general Bayesian linear model, including the posterior distribution of the parameters using a three stage model.
This document summarizes three common random graph models used to model complex real-world networks: Erdos-Renyi graphs, Watts-Strogatz small-world networks, and Barabasi-Albert scale-free networks. Erdos-Renyi graphs use a simple random edge placement process that can result in disconnected graphs or ones with small clustering. Watts-Strogatz networks address this by rewiring edges in a lattice, creating small-world properties but not realistic degree distributions. Barabasi-Albert networks use a preferential attachment mechanism where new nodes attach preferentially to higher degree nodes, producing power-law degree distributions seen in many real networks.
In this presentation we show that there exist graphs which greedy routing thak long time. This show that Small world is a Social phenomena and not mathematical one.
Some random graphs for network models - Birgit PlötzenederBirgit Plötzeneder
The document summarizes several random graph models:
1) Erdös and Renyi proposed connecting nodes with probability p, resulting in bell-shaped degree distributions.
2) Watts and Strogatz modeled small-world networks by rewiring edges in a ring lattice with probability p, finding short paths like social networks.
3) Barabasi and Albert grew networks by preferentially attaching new nodes to popular existing nodes, producing scale-free networks with power-law degree distributions and hubs.
Che cos'è una rete sociale, come nasce, a che cosa serve, come si trasforma in una rete creativa...
Il volume di Giuseppe RIva "I social network" pubblicato dal Mulino, Bologna.
A high-level overview of social network analysis using gephi with your exported Facebook friends network. See more network analysis at http://allthingsgraphed.com.
This document summarizes the method of variational formulation for linear and nonlinear problems. It introduces Gateaux derivatives and symmetry conditions, and defines variational formulations in both the restricted and extended senses. It provides an example applying these concepts to a first-order nonlinear differential equation. The key points are:
1) Gateaux derivatives generalize the concept of derivatives to nonlinear operators.
2) A variational principle exists if the Gateaux differential is symmetric.
3) Variational problems can be formulated in both a restricted sense, where the solutions are critical points of a functional, and an extended sense, where an equivalent functional exists.
4) An example applies these concepts to derive a variational formulation for
Reading the Lindley-Smith 1973 paper on linear Bayes estimatorsChristian Robert
The document outlines a seminar on Bayes estimates for the linear model. It introduces the linear model and Bayesian methods. It then discusses exchangeability, providing an example of an exchangeable distribution. It also discusses the general Bayesian linear model, including the posterior distribution of the parameters using a three stage model.
This document summarizes three common random graph models used to model complex real-world networks: Erdos-Renyi graphs, Watts-Strogatz small-world networks, and Barabasi-Albert scale-free networks. Erdos-Renyi graphs use a simple random edge placement process that can result in disconnected graphs or ones with small clustering. Watts-Strogatz networks address this by rewiring edges in a lattice, creating small-world properties but not realistic degree distributions. Barabasi-Albert networks use a preferential attachment mechanism where new nodes attach preferentially to higher degree nodes, producing power-law degree distributions seen in many real networks.
In this presentation we show that there exist graphs which greedy routing thak long time. This show that Small world is a Social phenomena and not mathematical one.
Some random graphs for network models - Birgit PlötzenederBirgit Plötzeneder
The document summarizes several random graph models:
1) Erdös and Renyi proposed connecting nodes with probability p, resulting in bell-shaped degree distributions.
2) Watts and Strogatz modeled small-world networks by rewiring edges in a ring lattice with probability p, finding short paths like social networks.
3) Barabasi and Albert grew networks by preferentially attaching new nodes to popular existing nodes, producing scale-free networks with power-law degree distributions and hubs.
Che cos'è una rete sociale, come nasce, a che cosa serve, come si trasforma in una rete creativa...
Il volume di Giuseppe RIva "I social network" pubblicato dal Mulino, Bologna.
A high-level overview of social network analysis using gephi with your exported Facebook friends network. See more network analysis at http://allthingsgraphed.com.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The Wishart and inverse-wishart distributionPankaj Das
The document discusses the Wishart and inverse-Wishart distributions which are used to model covariance matrices. It provides mathematical background on how the Wishart distribution arises from sampling covariance matrices from multivariate normal distributions. It also describes key properties of the Wishart distribution including its probability density function and how it relates to the chi-squared distribution when the dimensionality is one. Estimation of covariance matrices plays an important role in multivariate statistics.
IJCER (www.ijceronline.com) International Journal of computational Engineerin...ijceronline
This document discusses period-doubling bifurcations and the route to chaos in an area-preserving discrete dynamical system. It highlights two objectives: 1) Evaluating period-doubling bifurcations using computer programs as the system parameter p is varied, obtaining the Feigenbaum universal constant and accumulation point beyond which chaos occurs; and 2) Confirming periodic behaviors by plotting time series graphs. The document provides background on Feigenbaum universality and describes the specific area-preserving map studied. It outlines the numerical method used to obtain periodic points via Newton's recurrence formula and describes applying this to the map's Jacobian matrix.
EVEN GRACEFUL LABELLING OF A CLASS OF TREESFransiskeran
A labelling or numbering of a graph G with q edges is an assignment of labels to the vertices of G that
induces for each edge uv a labelling depending on the vertex labels f(u) and f(v). A labelling is called a
graceful labelling if there exists an injective function f: V (G) → {0, 1,2,......q} such that for each edge xy,
the labelling │f(x)-f(y)│is distinct. In this paper, we prove that a class of Tn trees are even graceful.
This document contains information about graphing algebraic equations:
1) It introduces Cartesian coordinates and using a graph to represent points that satisfy an equation in two variables.
2) It shows that a linear equation represents a straight line on a graph and that simultaneous linear equations have a single intersection point.
3) Quadratic and higher degree equations represent curves on a graph, with the number of intersections with the x-axis equal to the degree of the equation. Intersections can be real, imaginary, or coincident.
4) The absolute term in an equation affects the position but not the shape of its graph. Shifting terms affects the position of intersections along the x-axis.
1) The document defines algebraic curves as geometric figures formed by the set of points satisfying a given equation relating x and y.
2) Key properties of algebraic curves include their extent (domain and range), symmetry, intercepts, and asymptotes. Symmetry can be tested by substituting -x or -y. Intercepts are where the curve crosses the axes. Asymptotes are lines a curve approaches but never touches.
3) Tracing a curve involves determining its region, testing for symmetry, finding intercepts and asymptotes, and plotting points to sketch the curve.
CONSTRUCTION OF NONASSOCIATIVE GRASSMANN ALGEBRAIRJET Journal
1) The document discusses the construction of a nonassociative Grassmann algebra. It begins by introducing Grassmann and symmetric algebras, which are generated by vector spaces and related by determinants and permanents.
2) A new multiplication rule is derived for a nonassociative Grassmann algebra formed by combining the exterior and interior products of Grassmann and symmetric algebras. Multiplication tables are given for dimensions 3, 7, and 15.
3) The algebra is formed on a Z-graded vector space without positive gradation. Previous work by Z. Oziewicz and C. Sitarczyk combining Grassmann and symmetric algebras is cited as motivation.
This document discusses generating pseudo-random sequences using hyperbolic automorphisms of tori. It begins by defining pseudo-random sequences and describing how chaotic dynamical systems can generate them. It then focuses on using hyperbolic automorphisms of the d-dimensional torus, which are ergodic but orbits starting from rational points are periodic. The document proves lower and upper bounds on the period length of such orbits and describes algorithms to generate pseudo-random vectors without computational error in integer arithmetic. It concludes by discussing statistical testing of the generated sequences.
Here are the steps to find the line of intersection of the two planes:
1) Write the equations of the planes in standard form:
Plane 1: x + 2y - z = 4
Plane 2: 2x - y + z = 1
2) Set the equations equal to each other and solve as a system of equations:
x + 2y - z = 4
2x - y + z = 1
3) Eliminate one variable:
Subtract the second equation from the first:
(x + 2y - z) - (2x - y + z) = 4 - 1
-x + y = 3
4) Substitute back into one of the
1) A plane in 3D space is defined by a point P0(x0, y0, z0) lying on the plane and a normal vector n = <a, b, c> orthogonal to the plane.
2) The standard equation of a plane is ax + by + cz + d = 0, where n = <a, b, c> is the normal vector.
3) Two planes intersect in a line. The angle between their normal vectors defines the angle between the planes.
The document summarizes the application of field theory methods to solve the Ising model of phase transitions in statistical physics. It shows that:
1) The self-consistent Curie-Weiss solution of the Ising model can be obtained from a certain functional equation and corresponds to the quasiclassical approximation in field theory.
2) Within this approximation, the appearance of spontaneous magnetization in the Ising model can be interpreted as the formation of bound states of some associated field problem.
3) Retaining only "tree" diagrams, which neglect quantum fluctuations, gives the self-consistent field approximation; this is valid when fluctuations and their interaction are weak.
This document discusses computing fundamental numbers of the variety of nodal cubics in projective 3-space (P3). It constructs several compactifications of this variety, which can be obtained as a sequence of blow-ups of a projective bundle Knod. The numbers computed include intersection numbers involving characteristic conditions like points, lines, and tangency to planes, as well as the condition that the node lies on a plane. The computations are carried out using the Wit symbolic computation system.
A Unifying theory for blockchain and AILonghow Lam
This document proposes a unifying theory connecting blockchain and artificial intelligence technologies. It introduces the Lam-Visser theory and how it fits within the Damhof Quadrants framework. The document provides definitions related to the main result, which states that there exists a minimal, ultra-connected, almost everywhere linear and generic solvable, semi-countable polytope if a certain condition is met. It then discusses applications of this theory to questions of associativity and the computation of analytically independent subalgebras.
This document discusses complex functions and their derivatives. It defines a complex function as a function f(z) that maps complex numbers to complex numbers. The derivative of a complex function is defined as the limit of the difference quotient, which may not always exist. Some simple functions like Re(z) are shown to not have complex derivatives. The usual rules of differentiation, such as the sum, product, quotient and chain rules, are shown to hold for complex differentiable functions.
1) Complex numbers can be represented in Cartesian (x + iy) or polar (r(cosθ + i sinθ)) form, with conversions between the two.
2) The derivative of a complex function f(z) is defined if the Cauchy-Riemann equations are satisfied.
3) A function is analytic if it is differentiable and its partial derivatives are continuous, implying the Cauchy-Riemann equations always hold. Analytic functions have properties like equality of second partial derivatives.
In many scientific areas, systems can be described as interaction networks where elements correspond to vertices and interactions to edges. A variety of problems in those fields can deal with network comparison and characterization.
The problem of comparing and characterizing networks is the task of measuring their structural similarity and finding characteristics which capture structural information. In order to analyze complex networks, several methods can be combined, such as graph theory, information theory, and statistics.
In this project, we present methods for measuring Shannon’s entropy of graphs.
Analysis and algebra on differentiable manifoldsSpringer
This chapter discusses tensor fields and differential forms on manifolds. It provides definitions of tensor fields, differential forms, vector bundles, and the exterior derivative. It also introduces the Lie derivative and interior product. The chapter contains examples of vector bundles like the Möbius strip. It aims to make the reader proficient with computations involving vector fields, differential forms, and other concepts. Problems are included to help develop skills in computing integral distributions and differential ideals.
This document discusses complex numbers and functions. It introduces complex numbers using Cartesian (x + iy) and polar (r(cosθ + i sinθ)) forms. It describes the Cauchy-Riemann conditions that must be satisfied for a function of a complex variable to be differentiable. A function is analytic if it satisfies the Cauchy-Riemann conditions and its partial derivatives are continuous. Analytic functions have properties like equality of second-order partial derivatives and establishing a relation between the real and imaginary parts.
This document provides definitions and concepts related to graphs. It defines a graph as a data structure consisting of vertices and edges linking vertices, which can represent objects and relationships. Graphs are a generalization of trees that allow any type of relationship between nodes. Common graph representations include adjacency lists and matrices. The document also discusses graph traversal methods, minimum spanning trees, and applications of graphs.
1. A Hermitian form is a numerical function that satisfies certain properties relating to addition and multiplication of vectors in a complex linear space.
2. A Hermitian form is symmetric if its value is unchanged when the order of the vectors is reversed.
3. A symmetric Hermitian form is positive definite if its value is always positive, excluding the zero vector.
4. A unitary space is a complex linear space equipped with a symmetric positive definite Hermitian bilinear form, known as the scalar product, that assigns a complex number to each pair of vectors according to certain axioms.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The Wishart and inverse-wishart distributionPankaj Das
The document discusses the Wishart and inverse-Wishart distributions which are used to model covariance matrices. It provides mathematical background on how the Wishart distribution arises from sampling covariance matrices from multivariate normal distributions. It also describes key properties of the Wishart distribution including its probability density function and how it relates to the chi-squared distribution when the dimensionality is one. Estimation of covariance matrices plays an important role in multivariate statistics.
IJCER (www.ijceronline.com) International Journal of computational Engineerin...ijceronline
This document discusses period-doubling bifurcations and the route to chaos in an area-preserving discrete dynamical system. It highlights two objectives: 1) Evaluating period-doubling bifurcations using computer programs as the system parameter p is varied, obtaining the Feigenbaum universal constant and accumulation point beyond which chaos occurs; and 2) Confirming periodic behaviors by plotting time series graphs. The document provides background on Feigenbaum universality and describes the specific area-preserving map studied. It outlines the numerical method used to obtain periodic points via Newton's recurrence formula and describes applying this to the map's Jacobian matrix.
EVEN GRACEFUL LABELLING OF A CLASS OF TREESFransiskeran
A labelling or numbering of a graph G with q edges is an assignment of labels to the vertices of G that
induces for each edge uv a labelling depending on the vertex labels f(u) and f(v). A labelling is called a
graceful labelling if there exists an injective function f: V (G) → {0, 1,2,......q} such that for each edge xy,
the labelling │f(x)-f(y)│is distinct. In this paper, we prove that a class of Tn trees are even graceful.
This document contains information about graphing algebraic equations:
1) It introduces Cartesian coordinates and using a graph to represent points that satisfy an equation in two variables.
2) It shows that a linear equation represents a straight line on a graph and that simultaneous linear equations have a single intersection point.
3) Quadratic and higher degree equations represent curves on a graph, with the number of intersections with the x-axis equal to the degree of the equation. Intersections can be real, imaginary, or coincident.
4) The absolute term in an equation affects the position but not the shape of its graph. Shifting terms affects the position of intersections along the x-axis.
1) The document defines algebraic curves as geometric figures formed by the set of points satisfying a given equation relating x and y.
2) Key properties of algebraic curves include their extent (domain and range), symmetry, intercepts, and asymptotes. Symmetry can be tested by substituting -x or -y. Intercepts are where the curve crosses the axes. Asymptotes are lines a curve approaches but never touches.
3) Tracing a curve involves determining its region, testing for symmetry, finding intercepts and asymptotes, and plotting points to sketch the curve.
CONSTRUCTION OF NONASSOCIATIVE GRASSMANN ALGEBRAIRJET Journal
1) The document discusses the construction of a nonassociative Grassmann algebra. It begins by introducing Grassmann and symmetric algebras, which are generated by vector spaces and related by determinants and permanents.
2) A new multiplication rule is derived for a nonassociative Grassmann algebra formed by combining the exterior and interior products of Grassmann and symmetric algebras. Multiplication tables are given for dimensions 3, 7, and 15.
3) The algebra is formed on a Z-graded vector space without positive gradation. Previous work by Z. Oziewicz and C. Sitarczyk combining Grassmann and symmetric algebras is cited as motivation.
This document discusses generating pseudo-random sequences using hyperbolic automorphisms of tori. It begins by defining pseudo-random sequences and describing how chaotic dynamical systems can generate them. It then focuses on using hyperbolic automorphisms of the d-dimensional torus, which are ergodic but orbits starting from rational points are periodic. The document proves lower and upper bounds on the period length of such orbits and describes algorithms to generate pseudo-random vectors without computational error in integer arithmetic. It concludes by discussing statistical testing of the generated sequences.
Here are the steps to find the line of intersection of the two planes:
1) Write the equations of the planes in standard form:
Plane 1: x + 2y - z = 4
Plane 2: 2x - y + z = 1
2) Set the equations equal to each other and solve as a system of equations:
x + 2y - z = 4
2x - y + z = 1
3) Eliminate one variable:
Subtract the second equation from the first:
(x + 2y - z) - (2x - y + z) = 4 - 1
-x + y = 3
4) Substitute back into one of the
1) A plane in 3D space is defined by a point P0(x0, y0, z0) lying on the plane and a normal vector n = <a, b, c> orthogonal to the plane.
2) The standard equation of a plane is ax + by + cz + d = 0, where n = <a, b, c> is the normal vector.
3) Two planes intersect in a line. The angle between their normal vectors defines the angle between the planes.
The document summarizes the application of field theory methods to solve the Ising model of phase transitions in statistical physics. It shows that:
1) The self-consistent Curie-Weiss solution of the Ising model can be obtained from a certain functional equation and corresponds to the quasiclassical approximation in field theory.
2) Within this approximation, the appearance of spontaneous magnetization in the Ising model can be interpreted as the formation of bound states of some associated field problem.
3) Retaining only "tree" diagrams, which neglect quantum fluctuations, gives the self-consistent field approximation; this is valid when fluctuations and their interaction are weak.
This document discusses computing fundamental numbers of the variety of nodal cubics in projective 3-space (P3). It constructs several compactifications of this variety, which can be obtained as a sequence of blow-ups of a projective bundle Knod. The numbers computed include intersection numbers involving characteristic conditions like points, lines, and tangency to planes, as well as the condition that the node lies on a plane. The computations are carried out using the Wit symbolic computation system.
A Unifying theory for blockchain and AILonghow Lam
This document proposes a unifying theory connecting blockchain and artificial intelligence technologies. It introduces the Lam-Visser theory and how it fits within the Damhof Quadrants framework. The document provides definitions related to the main result, which states that there exists a minimal, ultra-connected, almost everywhere linear and generic solvable, semi-countable polytope if a certain condition is met. It then discusses applications of this theory to questions of associativity and the computation of analytically independent subalgebras.
This document discusses complex functions and their derivatives. It defines a complex function as a function f(z) that maps complex numbers to complex numbers. The derivative of a complex function is defined as the limit of the difference quotient, which may not always exist. Some simple functions like Re(z) are shown to not have complex derivatives. The usual rules of differentiation, such as the sum, product, quotient and chain rules, are shown to hold for complex differentiable functions.
1) Complex numbers can be represented in Cartesian (x + iy) or polar (r(cosθ + i sinθ)) form, with conversions between the two.
2) The derivative of a complex function f(z) is defined if the Cauchy-Riemann equations are satisfied.
3) A function is analytic if it is differentiable and its partial derivatives are continuous, implying the Cauchy-Riemann equations always hold. Analytic functions have properties like equality of second partial derivatives.
In many scientific areas, systems can be described as interaction networks where elements correspond to vertices and interactions to edges. A variety of problems in those fields can deal with network comparison and characterization.
The problem of comparing and characterizing networks is the task of measuring their structural similarity and finding characteristics which capture structural information. In order to analyze complex networks, several methods can be combined, such as graph theory, information theory, and statistics.
In this project, we present methods for measuring Shannon’s entropy of graphs.
Analysis and algebra on differentiable manifoldsSpringer
This chapter discusses tensor fields and differential forms on manifolds. It provides definitions of tensor fields, differential forms, vector bundles, and the exterior derivative. It also introduces the Lie derivative and interior product. The chapter contains examples of vector bundles like the Möbius strip. It aims to make the reader proficient with computations involving vector fields, differential forms, and other concepts. Problems are included to help develop skills in computing integral distributions and differential ideals.
This document discusses complex numbers and functions. It introduces complex numbers using Cartesian (x + iy) and polar (r(cosθ + i sinθ)) forms. It describes the Cauchy-Riemann conditions that must be satisfied for a function of a complex variable to be differentiable. A function is analytic if it satisfies the Cauchy-Riemann conditions and its partial derivatives are continuous. Analytic functions have properties like equality of second-order partial derivatives and establishing a relation between the real and imaginary parts.
This document provides definitions and concepts related to graphs. It defines a graph as a data structure consisting of vertices and edges linking vertices, which can represent objects and relationships. Graphs are a generalization of trees that allow any type of relationship between nodes. Common graph representations include adjacency lists and matrices. The document also discusses graph traversal methods, minimum spanning trees, and applications of graphs.
1. A Hermitian form is a numerical function that satisfies certain properties relating to addition and multiplication of vectors in a complex linear space.
2. A Hermitian form is symmetric if its value is unchanged when the order of the vectors is reversed.
3. A symmetric Hermitian form is positive definite if its value is always positive, excluding the zero vector.
4. A unitary space is a complex linear space equipped with a symmetric positive definite Hermitian bilinear form, known as the scalar product, that assigns a complex number to each pair of vectors according to certain axioms.
1. ON THE EVOLUTION OF RANDOM GRAPHS
P. ERD& and A. RBNYI
Institute of h4fathematics
Hmgarian Academy of Sciences, Hungary
1. Definition of a random graph
Let E,, .V denote the set of all graphs having n given labelled vertices VI, L’s;,.,
Vn and N edges. The graphs considered are supposed to be not oriented, without
parallel edges and without slings (such graphs are sometimes called linear graphs).
Thus a graph belonging to the set En, N is obtained by choosing N out of the
possible (5) edges between
the points VI, VZ, ..., Vn, and therefore the number of
n
elements of En, ?V is equal to 2 . A random graph r,, N can be defined as an
(’ AT‘>
element of En, N chosen at random, so that each of the elements of E,, N have the
same probability to be chosen, namely 1 ‘I;l . There is however an other slightly
/( >
different point of view, which has some advantages. We may conszder the forma-
tion of u random graph as a stochastic process defined as follows : At time t=l
we choose one out of the (;) p ossible edges connecting the points VI, VZ,..., V,,
each of these edges having the same probability to be chosen ; let this edge be denoted
by el. At time t=2 we choose one of the possible (z) -1 edges, different from er,
all these being equiprobable. Continuing this process at time t=k+l we choose
one of the (a) 4 p ossible edges different from the edges er, ez, ..., ek already
chosen, each of the remaining edges being equiprobable, i.e. having the probability
1 /I(;)-k). We d enote by r,, .V the graph consisting of the vertices VI, Vt, .. .,
LTfi and the edges el, e2, ‘.., eN.
11 Other not equivalent but closely connected notions of random graphs are as follows:
1) Ve may define a random graph i’z, G by dropping the restriction that there should
be no parallel edges; thus we may suppose that e,+t may be equal with probability
1 /(z) with each of the [z) edg es, independently of whether they are contained in the
sequence of edges e,, e?, .‘., e,t or not. These randum graphs are considered in the paper
131. 2) T%‘e may decide with respect to each of the (?J) edges, whether they should form
part of the random graph considered or not, the probability of including a given edge
being p= lV/:( i) for each edge and the decisions concerning different edges being in-
dependent. We denote the random graph thus obtained by rzf,%,. These random graphs
have been considered in the paper [4J
2. 344 TH~~ORIE DE L'IXFORMATION
The two definitions are clearIy equivalent”. According to the second definition
the number of edges of a random graph is interpreted as time, and according to
this interpretation we may investigate the evolution of a random graph, i.e. the
step-by-step unravelling of the structure of r,, 1~ when N increases.
The evolution of random graphs may be considered as a (rather simplified)
model of the evolution of certain real communication-nets, e. g. the railway-,
road- or electric network system of a country or some other unit, or of the growth
of structures of anorganic or organic matter, or even of the development of social
relations. Of course, if one aims at describing such a real situation, our model of
a random graph should be replaced by a more complicated but more realistic model.
The following possible lines of generalization of the considered stochastic process
of the formation of a random graph should be mentioned here :
a) One may distinguish different sorts of vertices, and !or edges-by a usual
terminology one may consider coloured vertices resp. edges.
b) One may attribute different probabilities to the different edges; this can be
done e.g. by attributing a weight, W, 2 0 to each of the !y) possib!e edges e so
that W=l and to suppose that ei is equal to the edge e with probability W, and
c
that after Ed; ez, ..., ek have been chosen, ek+l is equal to any edge e not occurring
among the edges el,e~, ..., ek with probability F where &= W, An
k c
eFeJ (j=l,2;-,k)
other alternative is to admit that the probability of choosing an edge e=(Vt, V,)
after k other edges have already been chosen, should depend on the number of
edges starting from the points Vt resp. V, which have already been chosen.
In what follows we consider only the simple random graph-formation process,
described above, i.e. we consider only the random graphs I‘,,,N.
Our main aim is to show through this special case that the evolution of a
random graph shows very clear-cut features. The theorems we have proved belong
to two classes. The theorems of the first class deal with the appearance of certain
subgraphs (e.g. trees, cycles of a given order etc.) or components, or other local
structural properties, and show that for many types of local structural properties fl
N(n)
a definite “ threshold ” A(n) can be given, so that if - 40 for n++m then
A(n)
the probability that the random graph r,, N~Q has the structural property A tends
-v(n)
to 0 for n--r-+00, while for - -t+ 00 for n-t+m the probability that r,,, WC%)
A(n)
has the structural property A tends to 1 for n ++m. In many cases still more
can be said : there exists a “threshold function ” for the property A, i.e. a probn-
N(n)
bility distribution function FA (x) so that if lim -=x the probability that I’,, ,vtn)
n++m A(n)
has the property A tends to FA (x) for n-++m.
The theorems of the second class are of similar type, only the properties 4
considered are not of a local character, but global properties of the graph rn, N
(e. g. connectivity, total number of components, etc.).
In the next 5 we briefly describe the process of evolution of the random graph
r n, N, The proofs,. which are completely elementary, and are based on the asymptotic
evaluation of combinatorial formulae and on some well-known general methods of
probability theory, are published in the papers [l] and [Z].
3. P. ERD~S AND A. RBNYI 345
2. The evolution of r,, N
If n is a fixed large positive integer and N is increasing from 1 to [y), the evolu-
tion of r,, ,LV passes through five clearly distinguishable phases. These phases
correspond to ranges of growth of the number N of edges, these ranges being
defined in terms of the number n of vertices.
P/KLSC 1. corresponds to the range N(n,) = o (s), For this phase it is charac-
teristic that r,, Ecn) consists almost surely Ci. e. with probability tending to 1 for
n --+a) exclusively of components which are trees. Trees of order k appear only
a-s e-2
when N(n) reaches the order of magnitude nk-1 (k=3, 4, .‘.j. If N(n)-pn k-1
with p>O, then the probability distribution of the number of components of I’,,, ~(~1
which are trees of order k tends for n--t+a to the Poisson distribution with mean
“alue l=(2P)“-’ k k-2 If AT(n)
w-,--++oo then the distribution of the number of
k! ’ yp-i
components which are trees of order k is approximately normal with mean
and with variance also equal to 154,. This result
holds also in the next two ranges, in fact it holds under the single condition that
M,‘i + c-2 for n-t+w.
Phase 2. corresponds to the range N(n)-cn with O<c<1/2.
In this case r,, xcn) already contains cycles of any fixed order with probability
tending to a positive limit: the distribution of the number of cycles of order k in
12 c)”
r,, TV is approximately a Poisson-distribution with mean value %k-. In this
range almost surely all components of 1 7a, Ncn) are either trees or components
consisting of an equal number of edges and vertices, i. e. components containing
exactly one cycle. The distribution of the number of components consisting of k
vertices and h edges tends for n ++bo to the Poisson distribution with mean value
In this phase though not all, but still almost
all (i. e. n--o in)) vertices belong to components which are trees. The mean number
of components is n-N(n)+O(l), i. e. in this range by adding a new edge the
number of components decreases by 1, except for a finite number of steps.
Phase 3. corresponds to the range N(n)-cn with c B lja. When N(n) passes
the threshold n/2, the structure of r,, N(~) changes abruptly. As a matter of fact
this sudden change of the structure of rn, NC%) is the most surprising fact discovered
by the investigation of the evolution of random graphs. While for N(n)-cz with
c<lk the greatest component of f,, NM is a tree and has (with probability
tending to 1 for n-++m) approximately $ log n- --%og
2 log n vertices, where
( !
a=2c-1-log2c, for N(n) - n/2, the greatest component has (with probability
tending to 1 for n-++ m) approximately n ~3 vertices and has a rather complex
structure. Moreover for N(n) - cn with c > l/2 the greatest component of rn, NC,,)
has (u-ith probability tending to 1 for n--t+- m) approximately G(c)n vertices, where
4. 346 TII~ORIE DE L'INFORMATIOS
(clearly G (l/2)=0
lim G (c)=1).and
c-i-m
Except this “ giant ” component, the other components are all relatively small,
most of them being trees, the total number of vertices belonging to components,
which are trees being almost surely n (l--G (c))+0 (n) for c 2 1,‘2.
As regards the mean number of components2), this is for lY(n) - c n with
x2 Cc)
c> l/Z asymptotically equal to $ X(c)-2 , where
>
X(c)=iJgy2 ce-qL2c(l-G(cj)
R=l
Theevolution of r,, ,vcn) in Phase 3, may be characterized by that the small
components (most of which are trees) melt, each after another, into the giant
component, the smaller components having the larger chance of “survival “; the
survival time of a tree of order k which is present in I’,, ,vcn) with -V(n) - cn,
c>lj2 is approximately exponentially distributed with mean value n/Zk.
P1lase 4. corresponds to the range N(n)-c n log n with c 5 l/2. In this phase
the graph almost surely becomes connected. If
k-l
(3) A+)=~ log nf - 2k n log log n+yn+o(n)
then there are with probability tending to 1 for n--t+m only trees of order 5 k
outside the giant component, the distribution of the number of trees of order k
e-W
having in the limit again a Poisson distribution with mean value -
k-k!’
Thus for k=I, i.e. for
(4) N(n) = f log n+yn+o(n)
rn,~~(,,) consists, with probability tending to 1 for n-++w, only of a connected
component containing n-O(l) points and a few isolated points, the distribution of
the number of these being approximately a Poisson distribution with mean value
e-*Y. Thus in case (4) the probability that the whole graph r,, .Qcn) is connected
tends to e-cm211 for ti+ +a and thus this probability approaches 1 as y increases.
This last result has been obtained by us already in 1958 (see [Z]). The proba-
bility of rz$ being connected has been investigated by E. N. Gilbert (see 141).
It should be mentioned that the investigation of r;f: can be reduced to that of
rn,N as f0110d): r,:: can be obtained by first choosing the value k of a random
variable 6 having the binomial distribution P(@ = k) = ((i))p” (l-~)(;)-~ where
2) The mean number of components of ,‘F, .v has been investigated in [S]. Our results for
Tn,.v are however more Ear reaching.
3) This idea has been used by J. Hcijek (51 in the theory of sampling from a finite population
who has shown in this way that the Lindeberg-type conditions given by us [6] for the
validity of the central limit theorem for samples from a finite population are not only
sufficient but also necessary.
5. P. ERDOS AX A. RBr;l-I 3-17
p=!k’/(‘) and then choosing 1’~. In this way one can show that the threshold and
the threshold function for connectivity of r$$ are the same as that of Pn,!v. It
should bc mcntioncd that this dots not follow from the inequalities given by
Gilbert [4].
PJrase 5. consists of the range lV(n)-(M. log n) W(n) where a(:!nj-+ 00. In
this range the whole graph is not only almost surely connected, but the orders of
all points are almost surely asymp:o:ically equal. Thus the graph becomes in this
phase “asymptotically regular”.
References
[l] P. Erdbs-A.Rknyi : On the evolution of random graphs, PubEicatiorr. 0-f the .Zlathemutxal
Institute oj+ the Hungarian Academy of Sciewes 5 (1960) (in print:.
I”] P. Erdk-A.R&yi : On random graphs, Publicationes Mathematicae, 6 (1959j 290-297
r3] T. L. Austin, Ii. E. Fagen, W. F. Penney and J. Riordan: The number of components in
random hnear graphs, Annals o-f IVfathematical Statistzcs, 30 (1959‘ 747-751
[ill E. N. Gdbert: Random graphs, .-lnlzal~ af ,Mathematzcal Statzstics, 30 (lYSY!, 1141-1144
[5] J HBjek: Limitmg distributions in simple random sampling from a finite population,
Publlcatlons oJ the .24athematical hstztute of the Hungarzan Academy o-f Scxnces 5 (1960)
I in print!.
[ii] P. Erdos and A. Rtnyl: On the central limit theorem fur samples from ;I finite pupulatlon,
Publicatmns oJ‘ the Mathematical Instztute of the FIuxgarzall .l~adem~ oj‘ SLte)lces,
4 (1959) 49-61.
Soit &,A l’ensemble de tous les graphes possedants IZ sommets dunnes et ayant
A- arcs. Nous considerons seulement des graphes non-orient& et bans boucles. Un
graph aleatoir J’,,,., est defini comme un element de l’ensemble E,, 1 choisi au hasard
tel que tous les elements de ET<,?; ont la m&me probabilite d’&re choisis.
Les auteurs considerent les proprietes probables de P7i.~ quand II et N tends
verj I’infini d’un tel faGon que E=lV(n) est une fonction don&e de 71.