{"history":[],"article":{"id":"93809c97-0ff2-46d6-a588-37f7800f87d3","title":"Random graph","slug":"random-graph","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), **random graph** is the general term to refer to [probability distributions](https://bliptext.com/articles/probability-distribution) over [graphs](https://bliptext.com/articles/graph-discrete-mathematics).\nRandom graphs may be described simply by a probability distribution, or by a [random process](https://bliptext.com/articles/random-process) which generates them.\nThe theory of random graphs lies at the intersection between [graph theory](https://bliptext.com/articles/graph-theory) and [probability theory](https://bliptext.com/articles/probability-theory).\nFrom a mathematical perspective, random graphs are used to answer questions about the properties of typical graphs.\nIts practical applications are found in all areas in which [complex networks](https://bliptext.com/articles/complex-network) need to be modeled – many random graph models are thus known, mirroring the diverse types of [complex networks](https://bliptext.com/articles/complex-network) encountered in different areas.\nIn a mathematical context, random graph refers almost exclusively to the [Erdős–Rényi random graph model](https://bliptext.com/articles/erd-s-r-nyi-model).\nIn other contexts, any graph model may be referred to as a random graph.\n\n# Models\nA random graph is obtained by starting with a set of n isolated vertices and adding successive edges between them at random.\nThe aim of the study in this field is to determine at what stage a particular property of the graph is likely to arise.\nDifferent **random graph models** produce different [probability distributions](https://bliptext.com/articles/probability-distribution) on graphs.\nMost commonly studied is the one proposed by [Edgar Gilbert](https://bliptext.com/articles/edgar-gilbert) but often called the [Erdős–Rényi model](https://bliptext.com/articles/erd-s-r-nyi-model), denoted G(n,p).\nIn it, every possible edge occurs independently with probability 0 < p < 1.\nThe probability of obtaining any one particular random graph with m edges is with the notation.\nA closely related model, also called the Erdős–Rényi model and denoted G(n,M), assigns equal probability to all graphs with exactly M edges.\nWith 0 ≤ M ≤ N, G(n,M) has elements and every element occurs with probability.\nThe G(n,M) model can be viewed as a snapshot at a particular time (M) of the **random graph process** \\tilde{G}_n, a [stochastic process](https://bliptext.com/articles/stochastic-process) that starts with n vertices and no edges, and at each step adds one new edge chosen uniformly from the set of missing edges.\nIf instead we start with an infinite set of vertices, and again let every possible edge occur independently with probability 0 < p < 1, then we get an object G called an **infinite random graph**.\nExcept in the trivial cases when p is 0 or 1, such a G [almost surely](https://bliptext.com/articles/almost-surely) has the following property:\n\"Given any n + m elements, there is a vertex c in V that is adjacent to each of and is not adjacent to any of.\"\nIt turns out that if the vertex set is [countable](https://bliptext.com/articles/countable) then there is, [up to](https://bliptext.com/articles/up-to) [isomorphism](https://bliptext.com/articles/graph-isomorphism), only a single graph with this property, namely the [Rado graph](https://bliptext.com/articles/rado-graph).\nThus any countably infinite **random graph** is almost surely the Rado graph, which for this reason is sometimes called simply the **random graph**.\nHowever, the analogous result is not true for uncountable graphs, of which there are many (nonisomorphic) graphs satisfying the above property.\nAnother model, which generalizes Gilbert's random graph model, is the **random dot-product model**.\nA random dot-product graph associates with each vertex a [real vector](https://bliptext.com/articles/real-vector).\nThe probability of an edge **u****v** between any **v**ertices **u** and **v** is some f**u**nction of the dot prod**u**ct **u** • **v** of their respecti**v**e **v**ectors.\nThe [network probability matrix](https://bliptext.com/articles/network-probability-matrix) models random graphs through edge probabilities, which represent the probability p_{i,j} that a given edge e_{i,j} exists for a specified time period.\nThis model is extensible to directed and undirected; weighted and unweighted; and static or dynamic graphs structure.\nFor M ≃ pN, where N is the maximal number of edges possible, the two most widely used models, G(n,M) and G(n,p), are almost interchangeable.\n[Random regular graphs](https://bliptext.com/articles/random-regular-graph) form a special case, with properties that may differ from random graphs in general.\nOnce we have a model of random graphs, every function on graphs, becomes a [random variable](https://bliptext.com/articles/random-variable).\nThe study of this model is to determine if, or at least estimate the probability that, a property may occur.\n\n# Terminology\nThe term 'almost every' in the context of random graphs refers to a sequence of spaces and probabilities, such that the error probabilities tend to zero.\n\n# Properties\nThe theory of random graphs studies typical properties of random graphs, those that hold with high probability for graphs drawn from a particular distribution.\nFor example, we might ask for a given value of n and p what the probability is that G(n,p) is [connected](https://bliptext.com/articles/connection-mathematics).\nIn studying such questions, researchers often concentrate on the asymptotic behavior of random graphs—the values that various probabilities converge to as n grows very large.\n[Percolation theory](https://bliptext.com/articles/percolation-theory) characterizes the connectedness of random graphs, especially infinitely large ones.\nPercolation is related to the robustness of the graph (called also network).\nGiven a random graph of n nodes and an average degree.\nNext we remove randomly a fraction 1-p of nodes and leave only a fraction p. There exists a critical percolation threshold below which the network becomes fragmented while above p_c a giant connected component exists.\nLocalized percolation refers to removing a node its neighbors, next nearest neighbors etc. until a fraction of 1-p of nodes from the network is removed.\nIt was shown that for random graph with Poisson distribution of degrees exactly as for random removal.\nRandom graphs are widely used in the [probabilistic method](https://bliptext.com/articles/probabilistic-method), where one tries to prove the existence of graphs with certain properties.\nThe existence of a property on a random graph can often imply, via the [Szemerédi regularity lemma](https://bliptext.com/articles/szemer-di-regularity-lemma), the existence of that property on almost all graphs.\nIn [random regular graphs](https://bliptext.com/articles/random-regular-graph), G(n,r-reg) are the set of r-regular graphs with r = r(n) such that n and m are the natural numbers, 3 \\le r < n, and rn = 2m is even.\nThe degree sequence of a graph G in G^n depends only on the number of edges in the sets\nIf edges, M in a random graph, G_M is large enough to ensure that almost every G_M has minimum degree at least 1, then almost every G_M is connected and, if n is even, almost every G_M has a perfect matching.\nIn particular, the moment the last isolated vertex vanishes in almost every random graph, the graph becomes connected.\nAlmost every graph process on an even number of vertices with the edge raising the minimum degree to 1 or a random graph with slightly more than edges and with probability close to 1 ensures that the graph has a complete matching, with exception of at most one vertex.\nFor some constant c, almost every labeled graph with n vertices and at least cn\\log(n) edges is [Hamiltonian](https://bliptext.com/articles/hamiltonian-cycle).\nWith the probability tending to 1, the particular edge that increases the minimum degree to 2 makes the graph Hamiltonian.\nProperties of random graph may change or remain invariant under graph transformations.\n[Mashaghi A.](https://bliptext.com/articles/alireza-mashaghi) et al., for example, demonstrated that a transformation which converts random graphs to their edge-dual graphs (or line graphs) produces an ensemble of graphs with nearly the same degree distribution, but with degree correlations and a significantly higher clustering coefficient.\n\n# Colouring\nGiven a random graph G of order n with the vertex V(G) = {1, ..., n}, by the [greedy algorithm](https://bliptext.com/articles/greedy-algorithm) on the number of colors, the vertices can be colored with colors 1, 2, ... (vertex 1 is colored 1, vertex 2 is colored 1 if it is not adjacent to vertex 1, otherwise it is colored 2, etc.).\nThe number of proper colorings of random graphs given a number of q colors, called its [chromatic polynomial](https://bliptext.com/articles/chromatic-polynomial), remains unknown so far.\nThe scaling of zeros of the chromatic polynomial of random graphs with parameters n and the number of edges m or the connection probability p has been studied empirically using an algorithm based on symbolic pattern matching.\n\n# Random trees\nA [random tree](https://bliptext.com/articles/random-[tree](https://bliptext.com/articles/tree-graph-theory)) is a [tree](https://bliptext.com/articles/tree-graph-theory) or [arborescence](https://bliptext.com/articles/arborescence-graph-theory) that is formed by a [stochastic process](https://bliptext.com/articles/stochastic-process).\nIn a large range of random graphs of order n and size M(n) the distribution of the number of tree components of order k is asymptotically [Poisson](https://bliptext.com/articles/poisson-distribution).\nTypes of random trees include [uniform spanning tree](https://bliptext.com/articles/uniform-spanning-tree), [random minimum spanning tree](https://bliptext.com/articles/random-minimum-spanning-tree), [random binary tree](https://bliptext.com/articles/random-binary-tree), [treap](https://bliptext.com/articles/treap), [rapidly exploring random tree](https://bliptext.com/articles/rapidly-exploring-random-tree), [Brownian tree](https://bliptext.com/articles/brownian-tree), and [random forest](https://bliptext.com/articles/random-forest).\n\n# Conditional random graphs\nConsider a given random graph model defined on the probability space and let be a real valued function which assigns to each graph in \\Omega a vector of m properties.\nFor a fixed, conditional random graphs are models in which the probability measure P assigns zero probability to all graphs such that '.\nSpecial cases are conditionally uniform random graphs, where P assigns equal probability to all the graphs having specified properties.\nThey can be seen as a generalization of the [Erdős–Rényi model](https://bliptext.com/articles/erd-s-r-nyi-model) G(n,M), when the conditioning information is not necessarily the number of edges M, but whatever other arbitrary graph property.\nIn this case very few analytical results are available and simulation is required to obtain empirical distributions of average properties.\n\n# History\nThe earliest use of a random graph model was by [Helen Hall Jennings](https://bliptext.com/articles/helen-hall-jennings) and [Jacob Moreno](https://bliptext.com/articles/jacob-moreno) in 1938 where a \"chance sociogram\" (a directed Erdős-Rényi model) was considered in studying comparing the fraction of reciprocated links in their network data with the random model.\nAnother use, under the name \"random net\", was by [Ray Solomonoff](https://bliptext.com/articles/ray-solomonoff) and [Anatol Rapoport](https://bliptext.com/articles/anatol-rapoport) in 1951, using a model of directed graphs with fixed out-degree and randomly chosen attachments to other vertices.\nThe [Erdős–Rényi model](https://bliptext.com/articles/erd-s-r-nyi-model) of random graphs was first defined by [Paul Erdős](https://bliptext.com/articles/paul-erd-s) and [Alfréd Rényi](https://bliptext.com/articles/alfr-d-r-nyi) in their 1959 paper \"On Random Graphs\" and independently by Gilbert in his paper \"Random graphs\".","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:19:10.343Z","updated_at":"2025-01-25T19:19:10.343Z","search_vector":"'graph':2 'random':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), **random graph** is the general term to refer to [probability distributions](https://bliptext.com/articles/probability-distribution) over [graphs](https://bliptext.com/articles/graph-discrete-mathematics).\nRandom graphs may be described simply by a probability distribution, or by a [random process](https://bliptext.com/articles/random-process) which generates them.\nThe theory of random graphs lies at the intersection between [graph theory](https://bliptext.com/articles/graph-theory) and [probability theory](https://bliptext.com/articles/probability-theory).\nFrom a mathematical perspective, random graphs are used to answer questions about the properties of typical graphs.\nIts practical applications are found in all areas in which [complex networks](https://bliptext.com/articles/complex-network) need to be modeled – many random graph models are thus known, mirroring the diverse types of [complex networks](https://bliptext.com/articles/complex-network) encountered in different areas.\nIn a mathematical context, random graph refers almost exclusively to the [Erdős–Rényi random graph model](https://bliptext.com/articles/erd-s-r-nyi-model).\nIn other contexts, any graph model may be referred to as a random graph.\n\n# Models\nA random graph is obtained by starting with a set of n isolated vertices and adding successive edges between them at random.\nThe aim of the study in this field is to determine at what stage a particular property of the graph is likely to arise.\nDifferent **random graph models** produce different [probability distributions](https://bliptext.com/articles/probability-distribution) on graphs.\nMost commonly studied is the one proposed by [Edgar Gilbert](https://bliptext.com/articles/edgar-gilbert) but often called the [Erdős–Rényi model](https://bliptext.com/articles/erd-s-r-nyi-model), denoted G(n,p).\nIn it, every possible edge occurs independently with probability 0 < p < 1.\nThe probability of obtaining any one particular random graph with m edges is with the notation.\nA closely related model, also called the Erdős–Rényi model and denoted G(n,M), assigns equal probability to all graphs with exactly M edges.\nWith 0 ≤ M ≤ N, G(n,M) has elements and every element occurs with probability.\nThe G(n,M) model can be viewed as a snapshot at a particular time (M) of the **random graph process** \\tilde{G}_n, a [stochastic process](https://bliptext.com/articles/stochastic-process) that starts with n vertices and no edges, and at each step adds one new edge chosen uniformly from the set of missing edges.\nIf instead we start with an infinite set of vertices, and again let every possible edge occur independently with probability 0 < p < 1, then we get an object G called an **infinite random graph**.\nExcept in the trivial cases when p is 0 or 1, such a G [almost surely](https://bliptext.com/articles/almost-surely) has the following property:\n\"Given any n + m elements, there is a vertex c in V that is adjacent to each of and is not adjacent to any of.\"\nIt turns out that if the vertex set is [countable](https://bliptext.com/articles/countable) then there is, [up to](https://bliptext.com/articles/up-to) [isomorphism](https://bliptext.com/articles/graph-isomorphism), only a single graph with this property, namely the [Rado graph](https://bliptext.com/articles/rado-graph).\nThus any countably infinite **random graph** is almost surely the Rado graph, which for this reason is sometimes called simply the **random graph**.\nHowever, the analogous result is not true for uncountable graphs, of which there are many (nonisomorphic) graphs satisfying the above property.\nAnother model, which generalizes Gilbert's random graph model, is the **random dot-product model**.\nA random dot-product graph associates with each vertex a [real vector](https://bliptext.com/articles/real-vector).\nThe probability of an edge **u****v** between any **v**ertices **u** and **v** is some f**u**nction of the dot prod**u**ct **u** • **v** of their respecti**v**e **v**ectors.\nThe [network probability matrix](https://bliptext.com/articles/network-probability-matrix) models random graphs through edge probabilities, which represent the probability p_{i,j} that a given edge e_{i,j} exists for a specified time period.\nThis model is extensible to directed and undirected; weighted and unweighted; and static or dynamic graphs structure.\nFor M ≃ pN, where N is the maximal number of edges possible, the two most widely used models, G(n,M) and G(n,p), are almost interchangeable.\n[Random regular graphs](https://bliptext.com/articles/random-regular-graph) form a special case, with properties that may differ from random graphs in general.\nOnce we have a model of random graphs, every function on graphs, becomes a [random variable](https://bliptext.com/articles/random-variable).\nThe study of this model is to determine if, or at least estimate the probability that, a property may occur.\n\n# Terminology\nThe term 'almost every' in the context of random graphs refers to a sequence of spaces and probabilities, such that the error probabilities tend to zero.\n\n# Properties\nThe theory of random graphs studies typical properties of random graphs, those that hold with high probability for graphs drawn from a particular distribution.\nFor example, we might ask for a given value of n and p what the probability is that G(n,p) is [connected](https://bliptext.com/articles/connection-mathematics).\nIn studying such questions, researchers often concentrate on the asymptotic behavior of random graphs—the values that various probabilities converge to as n grows very large.\n[Percolation theory](https://bliptext.com/articles/percolation-theory) characterizes the connectedness of random graphs, especially infinitely large ones.\nPercolation is related to the robustness of the graph (called also network).\nGiven a random graph of n nodes and an average degree.\nNext we remove randomly a fraction 1-p of nodes and leave only a fraction p. There exists a critical percolation threshold below which the network becomes fragmented while above p_c a giant connected component exists.\nLocalized percolation refers to removing a node its neighbors, next nearest neighbors etc. until a fraction of 1-p of nodes from the network is removed.\nIt was shown that for random graph with Poisson distribution of degrees exactly as for random removal.\nRandom graphs are widely used in the [probabilistic method](https://bliptext.com/articles/probabilistic-method), where one tries to prove the existence of graphs with certain properties.\nThe existence of a property on a random graph can often imply, via the [Szemerédi regularity lemma](https://bliptext.com/articles/szemer-di-regularity-lemma), the existence of that property on almost all graphs.\nIn [random regular graphs](https://bliptext.com/articles/random-regular-graph), G(n,r-reg) are the set of r-regular graphs with r = r(n) such that n and m are the natural numbers, 3 \\le r < n, and rn = 2m is even.\nThe degree sequence of a graph G in G^n depends only on the number of edges in the sets\nIf edges, M in a random graph, G_M is large enough to ensure that almost every G_M has minimum degree at least 1, then almost every G_M is connected and, if n is even, almost every G_M has a perfect matching.\nIn particular, the moment the last isolated vertex vanishes in almost every random graph, the graph becomes connected.\nAlmost every graph process on an even number of vertices with the edge raising the minimum degree to 1 or a random graph with slightly more than edges and with probability close to 1 ensures that the graph has a complete matching, with exception of at most one vertex.\nFor some constant c, almost every labeled graph with n vertices and at least cn\\log(n) edges is [Hamiltonian](https://bliptext.com/articles/hamiltonian-cycle).\nWith the probability tending to 1, the particular edge that increases the minimum degree to 2 makes the graph Hamiltonian.\nProperties of random graph may change or remain invariant under graph transformations.\n[Mashaghi A.](https://bliptext.com/articles/alireza-mashaghi) et al., for example, demonstrated that a transformation which converts random graphs to their edge-dual graphs (or line graphs) produces an ensemble of graphs with nearly the same degree distribution, but with degree correlations and a significantly higher clustering coefficient.\n\n# Colouring\nGiven a random graph G of order n with the vertex V(G) = {1, ..., n}, by the [greedy algorithm](https://bliptext.com/articles/greedy-algorithm) on the number of colors, the vertices can be colored with colors 1, 2, ... (vertex 1 is colored 1, vertex 2 is colored 1 if it is not adjacent to vertex 1, otherwise it is colored 2, etc.).\nThe number of proper colorings of random graphs given a number of q colors, called its [chromatic polynomial](https://bliptext.com/articles/chromatic-polynomial), remains unknown so far.\nThe scaling of zeros of the chromatic polynomial of random graphs with parameters n and the number of edges m or the connection probability p has been studied empirically using an algorithm based on symbolic pattern matching.\n\n# Random trees\nA [random [tree](https://bliptext.com/articles/tree-graph-theory)](https://bliptext.com/articles/random-[tree](https://bliptext.com/articles/tree-graph-theory)) is a [tree](https://bliptext.com/articles/tree-graph-theory) or [arborescence](https://bliptext.com/articles/arborescence-graph-theory) that is formed by a [stochastic process](https://bliptext.com/articles/stochastic-process).\nIn a large range of random graphs of order n and size M(n) the distribution of the number of tree components of order k is asymptotically [Poisson](https://bliptext.com/articles/poisson-distribution).\nTypes of random trees include [uniform spanning tree](https://bliptext.com/articles/uniform-spanning-tree), [random minimum spanning tree](https://bliptext.com/articles/random-minimum-spanning-tree), [random binary tree](https://bliptext.com/articles/random-binary-tree), [treap](https://bliptext.com/articles/treap), [rapidly exploring random tree](https://bliptext.com/articles/rapidly-exploring-random-tree), [Brownian tree](https://bliptext.com/articles/brownian-tree), and [random forest](https://bliptext.com/articles/random-forest).\n\n# Conditional random graphs\nConsider a given random graph model defined on the probability space and let be a real valued function which assigns to each graph in \\Omega a vector of m properties.\nFor a fixed, conditional random graphs are models in which the probability measure P assigns zero probability to all graphs such that '.\nSpecial cases are conditionally uniform random graphs, where P assigns equal probability to all the graphs having specified properties.\nThey can be seen as a generalization of the [Erdős–Rényi model](https://bliptext.com/articles/erd-s-r-nyi-model) G(n,M), when the conditioning information is not necessarily the number of edges M, but whatever other arbitrary graph property.\nIn this case very few analytical results are available and simulation is required to obtain empirical distributions of average properties.\n\n# History\nThe earliest use of a random graph model was by [Helen Hall Jennings](https://bliptext.com/articles/helen-hall-jennings) and [Jacob Moreno](https://bliptext.com/articles/jacob-moreno) in 1938 where a \"chance sociogram\" (a directed Erdős-Rényi model) was considered in studying comparing the fraction of reciprocated links in their network data with the random model.\nAnother use, under the name \"random net\", was by [Ray Solomonoff](https://bliptext.com/articles/ray-solomonoff) and [Anatol Rapoport](https://bliptext.com/articles/anatol-rapoport) in 1951, using a model of directed graphs with fixed out-degree and randomly chosen attachments to other vertices.\nThe [Erdős–Rényi model](https://bliptext.com/articles/erd-s-r-nyi-model) of random graphs was first defined by [Paul Erdős](https://bliptext.com/articles/paul-erd-s) and [Alfréd Rényi](https://bliptext.com/articles/alfr-d-r-nyi) in their 1959 paper \"On Random Graphs\" and independently by Gilbert in his paper \"Random graphs\".","revisionCount":0},"hasMore":false,"nextCursor":null}