Random graph

Inmathematics,random graphis the general term to refer toprobability distributionsovergraphs.Random graphs may be described simply by a probability distribution, or by arandom processwhich generates them.^[1]^[2]The theory of random graphs lies at the intersection betweengraph theoryandprobability theory.From a mathematical perspective, random graphs are used to answer questions about the properties oftypicalgraphs. Its practical applications are found in all areas in whichcomplex networksneed to be modeled – many random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas. In a mathematical context,random graphrefers almost exclusively to theErdős–Rényi random graph model.In other contexts, any graph model may be referred to as arandom graph.

Models

A random graph is obtained by starting with a set ofnisolated vertices and adding successive edges between them at random. The aim of the study in this field is to determine at what stage a particular property of the graph is likely to arise.^[3]Differentrandom graph modelsproduce differentprobability distributionson graphs. Most commonly studied is the one proposed byEdgar Gilbert,denotedG(n,p), in which every possible edge occurs independently with probability 0 <p< 1. The probability of obtainingany one particularrandom graph withmedges is $p^{m}(1-p)^{N-m}$ with the notation $N={\tbinom {n}{2}}$ .^[4]

A closely related model, theErdős–Rényi modeldenotedG(n,M), assigns equal probability to all graphs with exactlyMedges. With 0 ≤M≤N,G(n,M) has ${\tbinom {N}{M}}$ elements and every element occurs with probability $1/{\tbinom {N}{M}}$ .^[3]The latter model can be viewed as a snapshot at a particular time (M) of therandom graph process ${\tilde {G}}_{n}$ ,which is astochastic processthat starts withnvertices and no edges, and at each step adds one new edge chosen uniformly from the set of missing edges.

If 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 objectGcalled aninfinite random graph.Except in the trivial cases whenpis 0 or 1, such aGalmost surelyhas the following property:

Given anyn+melements $a_{1},\ldots,a_{n},b_{1},\ldots,b_{m}\in V$ ,there is a vertexcinVthat is adjacent to each of $a_{1},\ldots,a_{n}$ and is not adjacent to any of $b_{1},\ldots,b_{m}$ .

It turns out that if the vertex set iscountablethen there is,up to isomorphism,only a single graph with this property, namely theRado graph.Thus any countably infinite random graph is almost surely the Rado graph, which for this reason is sometimes called simply therandom graph.However, the analogous result is not true for uncountable graphs, of which there are many (nonisomorphic) graphs satisfying the above property.

Another model, which generalizes Gilbert's random graph model, is therandom dot-product model.A random dot-product graph associates with each vertex areal vector.The probability of an edgeuvbetween any verticesuandvis some function of thedot productu•vof their respective vectors.

Thenetwork probability matrixmodels 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. This model is extensible to directed and undirected; weighted and unweighted; and static or dynamic graphs structure.

ForM≃pN,whereNis the maximal number of edges possible, the two most widely used models,G(n,M) andG(n,p), are almost interchangeable.^[5]

Random regular graphsform a special case, with properties that may differ from random graphs in general.

Once we have a model of random graphs, every function on graphs, becomes arandom variable.The study of this model is to determine if, or at least estimate the probability that, a property may occur.^[4]

Terminology

The term 'almost every' in the context of random graphs refers to a sequence of spaces and probabilities, such that theerror probabilitiestend to zero.^[4]

Properties

The theory of random graphs studies typical properties of random graphs, those that hold with high probability for graphs drawn from a particular distribution. For example, we might ask for a given value of $n$ and $p$ what the probability is that $G(n,p)$ isconnected.In 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.Percolation theorycharacterizes the connectedness of random graphs, especially infinitely large ones.

Percolation is related to the robustness of the graph (called also network). Given a random graph of $n$ nodes and an average degree $\langle k\rangle$ .Next we remove randomly a fraction $1-p$ of nodes and leave only a fraction $p$ .There exists a critical percolation threshold $p_{c}={\tfrac {1}{\langle k\rangle }}$ below which the network becomes fragmented while above $p_{c}$ a giant connected component exists.^[1]^[5]^[6]^[7]^[8]

Localized 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. It was shown that for random graph with Poisson distribution of degrees $p_{c}={\tfrac {1}{\langle k\rangle }}$ exactly as for random removal.

Random graphs are widely used in theprobabilistic method,where one tries to prove the existence of graphs with certain properties. The existence of a property on a random graph can often imply, via theSzemerédi regularity lemma,the existence of that property on almost all graphs.

Inrandom regular graphs, $G(n,r-reg)$ are the set of $r$ -regular graphs with $r=r(n)$ such that $n$ and $m$ are the natural numbers, $3\leq r<n$ ,and $rn=2m$ is even.^[3]

The degree sequence of a graph $G$ in $G^{n}$ depends only on the number of edges in the sets^[3]

V_{n}^{(2)}=\left\{ij\:\ 1\leq j\leq n,i\neq j\right\}\subset V^{(2)},\qquad i=1,\cdots,n.

If 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. In particular, the moment the last isolated vertex vanishes in almost every random graph, the graph becomes connected.^[3]

Almost 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 ${\tfrac {n}{4}}\log(n)$ edges and with probability close to 1 ensures that the graph has a complete matching, with exception of at most one vertex.

For some constant $c$ ,almost every labeled graph with $n$ vertices and at least $cn\log(n)$ edges isHamiltonian.With the probability tending to 1, the particular edge that increases the minimum degree to 2 makes the graph Hamiltonian.

Properties of random graph may change or remain invariant under graph transformations.Mashaghi A.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.^[9]

Colouring

Given a random graphGof ordernwith the vertexV(G) = {1,...,n}, by thegreedy algorithmon 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.).^[3] The number of proper colorings of random graphs given a number ofqcolors, called itschromatic polynomial,remains unknown so far. The scaling of zeros of the chromatic polynomial of random graphs with parametersnand the number of edgesmor the connection probabilityphas been studied empirically using an algorithm based on symbolic pattern matching.^[10]

Random trees

Arandom treeis atreeorarborescencethat is formed by astochastic process.In a large range of random graphs of ordernand sizeM(n) the distribution of the number of tree components of orderkis asymptoticallyPoisson.Types of random trees includeuniform spanning tree,random minimal spanning tree,random binary tree,treap,rapidly exploring random tree,Brownian tree,andrandom forest.

Conditional random graphs

Consider a given random graph model defined on the probability space $(\Omega,{\mathcal {F}},P)$ and let ${\mathcal {P}}(G):\Omega \rightarrow R^{m}$ be a real valued function which assigns to each graph in $\Omega$ a vector ofmproperties. For a fixed $\mathbf {p} \in R^{m}$ ,conditional random graphsare models in which the probability measure $P$ assigns zero probability to all graphs such that ' ${\mathcal {P}}(G)\neq \mathbf {p}$ .

Special cases areconditionally uniform random graphs,where $P$ assigns equal probability to all the graphs having specified properties. They can be seen as a generalization of theErdős–Rényi modelG(n,M), when the conditioning information is not necessarily the number of edgesM,but whatever other arbitrary graph property ${\mathcal {P}}(G)$ .In this case very few analytical results are available and simulation is required to obtain empirical distributions of average properties.

History

The earliest use of a random graph model was byHelen Hall JenningsandJacob Morenoin 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.^[11]Another use, under the name "random net", was byRay SolomonoffandAnatol Rapoportin 1951, using a model of directed graphs with fixed out-degree and randomly chosen attachments to other vertices.^[12]

TheErdős–Rényi modelof random graphs was first defined byPaul ErdősandAlfréd Rényiin their 1959 paper "On Random Graphs"^[8]and independently by Gilbert in his paper "Random graphs".^[6]

References

^^a ^bBollobás, Béla (2001).Random Graphs(2nd ed.). Cambridge University Press.
^Frieze, Alan; Karonski, Michal (2015).Introduction to Random Graphs.Cambridge University Press.
^^a ^b ^c ^d ^e ^fBéla Bollobás,Random Graphs,1985, Academic Press Inc., London Ltd.
^^a ^b ^cBéla Bollobás,Probabilistic Combinatorics and Its Applications,1991, Providence, RI: American Mathematical Society.
^^a ^bBollobas, B.and Riordan, O.M. "Mathematical results on scale-free random graphs" in "Handbook of Graphs and Networks" (S. Bornholdt and H.G. Schuster (eds)), Wiley VCH, Weinheim, 1st ed., 2003
^^a ^bGilbert, E. N.(1959), "Random graphs",Annals of Mathematical Statistics,30(4): 1141–1144,doi:10.1214/aoms/1177706098.
^Newman, M. E. J. (2010).Networks: An Introduction.Oxford.
^^a ^bErdős, P.Rényi, A(1959) "On Random Graphs I" in Publ. Math. Debrecen 6, p. 290–297[1]Archived2020-08-07 at theWayback Machine
^Ramezanpour, A.; Karimipour, V.; Mashaghi, A. (2003). "Generating correlated networks from uncorrelated ones".Phys. Rev. E.67(46107): 046107.arXiv:cond-mat/0212469.Bibcode:2003PhRvE..67d6107R.doi:10.1103/PhysRevE.67.046107.PMID 12786436.S2CID 33054818.
^Van Bussel, Frank; Ehrlich, Christoph; Fliegner, Denny; Stolzenberg, Sebastian; Timme, Marc (2010). "Chromatic Polynomials of Random Graphs".J. Phys. A: Math. Theor.43(17): 175002.arXiv:1709.06209.Bibcode:2010JPhA...43q5002V.doi:10.1088/1751-8113/43/17/175002.S2CID 15723612.
^Moreno, Jacob L; Jennings, Helen Hall (Jan 1938)."Statistics of Social Configurations"(PDF).Sociometry.1(3/4): 342–374.doi:10.2307/2785588.JSTOR 2785588.
^Solomonoff, Ray; Rapoport, Anatol (June 1951). "Connectivity of random nets".Bulletin of Mathematical Biophysics.13(2): 107–117.doi:10.1007/BF02478357.

[Random_Graphs-1] Bollobás, Béla (2001).Random Graphs(2nd ed.). Cambridge University Press.

[Introduction_to_Random_graphs-2] Frieze, Alan; Karonski, Michal (2015).Introduction to Random Graphs.Cambridge University Press.

[Random_Graphs2-3] Béla Bollobás,Random Graphs,1985, Academic Press Inc., London Ltd.

[Random_Graphs3-4] Béla Bollobás,Probabilistic Combinatorics and Its Applications,1991, Providence, RI: American Mathematical Society.

[Handbook-5] Bollobas, B.and Riordan, O.M. "Mathematical results on scale-free random graphs" in "Handbook of Graphs and Networks" (S. Bornholdt and H.G. Schuster (eds)), Wiley VCH, Weinheim, 1st ed., 2003

[Random_graphs-6] Gilbert, E. N.(1959), "Random graphs",Annals of Mathematical Statistics,30(4): 1141–1144,doi:10.1214/aoms/1177706098.

[7] Newman, M. E. J. (2010).Networks: An Introduction.Oxford.

[On_Random_Graphs-8] Erdős, P.Rényi, A(1959) "On Random Graphs I" in Publ. Math. Debrecen 6, p. 290–297[1]Archived2020-08-07 at theWayback Machine

[9] Ramezanpour, A.; Karimipour, V.; Mashaghi, A. (2003). "Generating correlated networks from uncorrelated ones".Phys. Rev. E.67(46107): 046107.arXiv:cond-mat/0212469.Bibcode:2003PhRvE..67d6107R.doi:10.1103/PhysRevE.67.046107.PMID 12786436.S2CID 33054818.

[Chromatic_Polynomials_of_Random_Graphs-10] Van Bussel, Frank; Ehrlich, Christoph; Fliegner, Denny; Stolzenberg, Sebastian; Timme, Marc (2010). "Chromatic Polynomials of Random Graphs".J. Phys. A: Math. Theor.43(17): 175002.arXiv:1709.06209.Bibcode:2010JPhA...43q5002V.doi:10.1088/1751-8113/43/17/175002.S2CID 15723612.

[11] Moreno, Jacob L; Jennings, Helen Hall (Jan 1938)."Statistics of Social Configurations"(PDF).Sociometry.1(3/4): 342–374.doi:10.2307/2785588.JSTOR 2785588.

[12] Solomonoff, Ray; Rapoport, Anatol (June 1951). "Connectivity of random nets".Bulletin of Mathematical Biophysics.13(2): 107–117.doi:10.1007/BF02478357.

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