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Graph generated by a random process
In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability
Random_graph
Two closely related models for generating random graphs
mathematical field of graph theory, the Erdős–Rényi models are two closely related models for generating random graphs and the evolution of a random network. These
Erdős–Rényi_model
A random r-regular graph is a graph selected from 𝒢n,r, which denotes the probability space of all r-regular graphs on n vertices, where 3≤r<n and nr
Random_regular_graph
Statistical models for network analysis
Exponential family random graph models (ERGMs) are a set of statistical models used to study the structure and patterns within networks, such as those
Exponential family random graph models
Exponential_family_random_graph_models
Search algorithm
methods do themselves converge to an optimum, though. Rapidly exploring random graph (RRG) and RRT*, a variant of RRT that converges towards an optimal solution
Rapidly_exploring_random_tree
Area of discrete mathematics
known as random graph theory, which has been a fruitful source of graph-theoretic results. Topological graph theory deals with the study of graphs as topological
Graph_theory
In graph theory, the mathematically simplest spatial network
In graph theory, a random geometric graph (RGG) is the mathematically simplest spatial network, namely an undirected graph constructed by randomly placing
Random_geometric_graph
Maximal subgraph whose vertices can reach each other
related to invariants of matroids, topological spaces, and matrices. In random graphs, a frequently occurring phenomenon is the incidence of a giant component
Component_(graph_theory)
Infinite graph containing all countable graphs
In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with
Rado_graph
Function type in graph theory
objects of exchangeable random graph models. Graphons are tied to dense graphs by the following pair of observations: the random graph models defined by graphons
Graphon
Academic field
offshoot of graph theory with Paul Erdős and Alfréd Rényi's eight famous papers on random graphs. For social networks the exponential random graph model or
Network_science
Appendix:Glossary of graph theory in Wiktionary, the free dictionary. This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes
Glossary_of_graph_theory
Sparse graph with strong connectivity
In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander
Expander_graph
Type of random graph
statistical mechanics, probability theory, graph theory, etc. the random cluster model is a random graph that generalizes and unifies the Ising model
Random_cluster_model
Maximum-entropy random graph models are random graph models used to study complex networks subject to the principle of maximum entropy under a set of structural
Maximum-entropy random graph model
Maximum-entropy_random_graph_model
Algorithm that employs a degree of randomness as part of its logic or procedure
he used a simple randomized construction to establish the existence of Ramsey graphs. He famously used a more sophisticated randomized algorithm in 1959
Randomized_algorithm
Process forming a path from many random steps
distances bounded. A random walk on a graph is a very special case of a Markov chain. Unlike a general Markov chain, random walk on a graph enjoys a property
Random_walk
American mathematician
areas of spectral graph theory, extremal graph theory and random graphs, in particular in generalizing the Erdős–Rényi model for graphs with general degree
Fan_Chung
Network whose degree distribution follows a power law
scale-free. Random graph – Graph generated by a random process Erdős–Rényi model – Two closely related models for generating random graphs Non-linear preferential
Scale-free_network
Formation of a gel from a mass of polymers
component arises. The structure of a gel network can be conceptualised as a random graph. This analogy is exploited to calculate the gel point and gel fraction
Gelation
Structural analysis of a network
In network science, a biased random walk on a graph is a time path process in which an evolving variable jumps from its current state to one of various
Biased_random_walk_on_a_graph
Hungarian mathematician (born 1943)
research is combinatorics, particularly graph theory. His chief interests are in extremal graph theory and random graph theory. In 1996 he resigned his university
Béla_Bollobás
Mathematical theory on behavior of connected clusters in a random graph
since then. In a slightly different mathematical model for obtaining a random graph, a site is "occupied" with probability p or "empty" (in which case its
Percolation_theory
Method of generating random small-world graphs
The Watts–Strogatz model is a random graph generation model that produces graphs with small-world properties, including short average path lengths and
Watts–Strogatz_model
Graph where most nodes are reachable in a small number of steps
networks were identified as a class of random graphs by Duncan Watts and Steven Strogatz in 1998. They noted that graphs could be classified according to two
Small-world_network
Graph of numbers differing by a square
would in random graphs. The Paley graph of order 9 is a locally linear graph, a rook's graph, and the graph of the 3-3 duoprism. The Paley graph of order
Paley_graph
Complete subgraph added to a random graph
planted clique problem is the algorithmic problem of distinguishing random graphs from graphs that have a planted clique. This is a variation of the clique
Planted_clique
Model for a random simple path
spanning tree, a model for a random tree. It is a case of the more general topic of random walks. Assume G is some graph and γ {\displaystyle \gamma }
Loop-erased_random_walk
Approximate nearest neighbor search algorithm
datasets. HNSW stores vectors in a graph. Each vector is a node, and links connect it to some nearby vectors. The graph has several layers: upper layers
Hierarchical navigable small world
Hierarchical_navigable_small_world
Large connected component of a random graph
component of a given random graph that contains a significant fraction of the entire graph's vertices. More precisely, in graphs drawn randomly from a probability
Giant_component
Social structure made up of a set of social actors
the addition of autonomous agents to the groups. Randomly distributed networks: Exponential random graph models of social networks became state-of-the-art
Social_network
Spectral graph theory concept
spectral graph theory, a Ramanujan graph is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are
Ramanujan_graph
Family of random graph models
In network science, the Configuration Model is a family of random graph models designed to generate networks from a given degree sequence. Unlike simpler
Configuration_model
Australian-American statistician
degeneracy mitigation, and simulation-based inference of exponential-family random graph models (ERGMs). The framework provides probabilistic representations
Mark_S._Handcock
Set of random variables
Markov property described by an undirected graph. In other words, a random field is said to be a Markov random field if it satisfies Markov properties.
Markov_random_field
Measurement of graph sparsity
In graph theory, a k-degenerate graph is an undirected graph in which every non-empty subgraph has at least one vertex of degree at most k {\displaystyle
Degeneracy_(graph_theory)
Basic concept of graph theory
mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that
Connectivity_(graph_theory)
Conjecture in probabilistic combinatorics
of mathematics that studies the behavior of connected clusters in a random graph. The conjecture is named after its analogy to a bunk bed structure. It
Bunkbed_conjecture
Probability inequality for dependent events
central tool of the probabilistic method, particularly in the theory of random graphs. When the events are only weakly dependent, the bounds show that the
Janson_inequality
Graph defined from a set of points in the Euclidean plane
graphs of infinite random point sets, the finite site percolation threshold gives the fraction of points needed to support connectivity: if a random subset
Gabriel_graph
Matrix representation of a graph
In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix, or discrete Laplacian
Laplacian_matrix
Standard hostname for a networked device's loopback interface
Models Topology Random graph Erdős–Rényi Barabási–Albert Bianconi–Barabási Fitness model Watts–Strogatz Exponential random (ERGM) Random geometric (RGG)
Localhost
Network with non-trivial topological features
network is a graph (network) with non-trivial topological features—features that do not occur in simple networks such as lattices or random graphs but often
Complex_network
Clustering and community detection algorithm
hypothetical randomized partition of communities). In the above image, our initial collection of unsorted nodes is represented by the graph on the left
Leiden_algorithm
Automated methods for the creation of mazes
the maze generation steps for a graph that is not on a rectangular grid. First, the computer creates a random planar graph G shown in blue, and its dual
Maze_generation_algorithm
Random graph model in applied mathematics
is a random graph model subject to the principle of maximum entropy under constraints on the expectation of the degree sequence of sampled graphs. Whereas
Soft_configuration_model
Mathematical theory for sol–gel processes
Random graph theory of gelation is a mathematical theory for sol–gel processes. The theory is a collection of results that generalise the Flory–Stockmayer
Random graph theory of gelation
Random_graph_theory_of_gelation
Scale-free network generation algorithm
that they have power-law (or scale-free) degree distributions, while random graph models such as the Erdős–Rényi (ER) model and the Watts–Strogatz (WS)
Barabási–Albert_model
Mathematician at Courant Institute of Mathematical Sciences
estimating the positions of phase transitions in statistical mechanics and random graph theory. Park entered Seoul National University in 2001 and received her
Jinyoung_Park_(mathematician)
Network for communications over distance
Models Topology Random graph Erdős–Rényi Barabási–Albert Bianconi–Barabási Fitness model Watts–Strogatz Exponential random (ERGM) Random geometric (RGG)
Telecommunications_network
Index of articles associated with the same name
graph in which each different tree is equally likely to be selected Random minimal spanning tree, spanning trees of a graph formed by choosing random
Random_tree
Concept in network science
stochastic block model is a generative model for random graphs. This model tends to produce graphs containing communities, subsets of nodes characterized
Stochastic_block_model
Experiments examining the average path length for social networks
random graph Personal network – Set of human contacts known to an individual Random walk – Process forming a path from many random steps Random graph –
Small-world_experiment
Length of a shortest cycle contained in the graph
In graph theory, the girth of an undirected graph is the length of a shortest cycle contained in the graph. If the graph does not contain any cycles (that
Girth_(graph_theory)
Graph obeys some properties of random graphs
In graph theory, a graph is said to be a pseudorandom graph if it obeys certain properties that random graphs obey with high probability. There is no concrete
Pseudorandom_graph
Type of graph in mathematical graph theory
for random walks on graphs". Random Structures & Algorithms. 6: 51–54. doi:10.1002/rsa.3240060106. Jonasson, Johan (March 2000). "Lollipop graphs are
Lollipop_graph
Knowledge base that represents semantic relations between concepts in a network
used as a form of knowledge representation. It is a directed or undirected graph consisting of vertices, which represent concepts, and edges, which represent
Semantic_network
Network representing spatial objects
network is a lattice or a random geometric graph (see figure in the right), where nodes are distributed uniformly at random over a two-dimensional plane;
Spatial_network
Unsolved problem in computational complexity theory
perform well on random graphs, a major drawback of these algorithms is their exponential time performance in the worst case. The graph isomorphism problem
Graph_isomorphism_problem
Conjecture in graph theory
the density of copies of H {\displaystyle H} in a graph is asymptotically minimized by a random graph, as one would expect a p | E ( H ) | {\displaystyle
Sidorenko's_conjecture
Task of computing complete subgraphs
proteins. Listing the cliques in a dependency graph is an important step in the analysis of certain random processes. In mathematics, Keller's conjecture
Clique_problem
Clustering and community detection algorithm
stochastic block model., it is prone to finding spurious communities in random graphs and has been shown to systematically overfit empirical data . The value
Louvain_method
Concept in graph theory
community structure. Many basic network models, for example, such as the random graph and the Barabási–Albert model, do not display community structure. Community
Community_structure
Class of graphs
In graph theory, a forcing graph is one whose density determines whether a graph sequence is quasi-random. The term was first coined by Chung, Graham,
Forcing_graph
Mixing property of Markov chains and graphs
of a directed graph, in which case it can be used to analyze how quickly random walks in the graph converge. The conductance of a graph is closely related
Conductance_(graph_theory)
Measure of network community structure
consistent, and finds communities in its own null model, i.e. fully random graphs, and therefore it cannot be used to find statistically significant community
Modularity_(networks)
Study of graphs defined by geometric means
Geometric graph theory in the broader sense is a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. In a stricter
Geometric_graph_theory
Python library for graphs and networks
more graphing algorithms and functions. Classes for graphs and digraphs. Conversion of graphs to and from several formats. Ability to construct random graphs
NetworkX
2001 book by Joel Spencer
The Strange Logic of Random Graphs is a book on zero-one laws for random graphs. It was written by Joel Spencer and published in 2001 by Springer-Verlag
The Strange Logic of Random Graphs
The_Strange_Logic_of_Random_Graphs
Randomized algorithm for minimum cuts
In computer science and graph theory, Karger's algorithm is a randomized algorithm to compute a minimum cut of a connected graph. It was invented by David
Karger's_algorithm
Variable representing a random phenomenon
word. A random sentence of given length N {\displaystyle N} may be represented as a vector of N {\displaystyle N} random words. A random graph on N {\displaystyle
Random_variable
Concept in social network theory
measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph. One measure for the presence
Triadic_closure
American mathematician
concerns stochastic processes on random graphs. He showed, with Shirshendu Chatterjee, that contact processes on random graphs with power-law degree distributions
Rick_Durrett
activity-driven model is a temporal network model in which each node has a randomly-assigned "activity potential", which governs how it links to other nodes
Activity-driven_model
modeling large biological networks (or graphs); it compares real-world networks against a series of random graph models with respect to a multitude of
GraphCrunch
Branch of discrete mathematics
property for a random discrete object, such as a random graph? For instance, what is the average number of triangles in a random graph? Probabilistic
Combinatorics
Degree Preserving Randomization is a technique used in Network Science that aims to assess whether or not variations observed in a given graph could simply
Degree-preserving randomization
Degree-preserving_randomization
probability). A HGG generalizes a random geometric graph (RGG) whose embedding space is Euclidean. Mathematically, a HGG is a graph G ( V , E ) {\displaystyle
Hyperbolic_geometric_graph
Logical formulation of graph properties
important classes of graphs. Other topics of research in the logic of graphs include investigations of the probability that a random graph has a property specified
Logic_of_graphs
Graph representing edges of another graph
In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges
Line_graph
Graph of connected web pages
webgraph strongly differs from the degree distribution of the classical random graph model, the Erdős–Rényi model: in the Erdős–Rényi model, there are very
Webgraph
Degree of connectedness within a graph
In graph theory and network analysis, indicators of centrality assign numbers or rankings to nodes within a graph corresponding to their network position
Centrality
Tree which includes all vertices of a graph
of graph theory, a spanning tree T of an undirected graph G is a subgraph that is a tree which includes all of the vertices of G. In general, a graph may
Spanning_tree
In mathematics, with negligible exceptions
reformulated as follows. The proportion of graphs with n vertices that are in A equals the probability that a random graph with n vertices (chosen with the uniform
Almost_all
Graph with at most one cycle per component
In graph theory, a pseudoforest is an undirected graph in which every connected component has at most one cycle. That is, it is a system of vertices and
Pseudoforest
Theory of gel formation
percolation processes. Flory–Stockmayer theory is a special case of random graph theory of gelation. Gelation occurs when a polymer forms large interconnected
Flory–Stockmayer_theory
analysis. They were originally proposed as alteration of Exponential Random Graph Models (ERGMs) to allow for the study of social influence. ERGMs are
Autologistic actor attribute models
Autologistic_actor_attribute_models
Process by which people befriend similar people
policies have a decreased influence on fertility rates in such populations. In graph representation learning, homophily means that nodes with the same label
Homophily
Set of edges without common vertices
In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. In
Matching_(graph_theory)
Class of artificial neural networks
Graph neural networks (GNNs) are artificial neural networks designed for tasks whose inputs are graphs. Because graphs usually do not have a canonical
Graph_neural_network
Computer networking concept
Models Topology Random graph Erdős–Rényi Barabási–Albert Bianconi–Barabási Fitness model Watts–Strogatz Exponential random (ERGM) Random geometric (RGG)
Broadcast, unknown-unicast and multicast traffic
Broadcast,_unknown-unicast_and_multicast_traffic
Tendency for similar nodes to be connected
of networks: the random graph of Erdős and Rényi BA Model (Barabási-Albert model) In the ER model, since edges are placed at random without regard to
Assortativity
Analysis of social structures using network and graph theory
process of investigating social structures through the use of networks and graph theory. It characterizes networked structures in terms of nodes (individual
Social_network_analysis
Partition of a graph whose components are reachable from all vertices
In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly
Strongly_connected_component
Probability saying
asymptotically almost surely composite, by the prime number theorem; and in random graph theory, the statement " G ( n , p n ) {\displaystyle G(n,p_{n})} is connected"
Almost_surely
Software which facilitates quantitative or qualitative analysis of social networks
Duncan (1999). Small worlds: the dynamics of networks between order and randomness. Princeton, N.J: Princeton University Press. ISBN 978-0-691-11704-1. OCLC 40602717
Social network analysis software
Social_network_analysis_software
Bivariegated graph Cage (graph theory) Cayley graph Circle graph Clique graph Cograph Common graph Complement of a graph Complete graph Cubic graph Cycle graph De
List_of_graph_theory_topics
Nonconstructive method for mathematical proofs
shown that such a graph exists for any g and k, and the proof is reasonably simple. Let n be very large and consider a random graph G on n vertices, where
Probabilistic_method
Threshold of percolation theory models
worlds it generally refers to simplified lattice models of random systems or networks (graphs), and the nature of the connectivity in them. The percolation
Percolation_threshold
Graph that can be embedded in the plane
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect
Planar_graph
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travel, tourism, insurance