In
network science
Network science is an academic field which studies complex networks such as telecommunication networks, computer networks, biological networks, cognitive and semantic networks, and social networks, considering distinct elements or actors repr ...
, a critical point is a value of
average degree
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...
, which separates random networks that have a
giant component
In network theory, a giant component is a connected component of a given random graph that contains a finite fraction of the entire graph's vertices.
Giant component in Erdős–Rényi model
Giant components are a prominent feature of the Erdő ...
from those that do not (i.e. it separates a network in a subcritical regime from one in a supercritical regime).
Considering a random network with an average degree
the critical point is
where the average degree is defined by the fraction of the number of edges (
) and nodes (
) in the network, that is
.
Subcritical regime
In a subcritical regime the network has no
giant component
In network theory, a giant component is a connected component of a given random graph that contains a finite fraction of the entire graph's vertices.
Giant component in Erdős–Rényi model
Giant components are a prominent feature of the Erdő ...
, only small clusters. In the special case of
the network is not connected at all. A random network is in a subcritical regime until the average degree exceeds the critical point, that is the network is in a subcritical regime as long as
.
Supercritical regime
In a supercritical regime, in contrary to the subcritical regime the network has a
giant component
In network theory, a giant component is a connected component of a given random graph that contains a finite fraction of the entire graph's vertices.
Giant component in Erdős–Rényi model
Giant components are a prominent feature of the Erdő ...
. In the special case of
the network is complete (see
complete graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices ...
). A random network is in a supercritical regime if the average degree exceeds the critical point, that is if
.
Example on different regimes

Consider a
speed dating
Speed dating is a formalized matchmaking process which has the purpose of encouraging eligible singles to meet large numbers of new potential partners in a very short period of time.
Organization
Usually advance registration is required for ...
event as an example, with the participants as the nodes of the network. At the beginning of the event, people do not know anyone else. In this case the network is in a subcritical regime, that is, there is no
giant component
In network theory, a giant component is a connected component of a given random graph that contains a finite fraction of the entire graph's vertices.
Giant component in Erdős–Rényi model
Giant components are a prominent feature of the Erdő ...
in the network (even if there are a couple of people, who know each other). After the first round of dates, everyone knows exactly one other person. There is still no giant component in the network, the average degree is
, that is, everyone knows one other person on average, meaning that the network is at the critical point. After the second round, the
average degree
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...
of the network exceeds the critical point, and the
giant component
In network theory, a giant component is a connected component of a given random graph that contains a finite fraction of the entire graph's vertices.
Giant component in Erdős–Rényi model
Giant components are a prominent feature of the Erdő ...
is present. In this specific case, the average degree is
. The network is in a supercritical regime.
See also
*
Graph theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
*
Percolation theory
In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnecte ...
*
Complex network
In the context of network theory, a complex 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 occur in networks representing real ...
*
Random graph
In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs ...
References
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Networks
Network theory