Bose–Einstein condensation in networks is a phase transition

Phase transition

A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....

 observed in complex network

Complex network

In the context of network theory, a complex network is a graph with non-trivial topological features—features that do not occur in simple networks such as lattices or random graphs but often occur in real graphs...

s that can be described with the same mathematical model

Mathematical model

A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...

 as that explaining Bose–Einstein condensation in physics.

Background

In physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, a Bose–Einstein condensate

Bose–Einstein condensate

A Bose–Einstein condensate is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near absolute zero . Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at...

 is a state of matter that occurs in certain gases at very low temperatures. Any elementary particle, atom, or molecule, can be classified as one of two types: a boson

Boson

In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

 or a fermion

Fermion

In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

. For example, an electron is a Fermion, while a photon or a helium

Helium

Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...

 atom is a Boson. In quantum mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, the energy of a (bound) particle is limited to set of discrete values, called energy levels. An important characteristic of a Fermion is that it obeys the Pauli exclusion principle

Pauli exclusion principle

The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

, which states that no two fermions may occupy the same energy level. Bosons, on the other hand, do not obey the exclusion principle, and any number can exist in the same energy level. As a result, at very low energies (or temperatures), a great majority of the Bosons in a Bose gas

Bose gas

An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Bose–Einstein statistics...

 can be crowded into the lowest energy state, creating a Bose–Einstein condensate.

A Bose-Einstein condensate is therefore a quantum phenomenon characteristic of boson particles. Nevertheless, a similar type of condensation transition can occur also in off-equilibrium classical systems and in particular, complex networks. In this context, a condensation phenomenon occurs when a distribution of a large number of elements in a large number of element classes becomes degenerate, i.e. instead of having an even distribution of elements in the classes, one class (or a few classes) become occupied by a finite fraction of all the elements of the system.

Condensation transitions occur in traffic jams, where long queues of cars are found, in wealth distribution models where a few people might have a finite fraction of all the wealth or in spin glass models. However, the condensation transition in these models cannot in general be mapped to a Bose-Einstein condensation.

A network is characterized by a set of nodes or vertices and a set of links between these nodes. In mathematics, graph theory

Graph theory

In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

 describes networks in general. The theory of random graphs deals in particular with stochastic networks (networks in which each link is present with a given probability p). A large class of networks that describe real complex systems like the Internet, the world wide web

World Wide Web

The World Wide Web is a system of interlinked hypertext documents accessed via the Internet...

, airport networks or the biological networks of molecular interactions, are described by random networks. Network theory

Network theory

Network theory is an area of computer science and network science and part of graph theory. It has application in many disciplines including statistical physics, particle physics, computer science, biology, economics, operations research, and sociology...

 is a recent field of research which investigates methods of characterizing and modeling real complex networks. In particular it has been found that many complex networks have universal feature like the small world

Small-world network

In mathematics, physics and sociology, a small-world network is a type of mathematical graph in which most nodes are not neighbors of one another, but most nodes can be reached from every other by a small number of hops or steps...

 property and a scale-free degree distribution. The scale-free degree distribution of networks can be caused by the "preferential attachment

Preferential attachment

A preferential attachment process is any of a class of processes in which some quantity, typically some form of wealth or credit, is distributed among a number of individuals or objects according to how much they already have, so that those who are already wealthy receive more than those who are not...

" mechanism.

History

In the late 1990s, Ginestra Bianconi was a graduate student, working with Prof. Albert-László Barabási

Albert-Laszlo Barabasi

Albert-László Barabási is a physicist, best known for his work in the research of network theory. He is the former Emil T...

, a noted network theorist. At his request, she began investigating the fitness model, a model in which the network evolves with the "preferential attachment" mechanism but in addition, each node has an intrinsic quality or fitness that describe its ability to acquire new links. For example, in the world wide web each web page has a different content, in social network

Social network

A social network is a social structure made up of individuals called "nodes", which are tied by one or more specific types of interdependency, such as friendship, kinship, common interest, financial exchange, dislike, sexual relationships, or relationships of beliefs, knowledge or prestige.Social...

s different people might have different social skills, in airport networks each airport is connected to cities with unevenly distributed economic activity, etc. It was found that that under certain conditions, a single node could acquire most, if not all of the links in the network, resulting in the network analog of a Bose–Einstein condensate

Bose–Einstein condensate

A Bose–Einstein condensate is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near absolute zero . Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at...

. In particular, a perfect analogy could be drawn between the mathematics of the network and the mathematics of a Bose gas if each node in the network were thought of as an energy level, and each link as a particle. These results have implications for any real situation involving random graphs, including the world wide web, social networks, and financial markets.

The concept

The result of the efforts of Bose and Einstein is the concept of a Bose gas

Bose gas

An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Bose–Einstein statistics...

, governed by the Bose–Einstein statistics

Bose–Einstein statistics

In statistical mechanics, Bose–Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium.-Concept:...

, which describes the statistical distribution of identical particles with integer spin, now known as bosons (Such as the photon and helium-4). In Bose–Einstein statistics any number of identical bosons can be in the same state. In particular given an energy state the number of non-interacting bosons in thermal equilibrium at temperature is given by the Bose occupation number

where the constant is determined by equation describing the conservation of the number of particles


with being the density of states of the system.

These last equation may lack a solution at low enough temperatures when for . In this case a critical temperature is found such that for the system is in a Bose-Einstein condensed phase and a finite fraction of the bosons are in the ground state.

The density of states depends on the dimensionality of the space. In particular therefore for only in dimensions d>2. Therefore a Bose-Einstein condensation of an ideal Bose gas can only occur for dimensions d>2.

For a uniform three-dimensional Bose gas consisting of non-interacting particles with no apparent internal degrees of freedom, the critical temperature is given by:


where:

	is	the critical temperature,
		the particle density,
		the mass per boson,
		Planck's constant,
		the Boltzmann constant, and
		the Riemann zeta function; .

Connection with network theory

The evolution of many complex systems, including the World Wide Web, business, and citation networks, is encoded in the dynamic web describing the interactions between the system’s constituents. Despite their irreversible and nonequilibrium nature these networks follow Bose statistics and can undergo Bose–Einstein condensation. Addressing the dynamical properties of these nonequilibrium systems within the framework of equilibrium quantum gases predicts that the “first-mover-advantage,” “fit-get-rich(FGR),” and “winner-takes-all” phenomena observed in competitive systems are thermodynamically distinct phases of the underlying evolving networks.
Starting from the fitness model, the mapping of a Bose gas to a network can be done by assigning an energy to each node, determined by its fitness through the relation

where . In particular when all the nodes have equal fitness, when instead nodes with different "energy" has very different fitness. We assume that the network evolves through a modified preferential attachment

Preferential attachment

A preferential attachment process is any of a class of processes in which some quantity, typically some form of wealth or credit, is distributed among a number of individuals or objects according to how much they already have, so that those who are already wealthy receive more than those who are not...

 mechanism. At each time a new node i with energy drawn from a probability distribution enters in the network and attach a new link to a node j chosen with probability:

In the mapping to a Bose gas, we assign to every new link linked by preferential attachment to node j a particle in the energy state

The continuum theory predicts that the rate at which links accumulate on node i with "energy " is given by

where indicating the number of links attached to node i that was added to the network at the time step . is the partition function

Partition function (statistical mechanics)

Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...

, defined as:

The solution of this differential equation is:

where the dynamic exponent satisfies , plays the role of the chemical potential, satisfying the equation

where is the probability that a node has "energy" and "fitness" . In the limit the occupation number, giving the number of links linked to nodes with "energy" , follows the familiar Bose statistics

.

The definition of the constant in the network models is surprisingly similar to the definition of the chemical potential in a Bose gas.
In particular for probabilities such that for at high enough value of we have a condensation phase transition in the network model. When this occurs, one node, the one with higher fitness acquires a finite fraction of all the links. The Bose-Einstein condensation in complex networks is therefore a topological phase transition below which the network has a star-like dominant structure.

Bose-Einstein phase transition in complex networks

The mapping of a Bose gas predicts the existence of two distinct phases as a function of the energy distribution. In the fit-get-rich phase, describing the case of uniform fitness, the fitter nodes acquire edges at a higher rate than older but less fit nodes. In the end the fittest node will have the most edges, but the richest node is not the absolute winner, since its share of the edges (i.e. the ratio of its edges to the total number of edges in the system) reduces to zero in the limit of large system sizes (Fig.2(b)). The unexpected outcome of this mapping is the possibility of Bose–Einstein condensation for , when the fittest node acquires a finite fraction of the edges and maintains this share of edges over time (Fig.2(c)).

A representative fitness distribution ρ(η) that leads to a condensations


with λ = 1.

However, the existence of the Bose–Einstein condensation or the fit-get-rich phase does not depend on the temperature or of the system but depends only on the functional form of the fitness distribution of the system. In the end, the falls out of all topologically important quantities. In fact it can be shown that Bose–Einstein condensation exists in the fitness model even without mapping to a Bose gas. A similar gelation can be seen in models with superlinear preferential attachment

Generalized scale-free model

There has been a burst of activity in the modeling of scale-free complex networks. The recipe of Barabási and Albert has been followed by several variations and generalizations and the revamping of previous mathematicalworks...

, however, it is not clear whether this is an accident or a deeper connection lies between this and the fitness model.

Bose-Einstein condensation in evolutionary models and ecological systems

In evolutionary models each species reproduces proportionally to its fitness. In the infinite alleles model, each mutation generates a new species with a random fitness. This model was studied by the statistician J. F. C. Kingman

John Kingman

Sir John Frank Charles Kingman, born on 28 August 1939 in Beckenham, Kent, is a British mathematician.He was N. M. Rothschild and Sons Professor of Mathematical Sciences and Director of the Isaac Newton Institute at the University of Cambridge from 2001 until 2006, when he was succeeded by Sir...

 and is known as the "house of cards" models . Depending on the fitness distribution, the model shows a condensation phase transition. Kingman did not realize that this phase transition could be mapped to a Bose-Einstein condensation. Recently the mapping of this model to a Bose-Einstein condensation was made in the context of a stochastic model for non-neutral ecologies . When the condensation phenomenon in an ecological system occurs, one species becomes dominant and strongly reduces the biodiversity of the system. This phase transition describes a basic stylized mechanism which is responsible for the large impact of invasive species in many ecological systems.

Memory understood as an equilibrium Bose gas

Frohlich is the source of the idea that quantum coherent waves could be generated in the biological neural network

Biological neural network

In neuroscience, a biological neural network describes a population of physically interconnected neurons or a group of disparate neurons whose inputs or signalling targets define a recognizable circuit. Communication between neurons often involves an electrochemical process...

. His studies claimed to show that with an oscillating charge in a thermal bath, large numbers of quanta may condense into a single state known as a Bose condensate. Already in 1970 Pascual-Leone had shown that memory experiments can be modelled by the Bose-Einstein occupancy model. From this and a large body of other empirical findings (based on studies of EEG

EEG

EEG commonly refers to electroencephalography, a measurement of the electrical activity of the brain.EEG may also refer to:* Emperor Entertainment Group, a Hong Kong-based entertainment company...

 and psychometrics

Psychometrics

Psychometrics is the field of study concerned with the theory and technique of psychological measurement, which includes the measurement of knowledge, abilities, attitudes, personality traits, and educational measurement...

) Weiss and Weiss draw the generalized conclusion that memory span

Memory span

In psychology and neuroscience, memory span is the longest list of items that a person can repeat back in correct order immediately after presentation on 50% of all trials. Items may include words, numbers, or letters. The task is known as digit span when numbers are used. Memory span is a common...

can be understood as the quantum number of a harmonic oscillator, where memory is to be mapped into an equilibrium bose gas.