Bianconi–Barabási model
This model can explain that nodes with different characteristics acquire links at different rates.
It predicts that a node's growth depends on its fitness and can calculate the degree distribution.
The Bianconi–Barabási model [1][2] is named after its inventors Ginestra Bianconi and Albert-László Barabási.
It assigns an intrinsic fitness value to each node, which embodies all the properties other than the degree.
This model explains that age is not the best predictor of a node's success, rather latecomers also have the chance to attract links to become a hub.
The Bianconi–Barabási model can reproduce the degree correlations of the Internet Autonomous Systems.
[5] This model can also show condensation phase transitions in the evolution of complex network.
[7] The fitness network begins with a fixed number of interconnected nodes.
In the second case, if the fitness distribution has an infinite domain, then the node with the highest fitness value will attract a large number of nodes and show a winners-take-all scenario.
[11] Recently it has been shown that the Bianconi–Barabási model can be interpreted as a limit case of the model for emergent hyperbolic network geometry [13] called Network Geometry with Flavor.
[14] The Bianconi–Barabási model can be also modified to study static networks where the number of nodes is fixed.
In physics, a Bose–Einstein condensate 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 or a fermion.
Bosons, on the other hand, do not obey the exclusion principle, and any number can exist in the same state.
Therefore, a Bose-Einstein condensation of an ideal Bose gas can only occur for dimensions d > 2.
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.
This mapping predicts that the Bianconi–Barabási model can undergo a topological phase transition in correspondence to the Bose–Einstein condensation of the Bose gas.
This phase transition is therefore called Bose-Einstein condensation in complex networks.
Consequently 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 a competitive systems are thermodynamically distinct phases of the underlying evolving networks.
[2] Starting from the Bianconi-Barabási model, the mapping of a Bose gas to a network can be done by assigning an energy εi to each node, determined by its fitness through the relation[2][16] where β = 1 / T .
We assume that the network evolves through a modified preferential attachment mechanism.
At each time a new node i with energy εi drawn from a probability distribution p(ε) 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 εj.
The continuum theory predicts that the rate at which links accumulate on node i with "energy" εi 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, defined as: The solution of this differential equation is: where the dynamic exponent
In the limit, t → ∞, the occupation number, giving the number of links linked to nodes with "energy" ε, follows the familiar Bose statistics The definition of the constant μ in the network models is surprisingly similar to the definition of the chemical potential in a Bose gas.
When this occurs, one node, the one with higher fitness acquires a finite fraction of all the links.
The mapping of a Bose gas predicts the existence of two distinct phases as a function of the energy distribution.
The unexpected outcome of this mapping is the possibility of Bose–Einstein condensation for T < TBE, when the fittest node acquires a finite fraction of the edges and maintains this share of edges over time (Fig.2(c)).
In fact, it can be shown that Bose–Einstein condensation exists in the fitness model even without mapping to a Bose gas.
