Fitness and flavor in d-dimensional complex networks

The world seems to be relational. Many, probably the majority of, natural and artificial systems show emergent behavior, i.e. properties that cannot be explained within a reductionist framework, rather they appear because of the relations between the elements of the system. All phenomena seem to show relational properties whether you look at a very small spatial/energy scale (fundamental particles, many-body systems, molecules, proteins. etc..) or at a very large one (planetary systems, systems of stars, galaxies, orbiting black holes). Many systems show some complex characteristics: what we call complex systems (CS). Someone says it is easier to tell what is not a complex system than what it is, however everyone agrees upon some of those characteristics. Among the others, CS are made of a large number of elements that in themselves can be simple, the interactions are non-linear, CS are far from equilibrium, show emerging behavior (and when you take only part of the original system seldom you get similar behavior, think about taking half the human body, you pick the half!). During the last decades, the field has grown so much to include brand new disciplines, Socio-physics and Econophysics among the others; no time but now could be more exciting for a complex scientist. When a system is relational and complex you talk about a complex network (CN). CN theory [15] has become a field in its own and, funny enough, it seems to growth non-linearly with hundreds of articles popping up every month. Of particular importance, and representative of many natural systems, are the so-called asymptotically-scale free networks, or simply scale-free networks. In all those 4 models a major role is played by the attachment rule, that is the way in which each element (node) of the system (network) gets new connections (links). The attachment probability for a node to win over the others could be proportional to its degree, i.e. how many connections it already has. If the system is geographically constrained, as for ecological systems, power grids, public transportation, social face-to-face interactions, there will be (usually an inverse) proportionality to the geographical distance between the nodes. In this case the importance of the distance between the nodes can be regulated by the introduction of a parameter (αA in what follows). Furthermore, if each node has any ability (or inability) to attract new nodes, a fitness parameter can be introduced for every node. The possible values that the fitness parameter can take, i.e., the fitness parameter distribution, and the importance of the distances between nodes, open the doors for different models to emerge. Among these the Barabási-Albert model where there is no dependence on the distances and all the nodes have the same ability to attract new ones (fitness parameter equal one for all nodes). A possible extension of this model is the well-known Bianconi-Barabási one where now the fitness parameter is (introduced and chosen) uniformly between zero and one. As I will explain better later, it turned out that these are particular cases of more general models where the dependence on the distance of the growing mechanism and therefore the role of dimensionality of the system is introduced. The cited Bianconi-Barabási model is also a classical example of the phenomenon in which the dynamics of a classical network is mathematically described by quantum statistics (QS). QS have been shown to emerge spontaneously in the description of growing network models with fitness of the nodes. The implications of this mapping are profound. In particular the mapping of the Bianconi-Barabási model with a Bose gas is able to predict a topological phase transition in the network in which the dynamics of the networks is not stationary anymore but instead, it is dominated by the sequence of nodes with high fitness that arrive in the network and eventually become super-hubs. Similarly, the mapping of a (so-called) growing Cayley tree with fitness of nodes to the Fermi-Dirac distribution [58] leads to the analytical description of Invasion Percolation on these structures. Recently these classical results of network theory have been related to the properties of growing simplicial complexes. A simplicial complex is a generalized network structure that allows the description of many-body interactions between a set of nodes. Network Geometry with Flavor (NGF) is a non-equilibrium model of growing simplicial complexes with fitness that has been proposed to study emergent network geometry. In fact, the NGFs evolve thanks to purely combinatorial rules that make no use of any embedding space, but when the same length is attributed to each link of the simplicial complex they are able to generate structures with an emergent hyperbolic geometry. The flavor of the NGF is a parameter that can change the topological nature of the simplicial complexes and their evolution. For different values of the flavor you can get manifolds or networks that grow by uniform attachment or still networks evolving according to a generalized preferential attachment rule.

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