Communication and correlation among communities

Abstract

Given a network and a partition in communities, we consider the issues "how communities influence each other" and "when two given communities do communicate". Specifically, we address these questions in the context of small-world networks, where an arbitrary quenched graph is given and long range connections are randomly added. We prove that, among the communities, a superposition principle applies and gives rise to a natural generalization of the effective field theory already presented in [Phys. Rev. E 78, 031102] (n = 1), which here (n > 1) consists in a sort of effective TAP (Thouless, Anderson and Palmer) equations in which each community plays the role of a microscopic spin. The relative susceptibilities derived from these equations calculated at finite or zero temperature, where the method provides an effective percolation theory, give us the answers to the above issues. Unlike the case n = 1, asymmetries among the communities may lead, via the TAP-like structure of the equations, to many metastable states whose number, in the case of negative short-cuts among the communities, may grow exponentially fast with n. As examples we consider the n Viana-Bray communities model and the n one-dimensional small-world communities model. Despite being the simplest ones, the relevance of these models in network theory, as e.g. in social networks, is crucial and no analytic solution were known until now. Connections between percolation and the fractal dimension of a network are also discussed. Finally, as an inverse problem, we show how, from the relative susceptibilities, a natural and efficient method to detect the community structure of a generic network arises.For a short presentation of the main result see arXiv:0812.0608.

Synchronization transition of identical phase oscillators in a directed small-world network

“…30 In a small world network, the transition is expected to be of mean field universality. 15,18 To examine this point we have performed a finite size scaling analysis in the vicinity of ␣ P at fixed shortcut density = 1.0. The initial condition was a stable partially synchronized state at ␣ = 0.52 and R Ϸ 0.75.…”

“…12 Here we study the transitions from incoherence to partial synchronization and to complete synchronization in a sparse, directed small-world network of identical phase oscillators. Since the formulation of the model, 13 small-world networks have been studied as a medium for dynamical processes, such as the spreading of epidemics, 14 the Ising model, [15][16][17] and also synchronization of nonidentical phase oscillators. 18 Nontrivial behavior in sparse networks of coupled dynamical systems is known to occur in Boolean networks 19,20 and has also been reported for leaky integrate-and-fire models with and without delay coupling.…”

The unreasonable effectiveness of tree-based theory for networks with clustering

“…Furthermore, if one considers a dynamical system on a real-world network rather than on a grossly simplified caricature of it, then theoretical results become almost barren. This has motivated a wealth of recent research concerning analytical results on networks with clustering [7,[11][12][13][14][15][16][17][18][19][20][21][22][23].…”

Criticality and Chaos in Systems of Communities

“…Eqs. (20) represent a particular case of the general result derived in [16] valid for n interacting communities at equilibrium. In particular, one can check that Eqs.…”