Author Archive
282. Communication among communities
This is a guest blog post by Massimo Ostilli from the Center of Statistical Mechanics and Complexity (INFM, Roma). Dmitry.
In recent times, in the network science, the problem of detecting the community structure of a given network (a random graph), has attracted more attention. The general idea behind the concept of community structure comes from the observation that, in many situations, real data show an intrinsic partition of the vertices of the graph into 
groups, called communities, such that between any two communities there is a number of bonds that is relatively small if compared with the number of bonds present in each community. The partition(s) can be used to build a higher-level meta-network where the meta-nodes are now the communities (cells, proteins, groups of people, tec…) and play important roles in unveiling the functional organization inside the network. Given an hypothetical community structure, one of the most important issue is to understand whether or not the communities exchange information and to what extent, and, more in general, what are their correlations. In a recent work, we have emphasized that such a problem cannot be faced through an analysis that takes into account only the network topology (that is, the detailed description of nodes and bonds) that, by its definition, neglects any kind of correlation among the nodes. Nodes, in fact, are the sites where some physical or abstract status manifests as a result of the status of the other nodes. The most elementary example is the case in which at any node there is a dichotomy variable taking values ON and OFF. This happens – for example – in an network in which individuals, in somehow equivalent, are asked to say YES or NO to some politic proposal. The fact that the individuals know, in part, the opinion of the others, makes the answer of each individual partly conditioned by the others, especially, but not only, by those that are near (neighbors) in the social space. Physicists immediately understand that – within the equilibrium statistical mechanics – such a system can be cast by defining a suitable disordered Ising model. In this approach, the temperature 
can be seen as a parameter describing the freedom of the vertices to assume a state independently of the state of the other vertices, while the Ising couplings 
between two vertices 
and 
belonging to the 
-th and 
-th community, respectively, as a tendency of the vertices to be positively or negatively correlated, according to the amplitude and to the sign of . At least in principle, if a Gibbs-Boltzmann 
distribution with some Hamiltonian 
has been assumed, one can obtain 
from the data of the given graph by isolating the two vertices 
from all vertices of the graph other then them, and by measuring the correlation function of the obtained isolated dimer.
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