Consider a weighted combination of two lattices (in the case of the picture they are two fully connected graph). If you link the spins of each party of variables in a ferromagnetic or disorder way, you obtain a bipartite spin system. These kind of models present remarkable behaviour, with particular reference to the free energy which gives rise to a degenerate self-consistence equation. The model can be studied also outside the fully connected interaction between the parties, using, for instance, a Bernoulli distributed bond dilution. Moreover, it can be shown that an analogical neural network can be mapped into a bipartite model, widening, in this way, the possible applications of these kind of systems.

J. Phys. A: Math. Theor. 44 245002

EPJB (2013) 86:332

J. Stat. Mech. (2011) P02027

Take a chain in which each site can accommodate any number of particles. The probability of transition between the sites can depend on many parameters and the various models are classified with respect to them. This family of models is often called in the literature as

JSP (2014) 154:432-465

JSP (2003) 113:389-410

J. Phys. A: Math. Theor. 47 095001

Consider a network where we associate to each edge a dichotomic variable, i.e. +1 or -1. In this way, we can build a signed adjacency matrix. Allowing the nodes to take +1 or -1 as well, it is interesting to determine the node configuration which maximizes the quadratic form associated to the graph. This can be achieved using many different techniques from Algebra, like spectral analysis, or Physics, like the Ginzburg-Landau functional. An important version of this problem is obtained when some values of the nodes are fixed a priori. This simple variation implies the breakdown of standard techniques and requires developing new optimization strategies.

Journal of Complex Networks, 3 (3):469-506

Information and Inference: A Journal of the IMA, 1 (1), pp. 2167

ACM Transactions on Sensor Networks, 8 (3), pp. 1-42

Statistical Mechanics (SM) is a tool which allows to describe physical situations in which many agents are present. The main idea is to divide people in two groups, immigrants and native citizens, and allow them to interact with each other. This corresponds to the typical scenario of a bipartite Curie-Weiss model, that is where two parties of spins are present. The above mentioned mathematical rationale is also able to manage different sociological indicators. Recently, typical SM techniques have been successfully applied to describe marriages among immigrants and native citizens in Spain, in order to understand possible trends of integration. Now, the idea is to apply those techniques to many other European States, even without considering the role of immigration: in this way it would be possible, in principle, to check the existence of cultural fractures among the native citizens of a country.

Europhysics Letters 89, 68001-68007

Nature Scientific Reports 3, 4174

New Journal of Physics 16, 103034