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SUPPORTING INFORMATION Computing global structural balance in large-scale signed social
 

Summary: SUPPORTING INFORMATION
Computing global structural balance in large-scale signed social
networks
G. Facchetti, G. Iacono, C. Altafini
September 14, 2011
1 Optimization algorithm: gauge transformations
Given the adjacency matrix J , our approach to solving (2) of the paper consists in finding a
(diagonal) signature matrix T = diag(), Bn
2 , such that TJ T has the least possible number
of negative signs. Any such T is a change of sign through a cut set of the graph of J , see example
in Fig. 1(c) of the paper. These operations are called switching equivalences in the theory of signed
graphs [31], or gauge transformations in the theory of frustrated spin systems [27], and correspond
to changes in the partial order of the orthants of Rn in the theory of monotone systems [25, 24].
A system is exactly balanced if and only if there exists a such that J = TJ T has all entries
0. Unlike the usual algorithms to find the ground states in spin systems, which explore the
space of spin configurations s, our optimization procedure works on the adjacency matrix J . Since
= T1, in terms of the energy function (1) the two concepts are equivalent: if we choose s = ,
from (4) it is easily seen that flipping spins is equivalent to applying the gauge transformation (3)
to J . In spin glass theory [7] this corresponds to the notion that frustration (for us global balance)
is an invariant of transformations such as (3).

  

Source: Altafini, Claudio - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)

 

Collections: Engineering; Mathematics