Linear and optimization Hamiltonians in clustered exponential random graph modeling

Abstract

Exponential random graph theory is the complex network analog of the canonical ensemble theory from statistical physics. While it has been particularly successful in modeling networks with specified degree distributions, a naïve model of a clustered network using a graph Hamiltonian linear in the number of triangles has been shown to undergo an abrupt transition into an unrealistic phase of extreme clustering via triangle condensation. Here we study a non-linear graph Hamiltonian that explicitly forbids such a condensation and show numerically that it generates an equilibrium phase with specified intermediate clustering.

Discontinuous percolation transitions in epidemic processes, surface depinning in random media, and Hamiltonian random graphs

“…A model with two control parameters (B and J) studied in [58] exhibits zoo of metastable states, but it seems that most of these states are still too extreme to be physical. A further step in this direction could be to fix not only the average degree, but to fix the entire degree distribution, either by soft [59] or by hard [60] constraints.…”

Heterogeneous clustered random graphs

“…One known pathology of linear Hamiltonians is that models based on them often exhibit degeneracies [2,14], leading Park and Yook [20] to propose a non-linear optimisation Hamiltonian for degree and clustering at an individual level:…”