On the flux phase conjecture at half-filling: an improved proof

We present a simplification of Lieb's proof of the flux phase conjecture for interacting fermion systems -- such as the Hubbard model --, at half filling on a general class of graphs. The main ingredient is a procedure which transforms a class of fermionic Hamiltonians into reflection positive form. The method can also be applied to other problems, which we briefly illustrate with two examples concerning the $t-V$ model and an extended Falicov-Kimball model.Comment: 23 pages, Latex, uses epsf.sty to include 3 eps figures, to appear in J. Stat. Phys., Dec. 199

1-Safe Petri nets and special cube complexes: equivalence and applications

Nielsen, Plotkin, and Winskel (1981) proved that every 1-safe Petri net $N$ unfolds into an event structure $\mathcal{E}_N$. By a result of Thiagarajan (1996 and 2002), these unfoldings are exactly the trace regular event structures. Thiagarajan (1996 and 2002) conjectured that regular event structures correspond exactly to trace regular event structures. In a recent paper (Chalopin and Chepoi, 2017, 2018), we disproved this conjecture, based on the striking bijection between domains of event structures, median graphs, and CAT(0) cube complexes. On the other hand, in Chalopin and Chepoi (2018) we proved that Thiagarajan's conjecture is true for regular event structures whose domains are principal filters of universal covers of (virtually) finite special cube complexes. In the current paper, we prove the converse: to any finite 1-safe Petri net $N$ one can associate a finite special cube complex ${X}_N$ such that the domain of the event structure $\mathcal{E}_N$ (obtained as the unfolding of $N$) is a principal filter of the universal cover $\widetilde{X}_N$ of $X_N$. This establishes a bijection between 1-safe Petri nets and finite special cube complexes and provides a combinatorial characterization of trace regular event structures. Using this bijection and techniques from graph theory and geometry (MSO theory of graphs, bounded treewidth, and bounded hyperbolicity) we disprove yet another conjecture by Thiagarajan (from the paper with S. Yang from 2014) that the monadic second order logic of a 1-safe Petri net is decidable if and only if its unfolding is grid-free. Our counterexample is the trace regular event structure $\mathcal{\dot E}_Z$ which arises from a virtually special square complex $\dot Z$. The domain of $\mathcal{\dot E}_Z$ is grid-free (because it is hyperbolic), but the MSO theory of the event structure $\mathcal{\dot E}_Z$ is undecidable

Detection of Core-Periphery Structure in Networks Using Spectral Methods and Geodesic Paths

We introduce several novel and computationally efficient methods for detecting "core--periphery structure" in networks. Core--periphery structure is a type of mesoscale structure that includes densely-connected core vertices and sparsely-connected peripheral vertices. Core vertices tend to be well-connected both among themselves and to peripheral vertices, which tend not to be well-connected to other vertices. Our first method, which is based on transportation in networks, aggregates information from many geodesic paths in a network and yields a score for each vertex that reflects the likelihood that a vertex is a core vertex. Our second method is based on a low-rank approximation of a network's adjacency matrix, which can often be expressed as a tensor-product matrix. Our third approach uses the bottom eigenvector of the random-walk Laplacian to infer a coreness score and a classification into core and peripheral vertices. We also design an objective function to (1) help classify vertices into core or peripheral vertices and (2) provide a goodness-of-fit criterion for classifications into core versus peripheral vertices. To examine the performance of our methods, we apply our algorithms to both synthetically-generated networks and a variety of networks constructed from real-world data sets.Comment: This article is part of EJAM's December 2016 special issue on "Network Analysis and Modelling" (available at https://www.cambridge.org/core/journals/european-journal-of-applied-mathematics/issue/journal-ejm-volume-27-issue-6/D245C89CABF55DBF573BB412F7651ADB