On graph induced symbolic systems

In this paper, we investigate shift spaces arising from a multidimensional graph G. In particular, we investigate nonemptiness and existence of periodic points for a multidimensional shift space. We derive sufficient conditions under which these questions can be answered affirmatively. We investigate the structure of the shift space using the generating matrices. We prove that any d-dimensional shift of finite type is finite if and only if it is conjugate to a shift generated through permutation matrices. We prove that if any triangular pattern of the form a b c can be extended to a 1 x 1 square then the two dimensional shift space possesses periodic points. We introduce the notion of an E-pair for a two dimensional shift space. Using the notion of an E-pair, we derive sufficient conditions for non-emptiness of the two dimensional shift space under discussion

Percolation and the hard-core lattice gas model

AbstractRecently a uniqueness condition for Gibbs measures in terms of disagreement percolation (a type of dependent percolation involving two realizations) has been obtained. In general this condition is sufficient but not necessary for uniqueness. In the present paper we study the hard-core lattice gas model which we abbreviate as hard-core model. This model is not only relevant in Statistical Physics, but was recently rediscovered in Operations Research in the context of certain communication networks.First we show that the uniqueness result mentioned above implies that the critical activity for the hard-core model on a graph is at least Pc(1 − Pc), where Pc is the critical probability for site percolation on that graph.Then, for the hard-core model on bi-partite graphs, we study the probability that a given vertex v is occupied under the two extreme boundary conditions, and show that the difference can be written in terms of the probability of having a ‘path of disagreement’ from v to the boundary. This is the key to a proof that, for this case, the uniqueness condition mentioned above is also necessary, i.e. roughly speaking, phase transition is equivalent with disagreement percolation in the product space.Finally, we discuss the hard-core model on Zd with two different values of the activity, one for the even, and one for the odd vertices. It appears that the question whether this model has a unique Gibbs measure, can, in analogy with the standard ferromagnetic Ising model, be reduced to the question whether the third central moment of the surplus of odd occupied vertices for a certain class of finite boxes is negative

Distinguishing Infections on Different Graph Topologies

The history of infections and epidemics holds famous examples where understanding, containing and ultimately treating an outbreak began with understanding its mode of spread. Influenza, HIV and most computer viruses, spread person to person, device to device, through contact networks; Cholera, Cancer, and seasonal allergies, on the other hand, do not. In this paper we study two fundamental questions of detection: first, given a snapshot view of a (perhaps vanishingly small) fraction of those infected, under what conditions is an epidemic spreading via contact (e.g., Influenza), distinguishable from a "random illness" operating independently of any contact network (e.g., seasonal allergies); second, if we do have an epidemic, under what conditions is it possible to determine which network of interactions is the main cause of the spread -- the causative network -- without any knowledge of the epidemic, other than the identity of a minuscule subsample of infected nodes? The core, therefore, of this paper, is to obtain an understanding of the diagnostic power of network information. We derive sufficient conditions networks must satisfy for these problems to be identifiable, and produce efficient, highly scalable algorithms that solve these problems. We show that the identifiability condition we give is fairly mild, and in particular, is satisfied by two common graph topologies: the grid, and the Erdos-Renyi graphs

Graph-wreath products and finiteness conditions

A notion of \emph{graph-wreath product} is introduced. We obtain sufficient conditions for these products to satisfy the topologically inspired finiteness condition type $\operatorname{F}_n$. Under various additional assumptions we show that these conditions are necessary. Our results generalise results of Cornulier about wreath products in case $n=2$. Graph-wreath products include classical permutational wreath products and semidirect products of right-angled Artin groups by groups of automorphisms amongst others.Comment: 12 page

Characterization of revenue equivalence

The property of an allocation rule to be implementable in dominant strategies by a unique payment scheme is called \emph{revenue equivalence}. In this paper we give a characterization of revenue equivalence based on a graph theoretic interpretation of the incentive compatibility constraints. The characterization holds for any (possibly infinite) outcome space and many of the known results are immediate consequences. Moreover, revenue equivalence can be identified in cases where existing theorems are silent