Completeness of classical spin models and universal quantum computation

We study mappings between distinct classical spin systems that leave the partition function invariant. As recently shown in [Phys. Rev. Lett. 100, 110501 (2008)], the partition function of the 2D square lattice Ising model in the presence of an inhomogeneous magnetic field, can specialize to the partition function of any Ising system on an arbitrary graph. In this sense the 2D Ising model is said to be "complete". However, in order to obtain the above result, the coupling strengths on the 2D lattice must assume complex values, and thus do not allow for a physical interpretation. Here we show how a complete model with real -and, hence, "physical"- couplings can be obtained if the 3D Ising model is considered. We furthermore show how to map general q-state systems with possibly many-body interactions to the 2D Ising model with complex parameters, and give completeness results for these models with real parameters. We also demonstrate that the computational overhead in these constructions is in all relevant cases polynomial. These results are proved by invoking a recently found cross-connection between statistical mechanics and quantum information theory, where partition functions are expressed as quantum mechanical amplitudes. Within this framework, there exists a natural correspondence between many-body quantum states that allow universal quantum computation via local measurements only, and complete classical spin systems.Comment: 43 pages, 28 figure

An Analysis of the Friendship Paradox and Derived Sampling Methods

The friendship paradox (FP) is the famous sampling-bias phenomenon that leads to the seemingly paradoxical truth that, on average, people’s friends have more friends than they do. Among the many far-reaching research findings the FP inspired is a sampling method that samples neighbors of vertices in a graph in order to acquire random vertices that are of higher expected degree than average. Our research examines the friendship paradox on a local level. We seek to quantify the impact of the FP on an individual vertex by defining the vertex’s “friendship index”, a measure of the extent to which the phenomenon affects the vertex, either positively or negatively. We extend this measure to create aggregate measures that are indicative of FP-characteristics of the graph as a whole. We then examine these measures experimentally on theoretical canonical graphs, synthetic graphs, and the graphs of real-world networks as a means of demonstrating their usefulness for revealing information about a graph. These analyses place a particular focus on the similarities and differences our metrics have with the well-known degree-homophily measure, assortativity. Having defined this metric and quantified information about the FP’s impact on graphs and vertices, we turn to one of the famous results of the paradox, the ability to sample neighbors of vertices instead of the vertices themselves in order to find high-degree vertices in a graph. We focus on an overlooked detail of this sampling method, the additional computational cost it incurs for each sampled vertex. We analyze this cost from a few perspectives, breaking it down into multiple costs that might apply in varied situations, and then create a strong model that enables a fair comparison between sampling methods to better quantify the value of this method versus naïve random sampling. As we define costs, certain tweaks that can be applied to the method become apparent. Some of these allow us to amortize the cost of a more computationally expensive step over more vertices and achieve superior results for the investment. This leads to a number of new versions of the sampling method which we present and analyze. We then perform an extensive study on a particularly novel tweak, ‘inclusive random sampling’. Whereas the original method of sampling neighbors blindly exchanges a vertex for its neighbor, inclusive sampling learns the degree of both the vertex and the neighbor and chooses the one of higher degree. We explore this idea by applying it to two FP-inspired sampling methods, the previously mentioned random neighbor sampling, and a lesser-known method that samples edges instead of vertices. We prove interesting theoretical bounds on these methods and show their applications to different graph types. We also show the strengths and weaknesses of the different inclusive and exclusive methods through experimentation on a variety of synthetic and real-world graphs. We explore the connection between these methods and assortativity which further elucidates the characteristics that make one method superior to another for a given graph

Graph Theory and Networks in Biology

In this paper, we present a survey of the use of graph theoretical techniques in Biology. In particular, we discuss recent work on identifying and modelling the structure of bio-molecular networks, as well as the application of centrality measures to interaction networks and research on the hierarchical structure of such networks and network motifs. Work on the link between structural network properties and dynamics is also described, with emphasis on synchronization and disease propagation.Comment: 52 pages, 5 figures, Survey Pape