Tuning Ranking in Co-occurrence Networks with General Biased Exchange-based Diffusion on Hyper-bag-graphs

Co-occurence networks can be adequately modeled by hyper-bag-graphs (hb-graphs for short). A hb-graph is a family of multisets having same universe, called the vertex set. An efficient exchange-based diffusion scheme has been previously proposed that allows the ranking of both vertices and hb-edges. In this article, we extend this scheme to allow biases of different kinds and explore their effect on the different rankings obtained. The biases enhance the emphasize on some particular aspects of the network

Causal sets from simple models of computation

Causality among events is widely recognized as a most fundamental structure of spacetime, and causal sets have been proposed as discrete models of the latter in the context of quantum gravity theories, notably in the Causal Set Programme. In the rather different context of what might be called the 'Computational Universe Programme' -- one which associates the complexity of physical phenomena to the emergent features of models such as cellular automata -- a choice problem arises with respect to the variety of formal systems that, in virtue of their computational universality (Turing-completeness), qualify as equally good candidates for a computational, unified theory of physics. This paper proposes Causal Sets as the only objects of physical significance and relevance to be considered under the 'computational universe' perspective, and as the appropriate abstraction for shielding the unessential details of the many different computationally universal candidate models. At the same time, we propose a fully deterministic, radical alternative to the probabilistic techniques currently considered in the Causal Set Programme for growing discrete spacetimes. We investigate a number of computation models by grouping them into two broad classes, based on the support on which they operate; in one case this is linear, like a tape or a string of symbols; in the other, it is a two-dimensional grid or a planar graph. For each model we identify the causality relation among computation events, implement it, and conduct a possibly exhaustive exploration of the associated causal set space, while examining quantitative and qualitative features such as dimensionality, curvature, planarity, emergence of pseudo-randomness, causal set substructures and particles.Comment: 33 pages, 47 figure

Counting thin subgraphs via packings faster than meet-in-the-middle time

Vassilevska and Williams (STOC 2009) showed how to count simple paths on $k$ vertices and matchings on $k/2$ edges in an $n$-vertex graph in time $n^{k/2+O(1)}$. In the same year, two different algorithms with the same runtime were given by Koutis and Williams~(ICALP 2009), and Bj\"orklund \emph{et al.} (ESA 2009), via $n^{st/2+O(1)}$-time algorithms for counting $t$-tuples of pairwise disjoint sets drawn from a given family of $s$-sized subsets of an $n$-element universe. Shortly afterwards, Alon and Gutner (TALG 2010) showed that these problems have $\Omega(n^{\lfloor st/2\rfloor})$ and $\Omega(n^{\lfloor k/2\rfloor})$ lower bounds when counting by color coding. Here we show that one can do better, namely, we show that the "meet-in-the-middle" exponent $st/2$ can be beaten and give an algorithm that counts in time $n^{0.45470382 st + O(1)}$ for $t$ a multiple of three. This implies algorithms for counting occurrences of a fixed subgraph on $k$ vertices and pathwidth $p\ll k$ in an $n$-vertex graph in $n^{0.45470382k+2p+O(1)}$ time, improving on the three mentioned algorithms for paths and matchings, and circumventing the color-coding lower bound. We also give improved bounds for counting $t$-tuples of disjoint $s$-sets for $s=2,3,4$. Our algorithms use fast matrix multiplication. We show an argument that this is necessary to go below the meet-in-the-middle barrier.Comment: Journal version, 26 pages. Compared to the SODA'14 version, it contains some new results: a) improved algorithms for counting t-tuples of disjoint s-sets for the special cases of s = 2, 3, 4 and b) new hardness argument

Quantum Gravity as an Information Network: Self-Organization of a 4D Universe

I propose a quantum gravity model in which the fundamental degrees of freedom are information bits for both discrete space-time points and links connecting them. The Hamiltonian is a very simple network model consisting of a ferromagnetic Ising model for space-time vertices and an antiferromagnetic Ising model for the links. As a result of the frustration between these two terms, the ground state self-organizes as a new type of low-clustering graph with finite Hausdorff dimension 4. The spectral dimension is lower than the Hausdorff dimension: it coincides with the Hausdorff dimension 4 at a first quantum phase transition corresponding to an IR fixed point while at a second quantum phase transition describing small scales space-time dissolves into disordered information bits. The large-scale dimension 4 of the universe is related to the upper critical dimension 4 of the Ising model. At finite temperatures the universe graph emerges without big bang and without singularities from a ferromagnetic phase transition in which space-time itself forms out of a hot soup of information bits. When the temperature is lowered the universe graph unfolds and expands by lowering its connectivity, a mechanism I have called topological expansion. The model admits topological black hole excitations corresponding to graphs containing holes with no space-time inside and with "Schwarzschild-like" horizons with a lower spectral dimension.Comment: Revised version, to appear in Physical Review

Adjacency and Tensor Representation in General Hypergraphs.Part 2: Multisets, Hb-graphs and Related e-adjacency Tensors

HyperBagGraphs (hb-graphs as short) extend hypergraphs by allowing the hyperedges to be multisets. Multisets are composed of elements that have a multiplicity. When this multiplicity has positive integer values, it corresponds to non ordered lists of potentially duplicated elements. We define hb-graphs as family of multisets over a vertex set; natural hb-graphs correspond to hb-graphs that have multiplicity function with positive integer values. Extending the definition of e-adjacency to natural hb-graphs, we define different way of building an e-adjacency tensor, that we compare before having a final choice of the tensor. This hb-graph e-adjacency tensor is used with hypergraphs

From the flat-space S-matrix to the Wavefunction of the Universe

The physical information encoded in the cosmological late-time wavefunction of the universe is tied to its singularity structure and its behaviour as such singularities are approached. One important singularity is identified by the vanishing of the total energy, where the wavefunction reduces to the physics of scattering in flat space. In this paper, we discuss the behaviour of the perturbative wavefunction as its other singularities are approached and the role played by the flat-space scattering, in the simplified context of the class of toy models admitting a first principle definition in terms of cosmological polytopes. The problems then translates into the analysis of the structure of its facets, one of which -- the scattering facet -- beautifully encodes the flat-space S-matrix. We show that all the boundaries of the cosmological polytope encode information about the flat-space physics. In particular, a subset of its facets turns out to have a similar structure as the scattering facet, with the vertices which can be grouped together to form lower dimensional scattering facets. The other facets admit one (and only one) triangulation in terms of products of lower dimensional scattering facets. As a consequence, the whole perturbative wavefunction can be represented as a sum of product of flat-space scattering amplitudes. Finally, we turn the table around and ask whether the knowledge of the flat-space scattering amplitudes suffices to reconstruct the wavefunction of the universe. We show that, at least for our class of toy models, this is indeed the case at tree level if we are also provided with a subset of symmetries that the wavefunction ought to satisfy. Once the tree cosmological polytopes are reconstructed, the loop ones can be obtained as a particular projection of them.Comment: 37 pages, figures in Tik

A theory of quantum gravity based on quantum computation

This paper proposes a method of unifying quantum mechanics and gravity based on quantum computation. In this theory, fundamental processes are described in terms of pairwise interactions between quantum degrees of freedom. The geometry of space-time is a construct, derived from the underlying quantum information processing. The computation gives rise to a superposition of four-dimensional spacetimes, each of which obeys the Einstein-Regge equations. The theory makes explicit predictions for the back-reaction of the metric to computational `matter,' black-hole evaporation, holography, and quantum cosmology.Comment: LaTeX; 58 pages; replaced to include the 4 figures (pdf); this version with more extensive discussion of quantum cosmolog

Galois Theory of Algorithms

Many different programs are the implementation of the same algorithm. The collection of programs can be partitioned into different classes corresponding to the algorithms they implement. This makes the collection of algorithms a quotient of the collection of programs. Similarly, there are many different algorithms that implement the same computable function. The collection of algorithms can be partitioned into different classes corresponding to what computable function they implement. This makes the collection of computable functions into a quotient of the collection of algorithms. Algorithms are intermediate between programs and functions: Programs $\twoheadrightarrow$ Algorithms $\twoheadrightarrow$ Functions. \noindent Galois theory investigates the way that a subobject sits inside an object. We investigate how a quotient object sits inside an object. By looking at the Galois group of programs, we study the intermediate types of algorithms possible and the types of structures these algorithms can have.Comment: 25 pages, 1 figure. Fixed an error, corrected typos, and added a section on homotopy theor

Navigability of Random Geometric Graphs in the Universe and Other Spacetimes

Random geometric graphs in hyperbolic spaces explain many common structural and dynamical properties of real networks, yet they fail to predict the correct values of the exponents of power-law degree distributions observed in real networks. In that respect, random geometric graphs in asymptotically de Sitter spacetimes, such as the Lorentzian spacetime of our accelerating universe, are more attractive as their predictions are more consistent with observations in real networks. Yet another important property of hyperbolic graphs is their navigability, and it remains unclear if de Sitter graphs are as navigable as hyperbolic ones. Here we study the navigability of random geometric graphs in three Lorentzian manifolds corresponding to universes filled only with dark energy (de Sitter spacetime), only with matter, and with a mixture of dark energy and matter as in our universe. We find that these graphs are navigable only in the manifolds with dark energy. This result implies that, in terms of navigability, random geometric graphs in asymptotically de Sitter spacetimes are as good as random hyperbolic graphs. It also establishes a connection between the presence of dark energy and navigability of the discretized causal structure of spacetime, which provides a basis for a different approach to the dark energy problem in cosmology.Comment: 15 pages, 10 figure

Definable decompositions for graphs of bounded linear cliquewidth

We prove that for every positive integer $k$, there exists an $\text{MSO}_1$-transduction that given a graph of linear cliquewidth at most $k$ outputs, nondeterministically, some cliquewidth decomposition of the graph of width bounded by a function of $k$. A direct corollary of this result is the equivalence of the notions of $\text{CMSO}_1$-definability and recognizability on graphs of bounded linear cliquewidth.Comment: 39 pages, 5 figures. The conference version of the manuscript appeared in the proceedings of LICS 201