Cristopher Moore Santa Fe Institute Friday, February 14, 2020 11:30Am Luddy Hall 1106

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Cristopher Moore Santa Fe Institute Friday, February 14, 2020 11:30am Luddy Hall 1106 The physics of statistical inference and community detection Abstract: There is a deep analogy between Bayesian inference and statistical physics. Whenever we try to fit a model to noisy data, we can think about the “energy landscape" of possible models, and look for phase transitions where the ground truth suddenly gets lost in this landscape. I’ll use this framework to describe a phase transition in community detection in networks, where communities suddenly become hard or impossible to find. I will discuss why and how this detectability transition occurs, look at related spectral algorithms, and give a hint of similar phase transitions in other inference problems. Biography: Cristopher Moore received his B.A. in Physics, Mathematics, and Integrated Science from Northwestern University, and his Ph.D. in Physics from Cornell. From 2000 to 2012 he was a professor at the University of New Mexico, with joint appointments in Computer Science and Physics. Since 2012, Moore has been a resident professor at the Santa Fe Institute; he has also held visiting positions at École Normale Superieure, École Polytechnique, Université Paris 7, the Niels Bohr Institute, Northeastern University, the University of Michigan, and Microsoft Research. He has published over 150 papers at the boundary between physics and computer science, ranging from quantum computing, to phase transitions in NP-complete problems, to the theory of social networks and efficient algorithms for analyzing their structure. He is an elected Fellow of the American Physical Society, the American Mathematical Society, and the American Association for the Advancement of Science. With Stephan Mertens, he is the author of The Nature of Computation from Oxford University Press. .

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Cristopher Moore
Cristopher Moore Santa Fe Institute Santa Fe, NM 87501 [email protected] August 23, 2021 1 Education Born March 12, 1968 in New Brunswick, New Jersey. Northwestern University, B.A. in Physics, Mathematics, and the Integrated Science Program, with departmental honors in all three departments, 1986. Cornell University, Ph.D. in Physics, 1991. Philip Holmes, advisor. Thesis: “Undecidability and Unpredictability in Dynamical Systems.” 2 Employment Professor, Santa Fe Institute 2012–present Professor, Computer Science Department with a joint appointment in the Department of Physics and Astronomy, University of New Mexico, Albuquerque 2008–2012 Associate Professor, University of New Mexico 2005–2008 Assistant Professor, University of New Mexico 2000–2005 Research Professor, Santa Fe Institute 1998–1999 City Councilor, District 2, Santa Fe 1994–2002 Postdoctoral Fellow, Santa Fe Institute 1992–1998 Lecturer, Cornell University Spring 1991 Graduate Intern, Niels Bohr Institute/NORDITA, Copenhagen Summers 1988 and 1989 Teaching Assistant, Cornell University Physics Department Fall 1986–Spring 1990 Computer programmer, Bio-Imaging Research, Lincolnshire, Illinois Summers 1984–1986 3 Other Appointments Visiting Researcher, Microsoft Research New England Fall 2019 Visiting Professor, Ecole´ Normale Sup´erieure October–November 2016 Visiting Professor, Northeastern University October–November 2015 Visiting Professor, University of Michigan, Ann Arbor September–October 2005 Visiting Professor, Ecole´ Normale Sup´erieure de Lyon June 2004 Visiting Professor,
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Minimum Circuit Size, Graph Isomorphism, and Related Problems∗
SIAM J. COMPUT. c 2018 Society for Industrial and Applied Mathematics Vol. 47, No. 4, pp. 1339{1372 MINIMUM CIRCUIT SIZE, GRAPH ISOMORPHISM, AND RELATED PROBLEMS∗ ERIC ALLENDERy , JOSHUA A. GROCHOWz , DIETER VAN MELKEBEEKx , CRISTOPHER MOORE{, AND ANDREW MORGANx Abstract. We study the computational power of deciding whether a given truth table can be described by a circuit of a given size (the minimum circuit size problem, or MCSP for short) and of the variant denoted as MKTP, where circuit size is replaced by a polynomially related Kolmogorov measure. Prior to our work, all reductions from supposedly intractable problems to MCSP/MKTP hinged on the power of MCSP/MKTP to distinguish random distributions from distributions pro- duced by hardness-based pseudorandom generator constructions. We develop a fundamentally dif- ferent approach inspired by the well-known interactive proof system for the complement of graph isomorphism (GI). It yields a randomized reduction with zero-sided error from GI to MKTP. We generalize the result and show that GI can be replaced by any isomorphism problem for which the underlying group satisfies some elementary properties. Instantiations include linear code equivalence, permutation group conjugacy, and matrix subspace conjugacy. Along the way we develop encodings of isomorphism classes that are efficiently decodable and achieve compression that is at or near the information-theoretic optimum; those encodings may be of independent interest. Key words. reductions between NP-intermediate problems, graph isomorphism, minimum circuit size problem, time-bounded Kolmogorov complexity AMS subject classifications. 68Q15, 68Q17, 68Q30 DOI. 10.1137/17M1157970 1. Introduction. Finding a circuit of minimum size that computes a given Boolean function constitutes the overarching goal in nonuniform complexity theory.
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RIBBON TILE INVARIANTS from SIGNED AREA Cristopher Moore
RIBBON TILE INVARIANTS FROM SIGNED AREA Cristopher Moore Computer Science Department and Department of Physics and Astronomy University of New Mexico Albuquerque NM mo orecsunmedu Igor Pak Department of Mathematics MIT Cambridge MA pakmathmitedu May Abstract Ribb on tiles are p olyomino es consisting of n squares laid out in a path each step of which go es north or east Tile invariants were rst intro duced in P where a full basis of invariants of ribb on tiles was conjectured Here we present a complete pro of of the conjecture which works by asso ciating ribb on tiles with certain p olygons in the complex plane and deriving invariants from the signed area of these p olygons Introduction Polyomino tilings have b een an ob ject of attention of serious mathematicians as well as amateurs for many decades G Recently however the interest in tiling problems has grown as some imp ortant ideas and techniques have b een intro duced In P the second author intro duced a tile counting group which app ears to enco de a large amount of information concerning the combinatorics of tilings He made a conjecture on the group structure and obtained several partial results A sp ecial case of the conjecture was later resolved in MP In this pap er we continue this study and complete the pro of of the conjecture Consider the set of ribbon tiles T dened as connected nsquare tiles with no n two squares in the same diagonal x y c as in the gures b elow It is easy to n1 see that jT j as each tile can b e asso ciated with a path of length n in n the square
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Hierarchical Structure and the Prediction of Missing Links in Networks
Hierarchical structure and the prediction of missing links in networks Aaron Clauset,1,3 Cristopher Moore,1,2,3 M. E. J. Newman3,4∗ 1Department of Computer Science and 2Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131, USA 3Santa Fe Institute, 1399 Hyde Park Rd., Santa Fe, NM 87501, USA 4Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109, USA ∗To whom correspondence should be addressed. E-mail: [email protected]. Networks have in recent years emerged as an invaluable tool for describing and quan- tifying complex systems in many branches of science (1–3). Recent studies suggest that networks often exhibit hierarchical organization, where vertices divide into groups that further subdivide into groups of groups, and so forth over multiple scales. In many cases these groups are found to correspond to known functional units, such as ecological niches in food webs, modules in biochemical networks (protein interaction networks, metabolic networks, or genetic regulatory networks), or communities in social networks (4–7). Here we present a general technique for inferring hierarchical structure from network data and demonstrate that the existence of hierarchy can simultaneously explain and quanti- tatively reproduce many commonly observed topological properties of networks, such as right-skewed degree distributions, high clustering coefﬁcients, and short path lengths. We further show that knowledge of hierarchical structure can be used to predict missing con- nections in partially known networks with high accuracy, and for more general network 1 structures than competing techniques (8). Taken together, our results suggest that hierar- chy is a central organizing principle of complex networks, capable of offering insight into many network phenomena.
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A Statistical Analysis of Lobbying Networks in Legislative Politics∗
Mapping Political Communities: A Statistical Analysis of Lobbying Networks in Legislative Politics∗ In Song Kimy Dmitriy Kuniskyz January 3, 2018 Abstract A vast literature demonstrates the significance for policymaking of lobbying by special inter- est groups. Yet, empirical studies of political representation have been limited by the difficulty of observing a direct connection between politicians and interest groups. We bridge the two with an original dataset of two distinct observable political behaviors: (1) sponsorship of con- gressional bills, and (2) lobbying on congressional bills. We develop a latent space network model to locate politicians and interest groups in a common \marketplace", where proximity implies a closer political connection or alignment of interests. In contrast to repeated find- ings of ideological latent dimensions in previous literature on such models, we find distinct issue-specific political communities of interest groups and politicians. To validate the existence and interpretation of the community structure, we apply stochastic block models and a bipar- tite link community model to explicitly model political actors' community memberships. We consistently find that the latent preference structure of politicians and interest groups is non- ideological and primarily corresponds to industry interests and topical congressional committee memberships. Our findings therefore provide evidence for the existence of powerful political networks in U.S. legislative politics that do not align with the ideological polarization observed in electoral politics. Keywords: Network analysis, lobbying, ideal point estimation, scaling, stochastic block model, link community model, community detection. ∗We thank J. Lawrence Broz, Devin Caughey, Nolan McCarty, Cristopher Moore, Michael Peress, Yunkyu Sohn, and Hye Young You for helpful comments.
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Quantum Fourier Sampling Is Guaranteed to Fail to Compute
Quantum Fourier Sampling is Guaranteed to Fail to Compute Automorphism Groups of Easy Graphs Omar Shehab∗ and Samuel J. Lomonaco Jr.† Department of Computer Science and Electrical Engineering University of Maryland Baltimore County, Baltimore, MD Abstract The quantum hidden subgroup approach is an actively studied approach to solve combina- torial problems in quantum complexity theory. With the success of the Shor’s algorithm, it was hoped that similar approach may be useful to solve the other combinatorial problems. One such problem is the graph isomorphism problem which has survived decades of eﬀorts using the hid- den subgroup approach. This paper provides a systematic approach to create arbitrarily large classes of classically eﬃciently solvable graph automorphism problems or easy graph automor- phism problems for which hidden subgroup approach is guaranteed to always fail irrespective of the size of the graphs no matter how many copies of coset states are used. As the graph isomorphism problem is believed to be at least as hard as the graph automorphism problem, the result of this paper entails that the hidden subgroup approach is also guaranteed to always fail for the arbitrarily large classes of graph isomorphism problems. Combining these two results, it is argued that the hidden subgroup approach is essentially a dead end and alternative quantum algorithmic approach needs to be investigated for the graph isomorphism and automorphism problems. arXiv:1705.00760v2 [quant-ph] 4 Jun 2017 ∗[email protected]. Corresponding author. †[email protected] 1 Contents 1 Introduction 2 1.1 Outlineofthepaper ............................... ..... 4 1.2 Keytechnicalideas............................... ...... 5 2 Historical context 6 3 Preliminaries 9 3.1 GraphTheory....................................
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Fearful Symmetries: Factoring, Graph Isomorphism, and Quantum Computing
Fearful Symmetries: Factoring, Graph Isomorphism, and Quantum Computing Cristopher Moore, University of New Mexico and the Santa Fe Institute Joint work with Alexander Russell, University of Connecticut Daniel Rockmore, Dartmouth Leonard J. Schulman, Caltech Physics Problems: come from Nature have solutions that are as simple, symmetric, and beautiful as possible (far more so than we have any right to expect) Fig. 1: Nature Computer Science Problems: are artiﬁcial are maliciously designed to be the worst possible may or may not have elegant solutions... ...or proofs (cf. Erdős) Fig. 2: The Adversary Beauty is Truth, Truth Beauty In 1928, Dirac saw that the simplest, most beautiful equation for the electron has two solutions. Four years later, the positron was found in the laboratory. Conservation is Symmetry dx ∂H dp ∂H = , = − dt ∂p dt ∂x perhaps you are more familiar with p = mv 2 and F = m a ; try with H = (1/2)mv + V (x) Conservation of momentum follows from translation invariance: dp ∂H moving entire world by dx = − = 0 doesn’t change energy dt ∂x Conservation is Symmetry Noether’s Theorem: symmetry implies conservation dθ ∂H dJ ∂H = , = − dt ∂J dt ∂θ Conservation of angular momentum follows from symmetry under rotation! In classical and quantum mechanics, all conservation laws are of this form. Relativity is Symmetry Physics is invariant under changes of coordinates to a moving frame: x 1 −v/c x → γ !ct" !−v/c 1 " !ct" at small velocities, Galileo: x → x − vt , t → t Symmetry in Computer Science? In Physics, ﬁnding the right symmetry often solves the problem.
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Hierarchical Block Structures and High-Resolution Model Selection in Large Networks
Hierarchical block structures and high-resolution model selection in large networks Tiago P. Peixoto∗ Institut für Theoretische Physik, Universität Bremen, Hochschulring 18, D-28359 Bremen, Germany Discovering and characterizing the large-scale topological features in empirical networks are crucial steps in understanding how complex systems function. However, most existing methods used to obtain the modular structure of networks suffer from serious problems, such as being oblivious to the statistical evidence supporting the discovered patterns, which results in the inability to separate actual structure from noise. In addition to this, one also observes a resolution limit on the size of communities, where smaller but well-defined clusters are not detectable when the network becomes large. This phenomenon occurs not only for the very popular approach of modularity optimization, which lacks built-in statistical validation, but also for more principled methods based on statistical inference and model selection, which do incorporate statistical validation in a formally correct way. Here we construct a nested generative model that, through a complete description of the entire network hierarchy at multiple scales, is capable of avoiding this limitation, and enables the detection of modular structure at levels far beyond those possible with current approaches. Even with this increased resolution, the method is based on the principle of parsimony, and is capable of separating signal from noise, and thus will not lead to the identification of spurious modules even on sparse networks. Furthermore, it fully generalizes other approaches in that it is not restricted to purely assortative mixing patterns, directed or undirected graphs, and ad hoc hierarchical structures such as binary trees.
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Limitations of Quantum Coset States for Graph Isomorphism
Limitations of quantum coset states for graph isomorphism Sean Hallgren∗ Cristopher Moore† Martin Rotteler¨ ‡ Alexander Russell§ Pranab Sen¶ Abstract It has been known for some time that graph isomorphism reduces to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum computation are obtained by solving in- stances of the HSP. A common feature of the resulting algorithms is the use of quantum coset states, which encode the hidden subgroup. An open question has been how hard it is to use these states to solve graph isomorphism. It was recently shown by Moore, Russell, and Schulman [MRS05] that only an exponentially small amount of information is available from one, or a pair of coset states. A potential source of power to exploit are entangled quantum measurements that act jointly on many states at once. We show that entangled quantum measurements on at least Ω(n log n) coset states are necessary to get useful information for the case of graph isomorphism, matching an information theoretic upper bound. This may be viewed as a negative result because in general it seems hard to implement a given highly entangled measurement. Our main theorem is very general and also rules out using joint mea- n surements on few coset states for some other groups, such as GL(n, Fpm ) and G where G is finite and satisfies a suitable property. 1 Introduction Most exponential speedups that have been achieved in quantum computing are obtained by solving some in- stances of the Hidden Subgroup Problem (HSP). In particular, the problems underlying Shor’s algorithms for factoring and discrete logarithm [Sho97], as well as Simon’s problem [Sim94], can be naturally generalized to the HSP: given a function f : G → S from a group G to a set S that is constant on left cosets of some sub- group H ≤ G and distinct on different cosets, find a set of generators for H.
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Cristopher Moore, Santa Fe Institute Joint Work with Aaron Clauset
Inference in networks Cristopher Moore, Santa Fe Institute joint work with Aaron Clauset, Mark Newman, Xiaoran Yan, Yaojia Zhu, Lenka Zdeborová, Florent Krzakala, Aurelien Decelle, Pan Zhang, Jean-Baptiste Rouquier, and Tiffany Pierce Saturday, May 19, 2012 Learning statistics Saturday, May 19, 2012 Learning statistics Saturday, May 19, 2012 Learning statistics Saturday, May 19, 2012 Learning statistics Saturday, May 19, 2012 What is structure? Saturday, May 19, 2012 What is structure? Structure is that which... Saturday, May 19, 2012 What is structure? Structure is that which... makes data different from noise: makes a network different from a random graph Saturday, May 19, 2012 What is structure? Structure is that which... makes data different from noise: makes a network different from a random graph helps us compress the data: describe the network succinctly Saturday, May 19, 2012 What is structure? Structure is that which... makes data different from noise: makes a network different from a random graph helps us compress the data: describe the network succinctly helps us generalize from data we’ve seen from data we haven’t seen: Saturday, May 19, 2012 What is structure? Structure is that which... makes data different from noise: makes a network different from a random graph helps us compress the data: describe the network succinctly helps us generalize from data we’ve seen from data we haven’t seen: from one part of the network to another, Saturday, May 19, 2012 What is structure? Structure is that which... makes data different from noise: makes a network different from a random graph helps us compress the data: describe the network succinctly helps us generalize from data we’ve seen from data we haven’t seen: from one part of the network to another, or from one network to others generated by the same process Saturday, May 19, 2012 What is structure? Structure is that which..
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Ribbon Tile Invariants from the Signed Area
Journal of Combinatorial Theory, Series A 98, 1–16 (2002) doi:10.1006/jcta.2001.3208, available online at http://www.idealibrary.comon Ribbon Tile Invariants from the Signed Area Cristopher Moore Department of Computer Science and Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131 E-mail: [email protected] and Igor Pak Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 E-mail: [email protected] Received August 29, 2000 Ribbon tiles are polyominoes consisting of n squares laid out in a path, each step of which goes north or east. Tile invariants were first introduced by the second author (2000, Trans. Amer. Math. Soc. 352, 5525–5561), where a full basis of invariants of ribbon tiles was conjectured. Here we present a complete proof of the conjecture, which works by associating ribbon tiles with certain polygons in the complex plane, and deriving invariants from the signed area of these polygons. © 2002 Elsevier Science (USA) Key Words: polyomino tilings; tile invariants; Conway group; height representa- tion. 1. INTRODUCTION Polyomino tilings have been an object of attention of serious mathema- ticians as well as amateurs for many decades [G]. Recently, however, the interest in tiling problems has grown as some important ideas and tech- niques have been introduced. In [P1], the second author introduced a tile counting group, which appears to encode a large amount of information concerning the combinatorics of tilings. He made a conjecture on the group structure, and obtained several partial results. A special case of the conjec- ture was later resolved in [MP].
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Statistical Network Analysis: Models, Issues, and New Directions
Statistical Network Analysis: Models, Issues, and New Directions Edited by: Edoardo M. Airoldi David M. Blei Stephen E. Fienberg Anna Goldenberg Eric P. Xing Alice X. Zheng Contents Preface : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ix Part I Invited Presentations 1 Structural Inference of Hierarchies in Networks Aaron Clauset, Cristopher Moore (University of New Mexico), Mark E. J. Newman (University of Michigan) : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 Heider vs Simmel: Emergent Features in Dynamic Structures David Krackhardt (Carnegie Mellon University), Mark S. Handcock (University of Washington) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17 3 Joint Group and Topic Discovery from Relations and Text Andrew McCallum, Xuerui Wang, and Natasha Mohanty (University of Massachusetts, Amherst) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 33 4 Statistical Models for Networks: A Brief Review of Some Recent Research Stanley Wasserman, Garry Robins, Douglas Steinley (Indiana University) 51 Part II Other Presentations 5 Combining Stochastic Block Models and Mixed Membership for Statistical Network Analysis Edoardo M. Airoldi (Carnegie Mellon University), David M. Blei (Princeton), Stephen E. Fienberg (Carnegie Mellon University), Eric P. Xing (Carnegie Mellon University) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 65 vi Contents 6 Exploratory Study of a New Model
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