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Common asset holdings are widely believed to have been the primary vector of contagion in the recent financial crisis. We develop a network approach to the amplification of financial contagion due to the combination of overlapping portfolios and leverage, and we show how it can be understood in terms of a generalized branching process. By studying a stylized model we estimate the circumstances under which systemic instabilities are likely to occur as a function of parameters such as leverage, market crowding, diversification, and market impact. Although diversification may be good for individual institutions, it can create dangerous systemic effects, and as a result financial contagion gets worse with too much diversification. Under our model there is a critical threshold for leverage; below it financial networks are always stable, and above it the unstable region grows as leverage increases. The financial system exhibits "robust yet fragile" behavior, with regions of the parameter space where contagion is rare but catastrophic whenever it occurs. Our model and methods of analysis can be calibrated to real data and provide simple yet powerful tools for macroprudential stress testing., Comment: 25 pages, 8 figures

A wide variety of methods have been used to compute percolation thresholds. In lattice percolation, the most powerful of these methods consists of microcanonical simulations using the union-find algorithm to efficiently determine the connected clusters, and (in two dimensions) using exact values from conformal field theory for the probability, at the phase transition, that various kinds of wrapping clusters exist on the torus. We apply this approach to percolation in continuum models, finding overlaps between objects with real-valued positions and orientations. In particular, we find precise values of the percolation transition for disks, squares, rotated squares, and rotated sticks in two dimensions, and confirm that these transitions behave as conformal field theory predicts. The running time and memory use of our algorithm are essentially linear as a function of the number of objects at criticality., Comment: 6 pages, 8 figures

The proliferation of models for networks raises challenging problems of model selection: the data are sparse and globally dependent, and models are typically high-dimensional and have large numbers of latent variables. Together, these issues mean that the usual model-selection criteria do not work properly for networks. We illustrate these challenges, and show one way to resolve them, by considering the key network-analysis problem of dividing a graph into communities or blocks of nodes with homogeneous patterns of links to the rest of the network. The standard tool for doing this is the stochastic block model, under which the probability of a link between two nodes is a function solely of the blocks to which they belong. This imposes a homogeneous degree distribution within each block; this can be unrealistic, so degree-corrected block models add a parameter for each node, modulating its over-all degree. The choice between ordinary and degree-corrected block models matters because they make very different inferences about communities. We present the first principled and tractable approach to model selection between standard and degree-corrected block models, based on new large-graph asymptotics for the distribution of log-likelihood ratios under the stochastic block model, finding substantial departures from classical results for sparse graphs. We also develop linear-time approximations for log-likelihoods under both the stochastic block model and the degree-corrected model, using belief propagation. Applications to simulated and real networks show excellent agreement with our approximations. Our results thus both solve the practical problem of deciding on degree correction, and point to a general approach to model selection in network analysis.

Preferential attachment is a popular model of growing networks. We consider a generalized model with random node removal, and a combination of preferential and random attachment. Using a high-degree expansion of the master equation, we identify a topological phase transition depending on the rate of node removal and the relative strength of preferential vs. random attachment, where the degree distribution goes from a power law to one with an exponential tail., Comment: The final publication is available at http://www.epj.org

With the growth of the Internet, there has been a push toward designing reliable algorithms that scale effectively in terms of latency, bandwidth and other computational resources. Scalability has been a serious problem especially with peer-to-peer (p2p) networks which may have sizes of more than a million nodes. An important problem in designing reliable algorithms is Byzantine agreement. For reasons of scalability, message complexity is a critical resource for this problem. Unfortunately, previous solutions to Byzantine agreement require each processor to send $O(n)$ messages, where $n$ is the total number of processors in the network. In this dissertation, we show that the Byzantine agreement problem can be solved with significantly less that a linear number of messages both in theory and in practice. We implement and test algorithms that solve the classical problem with each processor sending only $\ ilde{O}(\sqrt{n})$ messages. Further, we consider the problem in the case where we assume the existence of a random beacon: a global source of random bits. We show that with this assumption, the required number of messages drops to $O(\log n)$, with small hidden constants. Our algorithms are Monte Carlo and succeed with high probability, that is probability $1-o(n^k)$ for some positive constant $k$. Our empirical results suggest that our algorithms may outperform classical solutions to Byzantine agreement for network of size larger than 30,000 nodes.

Random instances of feedforward Boolean circuits are studied both analytically and numerically. Evaluating these circuits is known to be a P-complete problem and thus, in the worst case, believed to be impossible to perform, even given a massively parallel computer, in time much less than the depth of the circuit. Nonetheless, it is found that for some ensembles of random circuits, saturation to a fixed truth value occurs rapidly so that evaluation of the circuit can be accomplished in much less parallel time than the depth of the circuit. For other ensembles saturation does not occur and circuit evaluation is apparently hard. In particular, for some random circuits composed of connectives with five or more inputs, the number of true outputs at each level is a chaotic sequence. Finally, while the average case complexity depends on the choice of ensemble, it is shown that for all ensembles it is possible to simultaneously construct a typical circuit together with its solution in polylogarithmic parallel time., Comment: 16 pages, 10 figures, matches published version

In many real-world networks, nodes have class labels, attributes, or variables that affect the network's topology. If the topology of the network is known but the labels of the nodes are hidden, we would like to select a small subset of nodes such that, if we knew their labels, we could accurately predict the labels of all the other nodes. We develop an active learning algorithm for this problem which uses information-theoretic techniques to choose which nodes to explore. We test our algorithm on networks from three different domains: a social network, a network of English words that appear adjacently in a novel, and a marine food web. Our algorithm makes no initial assumptions about how the groups connect, and performs well even when faced with quite general types of network structure. In particular, we do not assume that nodes of the same class are more likely to be connected to each other---only that they connect to the rest of the network in similar ways., Comment: 9 pages, 7 figures, KDD 2011: The 17th ACM SIGKDD Conference on Knowledge Discovery and Data Mining

In this paper we extend our previous work on the stochastic block model, a commonly used generative model for social and biological networks, and the problem of inferring functional groups or communities from the topology of the network. We use the cavity method of statistical physics to obtain an asymptotically exact analysis of the phase diagram. We describe in detail properties of the detectability/undetectability phase transition and the easy/hard phase transition for the community detection problem. Our analysis translates naturally into a belief propagation algorithm for inferring the group memberships of the nodes in an optimal way, i.e., that maximizes the overlap with the underlying group memberships, and learning the underlying parameters of the block model. Finally, we apply the algorithm to two examples of real-world networks and discuss its performance., Comment: Typos in eq. (40) on p. 13 fixed

A variety of tasks are typically performed after a phylogenetic reconstruction proper -- tasks which fall under the category phylogenetic post-analysis. In this dissertation, we present novel approaches and efficient algorithms for three post-analysis tasks: taking distances between (typically, all pairs in a set of) trees, bootstrapping, and building consensus trees. For instance, it is often the case that reconstruction finds multiple plausible trees. One basic way of addressing this situation is to take distances between pairs of trees, in order to gain an understanding of the extent to which the trees disagree. The most frequently employed manner for computing the distance between a tree pair is the Robinson-Foulds metric, a natural dissimilarity measure between a pair of phylogenetic trees. We present a novel family of algorithms for efficiently computing the Robinson-Foulds metric. Bootstrapping is a post-analysis technique for drawing support values on tree edges, and is often used for assessing the extent to which the underlying data (e.g., molecular sequences) supports a reconstructed tree. The basis of the approach is to reconstruct many trees, called replicates, based on random subsampling of the original data. However, to date, there has been little treatment in phylogeny regarding the question of how many bootstrap replicates to generate. We propose bootstopping criteria which are designed to provide on-the-fly (i.e., runtime) guidance for determining when enough bootstrap replicates have been reconstructed. Another common post-analysis task is to build a consensus tree, a summary tree that attempts to capture the information agreed upon by bootstrap replicates. Unfortunately, the most popular consensus methods are susceptible to confusion by rogue taxa, i.e., taxa that cannot be placed with assurance anywhere within the tree. We present novel theory and efficient algorithms to identify rogue taxa, as well as a novel technique for interpreting the resul

As we add rigid bars between points in the plane, at what point is there a giant (linear-sized) rigid component, which can be rotated and translated, but which has no internal flexibility? If the points are generic, this depends only on the combinatorics of the graph formed by the bars. We show that if this graph is an Erdos-Renyi random graph G(n,c/n), then there exists a sharp threshold for a giant rigid component to emerge. For c < c_2, w.h.p. all rigid components span one, two, or three vertices, and when c > c_2, w.h.p. there is a giant rigid component. The constant c_2 \approx 3.588 is the threshold for 2-orientability, discovered independently by Fernholz and Ramachandran and Cain, Sanders, and Wormald in SODA'07. We also give quantitative bounds on the size of the giant rigid component when it emerges, proving that it spans a (1-o(1))-fraction of the vertices in the (3+2)-core. Informally, the (3+2)-core is maximal induced subgraph obtained by starting from the 3-core and then inductively adding vertices with 2 neighbors in the graph obtained so far., Comment: To appear in SODA'11. Added proofs omitted from the proceedings version

We study truthful mechanisms for hiring a team of agents in three classes of set systems: Vertex Cover auctions, k-flow auctions, and cut auctions. For Vertex Cover auctions, the vertices are owned by selfish and rational agents, and the auctioneer wants to purchase a vertex cover from them. For k-flow auctions, the edges are owned by the agents, and the auctioneer wants to purchase k edge-disjoint s-t paths, for given s and t. In the same setting, for cut auctions, the auctioneer wants to purchase an s-t cut. Only the agents know their costs, and the auctioneer needs to select a feasible set and payments based on bids made by the agents. We present constant-competitive truthful mechanisms for all three set systems. That is, the maximum overpayment of the mechanism is within a constant factor of the maximum overpayment of any truthful mechanism, for every set system in the class. The mechanism for Vertex Cover is based on scaling each bid by a multiplier derived from the dominant eigenvector of a certain matrix. The mechanism for k-flows prunes the graph to be minimally (k+1)-connected, and then applies the Vertex Cover mechanism. Similarly, the mechanism for cuts contracts the graph until all s-t paths have length exactly 2, and then applies the Vertex Cover mechanism.

Celebrated work of Jerrum, Sinclair, and Vigoda has established that the permanent of a {0,1} matrix can be approximated in randomized polynomial time by using a rapidly mixing Markov chain. A separate strand of the literature has pursued the possibility of an alternate, purely algebraic, polynomial-time approximation scheme. These schemes work by replacing each 1 with a random element of an algebra A, and considering the determinant of the resulting matrix. When A is noncommutative, this determinant can be defined in several ways. We show that for estimators based on the conventional determinant, the critical ratio of the second moment to the square of the first--and therefore the number of trials we need to obtain a good estimate of the permanent--is (1 + O(1/d))^n when A is the algebra of d by d matrices. These results can be extended to group algebras, and semi-simple algebras in general. We also study the symmetrized determinant of Barvinok, showing that the resulting estimator has small variance when d is large enough. However, for constant d--the only case in which an efficient algorithm is known--we show that the critical ratio exceeds 2^{n} / n^{O(d)}. Thus our results do not provide a new polynomial-time approximation scheme for the permanent. Indeed, they suggest that the algebraic approach to approximating the permanent faces significant obstacles. We obtain these results using diagrammatic techniques in which we express matrix products as contractions of tensor products. When these matrices are random, in either the Haar measure or the Gaussian measure, we can evaluate the trace of these products in terms of the cycle structure of a suitably random permutation. In the symmetrized case, our estimates are then derived by a connection with the character theory of the symmetric group.

It is known that any quantum algorithm for Graph Isomorphism that works within the framework of the hidden subgroup problem (HSP) must perform highly entangled measurements across (n log n) coset states. One of the only known models for how such a measurement could be carried out efficiently is Kuperberg's algorithm for the HSP in the dihedral group, in which quantum states are adaptively combined and measured according to the decomposition of tensor products into irreducible representations. This quantum sieve starts with coset states, and works its way down towards representations whose probabilities differ depending on, for example, whether the hidden subgroup is trivial or nontrivial. In this paper we show that no such approach can produce a polynomial-time quantum algorithm for Graph Isomorphism. Specifically, we consider the natural reduction of Graph Isomorphism to the HSP over the wreath product Sn - Z2. Using a recently proved bound on the irreducible characters of Sn, we show that no algorithm in this family can solve Graph Isomorphism in less than e(n) time, no matter what adaptive rule it uses to select and combine quantum states. In particular, algorithms of this type can offer essentially no improvement over the best known classical algorithms, which run in time eO((n log n))., Presented at the Annual ACM Symposium (39th) held in San Diego, CA on 11-13 June 2007.

Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which an unknown subgroup $H$ of a group $G$ must be determined from a quantum state $\psi$ over $G$ that is uniformly supported on a left coset of $H$. These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of $\psi$ is computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation, in a suitably chosen basis, as well as its name) occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups $H$ of the $q$-hedral groups, i.e., semidirect products ${\mathbb Z}_q \ltimes {\mathbb Z}_p$, where $q \mid (p-1)$, and in particular the affine groups $A_p$, can be information-theoretically reconstructed using the strong standard method. Moreover, if $|H| = p/ {\rm polylog}(p)$, these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation. We compare our algorithms to two weaker methods that have been discussed in the literature—the “forgetful” abelian method, and measurement in a random basis—and show that both of these are weaker than the strong standard method. Thus, at least for some families of groups, it is crucial to use the full power of representation theory and nonabelian Fourier analysis, namely, to measure the high-dimensional representations in an adapted basis that respects the group's subgroup structure. We apply our algorithm f

We resolve the question of whether Fourier sampling can efficiently solve the hidden subgroup problem in general groups. Specifically, we show that the hidden subgroup problem in the symmetric group cannot be efficiently solved by strong Fourier sampling. Indeed we prove the stronger statement that no measurement of a single coset state can reveal more than an exponentially small amount of information about the identity of the hidden subgroup, in the special case relevant to the Graph Isomorphism problem.

The evaluation of incomplete satisfiability solvers depends critically on the availability of hard satisfiable instances. A plausible source of such instances consists of random k-SAT formulas whose clauses are chosen uniformly from among all clauses satisfying some randomly chosen truth assignment A. Unfortunately, instances generated in this manner tend to be relatively easy and can be solved efficiently by practical heuristics. Roughly speaking, as the formula's density increases, for a number of different algorithms, A acts as a stronger and stronger attractor. Motivated by recent results on the geometry of the space of satisfying truth assignments of random k-SAT and NAE-k-SAT formulas, we introduce a simple twist on this basic model, which appears to dramatically increase its hardness. Namely, in addition to forbidding the clauses violated by the hidden assignment A, we also forbid the clauses violated by its complement, so that both A and complement of A are satisfying. It appears that under this "symmetrization'' the effects of the two attractors largely cancel out, making it much harder for algorithms to find any truth assignment. We give theoretical and experimental evidence supporting this assertion., Comment: Preliminary version appeared in AAAI 2004

To test incomplete search algorithms for constraint satisfaction problems such as 3-SAT, we need a source of hard, but satisfiable, benchmark instances. A simple way to do this is to choose a random truth assignment A, and then choose clauses randomly from among those satisfied by A. However, this method tends to produce easy problems, since the majority of literals point toward the ``hidden'' assignment A. Last year, Achlioptas, Jia and Moore proposed a problem generator that cancels this effect by hiding both A and its complement. While the resulting formulas appear to be just as hard for DPLL algorithms as random 3-SAT formulas with no hidden assignment, they can be solved by WalkSAT in only polynomial time. Here we propose a new method to cancel the attraction to A, by choosing a clause with t > 0 literals satisfied by A with probability proportional to q^t for some q < 1. By varying q, we can generate formulas whose variables have no bias, i.e., which are equally likely to be true or false; we can even cause the formula to ``deceptively'' point away from A. We present theoretical and experimental results suggesting that these formulas are exponentially hard both for DPLL algorithms and for incomplete algorithms such as WalkSAT., Comment: 6 pages, 7 figures

Understanding the structure of the Internet graph is a crucial step for building accurate network models and designing efficient algorithms for Internet applications. Yet, obtaining its graph structure is a surprisingly difficult task, as edges cannot be explicitly queried. Instead, empirical studies rely on traceroutes to build what are essentially single-source, all-destinations, shortest-path trees. These trees only sample a fraction of the network's edges, and a recent paper by Lakhina et al. found empirically that the resuting sample is intrinsically biased. For instance, the observed degree distribution under traceroute sampling exhibits a power law even when the underlying degree distribution is Poisson. In this paper, we study the bias of traceroute sampling systematically, and, for a very general class of underlying degree distributions, calculate the likely observed distributions explicitly. To do this, we use a continuous-time realization of the process of exposing the BFS tree of a random graph with a given degree distribution, calculate the expected degree distribution of the tree, and show that it is sharply concentrated. As example applications of our machinery, we show how traceroute sampling finds power-law degree distributions in both delta-regular and Poisson-distributed random graphs. Thus, our work puts the observations of Lakhina et al. on a rigorous footing, and extends them to nearly arbitrary degree distributions., Comment: Long-format version (19 pages); includes small correction to section 6.1

A great deal of effort has been spent measuring topological features of the Internet. However, it was recently argued that sampling based on taking paths or traceroutes through the network from a small number of sources introduces a fundamental bias in the observed degree distribution. We examine this bias analytically and experimentally. For Erdos-Renyi random graphs with mean degree c, we show analytically that traceroute sampling gives an observed degree distribution P(k) ~ 1/k for k < c, even though the underlying degree distribution is Poisson. For graphs whose degree distributions have power-law tails P(k) ~ k^-alpha, traceroute sampling from a small number of sources can significantly underestimate the value of \alpha when the graph has a large excess (i.e., many more edges than vertices). We find that in order to obtain a good estimate of alpha it is necessary to use a number of sources which grows linearly in the average degree of the underlying graph. Based on these observations we comment on the accuracy of the published values of alpha for the Internet., Comment: 4 pages, 3 figures; supercedes cond-mat/0407339 and contains scaling results on the accuracy of multi-source traceroute studies