In the field of distributed consensus and blockchains, the synchronous communication model assumes that all messages between honest parties are delayed at most by a known constant $Delta$. Recent literature establishes that the longest-chain blockcha
in protocol is secure under the synchronous model. However, for a fixed mining rate, the security guarantees degrade with $Delta$. We analyze the performance of the longest-chain protocol under the assumption that the communication delays are random, independent, and identically distributed. This communication model allows for distributions with unbounded support and is a strict generalization of the synchronous model. We provide safety and liveness guarantees with simple, explicit bounds on the failure probabilities. These bounds hold for infinite-horizon executions and decay exponentially with the security parameter. In particular, we show that the longest-chain protocol has good security guarantees when delays are sporadically large and possibly unbounded, which is reflective of real-world network conditions.
Recovery of the causal structure of dynamic networks from noisy measurements has long been a problem of intense interest across many areas of science and engineering. Many algorithms have been proposed, but there is no work that compares the performa
nce of the algorithms to converse bounds in a non-asymptotic setting. As a step to address this problem, this paper gives lower bounds on the error probability for causal network support recovery in a linear Gaussian setting. The bounds are based on the use of the Bhattacharyya coefficient for binary hypothesis testing problems with mixture probability distributions. Comparison of the bounds and the performance achieved by two representative recovery algorithms are given for sparse random networks based on the ErdH{o}s-Renyi model.
The theory of mean field games is a tool to understand noncooperative dynamic stochastic games with a large number of players. Much of the theory has evolved under conditions ensuring uniqueness of the mean field game Nash equilibrium. However, in so
me situations, typically involving symmetry breaking, non-uniqueness of solutions is an essential feature. To investigate the nature of non-unique solutions, this paper focuses on the technically simple setting where players have one of two states, with continuous time dynamics, and the game is symmetric in the players, and players are restricted to using Markov strategies. All the mean field game Nash equilibria are identified for a symmetric follow the crowd game. Such equilibria correspond to symmetric $epsilon$-Nash Markov equilibria for $N$ players with $epsilon$ converging to zero as $N$ goes to infinity. In contrast to the mean field game, there is a unique Nash equilibrium for finite $N.$ It is shown that fluid limits arising from the Nash equilibria for finite $N$ as $N$ goes to infinity are mean field game Nash equilibria, and evidence is given supporting the conjecture that such limits, among all mean field game Nash equilibria, are the ones that are stable fixed points of the mean field best response mapping.
A message passing algorithm is derived for recovering communities within a graph generated by a variation of the Barab{a}si-Albert preferential attachment model. The estimator is assumed to know the arrival times, or order of attachment, of the verti
ces. The derivation of the algorithm is based on belief propagation under an independence assumption. Two precursors to the message passing algorithm are analyzed: the first is a degree thresholding (DT) algorithm and the second is an algorithm based on the arrival times of the children (C) of a given vertex, where the children of a given vertex are the vertices that attached to it. Comparison of the performance of the algorithms shows it is beneficial to know the arrival times, not just the number, of the children. The probability of correct classification of a vertex is asymptotically determined by the fraction of vertices arriving before it. Two extensions of Algorithm C are given: the first is based on joint likelihood of the children of a fixed set of vertices; it can sometimes be used to seed the message passing algorithm. The second is the message passing algorithm. Simulation results are given.
A variation of the preferential attachment random graph model of Barabasi and Albert is defined that incorporates planted communities. The graph is built progressively, with new vertices attaching to the existing ones one-by-one. At every step, the i
ncoming vertex is randomly assigned a label, which represents a community it belongs to. This vertex then chooses certain vertices as its neighbors, with the choice of each vertex being proportional to the degree of the vertex multiplied by an affinity depending on the labels of the new vertex and a potential neighbor. It is shown that the fraction of half-edges attached to vertices with a given label converges almost surely for some classes of affinity matrices. In addition, the empirical degree distribution for the set of vertices with a given label converges to a heavy tailed distribution, such that the tail decay parameter can be different for different communities. Our proof method may be of independent interest, both for the classical Barabasi -Albert model and for other possible extensions.
We study a semidefinite programming (SDP) relaxation of the maximum likelihood estimation for exactly recovering a hidden community of cardinality $K$ from an $n times n$ symmetric data matrix $A$, where for distinct indices $i,j$, $A_{ij} sim P$ if
$i, j$ are both in the community and $A_{ij} sim Q$ otherwise, for two known probability distributions $P$ and $Q$. We identify a sufficient condition and a necessary condition for the success of SDP for the general model. For both the Bernoulli case ($P={{rm Bern}}(p)$ and $Q={{rm Bern}}(q)$ with $p>q$) and the Gaussian case ($P=mathcal{N}(mu,1)$ and $Q=mathcal{N}(0,1)$ with $mu>0$), which correspond to the problem of planted dense subgraph recovery and submatrix localization respectively, the general results lead to the following findings: (1) If $K=omega( n /log n)$, SDP attains the information-theoretic recovery limits with sharp constants; (2) If $K=Theta(n/log n)$, SDP is order-wise optimal, but strictly suboptimal by a constant factor; (3) If $K=o(n/log n)$ and $K to infty$, SDP is order-wise suboptimal. The same critical scaling for $K$ is found to hold, up to constant factors, for the performance of SDP on the stochastic block model of $n$ vertices partitioned into multiple communities of equal size $K$. A key ingredient in the proof of the necessary condition is a construction of a primal feasible solution based on random perturbation of the true cluster matrix.
The principal submatrix localization problem deals with recovering a $Ktimes K$ principal submatrix of elevated mean $mu$ in a large $ntimes n$ symmetric matrix subject to additive standard Gaussian noise. This problem serves as a prototypical exampl
e for community detection, in which the community corresponds to the support of the submatrix. The main result of this paper is that in the regime $Omega(sqrt{n}) leq K leq o(n)$, the support of the submatrix can be weakly recovered (with $o(K)$ misclassification errors on average) by an optimized message passing algorithm if $lambda = mu^2K^2/n$, the signal-to-noise ratio, exceeds $1/e$. This extends a result by Deshpande and Montanari previously obtained for $K=Theta(sqrt{n}).$ In addition, the algorithm can be extended to provide exact recovery whenever information-theoretically possible and achieve the information limit of exact recovery as long as $K geq frac{n}{log n} (frac{1}{8e} + o(1))$. The total running time of the algorithm is $O(n^2log n)$. Another version of the submatrix localization problem, known as noisy biclustering, aims to recover a $K_1times K_2$ submatrix of elevated mean $mu$ in a large $n_1times n_2$ Gaussian matrix. The optimized message passing algorithm and its analysis are adapted to the bicluster problem assuming $Omega(sqrt{n_i}) leq K_i leq o(n_i)$ and $K_1asymp K_2.$ A sharp information-theoretic condition for the weak recovery of both clusters is also identified.
Community detection is considered for a stochastic block model graph of n vertices, with K vertices in the planted community, edge probability p for pairs of vertices both in the community, and edge probability q for other pairs of vertices. The ma
in focus of the paper is on weak recovery of the community based on the graph G, with o(K) misclassified vertices on average, in the sublinear regime $n^{1-o(1)} leq K leq o(n).$ A critical parameter is the effective signal-to-noise ratio $lambda=K^2(p-q)^2/((n-K)q)$, with $lambda=1$ corresponding to the Kesten-Stigum threshold. We show that a belief propagation algorithm achieves weak recovery if $lambda>1/e$, beyond the Kesten-Stigum threshold by a factor of $1/e.$ The belief propagation algorithm only needs to run for $log^ast n+O(1) $ iterations, with the total time complexity $O(|E| log^*n)$, where $log^*n$ is the iterated logarithm of $n.$ Conversely, if $lambda leq 1/e$, no local algorithm can asymptotically outperform trivial random guessing. Furthermore, a linear message-passing algorithm that corresponds to applying power iteration to the non-backtracking matrix of the graph is shown to attain weak recovery if and only if $lambda>1$. In addition, the belief propagation algorithm can be combined with a linear-time voting procedure to achieve the information limit of exact recovery (correctly classify all vertices with high probability) for all $K ge frac{n}{log n} left( rho_{rm BP} +o(1) right),$ where $rho_{rm BP}$ is a function of $p/q$.
We study the problem of recovering a hidden community of cardinality $K$ from an $n times n$ symmetric data matrix $A$, where for distinct indices $i,j$, $A_{ij} sim P$ if $i, j$ both belong to the community and $A_{ij} sim Q$ otherwise, for two know
n probability distributions $P$ and $Q$ depending on $n$. If $P={rm Bern}(p)$ and $Q={rm Bern}(q)$ with $p>q$, it reduces to the problem of finding a densely-connected $K$-subgraph planted in a large Erdos-Renyi graph; if $P=mathcal{N}(mu,1)$ and $Q=mathcal{N}(0,1)$ with $mu>0$, it corresponds to the problem of locating a $K times K$ principal submatrix of elevated means in a large Gaussian random matrix. We focus on two types of asymptotic recovery guarantees as $n to infty$: (1) weak recovery: expected number of classification errors is $o(K)$; (2) exact recovery: probability of classifying all indices correctly converges to one. Under mild assumptions on $P$ and $Q$, and allowing the community size to scale sublinearly with $n$, we derive a set of sufficient conditions and a set of necessary conditions for recovery, which are asymptotically tight with sharp constants. The results hold in particular for the Gaussian case, and for the case of bounded log likelihood ratio, including the Bernoulli case whenever $frac{p}{q}$ and $frac{1-p}{1-q}$ are bounded away from zero and infinity. An important algorithmic implication is that, whenever exact recovery is information theoretically possible, any algorithm that provides weak recovery when the community size is concentrated near $K$ can be upgraded to achieve exact recovery in linear additional time by a simple voting procedure.
Resolving a conjecture of Abbe, Bandeira and Hall, the authors have recently shown that the semidefinite programming (SDP) relaxation of the maximum likelihood estimator achieves the sharp threshold for exactly recovering the community structure unde
r the binary stochastic block model of two equal-sized clusters. The same was shown for the case of a single cluster and outliers. Extending the proof techniques, in this paper it is shown that SDP relaxations also achieve the sharp recovery threshold in the following cases: (1) Binary stochastic block model with two clusters of sizes proportional to network size but not necessarily equal; (2) Stochastic block model with a fixed number of equal-sized clusters; (3) Binary censored block model with the background graph being ErdH{o}s-Renyi. Furthermore, a sufficient condition is given for an SDP procedure to achieve exact recovery for the general case of a fixed number of clusters plus outliers. These results demonstrate the versatility of SDP relaxation as a simple, general purpose, computationally feasible methodology for community detection.
