General comments: (i) The authors only solve a new special kind of higher order consistency constraints, generalizing soft PN-potentials, but not a truly general class of constraints, as indicated in the title or in the abstract. In case of MAP-inference, which is normally desired, the goal is to obtain a single assignment which satisfies all given linear constraints. The relaxed model the authors optimize is simply a byproduct of looking for marginals instead of MAP-assignments (the added entropy is responsible for this). In case of vanishing entropy one gets the same model. Hence there certainly remains the disadvantage of a parameter in the PN-potential, but now hidden in the entropy.

Inference in Markov random fields subject to consistency structure is a fundamental problem that arises in many real-life applications. In order to enforce consistency, classical approaches utilize consistency potentials or encode constraints over feasible instances. Unfortunately this comes at the price of a tremendous computational burden. In this paper we suggest to tackle consistency by incorporating constraints on beliefs. This permits derivation of a closed-form message-passing algorithm which we refer to as the Constraints Based Convex Belief Propagation (CBCBP). Experiments show that CBCBP outperforms the conventional consistency potential based approach, while being at least an order of magnitude faster.

Major comments: - As mentioned by the authors in the introduction, the SBM is widely used throughout many areas, and not only for detecting communities. The authors introduce no condition on the parameter matrix Q so that their setup includes (in principle) cases as diverse as communities, anti-communities (graphs of a bipartite type) and any other type of connectivity structure. However, I believe that their algorithm (based on belief propagation and thus on the community structure) will reveal only communities. If I am right, they should specify where this underlying assumption comes into play. Is it only that definition 3 becomes useless when the structure is not the one of communities?

The stochastic block model (SBM) has long been studied in machine learning and network science as a canonical model for clustering and community detection. In the recent years, new developments have demonstrated the presence of threshold phenomena for this model, which have set new challenges for algorithms. For the detection problem in symmetric SBMs, Decelle et al. conjectured that the so-called Kesten-Stigum (KS) threshold can be achieved efficiently. This was proved for two communities, but remained open for three and more communities. We prove this conjecture here, obtaining a general result that applies to arbitrary SBMs with linear size communities. The developed algorithm is a linearized acyclic belief propagation (ABP) algorithm, which mitigates the effects of cycles while provably achieving the KS threshold in O(n ln n) time. This extends prior methods by achieving universally the KS threshold while reducing or preserving the computational complexity. ABP is also connected to a power iteration method on a generalized nonbacktracking operator, formalizing the spectral-message passing interplay described in Krzakala et al., and extending results from Bordenave et al.

The authors develop an very interesting method to the hard problem of computing the partition function of GM. In particular, there is no method to compute Z for general graphs accurately and log(Z) is a particularly interesting component in many fields: it is related to the likelihood in Machine Learning, it gives all the statistical information of a system in statistical mechanic, ... I think therefore that any effort toward that direction is an important contribution. Concerning the work presented here, I have some questions and concerns that I would like the authors to answer: 1) the authors mentioned that BP in general fail to converge on general graph but that more involved convergent alternative exists. I think the author should be more specific, precising if all the convergent algorithms do provide the same estimation of the BP partition function and it they are indeed linear in the system size. The 2-reg loop can be proven to converge in polynomial time but the full loop does not have these garranty.

The community detection problem requires to cluster the nodes of a network into a small number of well-connected'communities'. There has been substantial recent progress in characterizing the fundamental statistical limits of community detection under simple stochastic block models. However, in real-world applications, the network structure is typically dynamic, with nodes that join over time. In this setting, we would like a detection algorithm to perform only a limited number of updates at each node arrival. While standard voting approaches satisfy this constraint, it is unclear whether they exploit the network information optimally.

Belief Propagation (BP) is an important message-passing algorithm for various reasoning tasks over graphical models, including solving the Constraint Optimization Problems (COPs). It has been shown that BP can achieve state-of-the-art performance on various benchmarks by mixing old and new messages before sending the new one, i.e., damping. However, existing methods on tuning a static damping factor for BP not only is laborious but also harms their performance. Moreover, existing BP algorithms treat each variable node's neighbors equally when composing a new message, which also limits their exploration ability. To address these issues, we seamlessly integrate BP, Gated Recurrent Units (GRUs), and Graph Attention Networks (GATs) within the massage-passing framework to reason about dynamic weights and damping factors for composing new BP messages. Our model, Deep Attentive Belief Propagation (DABP), takes the factor graph and the BP messages in each iteration as the input and infers the optimal weights and damping factors through GRUs and GATs, followed by a multi-head attention layer.

Max-product Belief Propagation (BP) is a popular message-passing algorithm for computing a Maximum-A-Posteriori (MAP) assignment over a distribution represented by a Graphical Model (GM). It has been shown that BP can solve a number of combinatorial optimization problems including minimum weight matching, shortest path, network flow and vertex cover under the following common assumption: the respective Linear Programming (LP) relaxation is tight, i.e., no integrality gap is present. However, when LP shows an integrality gap, no model has been known which can be solved systematically via sequential applications of BP. In this paper, we develop the first such algorithm, coined Blossom-BP, for solving the minimum weight matching problem over arbitrary graphs. Each step of the sequential algorithm requires applying BP over a modified graph constructed by contractions and expansions of blossoms, i.e., odd sets of vertices.

We propose an original particle-based implementation of the Loopy Belief Propagation (LPB) algorithm for pairwise Markov Random Fields (MRF) on a continuous state space. This is achieved by considering proposal distributions in the exponential family whose parameters are updated iterately in an Expectation Propagation (EP) framework. The proposed particle scheme provides consistent estimation of the LBP marginals as the number of particles increases. We demonstrate that it provides more accurate results than the Particle Belief Propagation (PBP) algorithm of Ihler and McAllester (2009) at a fraction of the computational cost and is additionally more robust empirically. The computational complexity of our algorithm at each iteration is quadratic in the number of particles.

As AI systems are integrated into social networks, there are AI safety concerns that AI-generated content may dominate the web, e.g. in popularity or impact on beliefs. To understand such questions, this paper proposes the Digital Ecosystem of Beliefs (Digico), the first evolutionary framework for controlled experimentation with multi-population interactions in simulated social networks. The framework models a population of agents which change their messaging strategies due to evolutionary updates following a Universal Darwinism approach, interact via messages, influence each other's beliefs through dynamics based on a contagion model, and maintain their beliefs through cognitive Lamarckian inheritance. Initial experiments with an abstract implementation of Digico show that: a) when AIs have faster messaging, evolution, and more influence in the recommendation algorithm, they get 80% to 95% of the views, depending on the size of the influence benefit; b) AIs designed for propaganda can typically convince 50% of humans to adopt extreme beliefs, and up to 85% when agents believe only a limited number of channels; c) a penalty for content that violates agents' beliefs reduces propaganda effectiveness by up to 8%. We further discuss implications for control (e.g. legislation) and Digico as a means of studying evolutionary principles.