We study cascades in social networks with the independent cascade (IC) model and the Susceptible-Infected-recovered (SIR) model. The well-studied IC model fails to capture the feature of node recovery, and the SIR model is a variant of the IC model with the node recovery feature. In the SIR model, by computing the probability that a node successfully infects another before its recovery and viewing this probability as the corresponding IC parameter, an equivalence between the two models is established, except that the events of the infections along different out-going edges of a node become dependent in the SIR model, whereas these events are independent in the IC model. In this paper, we thoroughly compare the two models and examine the effect of this extra dependency in the SIR model. By a carefully designed coupling argument, we show that the seeds in the IC model have a stronger influence spread than their counterparts in the SIR model, and sometimes it can be significantly stronger. Specifically, we prove that, given the same network, the same seed sets, and the parameters of the two models being set based on the above-mentioned equivalence, the expected number of infected nodes at the end of the cascade for the IC model is weakly larger than that for the SIR model, and there are instances where this dominance is significant. We also study the influence maximization problem (the optimization problem of selecting a set of nodes as initial seeds in a social network to maximize their influence) with the SIR model. We show that the above-mentioned difference in the two models yields different seed-selection strategies, which motivates the design of influence maximization algorithms specifically for the SIR model. We design efficient approximation algorithms with theoretical guarantees by adapting the reverse-reachable-set-based algorithms, commonly used for the IC model, to the SIR model.

Let Φ be a random k-CNF formula on n variables and m clauses, where each clause is a disjunction of k literals chosen independently and uniformly. Our goal is, for most Φ, to (approximately) uniformly sample from its solution space. Let α = m/n be the density. The previous best algorithm runs in time npoly(k,α) for any α ≲ 2k/300 [Galanis, Goldberg, Guo, and Yang, SIAM J. Comput.'21]. In contrast, our algorithm runs in almost-linear time for any α ≲ 2k/3.

We give the first efficient algorithm to approximately count the number of solutions in the random $k$-SAT model when the density of the formula scales exponentially with $k$. The best previous counting algorithm for the permissive version of the model was due to Montanari and Shah and was based on the correlation decay method, which works up to densities $(1+o_k(1))\frac{2\log k}{k}$, the Gibbs uniqueness threshold for the model. Instead, our algorithm harnesses a recent technique by Moitra to work for random formulas with much higher densities. The main challenge in our setting is to account for the presence of high-degree variables whose marginal distributions are hard to control and which cause significant correlations within the formula.

We give the first efficient algorithm to approximately count the number of solutions in the random $k$-SAT model when the density of the formula scales exponentially with $k$. The best previous counting algorithm for the permissive version of the model was due to Montanari and Shah and was based on the correlation decay method, which works up to densities $(1+o_k(1))\frac{2\log k}{k}$, the Gibbs uniqueness threshold for the model. Instead, our algorithm harnesses a recent technique by Moitra to work for random formulas. The main challenge in our setting is to account for the presence of high-degree variables whose marginal distributions are hard to control and which cause significant correlations within the formula.

We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the so-called uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers $q\geq 3$ and $\Delta\geq 3$, we develop algorithms that produce samples within error $o(1)$ from the $q$-state Potts model on random $\Delta$-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases. The algorithm for the antiferromagnetic Potts model is based on iteratively adding the edges of the graph and resampling a bichromatic class that contains the endpoints of the newly added edge. Key to the algorithm is how to perform the resampling step efficiently since bichromatic classes can potentially induce linear-sized components. To this end, we exploit the tree uniqueness to show that the average growth of bichromatic components is typically small, which allows us to use correlation decay algorithms for the resampling step. While the precise uniqueness threshold on the tree is not known for general values of $q$ and $\Delta$ in the antiferromagnetic case, our algorithm works throughout uniqueness regardless of its value. In the case of the ferromagnetic Potts model, we are able to simplify the algorithm significantly by utilizing the random-cluster representation of the model. In particular, we demonstrate that a percolation-type algorithm succeeds in sampling from the random-cluster model with parameters $p,q$ on random $\Delta$-regular graphs for all values of $q\geq 1$ and $p

We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers $q\geq 3$ and $Δ\geq 3$, we develop algorithms that produce samples within error $o(1)$ from the $q$-state Potts model on random $Δ$-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases. The algorithm for the antiferromagnetic Potts model is based on iteratively adding the edges of the graph and resampling a bichromatic class that contains the endpoints of the newly added edge. Key to the algorithm is how to perform the resampling step efficiently since bichromatic classes may induce linear-sized components. To this end, we exploit the tree uniqueness to show that the average growth of bichromatic components is typically small, which allows us to use correlation decay algorithms for the resampling step. While the precise uniqueness threshold on the tree is not known for general values of $q$ and $Δ$ in the antiferromagnetic case, our algorithm works throughout uniqueness regardless of its value. In the case of the ferromagnetic Potts model, we simplify the algorithm significantly by utilising the random-cluster representation of the model. In particular, we show that a percolation-type algorithm succeeds in sampling from the random-cluster model with parameters $p,q$ on random $Δ$-regular graphs for all values of $q\geq 1$ and $p

We study the complexity of approximate counting Constraint Satisfaction Problems (#CSPs) in a bounded degree setting. Specifically, given a Boolean constraint language Gamma and a degree bound Delta, we study the complexity of #CSP_Delta(Gamma), which is the problem of counting satisfying assignments to CSP instances with constraints from Gamma and whose variables can appear at most Delta times. Our main result shows that: (i) if every function in Gamma is affine, then #CSP_Delta(Gamma) is in FP for all Delta, (ii) otherwise, if every function in Gamma is in a class called IM_2, then for all sufficiently large Delta, #CSP_Delta(Gamma) is equivalent under approximation-preserving (AP) reductions to the counting problem #BIS (the problem of counting independent sets in bipartite graphs) (iii) otherwise, for all sufficiently large Delta, it is NP-hard to approximate the number of satisfying assignments of an instance of #CSP_Delta(Gamma), even within an exponential factor. Our result extends previous results, which apply only in the so-called "conservative" case.

Graph coloring is arguably the most exhaustively studied problem in the area of approximate counting. It is conjectured that there is a fully polynomial-time (randomized) approximation scheme (FPTAS/FPRAS) for counting the number of proper colorings as long as q ≥ Δ + 1, where q is the number of colors and Δ is the maximum degree of the graph. The bound of q = Δ + 1 is the uniqueness threshold for Gibbs measure on Δ-regular infinite trees. However, the conjecture remained open even for any fixed Δ > 3 (The cases of Δ = 1, 2 are trivial). In this paper, we design an FP- TAS for counting the number of proper four-colorings on graphs with maximum degree three and thus confirm the conjecture in the case of Δ = 3. This is the first time to achieve this optimal bound of q = Δ + 1. Previously, the best FPRAS requires and the best deterministic FPTAS requires q > 2.581Δ + 1 for general graphs. In the case of Δ = 3, the best previous result is an FPRAS for counting proper 5-colorings. We note that there is a barrier to go beyond q = Δ + 2 for single-site Glauber dynamics based FPRAS and we overcome this by correlation decay approach. Moreover, we develop a number of new techniques for the correlation decay approach which can find applications in other approximate counting problems.

Hardcore and Ising models are two most important families of two state spin systems in statistic physics. Partition function of spin systems is the center concept in statistic physics which connects microscopic particles and their interactions with their macroscopic and statistical properties of materials such as energy, entropy, ferromagnetism, etc. If each local interaction of the system involves only two particles, the system can be described by a graph. In this case, fully polynomial-time approximation scheme (FPTAS) for computing the partition function of both hardcore and anti-ferromagnetic Ising model was designed up to the uniqueness condition of the system. These result are the best possible since approximately computing the partition function beyond this threshold is NP-hard. In this paper, we generalize these results to general physics systems, where each local interaction may involves multiple particles. Such systems are described by hypergraphs. For hardcore model, we also provide FPTAS up to the uniqueness condition, and for anti-ferromagnetic Ising model, we obtain FPTAS under a slightly stronger condition.