Belief propagation for networks with loops

Kirkley, Alec; Cantwell, George T.; Newman, M. E. J.

doi:10.1126/sciadv.abf1211

Cited by 38 publications

(26 citation statements)

References 53 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the whole, our findings demonstrate that network heterogeneities, here expressed by degree fluctuations, play a surprisingly important role in the high-connectivity limit of spin models. Other types of topological features, such as modular structure and the presence of loops [97][98][99], should as well play a fundamental role in the high-connectivity behavior. While spin models on sparse networks pose many technical challenges [44,45,92], the mean-field theory of fully-connected models has a simpler formal structure [7], at the cost of completely neglecting the network structure.…”

Section: Discussionmentioning

confidence: 99%

Mean-field theory of vector spin models on networks with arbitrary degree distributions

Metz

Peron

2022

J. Phys. Complex.

View full text Add to dashboard Cite

Understanding the relationship between the heterogeneous structure of complex networks and cooperative phenomena occurring on them remains a key problem in network science. Mean-ﬁeld theories of spin models on networks constitute a fundamental tool to tackle this problem and a cornerstone of statistical physics, with an impressive number of applications in condensed matter, biology, and computer science. In this work we derive the mean-ﬁeld equations for the equilibrium behavior of vector spin models on high-connectivity random networks with an arbitrary degree distribution and with randomly weighted links. We demonstrate that the high-connectivity limit of spin models on networks is not universal in that it depends on the full degree distribution. Such nonuniversal behavior is akin to a remarkable mechanism that leads to the breakdown of the central limit theorem when applied to the distribution of effective local ﬁelds. Traditional mean-ﬁeld theories on fully-connected models, such as the Curie-Weiss, the Kuramoto, and the Sherrington-Kirkpatrick model, are only valid if the network degree distribution is highly concentrated around its mean degree. We obtain a series of results that highlight the importance of degree ﬂuctuations to the phase diagram of mean-ﬁeld spin models by focusing on the Kuramoto model of synchronization and on the Sherrington-Kirkpatrick model of spin-glasses. Numerical simulations corroborate our theoretical ﬁndings and provide compelling evidence that the present mean-ﬁeld theory describes an intermediate regime of connectivity, in which the average degree $c$ scales as a power $c \propto N^{b}$ ($b < 1$) of the total number $N \gg 1$ of spins. Our ﬁndings put forward a novel class of spin models that incorporate the effects of degree ﬂuctuations and, at the same time, are amenable to exact analytic solutions.

show abstract

Section: Discussionmentioning

confidence: 99%

Mean-field theory of vector spin models on networks with arbitrary degree distributions

Metz

Peron

2022

J. Phys. Complex.

View full text Add to dashboard Cite

show abstract

“…Our factor graph is admittedly not a tree. One possibility is to modify the belief propagation algorithm to account for loops [36]. We instead follow the standard argument that:…”

Section: Define the Block Matricesmentioning

confidence: 99%

Nonbacktracking spectral clustering of nonuniform hypergraphs

Chodrow¹,

Eikmeier²,

Haddock³

2022

Preprint

View full text Add to dashboard Cite

Spectral methods offer a tractable, global framework for clustering in graphs via eigenvector computations on graph matrices. Hypergraph data, in which entities interact on edges of arbitrary size, poses challenges for matrix representations and therefore for spectral clustering. We study spectral clustering for nonuniform hypergraphs based on the hypergraph nonbacktracking operator. After reviewing the definition of this operator and its basic properties, we prove a theorem of Ihara-Bass type to enable faster computation of eigenpairs. We then propose an alternating algorithm for inference in a hypergraph stochastic blockmodel via linearized belief-propagation, offering proofs that both formalize and extend several previous results. We perform experiments in real and synthetic data that underscore the benefits of hypergraph methods over graph-based ones when interactions of different sizes carry different information about cluster structure. Through an analysis of our algorithm, we pose several conjectures about the limits of spectral methods and detectability in hypergraph stochastic blockmodels writ large.

show abstract

“…Instead, for a given pair of nodes, the states of the joint predecessors are not independent of one another, even on a tree. The cavity method has recently been applied in the presence of node correlations on loopy graphs to investigate percolation, random matrix spectra [62], and the equilibrium state of Ising spin models [63]. The authors explicitly include the contribution from short loops, e.g., triangles, which break the assumption of independence of states the cavity method relies on.…”

Section: Pairwise Correlation For the Linear Threshold Model On Fully...mentioning

confidence: 99%

Uncovering the non-equilibrium stationary properties in sparse Boolean networks

Torrisi,

Kühn,

Annibale

2022

Preprint

View full text Add to dashboard Cite

Dynamic processes of interacting units on a network are out of equilibrium in general. In the case of a directed tree, the dynamic cavity method provides an efficient tool that characterises the dynamic trajectory of the process for the linear threshold model. However, because of the computational complexity of the method, the analysis has been limited to systems where the largest number of neighbours is small. We devise an efficient implementation of the dynamic cavity method which substantially reduces the computational complexity of the method for systems with discrete couplings. Our approach opens up the possibility to investigate the dynamic properties of networks with fat-tailed degree distribution. We exploit this new implementation to study properties of the non-equilibrium steady-state. We extend the dynamical cavity approach to calculate the pairwise correlations induced by different motifs in the network. Our results suggest that just two basic motifs of the network are able to accurately describe the entire statistics of observed correlations. Finally, we investigate models defined on networks containing bi-directional interactions. We observe that the stationary state associated with networks with symmetric or anti-symmetric interactions is biased towards the active or inactive state respectively, even if independent interaction entries are drawn from a symmetric distribution. This phenomenon, which can be regarded as a form of spontaneous symmetry-breaking, is peculiar to systems formulated in terms of Boolean variables, as opposed to Ising spins.

show abstract

Belief propagation for networks with loops

Cited by 38 publications

References 53 publications

Mean-field theory of vector spin models on networks with arbitrary degree distributions

Mean-field theory of vector spin models on networks with arbitrary degree distributions

Nonbacktracking spectral clustering of nonuniform hypergraphs

Uncovering the non-equilibrium stationary properties in sparse Boolean networks

Contact Info

Product

Resources

About