1,837 research outputs found
Community detection and stochastic block models: recent developments
The stochastic block model (SBM) is a random graph model with planted
clusters. It is widely employed as a canonical model to study clustering and
community detection, and provides generally a fertile ground to study the
statistical and computational tradeoffs that arise in network and data
sciences.
This note surveys the recent developments that establish the fundamental
limits for community detection in the SBM, both with respect to
information-theoretic and computational thresholds, and for various recovery
requirements such as exact, partial and weak recovery (a.k.a., detection). The
main results discussed are the phase transitions for exact recovery at the
Chernoff-Hellinger threshold, the phase transition for weak recovery at the
Kesten-Stigum threshold, the optimal distortion-SNR tradeoff for partial
recovery, the learning of the SBM parameters and the gap between
information-theoretic and computational thresholds.
The note also covers some of the algorithms developed in the quest of
achieving the limits, in particular two-round algorithms via graph-splitting,
semi-definite programming, linearized belief propagation, classical and
nonbacktracking spectral methods. A few open problems are also discussed
Network centrality: an introduction
Centrality is a key property of complex networks that influences the behavior
of dynamical processes, like synchronization and epidemic spreading, and can
bring important information about the organization of complex systems, like our
brain and society. There are many metrics to quantify the node centrality in
networks. Here, we review the main centrality measures and discuss their main
features and limitations. The influence of network centrality on epidemic
spreading and synchronization is also pointed out in this chapter. Moreover, we
present the application of centrality measures to understand the function of
complex systems, including biological and cortical networks. Finally, we
discuss some perspectives and challenges to generalize centrality measures for
multilayer and temporal networks.Comment: Book Chapter in "From nonlinear dynamics to complex systems: A
Mathematical modeling approach" by Springe
Dynamical Phase Transitions in Graph Cellular Automata
Discrete dynamical systems can exhibit complex behaviour from the iterative
application of straightforward local rules. A famous example are cellular
automata whose global dynamics are notoriously challenging to analyze. To
address this, we relax the regular connectivity grid of cellular automata to a
random graph, which gives the class of graph cellular automata. Using the
dynamical cavity method (DCM) and its backtracking version (BDCM), we show that
this relaxation allows us to derive asymptotically exact analytical results on
the global dynamics of these systems on sparse random graphs. Concretely, we
showcase the results on a specific subclass of graph cellular automata with
``conforming non-conformist'' update rules, which exhibit dynamics akin to
opinion formation. Such rules update a node's state according to the majority
within their own neighbourhood. In cases where the majority leads only by a
small margin over the minority, nodes may exhibit non-conformist behaviour.
Instead of following the majority, they either maintain their own state, switch
it, or follow the minority. For configurations with different initial biases
towards one state we identify sharp dynamical phase transitions in terms of the
convergence speed and attractor types. From the perspective of opinion dynamics
this answers when consensus will emerge and when two opinions coexist almost
indefinitely.Comment: 15 page
Non-backtracking Walk Centrality for Directed Networks
The theory of zeta functions provides an expression for the generating function of nonbacktracking walk counts on a directed network. We show how this expression can be used to produce a centrality measure that eliminates backtracking walks at no cost. We also show that the radius of convergence of the generating function is related to the spectrum of a three-by-three block matrix involving the original adjacency matrix. This gives a means to choose appropriate values of the attenuation parameter. We find that three important additional benefits arise when we use this technique to eliminate traversals around the network that are unlikely to be of relevance. First, we obtain a larger range of choices for the attenuation parameter. Second, a natural approach for determining a suitable parameter range is invariant under the removal of certain types of nodes, we can gain computational efficiencies through reducing the dimension of the resulting eigenvalue problem. Third, the dimension of the linear system defining the centrality measures may be reduced in the same manner. We show that the new centrality measure may be interpreted as standard Katz on a modified network, where self loops are added, and where nonreciprocal edges are augmented with negative weights. We also give a multilayer interpretation, where negatively weighted walks between layers compensate for backtracking walks on the only non-empty layer. Studying the limit as the attenuation parameter approaches its upper bound also allows us to propose a generalization of eigenvector-based nonbacktracking centrality measure to this directed network setting. In this context, we find that the two-by-two block matrix arising in previous studies focused on undirected networks must be extended to a new three-by-three block structure to allow for directed edges. We illustrate the centrality measure on a synthetic network, where it is shown to eliminate a localization effect present in standard Katz centrality. Finally, we give results for real networks
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Expanders with Symmetry: Constructions and Applications
Expanders are sparse yet well-connected graphs with numerous theoretical and practical uses. Symmetry is a valuable structure for expanders as it enables efficient algorithms and a richer set of applications. This thesis studies expanders with symmetry, giving new constructions and applications. We extend expander construction techniques to work with symmetry and give explicit constructions of expanders with varying quality of expansion and symmetries of various groups. In particular, we construct graphs with large Abelian group symmetries via the technique of \textit{graph lifts}. We also give a generic amplification procedure that converts a weak expander to an almost optimal one while preserving symmetries. This procedure is obtained by generalizing prior amplification techniques that work for Cayley graphs over Abelian groups to Cayley graphs over any finite group. In particular, we obtain almost-Ramanujan expanders over every non-abelian finite simple group. We then explore the utility of having both symmetry and expansion simultaneously. We obtain explicit quantum LDPC codes of almost linear distance and \textit{good} classical quasi-cyclic codes with varying circulant sizes using prior results and our constructions of graphs with Abelian symmetries. We show how our generic amplification machinery boosts various structured expander-like objects: \textit{quantum expanders}, \textit{dimension expanders}, and \textit{monotone expanders}. Finally, we prove a structural result about expanding Cayley graphs, showing that they satisfy a \enquote{degree-2} variant of the \textit{expander mixing lemma}. As an application of this, we give a randomness-efficient query algorithm for \textit{homomorphism testing} of unitary-valued functions on finite groups and a derandomized version of the celebrated Babai--Nikolov--Pyber (BNP) lemma
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