We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

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Graphical Models In Applied Multivariate Statistics

Introduction to Probability and its Applications. By R. L. Scheaffer. ISBN 0 534 91970 7. PWS–Kent, Boston, 1990. 364 pp. £16.95.

Introduction to Probability and its Applications

White matter structural connections are likely to support flow of functional activation or functional connectivity. While the relationship between structural and functional connectivity profiles, here called SC-FC coupling, has been studied on a whole-brain, global level, few studies have investigated this relationship at a regional scale. Here we quantify regional SC-FC coupling in healthy young adults using diffusion-weighted MRI and resting-state functional MRI data from the Human Connectome Project and study how SC-FC coupling may be heritable and varies between individuals. We show that regional SC-FC coupling strength varies widely across brain regions, but was strongest in highly structurally connected visual and subcortical areas. We also show interindividual regional differences based on age, sex and composite cognitive scores, and that SC-FC coupling was highly heritable within certain networks. These results suggest regional structure-function coupling is an idiosyncratic feature of brain organisation that may be influenced by genetic factors. The relationship between the human structural and functional connectome is still not well established. Here the authors show the interindividual variability that exists in regional coupling of structural and functional connectivity across the brain, and that this is heritable.

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Heritability and interindividual variability of regional structure-function coupling.

Null models in network neuroscience

Patterns of brain structural connectivity (SC) and functional connectivity (FC) are known to be related. In SC-FC comparisons, FC has classically been evaluated from correlations between functional...

/pdf/revisiting-correlation-based-functional-connectivity-and-its-27bc91k4kc.pdf

Revisiting correlation-based functional connectivity and its relationship with structural connectivity

This work examines a distributed learning problem where the agents of a network form their beliefs about certain hypotheses of interest Each agent collects streaming (private) data and updates continually its belief by means of a diffusion strategy, which blends the agent's data with the beliefs of its neighbors We focus on weakly-connected graphs, where the network is partitioned into sending and receiving sub-networks, and we allow for heterogeneous models across the agents First, we examine what agents learn ( social learning ) and provide an analytical characterization for the long-term beliefs at the agents Among other effects, the analysis predicts when a leader-follower behavior is possible, where some sending agents control the beliefs of the receiving agents by forcing them to choose a particular and possibly fake hypothesis Second, we consider the dual or reverse learning problem that reveals how agents learn: given the beliefs collected at a receiving agent, we would like to discover the influence that any sending sub-network might have exerted on this receiving agent ( topology learning ) An unexpected interplay between social and topology learning emerges: given $H$ hypotheses and $S$ sending sub-networks, topology learning can be feasible when $H\geq S$  The latter being only a necessary condition, we then examine the feasibility of topology learning for two useful classes of problems The analysis reveals that a critical element to enable topology learning is a sufficient degree of diversity in the statistical models of the sending sub-networks

/pdf/interplay-between-topology-and-social-learning-over-weak-22ub0lyowt.pdf

Interplay Between Topology and Social Learning Over Weak Graphs

This article studies the problem of reconstructing the topology of a network of interacting agents via observations of the state-evolution of the agents. We focus on the large-scale network setting with the additional constraint of partial observations, where only a small fraction of the agents can be feasibly observed. The goal is to infer the underlying subnetwork of interactions and we refer to this problem as local tomography . In order to study the large-scale setting, we adopt a proper stochastic formulation where the unobserved part of the network is modeled as an Erdős-Renyi random graph, while the observable subnetwork is left arbitrary. The main result of this work is to establish that, under this setting, local tomography is actually possible with high probability, provided that certain conditions on the network model are met (such as stability and symmetry of the network combination matrix). Remarkably, such conclusion is established under the low-observability regime , where the cardinality of the observable subnetwork is fixed, while the size of the overall network scales to infinity.

Local Tomography of Large Networks Under the Low-Observability Regime

We consider a distributed social learning problem where a network of agents is interested in selecting one among a finite number of hypotheses. The data collected by the agents might be heterogeneous, meaning that different sub-networks might observe data generated by different hypotheses. For example, some sub-networks might be receiving (or even intentionally generating) data from a fake hypothesis and will bias the rest of the network via social influence. This work focuses on a two-step diffusion algorithm where each agent: i) first updates individually its belief functio. using its privat. data; ii) then computes a new belief function by exponentiatin. a linear combination of the log-beliefs of its neighbors. We obtain analytical formulas that reveal how the agents’ detection capability and the network topology interplay to influence the asymptotic beliefs of the agents. Some interesting behaviors arise, such as the "mind-control" effect or the "truth-is-somewhere-in-between" effect.

Exponential Collapse of Social Beliefs over Weakly-connected Heterogeneous Networks

In this work we consider the problem of learning an Erdős-Renyi graph over a diffusion network when: i) data from only a limited subset of nodes are available (partial observation); ii) and the inferential goal is to discover the graph of interconnections linking the accessible nodes (local structure learning). We propose three matrix estimators, namely, the Granger, the one-lag correlation, and the residual estimators, which, when followed by a universal clustering algorithm, are shown to retrieve the true subgraph in the limit of large network sizes. Remarkably, it is seen that a fundamental role is played by the uniform concentration of node degrees, rather than by sparsity.

Augusto Santos

Papers

Revisiting correlation-based functional connectivity and its relationship with structural connectivity

Interplay Between Topology and Social Learning Over Weak Graphs

Local Tomography of Large Networks Under the Low-Observability Regime

Exponential Collapse of Social Beliefs over Weakly-connected Heterogeneous Networks

Graph Learning with Partial Observations: Role of Degree Concentration