1	Discussion on Statistical models for networks Tian Zheng Department of Statistics, Columbia University
2	This discussion Based on papers found online and powerpoint files from these presenting authors; Also based on my understanding of these papers/presentations slides. Some texts and figures are “borrowed” from these papers and presentation slides.
3	Network Network: sets of objects (nodes or vertices) connected by relations (edges or arcs). Study of networks Represent relations and dependencies between nodes; Understand the causes, consequences, dynamic trends and stochastic mechanisms of networks; In most applications, networks are studied using graphical models. In some context, hypergraph is also used.
4	Data Data with an explicit network structure Social network Computer network Explore dependence (as a network) structure among observed individuals Gene expression analysis Infer interactions (as a network) among objects towards an outcome Identify disease predisposing gene-gene interactions
5	Types of networks Ma’ayan, Blitzer and Iyengar (2005) Annu. Rev. Biophys. Biomol. Struct. 34:319-349 Type I: undirected Type II: directed Type III: directed and weighted Type IV: directed/weighted with spatial specifications.
6	Exponentially parameterized random graph models Handcock MS (2003) “Assessing Degeneracy in Statistical Models for Social Networks” X is a random graph, x is an random instance of X. t(x) are statistics based on x. θ is the model parameter. c(θ) is the normalizing function. Computing c(θ) by enumerating all possible x is infeasible for graphs with many nodes.
7	Exponentially parameterized random graph models c(θ) can be estimated using MCMC methods Based on M sampled graphs, , from . Model degeneracy occurs if puts a substantial proportion of its probability mass on a small number of graphs on the boundary.
8	Recommendations Mean-parameterization and mixed parameterization are easy to interpret and estimate, and tend to have better stability. Use nondegeneracy priors in MCMC computation.
9	Information Processing in Cellular Networks (Iyengar and Ma’ayan) Network: based on the biochemistry, cell biology and cell physiology literature. Studied step-wise signal propagation (reactions in chemical space) Originate from nodes representing ligands A link: direct interaction between two nodes (chemical reactions) Signal: the number links per step Generate a series of subnetworks
10	Information Processing in Cellular Networks (Iyengar and Ma’ayan) The series of subnetworks can be used to study the characteristics of the signal propagation in the cellular network. Such characteristics are studied via regulatory motifs. “A motif is a group of interacting components capable of signal processing.” Modular small subgraphs? Shuffled networks were used to evaluate enrichments of motifs.
11	Information Processing in Cellular Networks (Iyengar and Ma’ayan) An exponential model can be defined as Probability for observing x depends on current step n and originating ligand i. Here t(x) may represent total signal, motif counts, and clustering efficient.
12	Dynamical analysis of networks in neutral systems (Brown) Data Voltage traces from an electrode are recorded. Each voltage trace, spike events and number of neurons are identified. Each spike is then assigned to a neuron. Network to be inferred: represented by the connectivity matrix. Goal: characterizing the relation between the stimulus and an ensemble of neurons the relation among the spiking activity of the neurons
13
14	Dynamical analysis of networks in neutral systems (Brown) Current model concerns inference of a network structured model to represent the spike train data observed on an ensemble of neurons. Some exponential random graph model can be assumed to study the stochastic mechanism of such networks. If MCMC methods are to be used, discussion on the nondegeneracy priors can also be applied.
15	Thanks!