Statistical Inference on Random Graphs: Comparative Power Analyses via Monte Carlo

Pao, Henry; Coppersmith, Glen; Priebe, Carey E.

doi:10.1198/jcgs.2010.09004

Cited by 18 publications

(35 citation statements)

References 11 publications

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“…Applying these kNN classifiers to a problem of considerable scientific interest -classifying human structural MR connectomes -we find that even with a relatively small sample size (s ≥ 20), the approximately graph-matched kNN algorithm performs nearly as well as the kNN algorithm using vertex labels, and slightly better than a kNN algorithm applied to a set of graph invariants proposed previously (Pao, Coppersmith, and Priebe 2011). This suggests that the asymptotics might apply even for very small sample sizes.…”

Section: Discussionmentioning

confidence: 74%

“…Specifically, letL ψ s be the misclassification rate for some classifier that operates on T s , that is, only has access to shuffled graphs. Consider the set of seven graph invariants studied in Pao, Coppersmith, and Priebe (2011): size, max degree, max eigenvalue, scan statistic, number of triangles, clustering coefficient, and average path length. Via Monte Carlo, Pao, Coppersmith, and Priebe were unable to find a uniformly most powerful graph invariant test statistic (Priebe, Coppersmith, and Rukhin 2010) for a particular hypothesis testing scenario on shuffled graphs.…”

Section: Comparing Asymptotic Performancesmentioning

confidence: 99%

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Shuffled Graph Classification: Theory and Connectome Applications

Vogelstein

Priebe

2015

J Classif

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We develop a formalism to address statistical pattern recognition of graph valued data. Of particular interest is the case of all graphs having the same number of uniquely labeled vertices. When the vertex labels are latent, such graphs are called shuffled graphs. Our formalism provides insight to trivially answer a number of open statistical questions including: (i) under what conditions does shuffling the vertices degrade classification performance and (ii) do universally consistent graph classifiers exist? The answers to these questions lead to practical heuristic algorithms with stateof-the-art finite sample performance, in agreement with our theoretical asymptotics. Applying these methods to classify sex and autism in two different human connectome classification tasks yields successful classification results in both applications.

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Section: Discussionmentioning

confidence: 74%

Section: Comparing Asymptotic Performancesmentioning

confidence: 99%

Shuffled Graph Classification: Theory and Connectome Applications

Vogelstein

Priebe

2015

J Classif

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“…In [9] various graph invariants (size, maximum degree, scan statistic, etc.) are considered for their power as test statistics and it is demonstrated via Monte Carlo that no single invariant is uniformly most powerful, while in [10] it is demonstrated that asymptotics can provide misleading comparative power analysis for size vs. maximum degree except for astronomically large graphs; see also [11] for a summary.…”

Section: Invariantsmentioning

confidence: 99%

Attribute Fusion in a Latent Process Model for Time Series of Graphs

Tang

Park²,

Lee

et al. 2013

IEEE Trans. Signal Process.

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We consider anomaly/change point detection given a time series of graphs with categorical attributes on the edges. Various attributed graph invariants are considered, and their power for detection as a function of a linear fusion parameter is presented.Index Terms-Anomaly Detection, Attributed Random Graphs, Fusion, Random Dot Product Graphs TIME SERIES OF ATTRIBUTED GRAPHSGiven a time series of attributed graphs G(t) = (V, ϕ(·, t)), t = 1, 2, · · · , where the vertex set V = [n] = {1, · · · , n} is fixed throughout and edge attribution functions ϕ(·, t) : V 2→ {0, · · · , K} are time-dependent, we wish to detect anomalies and/or change points. Let us consider vertices to represent "actors," and an edge between vertex u and vertex v at time t (uv ∈ E(t), where the edge set E(t) is given by E(t) = {uv ∈ V 2 : ϕ(uv, t) > 0}) represents the existence of a communications event between actors u and v during the time period (t − 1, t]. Categorical edge attributes ϕ(uv, t), when non-zero, represent some mode of the communication event between actors u and v during (t − 1, t]; for instance, a topic label derived from the content of the communication. We will not consider directed edges or hypergraphs (hyper-edges consisting of more than two vertices) or multi-graphs (more than one edge between any two vertices at any time t) or self-loops (an edge from a vertex to itself) or weighted edges, although all of these generalizations of simple attributed graphs may be relevant for specific applications.The specific anomaly we will consider is the "chatter" alternative -a small (unspecified) subset of vertices with altered communication behavior during some time period in an otherwise stationary setting, as depicted in Figure 1. This figure notionally depicts the entire vertex set V behaving in some null state for t = 1, · · · , t * − 1; then, at time t in their null state throughout.) Our statistical inference task is to determine whether or not there has emerged a chatter group at some time t = t * . Fig. 1. Notional depiction of a time series of graphs in which the entire vertex set V behaves in some null state for t = 1, · · · , t * − 1 and then, at time t * , a subset of vertices V A exhibits a change in behavior.The latent process model for time series of attributed graphs presented in [1] provides for precisely this temporal structure. Each vertex is governed by a continuous time, finite state stochastic process {X v (t)} v∈V , with the state-space given by {0, 1, · · · , K}. The probability of edge uv at time t is determined by the inner product of the sub-probability vectors specified by p w,k (t) = t t−1The attribute ϕ(uv, t) for edge uv at time t, given that there is indeed an edge, is given byFor the scenario depicted in Figure 1, the vertex processes {X v (t)} v∈V A are stationary until time t * − 1 and then undergo a change point, while the processes {X v (t)} v∈V \V A remain stationary throughout all time.Our latent process model produces a dependent time series of attributed graphs G(t), each of which i...

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“…The graph scan statistic is in fact quite powerful for some n p m s, as demonstrated in Pao et al (2011). In particular, Rukhin (2009) has shown that renders the maximum degree asymptotically inadmissible for a large portion of our parameter space.…”

Section: Resultsmentioning

confidence: 96%

“…This "graph scan statistic" warrants study for its use in inference tests for anomaly detection in random social networks (see Pao et al, 2011;Priebe, 2004;Priebe et al, 2005).…”

Section: Introductionmentioning

confidence: 99%

On the Limiting Distribution of a Graph Scan Statistic

Rukhin¹,

Priebe²

2012

Communications in Statistics - Theory and Methods

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Let p ∈ 0 1 and let n ∈ . Moreover, let s ∈ p 1 , let m = m n be a positive, integer-valued function of n with m = O n k k+1 for some positive integer k. We define ER n p to be the random graph model on n vertices where each edge is independently included in the graph with probability p. We also define n p m s to be the random graph model on n vertices where edges are included independently; however, there is a fixed subset of m vertices for which each of the m 2 edges are included with probability s (and the remaining n 2 − m 2 edges are each included with probability p). Let G = V E denote a graph on n vertices. For any vertex v ∈ V , let N v denote the set of neighbors of v along with v itself, and let v denote the induced subgraph on this neighborhood of vertices. The aim of this article is to determine the limiting distribution (n → ) of the graph scan statistic = max v∈V E vWe prove that is asymptotically Gumbel in both the ER n p and n p m s random graph models.

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Statistical Inference on Random Graphs: Comparative Power Analyses via Monte Carlo

Cited by 18 publications

References 11 publications

Shuffled Graph Classification: Theory and Connectome Applications

Shuffled Graph Classification: Theory and Connectome Applications

Attribute Fusion in a Latent Process Model for Time Series of Graphs

On the Limiting Distribution of a Graph Scan Statistic

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