Int. Statistical Inst.: Proc. 58th World Statistical Congress, 2011, Dublin (Session CPS014) p.4149

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1 Int. Statistical Inst.: Proc. 58th orld Statistical Congress, 011, Dublin (Session CPS014) p.4149 Invariant heory for Hypothesis esting on Graphs Priebe, Carey Johns Hopkins University, Applied Mathematics Statistics 3400 North Charles Street Baltimore, Maryl , USA Rukhin, Andrey Naval Surface arfare Center, Sensor Fusion Department Frontage Road Dahlgren, Virginia 448, USA 1 Introduction Following the setting outlined in Priebe et al. 011 we aim to detect anomlies within attributed graphs. In particular, let V = {1,..., n} be the fixed set of vertices φ : ( V {0, 1,..., K} be an edge-attribution function. he graph on V is defined to be G = (V, E φ ) where (u, v) E φ φ(u, v) > 0. e say that the edge (u, v) has attribute c {1,..., K} if φ(u, v) = c. One can view the categorical edge attributes as some mode of the communication event between actors u v (e.g., a topic label derived from the content of the communication). he specific anomaly we aim to detect is the chatter alternative a small (unspecified) subset of vertices with altered communication behavior in an otherwise homogeneous setting. Our inference task is to determine whether or not a graph (V, E φ ) includes a subset of vertices M = {v 1, v,..., v m } whose edge-connectivity within the subset exhibits a different behaviour than that found among the remaining vertices in the graph. o this end we consider the problem of detecting chatter anomalies in a graph using hypothesis testing on a fusion of attributed graph invariants. In particular, the focus of this paper is analyzing comparing the inferential power of the linear attribute fusion of the attributed q-clique invariant q (G) = h ( ) u 1,..., u q ; c 1,..., c K, c 1,...,c K P (( q,k) 1,...,c K (u 1,...,u q ) ( V q) where the sum is over the collection of partitions P ( ( ( q, K) of q into K non-negative parts, = {w i } i P (( q,k) are the fusion weights, the summ h ( ) u 1,..., u q ; c 1,..., c K indicates the event that the vertices u 1,..., u q are elements of a q-clique with c r edges of color r. Specifically, we consider the cases q = which yields the size fusion q = 3 which yields the triangle fusion 3. Our rom graph model is motivated by the time series model found in Lee Priebe forthcoming: for each vertex v V we assign a latent variable X v = (X1 v,..., Xv d ) drawn independently of all other vertices from some d-dimensional distribution. he edge-attribution function will be a rom variable where the probability of an edge (u, v) having attribute c is defined to be a some predetermined function of the inner product of the latent variables. e assume that the edge attributes, conditioned on the latent variables, are independent. In this paper, we will assume that X v Dirichlet (λ v 0,..., λv K ) P {φ(u, v) = c} = Xc u Xc v

2 Int. Statistical Inst.: Proc. 58th orld Statistical Congress, 011, Dublin (Session CPS014) p.4150 for all (u, v) ( V all c {1,..., K}. his model choice is analogous to the first second approximations found in Lee Priebe forthcoming: if we write λ v = (λ v 0,..., λv K ) = (1 + rxv 0,..., 1 + rxv K ) for some fixed (x v 0,..., xv K ) in the unit simplex non-negative real r, then r yields the first approximation model (i.e., the independent edge model ). e mention that our approach herein differs from the second approximation in Lee Priebe forthcoming; their second approximation yields a inner product model with truncated Gaussian latent variables. Related work may be found in Bollobas et al. 007 Section 16.4 references therein. e also direct the interested reader to Priebe et al. 011 in which the authors study other linear attribute fusion invariants; in particular, the authors consider maxd (G) = max v scan (G) = max v K K u Nv u,x Nv I{φ(u, v) = c} I{φ(u, x) = c}, where Nv = {u (u, v) E} {v} is the closed neighborhood of vertex v in the graph. Finally, we add that we will restrict ourselves to simple undirected graphs. e will not consider hyper-graphs (hyper-edges consisting of more than two vertices), multi-graphs (more than one edge between any two vertices), self-loops (an edge from a vertex to itself), or weighted edges. Notation For each positive integer l we use the notation l = {1,..., l}. For each v V we assign a latent position vector X v = (X0 v,..., Xv K ) Dirichlet(λv ) for some fixed parameter vector λ v R K+1 +. e also assume that the latent positions are independent. Our null hypothesis assumes a version of homogeneity among the vertices; specifically, H 0 : X v = (X v 0,..., X v K) Dirichlet(λ) for all v V for some Dirichlet parameter vector λ = (λ 0,..., λ K ). Our alternative hypothesis incorporates the anomaly feature described in the preceding section as follows. Assume m = m(n) < n satisfies the m(n) following two conditions: lim n m(n) = lim n n = 0. Our alternative hypothesis is defined to be H 1 : X v (Y0 v =,..., Y K v ) iid Dir(η) i m, (X 0 v,..., Xv K ) iid Dir(λ) i n m. for some fixed Dirichlet parameter vector η = (η 0,..., η K ) the same λ = (λ 0,..., λ K ) from the null hypothesis. For convenience, we also define Λ = 0 c K λ c H = 0 c K η c. e define ε c = I {φ (u, v) = c} (u,v) ( V to be the size (i.e., number of -cliques) of attribute c in the graph. Similarly, for the number of triangles (i.e., 3-cliques) we write τ c, τ b,c, τ b,c,d to denote the number of 3-cliques with three c-colored edges, two b-colored one c-colored edge, one edge of each of three edge-colors b, c, d, respectively.

3 Int. Statistical Inst.: Proc. 58th orld Statistical Congress, 011, Dublin (Session CPS014) p.4151 Before proceeding, we highlight a relevant property of the mixed moments of the Dirichlet distribution (see Johnson Kotz 197): if r 1,..., r s are non-negative (X 1,..., X s ) Dirichlet(θ 1,..., θ s ) then s (1) E X r i = Γ ( s i=1 θ i) s j=1 Γ (θ j + r j ) ( s i=1 Γ (θ s i) Γ j=1 (θ j + r j )) i=1 where Γ denotes Euler s stard Gamma function. ith this property one can compute the exact moments of the Hajek projection of 3 under either hypothesis. e write ν (i) to denote the i-th moment of X c in the null latent vector ν (i,j) (b,c) to denote the joint (i, j)-moment of (X b, X c ). Similarly, we ll write µ (i) to denote the i-th moment of Y c in the anomalous latent vector µ (i,j) (b,c) to denote the joint (i, j)-moment of (Y b, Y c ). 3 Analysis e will appeal to Hajek s Projection method, detailed in Nowicki ierman 1988, in order to demonstrate the asymptotic normality of the fusion invariants in this article. his approach is outlined as follows: e define the projection of the fusion to be the centered sum of independent rom variables = v V E X v (n 1)E. For both the size triangle fusion we aim to show that E V ar ( ) = V ar ( ) + E V ar ( ) D N(0, 1). o this end, one appeals to Chebyshev s Inequality to show that V ar ( ) = o(v ar ( )) (see Nowicki ierman 1988 for the detailed argument). Specifically, if { } P V ar ( ) ɛ V ar ( ) ɛ V ar ( 0 ) for any positive ɛ, then V ar ( ) + E V ar ( ) D N(0, 1) by applying the Central Limit heorem to the normalized sum of independent rom variables in the second term of the left-h side. 3.1 he Attributed Size Fusion For each c K define ε c = I {φ (u, v) = c} (u,v) ( V to be the number of edges of color c in the graph. he linear attribute fusion with parameter = (w 1,..., w K ) is defined to be = K ε c.

4 Int. Statistical Inst.: Proc. 58th orld Statistical Congress, 011, Dublin (Session CPS014) p he Attributed Size Fusion under H 0 e present all relevant terms within the Hajek Projection of the attributed size fusion under the null hypothesis. he expectation of under the null is given by E 0 = ( n ) K. E 0 X a for any fixed a V is given by E 0 X a = (n 1) K e can now evaluate under the null: = a n E X a (n 1)E = (n 1) a n 1 c K X (a) c + X c (a) ( ) K n 1 ( ) n K.. he variance of this sum of independent identically distributed rom variables is V ar 0 ( ) = Θ ( n 3). ( As V ar 0 ) ( n (+1) E I {φ (u, v) > 0} = o(v ar 0 ( )) by the Cauchy-Schwarz Inequality (see Nowicki ierman 1988 for full details), we have the desired convergence to the stard normal distribution he Attributed Size Fusion under H 1 For the alternative we write the edge attribution function as Yc u Yc v u, v m, P {φ (u, v) = c} = Yc u Xc v u m, v n m, Xc u Xc v u, v n m. e perform a similar but more involved analysis to deduce the limiting distribution of the attribued size fusion of the graph under these conditions, yielding V ar 1 ( ) = Θ ( n 3) as desired. V ar 1 ( ) = Θ ( n = o(v ar 1 ( )) Asymptotic Power Analysis of the Attributed Size Fusion Returning to the context of hypothesis testing, assume we are interested in performing an α-level hypothesis test to determine whether or not the graph includes an anomalous set of m vertices whose underlying latent distribution differs from the the null component of the graph. e define β = { } lim n P 1 > c α where cα = c(α, n) is the α-level critical value of the test.

5 Int. Statistical Inst.: Proc. 58th orld Statistical Congress, 011, Dublin (Session CPS014) p.4153 Fix c K. he difference of the corresponding terms the hypotheses means can be written as ( )( ) E 1 ε c E 0 ε c = D 1 + D m n m ( ) ( ) ( = 1 1 µ (1) m ν(1) + µ (1) ) (here D i corresponds to the edge-count that includes edges with exactly i anomalous vertices). he reader can verify that V ar 1( ) V ar 0 ( ) 1. Moreover, given that the limiting distribution (under the null) is normal, we write c α = z α V ar0 ( ) + E 0 thus β = P { Z > z α lim n ( )} E1 E0 V ar0 ( ). Recall that V ar 0 ( ) = Θ(n 3 ); thus, if D 1 0 c (i.e. there is signal in the null-to-anomaly connectivity) then the limiting power β > α when m(n m) n 3 0 or, equivalently, when m = Ω( n) (similarly, if m = ω( n) then β 1). Furthermore, if c D 1 = 0 (i.e. there is no signal in the null-to-anomaly connectivity) the limiting power β > α when c D 0 m 0 (which is equivalent to m = Ω( 4 n 3 )). Moreover, if m = ω( 4 n 3 ) n 3 under these conditions then β 1. It follows that the optimal choice of weights (w 1,..., w K ) is the one which maximizes the expression ( ) E1 E0 lim n V ar0 ( ) in either of the two above-mentioned cases. 3. he Attributed Number of riangles Fusion e begin by writing τ = c K τ c + b c τ b,c + τ b,c,d. e denote the number-of-triangles fusion invariant to be = w b,c τ b,c + w b,c,d τ b,c,d. 3 c K τ c + b c Similar to what was done in the previous section, we obtain E 0 = ( ) n 3 c K ν ( 3 + b c w b,c 3ν ( (b) ν (1,1) (b,c) + w b,c,d 3ν (1,1) (b,c) ν(1,1) (b,d) ν(1,1) (c,d)

6 Int. Statistical Inst.: Proc. 58th orld Statistical Congress, 011, Dublin (Session CPS014) p.4154 ( ( ) ) n 1 V ar 0 ( ) = Θ n under the null ( 3 E 1 = Θ i=0 ( m i )( n m 3 i ( ( ) ) n 1 V ar 1 ( ) = Θ n ) ) ( under the alternative. Again, V ar 0 3 ) = o(v ar 0 ( ( )) V ar 1 3 ) = o(v ar 1 ( )). As in the case with the attributed size fusion, we are interested in performing an α-level hypothesis test. he terms within the difference in means can be expressed as E 1 E 0 = D (b,c) + D (b,c,d) D + c K b c ( 3 ( m i = Θ i=1 )( n m 3 i ) δ i (H, Λ) where δ i (H, Λ) is the mixed-moments difference when there are i anomalous vertices in a 3-clique. he reader can verify that V ar 1( ) V ar 0 ( ) 1. Since V ar 0 ( ) = Θ(n 5 ), we have that the limiting power β 3 > α when when m = Ω( i n i 1 ) the corresponding mixed-moments expression δ i (H, Λ) is non-zero. ) 4 Conclusion e have presented preliminary results for linear attribute fusion for clique sizes q = 3 in terms of inferential power when detecting the prescribed anomaly within our model. In general, the most powerful choice of q depends on m as a function of n on the Dirichlet parameter vectors λ η through the mixed moments. References B. Bollobas, S. Janson, O. Riordan. he phase transition in inhomogeneous rom graphs. Rom Structures Algorithm, 31:3 1, 007. N. Johnson S. Kotz. Distributions in Statistics: Continuous Multivariate Distributions. iley, New York, 197. N. Lee C. E. Priebe. A Latent Process Model for ime Series of Attributed Rom Graphs. Statistical Inference for Stochastic Processes, forthcoming. K. Nowicki J. ierman. Subgraph counts in rom graphs using incomplete u-statistics methods. Discrete Mathematics, 7:99 310, C. E. Priebe, N. Lee, Y. Park, M. ang. Attribute fusion in a latent process model for time series of graphs. he 011 IEEE orkshop on Statistical Signal Processing (SSP011), 011.