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The Theory of Zeta Graphs with an Application to Random Networks Christopher Ré Stanford

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Social Network Data Social network data is ubiquitous and high value. Since 2000, many studies of the dynamics of these graphs, Watts-Strogatz, Preferential Attachment, etc. Design new random graph models to capture some new aspect of an observed network. Above is not the goal of this work…

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What’s the matter with Erdös-Rényi? G(N,p) does not match real-world graphs (degree distribution, diameter) G(N,p) does not match real-world graphs (degree distribution, diameter) But we have a beautiful theory of G(N,p) (zero-one laws, the “movie”, threshold phenomenon, ….) Much of this work enabled by simple, declarative G(N,p). Find an ER-like model to replace generative models for DB theory-style theorems? May lead to rigorous hypothesis testing for these models (key question in motifs).

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Which model should we study? “At each time step, a new vertex is added. Then, with probability δ, two vertices are chosen uniformly at random and joined by an undirected edge.” – CHKNS Many models. For this study: simple & popular. Callway, Hopcroft, Kleinberg, Newman, Strogatz (CHKNS) CHKNS captures one salient aspect of many models: Arrival order of node affect its properties. NB: Does not capture all phenomenon of interest.

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Zeta Graphs Simple model to capture “arrival order” NB: We’ll use a directed variant, all queries are binary since its easier to describe.

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Zeta graphs Bare bones model to break symmetry: 1 connects to many nodes (~ log N). N connects to 1 node (in expectation) Bare bones model to break symmetry: 1 connects to many nodes (~ log N). N connects to 1 node (in expectation) ER-like: Edges are present independently. Zeta graphs are a family of sets of graphs indexed by N Fixed node set: [N] = {1,…,N} (Index ≈ arrival order) Stochastic edge set (independent edges)

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Informal Main Result Conjunctive Graph Queries cannot distinguish between Zeta graphs and CHKNS as N to ∞. 1. Determine the Theory of Zeta Graphs 2. Show the Theory of CHKNS is sandwiched between two “slices” of Zeta Graphs. Here, Theory is set of CQs with probability 1

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1 st Technical Challenge: Graph Patterns

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Our goal for this section Given (1) a Language of Boolean queries L, and (2) a family of probability models M(1), M(2), …,M(N) check if lim N to ∞ Pr M(N) [q] = 1 for q in L Given (1) a Language of Boolean queries L, and (2) a family of probability models M(1), M(2), …,M(N) check if lim N to ∞ Pr M(N) [q] = 1 for q in L For the talk: (1)L will be “graph patterns” positive conjunctive queries over binary relations. (2)The family of probability models M(N)= “Theory” Th(L,M) = { q in L : lim N to ∞ Pr M(N) [q] = 1 }

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Boolean Query Answering on ER Graphs (2) Compute expected number of tuples. (1) Form “full query” corresponding to q. (3) Use Janson’s Inequality to relate E[Q] to Pr[q]

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Recall: Classical Janson’s Inequality A classical sufficient condition for Pr[q] to 1. A Q(c) and Q(d) properly overlap if they are not identical, but they share at least one identical subgoal see Alon & Spencer, Random Graphs A corollary of Janson’s inequality is:

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Boolean Query Answering on ER Graphs (2) Compute expected number of tuples. (1) Form “full query” corresponding to q. (3) Use Janson’s Inequality to relate E[Q] to Pr[q] What changes for Zeta graphs?

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Computing Expectation

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Multiple Valued Zeta (MVZ) Functions Only use integer s i in this talk MVZs show up in some strange places…

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Order Matters: Paths of Length 2 If x < y < z If x < z < y So in our “atoms” variables will be totally ordered

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Why Multiple-Valued Zeta (MVZ)? Well-studied special function. We get for free: 1.Asymptotics [Costermans et al. 2005] 2.Algebraic Identities [Zudilin & Zudilin 2003] 3.Fancy sounding function (not helpful)

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Asymptotic Estimates for MVZs This is a small variation of Costermans et al. result. (expected # of edges) (expected # of triangles) (expected # of K 4 )

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Indicates shared identical goal Pr[2 Paths] Consider pairs of properly overlapping 2 paths. And others o(E[Q] 2 ) and since E[Q] = (1), Pr[Q] = 1 – o(1) …

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Two cycles you’re out! r cycle r cycle s cycle s cycle (1) For all r, s ≥ 2, Pr M(N) [ B(r,s) ] 0 as N to ∞, i.e., no bicycles. B(r,s) (2) Any connected graph q with at most one cycle appears with probability 1. 1 st result: Two Parts: (A) Any individual pattern, check E, and (B) Different “orderings” are non-negatively correlated.

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Back to CHKNS

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Central Message How different is CHKNS from the family of Zeta graphs? Up to CQs, the answer is not at all.

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Key Technical Issues 1. CHKNS Edge probabilities have a painful form. – But can be sandwiched by “Zeta slices” 2. CHKNS Edges are correlated! - Develop bounds on correlations 3. Show that CHKNS can be essentially embedded in a part of Zeta graphs. Goal: Establish that Th(“Graph Patterns”, CHKNS ) = Th(“Graph Patterns”, Zeta Graphs)

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Other Related Work Graph Models. Huge amounts. Volumes! [Lynch 05]: Conditions on a skewed degree distribution, but symmetrizes labels. Proves a 0-1 law for all of FO! Zeta graphs and CHKNS have no 0-1 law. Inspired by this paper!

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Future Work & Conclusion “Conjunctive” theory of simple random graph models with order. Does a simpler model capture CHKNS? Could one capture Albert & Barabasi’s preferential attachment model? Richer Languages?

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Expectations for Ordered Graphs Since sensitive to order, consider graph patterns with order among variables. Then expectation has a semi-closed form. This function has an MVZ

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Computing Expectations of General CQs If variables in Q are totally ordered, then we can compute E[Q] using MVZs. Obvious algorithm: given a query, add in equality and inequality in all possible ways. This takes exponential time in Q (#P-hard).

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