the theory of zeta graphs with an application to random networks christopher ré stanford
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The Theory of Zeta Graphs with an Application to Random Networks
Christopher RéStanford
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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)
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 GraphsSimple 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)
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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1st Technical Challenge:Graph Patterns
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Our goal for this sectionGiven(1) a Language of Boolean queries L, and (2) a family of probability models M(1), M(2), …,M(N) check if limN to ∞ PrM(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 : limN to ∞ PrM(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 InequalityA 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 si 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.
0 1 1
00 2
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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 MVZsThis is a small variation of Costermans et al. result.
(expected # of edges)
(expected # of triangles)
(expected # of K4)
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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] = w(1), Pr[Q] = 1 – o(1)
0
0
0
0
1
2 1
1 1
1 1
…
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Two cycles you’re out!
rcycle
scycle
(1) For all r, s ≥ 2, PrM(N)[ B(r,s) ] < 1 – e for some fixed e > 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.
1st 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).