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Motivation, Background Main Results Proof Sketches

On Symmetry of Uniform and Preferential Attachment Graphs

Abram Magner, Svante Janson, Giorgos Kollias, Wojciech Szpankowski June 12, 2014

Abram Magner, Svante Janson, Giorgos Kollias, Wojciech Szpankowski On Symmetry of Uniform and Preferential Attachment Graphs

Motivation, Background Main Results Proof Sketches

Symmetry

A graph has a symmetry if there are two nodes that have the same global view of the (unlabeled) graph. Definition (Automorphism, automorphism group) An automorphism of a graph G on n vertices is a permutation π : [n] → [n] such that {u, v} ∈ E(G) ⇐ ⇒ {π(u), π(v)} ∈ E(G). The set of automorphisms of G with the operation of function composition is called the automorphism group Aut(G) of G. An automorphism is an isomorphism that results in the same labeled graph.

Abram Magner, Svante Janson, Giorgos Kollias, Wojciech Szpankowski On Symmetry of Uniform and Preferential Attachment Graphs

Motivation, Background Main Results Proof Sketches

Application: Counting structures

Motivating question: How many unlabeled graphs of size n are there? Unlabeled d-regular graphs? Size of the isomorphism class of G: |S(G)| =

n! |Aut(G)|.

Figure : Aut(G1) = {ID, (1432), (13), (24), (12)(34), (13)(24), (14)(23), (1234)}, Aut(G2) = {ID, (13), (24), (13)(24)}.

Erd˝

• s and R´

enyi: Almost all graphs are asymmetric (so sizes of isomorphism classes are almost all n!).

Abram Magner, Svante Janson, Giorgos Kollias, Wojciech Szpankowski On Symmetry of Uniform and Preferential Attachment Graphs

Motivation, Background Main Results Proof Sketches

Application: Structure compression

Motivating question: How much more can we compress a graph if we can throw away the labels? Choi and Szpankowski (2012): Theorem Provided all isomorphic graphs are equiprobable, HG = HS + log n! −

• s∈S

P(s) log |Aut(s)|. Here, HG is the entropy of the distribution on labeled graphs G, and HS is the entropy of the distribution on structures S induced by G. So more symmetry = ⇒ less of a difference in compressibility between labeled and unlabeled graphs.

Abram Magner, Svante Janson, Giorgos Kollias, Wojciech Szpankowski On Symmetry of Uniform and Preferential Attachment Graphs

Motivation, Background Main Results Proof Sketches

Defect: A tool for proving asymmetry

Let N(u) denote the neighbors of u. Defect of a vertex: Dπ(u) := |N(π(u))∆π(N(u))|. Defect of a permutation: Dπ(G) := maxu∈V (G) Dπ(u). Defect of a graph: D(G) := minπ=ID Dπ(G). Defect-based criterion for asymmetry: G is asymmetric ⇐ ⇒ D(G) > 0 ⇐ ⇒ ∀π = ID, ∃us.t.Dπ(u) > 0.

Figure : Dπ(2) = |{2, 4}∆{1, 2, 4, 5}| = 2, Dπ(G) ≥ 2, D(G) = 0.

Abram Magner, Svante Janson, Giorgos Kollias, Wojciech Szpankowski On Symmetry of Uniform and Preferential Attachment Graphs

Motivation, Background Main Results Proof Sketches

Prior work

Asymmetric Graphs (Erd˝

• s and R´

enyi, 1963): For fixed p, G(n, p) is asymmetric with high probability. The Asymptotic Number of Unlabelled Regular Graphs, (Bollob´ as, 1982): Random d-regular graphs (for d ≥ 3 fixed) are asymptotically asymmetric. On the Asymmetry of Random Regular Graphs and Random Graphs (Kim, Sudakov, Vu, 2002): If p ≫ log n

n

and 1 − p ≫ log n

n , we have, almost surely,

D(G(n, p)) = (2 − o(1))np(1 − p). If log n ≪ d and n − d ≫ log n, then, almost surely, D(G(n, d)) = (2 + o(1))d

• 1 − d

n

• .

Abram Magner, Svante Janson, Giorgos Kollias, Wojciech Szpankowski On Symmetry of Uniform and Preferential Attachment Graphs

Motivation, Background Main Results Proof Sketches

Preferential attachment: motivation

Networks in the real world exhibit a power law degree distribution. Barab´ asi & Albert: This could arise from a rich get richer mechanism. Bollob´ as & Riordan: BA’s description is mathematically

• imprecise. The preferential attachment model is born!

Abram Magner, Svante Janson, Giorgos Kollias, Wojciech Szpankowski On Symmetry of Uniform and Preferential Attachment Graphs

Motivation, Background Main Results Proof Sketches

Preferential attachment: the model

A preferential attachment graph PA(n, m) on n vertices, with m choices per vertex:

Abram Magner, Svante Janson, Giorgos Kollias, Wojciech Szpankowski On Symmetry of Uniform and Preferential Attachment Graphs

Motivation, Background Main Results Proof Sketches

Main results: symmetry

Theorem (Symmetry results for m = 1, 2) Fix m = 1, 2, and let Gn∼UA(n, m) or Gn∼PA(n, m). Then there exists a constant C > 0 such that, for n sufficiently large, Pr[|Aut(Gn)| > 1] > C. For the uniform attachment model, the result for m = 1 can be strengthened to symmetry with high probability.

Abram Magner, Svante Janson, Giorgos Kollias, Wojciech Szpankowski On Symmetry of Uniform and Preferential Attachment Graphs

Motivation, Background Main Results Proof Sketches

Asymmetry conjecture

Conjecture For fixed m ≥ 3, and either Gn∼PA(n, m) or Gn∼UA(n, m), lim

n→∞ Pr[|Aut(Gn)| = 1] = 1.

Proving asymmetry is challenging (it is a global property of a graph, while symmetry is local). There is ample empirical evidence for the conjecture, and we add some more via defect.

Abram Magner, Svante Janson, Giorgos Kollias, Wojciech Szpankowski On Symmetry of Uniform and Preferential Attachment Graphs

Motivation, Background Main Results Proof Sketches

Empirical evidence for asymmetry

200 400 600 800 1000 Graph size (number of vertices) 5 10 15 20 25 30 35 Numerical defect

Numerical defect vs graph size for the Uniform Attachment model

m=1 m=2 m=3 m=4

Figure : Empirical graph defect for graphs up to 1000 nodes.

Abram Magner, Svante Janson, Giorgos Kollias, Wojciech Szpankowski On Symmetry of Uniform and Preferential Attachment Graphs

Motivation, Background Main Results Proof Sketches

A bit of notation

Definition Let π be a nontrivial permutation in Sn, and let u ∈ [n]. Then we define ω(π, u) = min{u, π(u)}, and ω(π) = min{v ∈ [n]|π(v) = v}.

Abram Magner, Svante Janson, Giorgos Kollias, Wojciech Szpankowski On Symmetry of Uniform and Preferential Attachment Graphs

Motivation, Background Main Results Proof Sketches

Main results: expected defect, weak asymmetry

Theorem (Expected vertex defect) Fix m ∈ N in the uniform attachment model. For n sufficiently large, π = ID, π ∈ Sn, and u ∈ [n] not fixed by π,

log

• n

max{ω(π, u) + 2, (2m + 2)}

• ≤ E[Dπ(u)] ≤ 1 + 4m
• 2 + log
• n

ω(π, u)

• .

Theorem (Weak asymmetry) Fix m ≥ 1 and consider a sequence of graphs in the uniform attachment model Gn∼UA(n, m). Let {πn}∞

n=1, πn ∈ Sn − {ID},

and, for each n, let un = ω(πn). Then Pr[Dπn(un) = 0] n→∞ − − − → 0.

Abram Magner, Svante Janson, Giorgos Kollias, Wojciech Szpankowski On Symmetry of Uniform and Preferential Attachment Graphs

Motivation, Background Main Results Proof Sketches

Proof sketch: m = 1, 2

Main idea: Lower bound the probability of a small subgraph that implies symmetry.

Abram Magner, Svante Janson, Giorgos Kollias, Wojciech Szpankowski On Symmetry of Uniform and Preferential Attachment Graphs

Motivation, Background Main Results Proof Sketches

Proof sketch: more details on m = 2

Goal: Consider two nodes that make the same choices and are unchosen. Probability that there are u, v > n

2 such that

{cu,1, cu,2} = {cv,1, cv,2} ⊂ [n/2]: bounded below by C > 0, by a birthday paradox argument: Θ(n2) birthdays, Θ(n2) people. Condition on lexicographic ordering of pairs of vertices > n/2 and on the event that two such vertices pick the same pair. Conclude that such a pair is unchosen with positive probability.

Abram Magner, Svante Janson, Giorgos Kollias, Wojciech Szpankowski On Symmetry of Uniform and Preferential Attachment Graphs

Motivation, Background Main Results Proof Sketches

Future work

1 Asymmetry for growing mt as a function of time: defect-based

proof works in the uniform attachment case, and possibly with preferential attachment.

2 Asymmetry for constant m ≥ 3? 3 Study the structure of the automorphism group for cases

where symmetry has positive probability.

Abram Magner, Svante Janson, Giorgos Kollias, Wojciech Szpankowski On Symmetry of Uniform and Preferential Attachment Graphs