The document discusses the concept of graphons, which are symmetric measurable functions used to model graphs, especially in the context of social media networks and computer networks. It explains the mathematical properties of graphons, their convergence, and how they can be used to generate new graphs, particularly in the study of sparse and dense graph sequences. Additionally, the document examines specific types of graphs, such as star graphs and Erdős-Rényi graphs, and discusses the limitations of using graphons for understanding sparse graphs.

1. WIMSIG 2024, 1st October 2024
Sevvandi Kandanaarachchi, Cheng Soon Ong
CSIRO’s Data61
Graphons of Line Graphs

2. Growing graphs/networks
Social media networks
Career networks – LinkedIn
Grow over time
Computer networks – WWW
Graphons can be used to model these
networks if certain conditions are satisfied.
Graphons have nice mathematical properties.

3. Star graphs 𝐾!,#
• Star graphs
• Adjacency matrix pixel pictures
ones - black
zeros - white
(Empirical graphons)

4. The problem
• Pixel pictures (empirical graphons) converge to zero
Empirical graphons Graphon

5. What are graphons?
• Graph limits
• Examples from What is a graphon? By Daniel Glasscock
Erdos-Renyi graphs
Growing uniform attachment graphs
Bi-partite graphs

6. Mathematically what is a graphon?
• Graphon is a symmetric measurable function 𝑊: [0,1]! → [0,1]
• A function defined on a unit square
• Empirical graphon: when you colour in the squares of the adjacency matrix and scale it to [0, 1]
• Empirical graphon of graph 𝐺, is 𝑊" .
• Graphs converging to graphons
• How is convergence defined?
• Graph homomorphisms and homomorphism densities
Empirical graphons Graphon

7. Graph homomorphism - convergence
• A graph homomorphism from 𝐹 to 𝐺 is a map 𝑓: 𝑉(𝐹) → 𝑉(𝐺) such that if 𝑢𝑣 ∈ 𝐸(𝐹) then
𝑓(𝑢)𝑓(𝑣) ∈ 𝐸(𝐺) (Maps edges to edges). Let 𝐻𝑜𝑚(𝐹, 𝐺) be the set of all such
homomorphisms and let hom(𝐹, 𝐺) = |𝐻𝑜𝑚(𝐹, 𝐺)|. Then homomorphism density is
defined as
• 𝑡(𝐹, 𝐺) =
#$%(',")
|+(")||"($)| , 𝑡(𝐹, 𝐺) ∈ ℝ
• 𝑡(𝐹, 𝑊) = ∫[-,.]|"($)| ∏
01∈3(')
𝑊(𝑥0, 𝑥1)𝑑𝑥
• A graph sequence {𝐺4}4 is said to be convergent if 𝑡 𝐹, 𝐺4 4 converges as 𝑛 goes to infinity
for any simple graph 𝐹.
• Cut metric convergence is the same as homomorphism density convergence

8. How can we use graphons?
• To generate new graphs (sample graphs)
• We can sample a graph with a larger number of nodes than currently seen - depicting the
graph at a future time point
• Suppose there are the 𝑛 nodes in your new graph {1,2, … , 𝑛}
• Sample 𝑛 random numbers from the interval [0,1] - {𝑟., 𝑟!, … , 𝑟4}
• For every (𝑟0, 𝑟
1) consider the value of the graphon 𝑊(𝑟0, 𝑟
1) = 𝑝 as a probability
• Toss a coin with probability 𝑝, and connect the edge between 𝑖 and 𝑗 if you get heads
𝑟0
𝑟&

9. The problem, again
• When 𝑊 = 0, sampled graphs are empty.
• Is it rare to get 𝑊 = 0 ? No, for sparse graphs 𝑊 = 0. Real-world graphs are sparse.
• This graphon can’t be used to understand sparse graphs.
Empirical graphons 𝑊"! Graphon 𝑊

10. Dense and sparse graphs
• Consider a graph sequence {𝐺4}4, where 𝐺4 has 𝑛 nodes and 𝑚 edges.
• Dense graphs: 𝑙𝑖𝑚 𝑖𝑛𝑓
4→6
7
4' = 𝑐 > 0
• Edges grow quadratically with nodes
• Graph sequences with strictly positive edge-density limits
• Sparse graphs: 𝑙𝑖𝑚
4→6
7
4' = 0
• Graph sequences with edge-density going to zero
• Edges grow subquadratically with nodes

11. Line graphs
• Edges Mapped to vertices
• Vertices are connected, if they share an edge
• The name line graphs - term came from Frank
Harary (motivated by Harary calling “vertices"
and “edges”, “points” and “lines”)
• Work originated by Hassler Whitney in 1932
• Other names used, interchange graphs,
edge-to-vertex dual, covering graph,
derivative, derived graph, adjoint, conjugate
Graph Line graph

12. Star graphs 𝐾!,#
• Star graphs
• Line graphs of star graphs
are complete (and dense)
• Empirical graphons of line graphs
are not zero

13. Multiple stars
• Star graphs
• Line graphs of stars (complete subgraphs)
• Empirical graphons of line graphs

14. Back to the problem
• Taking line graphs worked for star graphs
• But will it work for all sparse graphs? No!
• Which type of sparse graphs will it work on?
• How about dense graphs? Will line graphs work for dense graphs?
Empirical graphons Graphon

15. Erdos-Renyi graphs 𝐺(𝑛, 𝑝)
• Recall: dense graphs: 𝑙𝑖𝑚 𝑖𝑛𝑓
4→6
7
4' = 𝑐 > 0
• For 𝐺(𝑛, 𝑝), this limit equals 𝑝
• For Erdos-Renyi graphs line graphs converge to the zero
graphon with probability 1 (Preprint has a Thm)
Erdos-Renyi pixel pictures
and graphon
Line graphs of Erdos-Renyi graphs

16. Intuition behind Erdos-Renyi line graphs
• Edge density is equivalent to the the non-zero area of the graphon

17. Back to the problem
• For Erdos-Renyi graphs, taking line graphs didn’t work
• But for star graphs, line graphs worked.
• What is the condition that will tell us that line graphs will work? (Give non-zero graphon)
Empirical graphons for sparse graphs Graphon

18. Square-degree property
• (Sum of degree squares) ≥ 𝑐( (square of sum of degrees) for 𝑐( > 0 and 𝑛 > 𝑁)
• Let {𝐺4}4 denote a sequence of graphs. We say that {𝐺4}4 exhibits the square-degree
property if there exists some 𝑐. > 0 and 𝑁- ∈ ℕ such that for all 𝑛 > 𝑁- we have
∑deg 𝑣0,4
!
≥ 𝑐. ∑deg 𝑣0,4
!
• We denote the set of graph sequences satisfying the square-degree property by 𝑆8

19. Where does square-degree come from?
• Let the line graph of graph 𝐺 be denoted by 𝐿(𝐺), and 𝐺 has 𝑛 nodes and 𝑚 edges
• Then edge density of line graph density(𝐿(𝐺)) =
#:;<:=
>?? @A==0B?: :;<:=
=
"
#
∑$DEFG$
#
H7
"
#
7(7H.)
• ∑deg 𝑣0,4
!
≥ 𝑐. ∑deg 𝑣0,4
!
= 𝑐.𝑚!
• Square-degree property <=> line graph edge density is non-zero
<=> graphon of line graphs have non-zero area
• Star graphs, superlinear preferential attachment graphs are square

20. {𝐺*}*
Dense
𝑊 ≠ 0
{𝐺*}* ∉ 𝑆+
𝑈 = 0
Eg: Erdos Renyi
graphs
{𝐺*}* ∉ 𝑆+
𝑈 = 0
e.g. Paths,
Cycles
e.g. Superlinear pref
attachment graphs,
Stars
Sparse
𝑊 = 0
{𝐺*}* ∈ 𝑆+
𝑈 ≠ 0
• {𝐺!}! → 𝑊
𝐻" = 𝐿(𝐺!), line graph of 𝐺!
{𝐻" }" → 𝑈
𝑊 and 𝑈 graphons
𝐷, the set of dense graph sequences,
𝑆 sparse graph sequences,
𝑆# square-degree property
For graphs satisfying the square degree property, we’re good

21. Map between {𝐺!}! and {𝐻"}"
• Say {𝐺4}4 → 𝑊 and {𝐻7}7 → 𝑈 where 𝐻7 = 𝐿(𝐺4) and 𝑊 and 𝑈 are graphons. 𝐷, the
set of dense graph sequences, 𝑆 sparse graph sequences, 𝑆8 square-degree property
satisfying graph sequences
{𝐺*}* ∈ 𝐷 ≡ 𝑊 ≠ 0
{𝐺*}* ∈ 𝑆𝑆+ ⟹ 𝑊 = 0
{𝐺*}* ∈ 𝑆+ ⟹ 𝑊 = 0
{𝐻,}, ∈ 𝑆 ≡ 𝑈 = 0
{𝐻,}, ∈ 𝐷 ≡ 𝑈 ≠ 0

22. Bird’s-eye view
• Graphons for Sparse graphs
• Work by Caron and Fox
• Many methods led by Borgs and Chayes
• Work by Janson and others
• These methods have complex mathematical machinery
• Why? Because standard construction for sparse graphs give graphons with point masses
with measure zero
• The difference in our work
• Line graphs enable us to use the standard path for dense graphons

23. Thank you!
Preprint:
https://web3.arxiv.org
/abs/2409.01656

24. Extra slides

25. Convergence - cut metric
• The cut norm of a graphon 𝑊 is defined as ∥ 𝑊 ∥□= 𝑠𝑢𝑝
J,K
∫J×K𝑊(𝑥, 𝑦)𝑑𝑥𝑑𝑦 where
supremum is taken over all measurable sets 𝑆 and 𝑇 of [0,1]
• Given two graphons 𝑊. and 𝑊!, the cut metric is defined as
𝛿□(𝑊., 𝑊!) = 𝑖𝑛𝑓
M
∥ 𝑊. − 𝑊
!
M
∥□
where 𝜑 is a measure preserving bijection from [0,1] to [0,1].
• Why 𝜑? Because nodes can be re-named or permuted.
• Convergence in homomorphism density is equivalent to convergence in cut-metric. (Borgs et
al 2011)
• If a graph sequence converges (cut metric or hom. density) we have a graphon.

26. Experiment:
Star graphs
• From 𝑊-!
(empirical graphon) sample
graph with more vertices, then get the
line graph 8
𝐻.
• From 𝑈/"
(empirical graphon) sample
graph with more vertices 8
𝐻0
• Compare with actual star graph 𝐻
Degree Cosine Edge Density Triangle Density
200 300 400 500 200 300 400 500 200 300 400 500
0
1
2
3
0.25
0.50
0.75
1.00
0.4
0.6
0.8
1.0
Nodes
Value
Graph
H
H
^
U
H
^
W
8
𝐺 8
𝐻. = L(;
𝐺)
8
𝐻0
sample
sample

27. ∑deg 𝑣$,#
%
≥ 𝑐! ∑deg 𝑣$,#
%
• Say vertex degrees are given by d(, d1, … , 𝑑* then
• Want ∑ 𝑑2
1
≥ 𝑐 ∑𝑑2
1
• Intuition: this can only happen when some degrees are very very big compared to others.
• Mixed terms on RHS need to be smaller than the squares on the LHS
• Examples: superlinear preferential attachment graphs and stars

28. Limitations of the graphon
• Graphs sampled from the graphon are either dense or empty graphs (consequence of
Aldous-Hoover theorem)
• If non-zero area of graphon > 0
• Sampled graphs are dense
• If non-zero area of graphon = 0 (graphon 𝑊 = 0 )
• Graphs are empty

29. Existing work on sparse graphs
• Work by Caron and Fox
• Kallenberg exchangeability and Levy measures
• Many methods led by Borgs and Chayes
• Rescaled and stretched graphons, Graphexes
• Work by Janson and others
• Edge exchangeable graphs
• These methods have complex mathematical machinery

30. How does the graphon come in?
• Theorem (Borgs et al. 2008)
• {𝐺4}4 converges iff there is a 𝑊 such that 𝑡(𝐹, 𝐺4) → 𝑡(𝐹, 𝑊) .
• And for every 𝑊 there is a convergent sequence of graphs like above.
Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T., & Vesztergombi, K. (2008). Convergent sequences of dense graphs
I: Subgraph frequencies, metric properties and testing. Advances in Mathematics, 219(6), 1801–1851.
https://doi.org/10.1016/j.aim.2008.07.008

31. Why do sparse graphs give zero graphon?
• Area of the graphon and edge density are closely related.
• Non-zero area of the graphon ≈ edge density of graph
• For sparse graphs, edge density -> 0

32. What happens when we use line graphs
• Certain sparse graphs give dense line graphs
• Then the graphon of line graphs is not zero for these types of sparse graphs
Sparse Dense

33. Square-degree graphs are sparse
• ∑deg𝑣0,4
!
≥ 𝑐.𝑚!
• deg𝑣0,4 ≤ 𝑛 − 1
• 𝑛(𝑛 − 1)! ≥ ∑deg𝑣0,4
!
≥ 𝑐.𝑚! ⟹
7#
43 ≤
.
N
• This implies 𝑙𝑖𝑚
4→6
7
4# = 0 (definition of Sparse)

34. Are all sparse graphs square-degree?
• No! Paths, Cycles, r-regular graph sequences.
• If {𝐺4}4 is dense, then {𝐺4}4 ∉ 𝑆8
Dense
graphs 𝐷
Square-
degree
𝑆+
Sparse graphs 𝑆𝑆8

35. Square-degree result
• Theorem: Let {𝐺4}4 ∈ 𝑆 be a sparse graph sequence. Let {𝐻7}7 be the corresponding
sequence of line graphs with 𝐻7 = 𝐿(𝐺4). Then {𝐺4}4 ∈ 𝑆8 ≡ {𝐻7}7 ∈ 𝐷 , i.e., {𝐺4}4
satisfies 𝑆𝑞 if and only if {𝐻7}7 is dense.
• 𝑆 - the set of sparse graph sequences
• 𝑆8 - the set of square-degree graph sequences
• 𝐷 - the set of dense graph sequences
• 𝑆𝑞 - square-degree condition

36. Examples of deterministic graphs
• 𝐺* , graph with 𝑛 nodes and 𝑚 edges
• Maybe show pictures of these
Graph Edge-density
Graph belongs
to
Line graph
edge density
Line graph
belongs to
Complete
Graphs
1 Dense graphs 4/(n+1) Sparse graphs
r-regular
graphs
r/(n-1) Sparse 4(r-1)/(rn-2) Sparse
Path 2/n Sparse 2/(n-2) Sparse
Cycles 2/(n-1) Sparse 2/(n-1) Sparse
Stars 2/n Sparse 1 Dense

37. Preferential Attachment Graphs
• New nodes connect to more connected nodes
• The probability Π(𝑖) that a new node connects to node 𝑖, which has degree 𝑘0 is given by
• Π(𝑖) =
O$
4
∑$ O$
4
• Maximum degree 𝑘7PQ
• Three regimes of 𝛼: sublinear 𝛼 < 1 , linear 𝛼 = 1, and superlinear 𝛼 > 1
• Sethuraman & Venkataramani (2019) show that for 𝛼 > 1, 𝑃 𝑙𝑖𝑚
4→6
.
4
𝑘7PQ = 1 = 1
Sethuraman, S., & Venkataramani, S. C. (2019). On the Growth of a Superlinear Preferential Attachment Scheme. Springer Proceedings in Mathematics and Statistics, 283,
243–265. https://doi.org/10.1007/978-3-030-15338-0_9

38. Superlinear pref. attachment
• Using this result we show
• Lemma: Let {𝐺4}4 denote a sequence of graphs growing by superlinear preferential
attachment with 𝛼 > 1. Then {𝐺4}4 ∈ 𝑆8 almost surely.
• Say {𝐺4}4 → 𝑊 and {𝐻7}7 → 𝑈 where 𝐻7 = 𝐿(𝐺4) and 𝑊 and 𝑈 are graphons.
• If {𝐺4}4 is a superlinear pref. attachment graph sequence, then 𝑈 ≠ 0

39. Thank you!