Random graph models with fixed degree sequences: choices...
Transcript of Random graph models with fixed degree sequences: choices...
![Page 1: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/1.jpg)
Random graph models with fixed degree sequences:
choices, consequences and irreducibilty proofs for
sampling
Joel Nishimura1, Bailey K Fosdick2, Daniel B Larremore3 and Johan Ugander4
1Arizona State Univ. 2Colorado State 3Univ. of Colorado 4Stanford Univ.
ASU Discrete Math Seminar 2018
![Page 2: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/2.jpg)
See paper for the numerous literature connections
![Page 3: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/3.jpg)
What is notable about a graph?
![Page 4: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/4.jpg)
Interpretation requires a Null Model
karate club
![Page 5: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/5.jpg)
Interpretation requires a Null Model
model
err
or
implementation difficulty and/or required understanding
replicated experiments
Erdős–Rényi
application specific simulation
fixed degree sequence
![Page 6: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/6.jpg)
Stub Matching
edges to
stubs
join 2
stubs
drop stub
labels
Stub-labeledVertex-labeled
![Page 7: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/7.jpg)
Self-loops
edges to
stubs
join 2
stubs
drop stub
labels
![Page 8: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/8.jpg)
Self-loops and Multiedges
• Self-loops and multiedges are asymptotically rare
(for reasonable degree sequences)
• Have been frequently been ignored, or simply deleted
• BUT – they can also have large impacts on finite sized null models
• AND – in null models which allow self-loops or multiedges, stub matching does not sample adjacency matrices uniformly at random
![Page 9: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/9.jpg)
Interpretation requires a Null Model
model
err
or
implementation difficulty and/or required understanding
replicated experiments
Erdős–Rényi
application specific simulation
fixed degree sequence
??
?
![Page 10: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/10.jpg)
multiedges self-loops
vertex-
labeled
simple
stub-
labeled
Stub labeled graphs are biased
against multiedges and self-loops
![Page 11: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/11.jpg)
Consider “1,2,2,1”Uniformly samples
from d and e are the
same
Uniform samples are
different
![Page 12: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/12.jpg)
There’s a choice of graphs – and it matters
![Page 13: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/13.jpg)
Example 1
• Geometer’s collaboration graph n=9,072m=22,577
• Nodes: computational geometry researchers.
• Edges: collaboration on a book or paper
• Degree assortativity
• Do high productivity authors coauthor with other high productivity authors?
![Page 14: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/14.jpg)
![Page 15: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/15.jpg)
Multiedges, self-loops, and labeling?
![Page 16: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/16.jpg)
Stub-labeling isn’t causal
• Node iStub 1: first paper
Stub 2: second paper
• Node iStub 1: first paper
Stub 2: second paper
• Node jStub 1: first paper
Stub 2: second paper
• Node jStub 1: first paper
Stub 2: second paper
Consider a collaboration network, and two potential stub
labelings:
![Page 17: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/17.jpg)
Vertex-Labeling is Causal
• Consider a collaboration network:
• Nodes: authors
• Edges: papers/books with unique title
• Suppose you order each edge’s arrival
• Each vertex labeled graph has m! edge orderings
• i.e. all adjacency matrices correspond to the same number of timelines where papers were produced in different orders.
![Page 18: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/18.jpg)
Example 2
• Swallow graph n=17
• Nodes: barn swallows.
• Edges: bird-bird interactions
• Trait assortativity (based on bird color)
• Do birds of a similar color interact together?
![Page 19: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/19.jpg)
![Page 20: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/20.jpg)
Example 3• South Indian village social support network n=782
• Nodes: villagers Edges: reported social support
• Community detection via modularity maximization
• Modularity has a built in stub-labeled Chung-Lu null model
• Do results change if we use vertex labeled model?
Chung Lu
estimation
# of edges
observed in
configuratio
n models
![Page 21: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/21.jpg)
![Page 22: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/22.jpg)
Sampling graphs uniformly at random…
… is surprisingly difficult (except pseudo-graphs)
![Page 23: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/23.jpg)
Sampling graphs uniformly at random
![Page 24: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/24.jpg)
Sampling via Markov chain Monte Carlo
G0 G1 G2 G3 G4 G5
Goal: A sequence of degree constrained graphs such that subsampling
from this sequence approximates a set of graphs drawn uniformly at
random.
![Page 25: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/25.jpg)
Double EdgeSwaps
![Page 26: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/26.jpg)
, the Graph of Graphs
![Page 27: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/27.jpg)
, the Graph of Graphs
![Page 28: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/28.jpg)
Dealing with Constraints
![Page 29: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/29.jpg)
Dealing with Constraints
no self-loops
![Page 30: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/30.jpg)
Dealing with Constraints
![Page 31: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/31.jpg)
MCMC requirements
1. Random walks can reach any graph -Irreducibility/GOG connected
2. Balanced transition probabilities-P(𝐺𝑖 → 𝐺𝑗) = P(𝐺𝑗 → 𝐺𝑖)-i.e. edges will be weighted but undirected
3. Markov chain is aperiodic -otherwise subsampling can be biased
NOTE: There are mixing time results for some degree sequences. There are also numerical methods to gauge convergence. I will not discuss either.
![Page 32: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/32.jpg)
Is the GoG periodic? Nope!
Or
![Page 33: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/33.jpg)
Stub-labeled
GoG
Vertex-labeled
GoG
GoG is an
undirected
simple graph
GoG is a directed
pseudograph
Are transition probabilities balanced?
![Page 34: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/34.jpg)
Stub-labeled
GoG
Vertex-labeled
GoG
GoG is an
undirected
simple graph
![Page 35: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/35.jpg)
Is the GoG connected?• Most difficult of the 3 questions
• Need special proof for each of choice of self-loops/multiedges
• Stub labeled GoG connectivity iff vertex labeled GoG Connectivity,
because the following swap permutes stubs:
![Page 36: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/36.jpg)
Connectivity of Graph of Pseudographs
start target diff
# of stubs per node
# gold = # maroon
![Page 37: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/37.jpg)
Connectivity of Graph of Pseudographs
can always find a graph one edge closer to target
swap
start target
![Page 38: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/38.jpg)
Connectivity on other GoGs?
![Page 39: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/39.jpg)
Disconnectivity of loopy graphs
Consider graphs with self-loops but no multiedges
There are no swaps between these graphs
Two directions for generalizations: cycles and cliques
![Page 40: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/40.jpg)
![Page 41: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/41.jpg)
Degree sequence: “2,2,…,2}Swaps can:
1. Merge two cycles into a larger cycle (or do the reverse).
2. Swap two edges inside a cycles, preserving cycle length
3. Make a self-loop & reduce cycle length by 1 (or do the reverse), but only for cycles of length 4 or more.
Swaps cannot make every edge a self-loop
This can be further generalized
![Page 42: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/42.jpg)
3) Vk are vertices
k distance from a
vertex in V0
1) Let V0 be
vertices without a
self-loop
2) Vertices in V1
have a neighbor in V0
A taxonomy of V
![Page 43: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/43.jpg)
Let Vk be
vertices k hops
from a vertex
without a
selfloop
![Page 44: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/44.jpg)
![Page 45: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/45.jpg)
Deg seq: “n+1,…n+1,n-1,…,n-1”
No swaps are possible
![Page 46: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/46.jpg)
![Page 47: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/47.jpg)
![Page 48: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/48.jpg)
Q1 and Q2 are exactly the problems
![Page 49: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/49.jpg)
Proof of 4.20 outline
increasing
number of
self-loops
connected components
graphs with a fixed
degree sequence
![Page 50: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/50.jpg)
increasing
number of
self-loops
connected components
graphs with most self-loops
in ‘yellow’ ‘m*-loopy’
graphs: graphs with
the most self-loops
![Page 51: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/51.jpg)
increasing
number of
self-loops
connected components
graphs with most self-loops
in ‘yellow’ ‘m*-loopy’
graphs: graphs with
the most self-loops
Note: connectivity of
follows from connectivity
of simple graphs and an
exchange lemma.
![Page 52: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/52.jpg)
increasing
number of
self-loops
connected components
graphs with most self-loops
in ‘yellow’ ‘m*-loopy’
graphs: graphs with
the most self-loops
The GoG is disconnected
iff there is some
component where:
U
![Page 53: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/53.jpg)
Zooming into
![Page 54: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/54.jpg)
Easy case:
Harder case:
![Page 55: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/55.jpg)
What do we know about ?
Maximum number of self-loops
implies no open wedges in V0.
No sequence of swaps can net
create open wedges in V0.
&
![Page 56: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/56.jpg)
Example: V4 is empty in any
Open Wedge
![Page 57: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/57.jpg)
![Page 58: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/58.jpg)
Q2
Q1
![Page 59: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/59.jpg)
is m*-loopy
Decreasing any degree in K0
leaves Vu1 with excess degree.
![Page 60: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/60.jpg)
is also m*-loopy
By an alternating cycle/path argument.
Thus
Q: Can a different swap connect loopy-graphs?
![Page 61: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/61.jpg)
Triangle swaps connect the GoG
![Page 62: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/62.jpg)
Bonus: other constraints
• Connected Graphs
• GoG known to be connected, but algorithms require complicated data-structures to track effect of edge changes.
• Graphs with the same clustering coefficients
• Or, triangle constraints
![Page 63: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/63.jpg)
Triangle MCMC constraints
• Total number of triangles
• Number of triangles incident at each node
Do these affect connectedness in simple graphs?
![Page 64: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/64.jpg)
Can we constrain number of triangles
![Page 65: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/65.jpg)
How about triangle sequence
![Page 66: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/66.jpg)
And more!
![Page 67: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/67.jpg)
Thanks for listening!
![Page 68: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/68.jpg)
![Page 69: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/69.jpg)
![Page 70: Random graph models with fixed degree sequences: choices ...andrzej/seminar/DiscreteMath2018_vInter.pdf · 1. Merge two cycles into a larger cycle (or do the reverse). 2. Swap two](https://reader035.fdocuments.in/reader035/viewer/2022081521/5ec7ebc3d705db2ca70f7456/html5/thumbnails/70.jpg)